plain.txt

Let $ G $ be a compact topological group and let $ int_G $ be an invariant vector - valued integral.
Let $ rho $ be a continuous group representation into the vector space $ V $ .
For a fixed $ vin V $ , consider the function $ f_v:Gto V $ such that $ gmapsto rho(g)v $ .
Define a linear transformation $ text{Av}_rho:Vto V $ by the formula $ text{Av}_rho(v) = frac{1}{text{vol}(G)}int_G rho(g)vtext dg $ .
Heuristically, we can say that this is the barycenter of the $ G $ - orbit of $ v $ .
Let $ mathcal C $ be a small groupoid and let $ x $ be an object of $ mathcal C $ .
The star of $ x $ , denoted $ text{St}(x) $ , is the set of objects of the category of objects under $ x $ .
That is, it is the set of morphisms with source $ x $ .
The group $ mathcal C(x,x) $ of automorphisms of $ x $ is written $ pi(mathcal C,x) $ .
A topological space $ X $ is a $ k $ - space if every compactly closed subspace of $ X $ is closed.
Any topological space $ X $ can be made a $ k $ - space by endowing it with a new topology in which a subspace is closed if and only if it is compactly closed in the original topology.
The resulting space is denoted $ kX $ .
The identity function $ kXto X $ is continuous.
If $ X $ is weak Hausdorff, then so is $ kX $ , so $ kX $ is compactly generated so that $ X $ and $ kX $ have exactly the same compact subsets.
Let $ A $ be an algebra over the field $ mathbb k $ .
The elements $ {a_i}_{iin I} $ generate $ A $ if every element $ alphain A $ can be written (not necessarily uniquely) as a polynomial in the $ a_i $ with coefficients in $ mathbb k $ .
Let $ Usubsetmathbb R^n $ be an open subset and let $ f:Utomathbb R $ .
The point $ a in U $ is a local minimum of $ f $ if there is a neighborhood $ V $ of $ a $ such that for all $ x in V $ , $ f(a)leq f(x) $ .
Consider a topological space $ X $ and consider the chain complex given by the groups of singular n - chains of $ X $ .
The $ n $ th homology group $ text{Ker}(partial_{n})/text{Im}(partial_{n+1}) $ is denoted $ H_n(X) $ and is called the $ n $ th singular homology group of $ X $ .
A Borel space is a measurable space $ (X, B) $ given by the Borel σ - algebra on $ X $ .
the aka on wikidata here is not correct
Let $ v_0,dots,v_n $ be an n - simplex.
We get from the order on the vertices a canonical linear homeomorphism from $ Delta^n $ to any other $ n $ - simplex that preserves the order of the vertices: $ (t_0,dots,t_n)mapsto sum_i t_iv_i $ .
The $ t_i $ are called the barycentric coordinates of the point $ sum_i t_iv_i in v_0,dots,v_n $ .
Let $ G $ be a group acting on the set $ X $ .
Then $ G $ also acts on the power set $ mathcal P(X) $ as follows:
If $ Yin mathcal P(X) $ , then $ Ysubset X $ .
Define $ gcdot Y ={gcdot ymid yin Y} $ .
The topological space constructed in the proof that every group is a fundamental group is called the homotopy cofiber of the map $ p $ therein.
A subset $ E $ of the metric space $ (X,rho) $ is totally bounded if for all $ varepsilon > 0 $ , there exists a finite set $ {x_i}_{i=1}^n $ of points in $ X $ such that $ E subseteq bigcup_{i=1}^n B(x_i, varepsilon), $ where $ B(x_i, varepsilon) = {x in X mid rho(x_i,x)< varepsilon} $ .
That is, for every $ varepsilon > 0 $ , $ E $ can be covered by a finite set of balls of radius $ varepsilon $ .
The conjugate matrix transpose of a complex $ mtimes n $ matrix $ A $ is the $ n times m $ matrix $ A^* $ given by taking the usual matrix transpose $ A^T $ and then setting $ A^* $ to be the matrix whose entries are the complex conjugate of each entry of $ A^T $ .
An algebra over a field $ F $ is a vector space $ U $ over $ F $ with a bilinear binary operation $ Utimes U to U $ that is not necessarily associative but that follows left and right distributivity.
One can loosen this definition and only require $ F $ to be a commutative ring.
Let $ Msubsetmathbb R^n $ be a compact, oriented embedded m - dimensional manifold with boundary.
Let $ U $ be an open set containing $ M $ and $ omega $ an m - form on $ U $ .
For each $ p in M $ , choose a smooth local parametrization $ phi_p: U_p to M $ compatible with the orientation on $ M $ .
Then the image $ phi_p(U_p) $ is an open subset of $ M $ , so there exists an open set $ V_p subset mathbb R^n $ such that $ phi_p(U_p) = V_p cap M $ because of the subspace topology.
Since $ M $ is compact, finitely many of these $ V_p $ cover $ M $ .
Call them $ V_i $ for $ 1leq i leq k $ .
Then $ V = bigcuplimits_{i=1}^k V_i $ is an open set containing $ M $ .
Let $ {psi_j} $ be a partition of unity subordinate to the $ V_i $ .
Then $ omega = sum_{j} psi_j omega $ , so define $ int_M omega = sum_{j} psi_jomega $ .
The Kronecker delta $ delta_{ij} $ is a function of the two variables $ i $ and $ j $ defined by $ delta_{ij} = } $
The width of a rectangle $ Q $ in $ mathbb R^n $ is the maximum of the lengths of the component intervals of $ Q $ : $ text{width}(Q)= max_{1leq i leq n} (b_i - a_i) $ .
The degree of a polynomial is the highest power of the variable in its expression as a formal sum $ a_nx^n + cdots + a_0 $ for $ a_i in R $ .
Let $ S $ be a bounded set in $ mathbb R^n $ and let $ f: S to mathbb R $ be a bounded function.
Define $ f_S: mathbb R^n to mathbb R $ by $ f_S(x)=} $ .
A group $ G $ acts freely on the set $ X $ if it has trivial stabilizer.
That is, if $ gx=x $ implies $ g=e $ for all $ xin X $ .
For $ A $ an $ mtimes n $ matrix and $ B $ an $ ntimes p $ matrix with entries $ (a_{ij}) $ and $ (b_{kell}) $ respectively, the product $ C=AB $ is the $ mtimes p $ matrix whose entries are given by $ c_{ij} = sum_{k=1}^n a_{ik}b_{kj} $ for $ 1 leq i leq m $ and $ 1 leq j leq m $ .
Note that this binary operation is not commutative, and $ BA $ may not even be defined even though $ AB $ is due to differing dimensions.
A graph $ X $ is connected if there is an edge path connecting every vertex of $ X $ to every other.
An automorphism of a Lie algebra $ L $ of the form $ text{exp}(text{ad} (x)) $ where $ text{ad} $ is the adjoint representation is called an inner automorphism of $ L $ .
Let $ A $ and $ B $ be subspaces of the topological space $ X $ such that $ X $ is the union of the interiors of $ A $ and $ B $ .
A Mayer - Vietoris sequence is an exact sequence of the form $ cdotsto H_n(Acap B)to H_n(A)oplus H_n(B) to H_n(X) to H_{n - 1}(Acap B) tocdots $ $ to H_0(X) to 0 $ .The arrows between $ H_n(Acap B) $ and $ H_n(A)oplus H_n(B) $ are generally labeled $ Phi $ , the arrows between $ H_n(A)oplus H_n(B) $ and $ H_n(X) $ are generally labeled $ Psi $ , and the arrows between degrees (the boundary homomorphisms) are as usual denoted $ partial $ .
Let $ R $ be a ring and $ {M_i}_{iin I} $ a family of left $ R $ - modules indexed by $ I $ .
The direct sum of the $ M_i $ , written $ bigoplus_{iin I} M_i, $ is the set of all sequences $ {a_i}_{iin I} $ where $ a_i in M_i $ and $ a_i=0 $ for "cofinitely many" indices.
That is, $ a_i neq 0 $ for finitely many $ i $ .
Let $ text{Form}(L) $ be a formal language of propositional logic.
Then a proof of $ S_kin text{Form}(L) $ from a subset $ Gammasubseteq text{Form}(L) $ is a finite list of statements $ S_1,dots, S_k $ such that for all $ S_i $ , at least one of the following is true: (i) $ S_i in Gamma $ ; (ii) $ S_i $ is an axiom; (iii) there exist $ j,ell < i $ such that $ S_ell = S_j rightarrow S_i $ , i.e. that $ S_i $ "is there by" modus ponens.
If there exists a proof from $ Gamma subseteq text{Form}(L) $ to $ phi in text{Form}(L) $ , we say that $ Gamma $ proves $ phi $ and we write $ Gammavdash phi $ .
A subset $ G $ of a topological space $ (X,mathcal T) $ is open if it is an element of $ mathcal T $ .
When $ X $ is a metric space, a subset $ A subset X $ is open if and only if for all $ a in A $ , there exists $ varepsilon > 0 $ such that $ B(a,varepsilon) = {x in X mid rho(a,x) < varepsilon}subset A $ .
A partition of a set $ X $ is a subset $ mathcal S $ of the power set $ mathcal P(X) $ such that (i) $ Ain mathcal S $ implies that $ Aneq emptyset $ ; (ii) For every $ xin X $ , there is a unique $ Ain mathcal S $ such that $ xin A $ .
(Equivalently, the union of the sets $ Ain mathcal S $ is $ X $ and the intersection of any two elements of $ mathcal S $ is empty.)
Let $ G $ be a matrix Lie group.
Then the Lie algebra of $ G $ is the set of all matrices $ X in M_n(mathbb C) $ such that the matrix exponential $ e^{tX} in G $ for all $ t in mathbb R $ .
The Riemann sphere is a model of the extended complex plane $ mathbb C cup {infty} $ .
Let $ U subset mathbb R^n $ be open.
A function $ f: Uto mathbb R^m $ is of class $ C^k $ if its derivatives $ Df, D^2f,dots, D^kf $ all exist and are continuous on $ U $ .
A multiplicative character on a topological group $ G $ is a homomorphism from $ G $ to the group of units of the field of complex numbers $ mathbb C^times $ .
Let $ X $ and $ Y $ be topological spaces.
Then a continuous function $ f:Xto Y $ is a homotopy equivalence if there exists a continuous function $ g:Yto X $ such that both $ fcirc g $ and $ gcirc f $ are homotopic to the respective identity functions on the domain.
The spaces $ X $ and $ Y $ are said to be homotopy equivalent or to have the same homotopy type .
Homotopy equivalence is an equivalence relation.
A permutation is a bijection from a set to itself.
A lattice is a partially ordered set in which every pair of elements has a unique infimum and a unique supremum.
If $ (X, leq) $ is a lattice, we call the supremum of $ x $ and $ y $ the join of $ x $ and $ y $ and denote it $ xlor y $ .
Similarly, we call the infimum of $ x $ and $ y $ their meet and denote it $ xland y $ .
These are in fact binary operations on $ X $ .
Moreover, they respect the partial order on $ X $ .
We can also define a lattice as a triple $ (L, lor,land) $ where $ L $ is a set and $ lor $ and $ land $ are two commutative, associative binary operations on $ L $ satisfying the following absorption laws for all $ a,b in L $ : (i) $ alor (aland b) = a $ ; (ii) $ aland (alor b) = a $ .
Immediately following these two laws we have that $ aland a = a $ and $ alor a = a $ .
It is easy to check that the poset notion of a lattice admits a description in the other form.
To see the converse, define a partial order $ leq $ on $ L $ by setting $ aleq b $ if $ a= aland b $ or $ aleq b $ if $ b = alor b $ for all $ a,bin L $ .
A simplicial complex is a ∆ - complex whose simplices are uniquely determined by their vertices.
That is, each n - simplex has $ n+1 $ distinct vertices and no other n - simplex has this same set of vertices.
Let $ V $ be a vector space over the field $ mathbb R $ of real numbers, and denote its basis by $ {v_k}_{k=1}^n $ .
The complexification of $ V $ is the vector space $ V_mathbb C $ over the field $ mathbb C $ of complex numbers with basis $ {v_k}_{k=1}^n $ .
An open cover $ {U} $ of a topological space $ B $ is locally finite if every point $ bin B $ is in only finitely many $ U $ .
Let $ G $ be a group acting on the sets $ X $ and $ Y $ .
A function $ f:Xto Y $ is $ G $ - equivariant if for all $ xin X $ and all $ gin G $ , $ f(gcdot x) = gcdot f(x) $ .
If the function $ f $ is bijective, then this function is called a $ G $ - isomorphism .
A binary operation $ cdot $ on a set $ A $ is a function $ Atimes Ato A $ .
A representation $ rho $ of a group $ G $ is called completely reducible or is said to have the complete reducibility property if it can be written as a direct sum of irreducible representations.
To define the Lebesgue integral for a general real - valued function, we must first define it for nonnegative functions.
Let $ (X,Sigma,mu) $ be a measure space and let $ f $ be a nonnegative, measurable, extended - real - valued function on $ X $ (that is, $ f(x) geq 0 $ for all $ x in X $ , and it may be the case that $ f(x) = +infty $ ).
Define $ int_X fdmu = supleft{int_X sdmu mid 0leq sleq f, s text{ simple}right} $ where the supremum is taken over the Lebesgue integrals of nonnegative simple functions $ s $ less than $ f $ .
Now let $ f $ be a measurable extended - real - valued function on $ X $ .
Then $ f $ may be written as $ f = f^+ - f^ - $ where $ f^+(x) =} quadtext{and}quad f^ - (x) = } $ so that $ f^+ $ and $ f^ - $ are both nonnegative and measurable.
Note that $ |f| = f^+ + f^ - $ .
The Lebesgue integral of the measurable function $ f $ exists if at least one of $ int f^+dmu $ and $ int f^ - dmu $ is finite.
In this case, define $ int f dmu = int f^+dmu - int f^ - dmu $ .
If $ int|f|dmu < infty $ , then $ f $ is Lebesgue integrable .
A sequence of homomorphisms $ cdots to A_{n+1}to A_n to A_{n - 1}to cdots $ where the arrow between $ A_{k+1} $ and $ A_{k} $ is labelled $ alpha_{k+1} $ is an exact sequence if the kernel of $ alpha_k $ is equal to the image of $ alpha_{k+1} $ .
That is, if $ text{Ker}(alpha_n) = text{Im}(alpha_{n+1}) $ .
Equivalently, an exact sequence is a chain complex whose homology groups are all trivial.
A binary operation $ cdot $ on the set $ A $ is commutative if for all $ a,bin A $ , $ ab = ba $ .
A Lie group is a group that is also a manifold.
Let $ (X,leq) $ be a poset and let $ Ssubseteq X $ .
The infimum of this subset, if it exists, is a lower bound $ a $ of $ S $ in $ X $ such that
for all lower bounds $ y $ of $ S $ in $ X $ , $ yleq a $ .
That is, the infimum is the greatest lower bound.
Let $ F $ be a field.
An absolute value on $ F $ is a map $ |cdot|:Fto mathbb R $ such that (i) $ |x| geq 0 $ for all $ xin F $ ; (ii) $ |x| = 0 $ if and only if $ x=0 $ ; (iii) $ |x+y| leq |x| + |y| $ for all $ x,yin F $ ; (iv) $ |xcdot y| = |x|cdot |y| $ for all $ x,yin F $ .
Together, $ 1 $ and $ 2 $ make up the condition that $ |cdot| $ be positive definite while $ 3 $ is just the triangle inequality.
Let $ R $ be a ring and $ M $ and $ N $ $ R $ - modules.
If $ f:Mto N $ is a module homomorphism, the kernel of $ f $ is the set $ {min Mmid f(m) = e_N}=text{ker}(f)subseteq M $
The compact symplectic group $ text{Sp}(n) $ is the group intersection of the complex symplectic group $ text{Sp}(2n,mathbb C) $ and the unitary group $ text{U}(2n) $ : $ text{Sp}(n) = text{Sp}(2n,mathbb C)captext{U}(2n) $ .
It is a matrix Lie group.
Let $ P $ be a partition of a set of the interval $ a,b $ .
Then arrange the elements of $ P $ in increasing order: $ a = t_0 < t_1< cdots< t_k = b $ .
The $ t_i, t_i+1 $ are the subintervals of $ a,b $ given by $ P $ .
The quadratic integers are algebraic integers of degree (ii) Equivalently, they are solutions to monic, integer - valued polynomials of degree (ii) That is, solutions to equations of the form $ x^2+bx+c=0 $ .
For each integer $ d $ , we have particular rings of quadratic integers given by $ mathcal O_d = } $ Moreover, these are Euclidean domains with norm $ Nleft(x+ysqrt{d}right) = x^2+y^2sqrt{d} $ .
If we define for $ alpha = x+ysqrt{d} $ its conjugate $ overline a = x - ysqrt d $ , then $ N(alpha) = alphaoverlinealpha $ .
Let $ C $ be a compact, rectifiable set in $ mathbb R^{n - 1} $ .
Let $ phi,psi: C tomathbb R $ be continuous functions such that $ phi(x) leq psi(x) $ for all $ x in C $ .
Then the set $ S = {(x,t)mid xin C, phi(x) leq t leq psi(x)} subset mathbb R^{n - 1}times mathbb R $ is called a simple region in $ mathbb R^n $ .
A point $ x in X $ a topological space is a limit point of the subset $ Asubset X $ if it is the limit of some sequence of elements of $ A $ .
Let $ (X,Sigma,mu) $ be a measure space and let $ {f_n}_{ninmathbb N} $ be a sequence of functions $ f_n: Xto mathbb R $ .
The sequence converges almost uniformly to $ f $ if for all $ varepsilon > 0 $ , there exists a set $ Ain Sigma $ of measure less than $ varepsilon $ such that $ f_n $ converges uniformly to $ f $ on $ Xsetminus A $ .
A face of an n - simplex $ v_0,dots,v_n $ is the subsimplex with vertices any nonempty subset of the $ v_i $ .
By convention, these are ordered according to their order in the parent simplex.
Let $ (X, Sigma, mu) $ be a measure space.
The Lebesgue integral of the characteristic function of a measurable set $ A in Sigma $ is defined to be $ int chi_Admu = mu(A) $ .
Let $ (V,rho) $ be a representation of the group $ G $ .
The $ G $ - invariant elements of $ V $ is $ V^G = {vin Vmid rho(g)v=v}, $ the set of vectors in $ V $ left unchanged by the action of $ G $ .
Let $ C $ be a category.
A morphism $ fin text{Hom}_C(A,B) $ is a monomorphism if for all objects $ Z $ of $ C $ and all morphisms $ alpha,betain text{Hom}_C(Z,A) $ , $ fcirc alpha = fcirc betaimplies alpha = beta $ .
Let $ L $ be a Lie algebra.
A vector subspace $ K $ of $ L $ is a subalgebra of $ L $ if $ xy in K $ whenever $ x,y in K $ .
Moreover, $ K $ is itself a Lie algebra.
A module over a ring $ M $ is semisimple if it may be written as the direct sum of simple modules.
A vector space $ V $ over the field $ F $ is a set together with two operations "addition" ( $ +: Vtimes V to V $ ) and "scalar multiplication" ( $ cdot: Ftimes V to V $ ) satisfying the following eight axioms: (i) Addition is associative (ii) Addition is commutative (iii) There exists a vector $ 0 in V $ such that $ 0 + v = v $ for all $ v in V $ .
( $ 0 $ is the additive identity in $ V $ ) (iv) For every $ v in V $ , there exists $ - v in V $ such that $ v + ( - v) = 0 $ .
( $ - v $ is the additive inverse of $ v $ ) (v) $ a(bv)= (ab)v $ for all $ a,b in F $ and $ v in V $ .
(vi) $ 1v = v $ for all $ v in V $ , where $ 1 $ is the multiplicative identity in $ F $ .
(vii) $ a(u+v) = au+av $ for all $ a in F $ , $ u,v in V $ .
(viii) $ (a+b)v = av+bv $ for all $ a,b in F $ , $ vin V $ .
Axioms (vii) and (viii) are distributive laws.
Let $ Dsubsetmathbb R^n $ be connected and open.
A function $ f:Dto mathbb R $ is harmonic if for all $ zin D $ and every
$ varepsilon > 0 $ with $ text{dist}(z,partial D) > varepsilon $ , then $ f(z) = MV(f,z,varepsilon) $ where $ partial D $ denotes the boundary of $ D $ and $ MV $ denotes the circular mean value.
Equivalently, a function $ f $ is harmonic if its Laplacian is 0.
Let $ Usubsetmathbb R^n $ be an open subset and let $ f:Utomathbb R $ .
The point $ a in U $ is a local maximum of $ f $ if there is a neighborhood $ V $ of $ a $ such that for all $ x in V $ , $ f(a)geq f(x) $ .
Euler's totient function , written $ phi(n) $ , is a function $ mathbb Ntomathbb N $ that gives the number of integers less than $ n $ that are coprime to $ n $ .
That is, the number of integers less than $ n $ whose greatest common divisor with $ n $ is (i)
Let $ p:mathcal Etomathcal B $ and $ p':mathcal E'to mathcal B $ be coverings of groupoids.
A map of coverings of groupoids (of $ mathcal B $ , specifically!) is a functor $ g:mathcal Etomathcal E' $ such that the following diagram of functors commutes:
The transpose of an $ mtimes n $ matrix $ A = (a_{ij}) $ is the $ ntimes m $ matrix $ A^T $ whose entries are given by $ a_{ji} $ .
Let $ (X,x_0) $ and $ (Y,y_0) $ be based topological spaces.
The wedge sum $ Xvee Y $ is the quotient of the disjoint union $ Xamalg Y $ given by identifying $ x_0 $ and $ y_0 $ to a single point.
A group is a set $ G $ together with a binary operation $ cdot:Gtimes G to G $ satisfying the following properties: (i) $ cdot $ is associative; (ii) there exists an identity element $ e in G $ such that $ gcdot e = g = ecdot g $ for all $ g in G $ .
(iii) for each $ g in G $ there exists an inverse element $ g^{ - 1} $ such that $ gcdot g^{ - 1} = e = g^{ - 1}cdot g $ .
Groups can be thought of as - associative magmas - semigroups with an identity and with inverses - monoids with inverses - categories with a single object
A component interval of a rectangle $ Q subset mathbb R^n $ is one of the intervals $ a_i,b_i $ that make up the product.
In the category of coverings of groupoids, let $ text{Aut}(mathcal E) subset text{Cov}(mathcal E,mathcal E) $ denote the group of automorphisms of $ mathcal E $ .
Note that there do exist maps $ g:mathcal Eto mathcal E $ that are not isomorphisms because in general, we may have one conjugate subgroup be a proper subgroup of another.
A function of class $ C^{infty} $ is smooth .
That is, a smooth function has continuous total derivative of all orders.
The circle group , usually denoted $ mathbb T $ , is the complex $ 1 $ - torus.
Equivalently, it is the multiplicative group of complex numbers with complex absolute value $ 1 $ .
Let $ (X,Sigma, mu) $ be a measure space.
The essential supremum $ alpha $ is the largest real number such that $ mu({xmid f(x) < alpha}) = 0 $ .
Let $ X $ be a based topological space and let $ I $ be the unit interval.
The reduced cone on $ X $ is the smash product $ CX = Xwedge I $ where $ I $ has as its basepoint $ 1 $ .
Explicitly, $ CX = (Xtimes I)/(({*}times I) cup (Xtimes{1})) $ .
The usual cone is done without identifying the line $ {*}times I $ through the basepoint of $ X $ to a point.
A rectangle $ Q $ in $ mathbb R^n $ is a product of intervals in $ mathbb R $ : $ Q = a_1,b_1times cdotstimes a_n,b_n $ .
Let $ X $ be a topological space.
An open cover of $ A $ is a collection of open subsets of $ X $ such that $ A $ lies in the union of the elements of the collection.
That is, an open covering is a collection $ {G_lambda}_{lambdainLambda} $ of subsets of $ X $ such that $ Asubset bigcuplimits_{lambdainLambda} G_lambda $ .
Let $ M $ be a manifold with boundary.
The boundary points of $ M $ are the points $ pin M $ for which there exists a chart $ (U,phi) $ such that $ phi(p) in partial H^m $ , the boundary of the closed half - space.
A function $ f:Ato B $ is surjective if for all $ b in B $ there exists $ ain A $ such that $ f(a)=b $ .
An element $ x $ of a ring $ R $ is nilpotent if there exists $ n in mathbb Z $ such that $ x^n =0 $ .
A loop is a path $ p $ such that $ p(0) = p(1) $ .
This point is called the basepoint of the loop.
A topological space is a set $ X $ together with a collection $ T $ of subsets of $ X $ .
The elements of $ T $ are called the open sets and they satisfy the following properties: (i) $ Xin T $ and $ emptyset in T $ .
(ii) Arbitrary unions of open sets are open.
(iii) Finite intersections of open sets are open.
The choice of $ T $ is called a topology on $ X $ .
Let $ A $ be a finite abelian group.
The exponent of $ A $ is the least common multiple of the orders of all of the elements of $ A $ .
Let $ (X,mathcal T) $ be a Hausdorff topological space and let $ Sigma $ be a σ - algebra containing $ mathcal T $ (so that $ Sigma $ is at least as fine as the Borel σ - algebra on $ X $ ).
A measure $ mu $ on $ Sigma $ is locally finite if for all $ p in X $ , there exists an open neighborhood $ N_p $ of $ p $ such that the $ mu $ - measure of $ N_p $ is finite.
That is, $ forall p in X, exists N_p in T text{ s.t. } p in N_p text{ and } |mu(N_p)| < infty $ .
Let $ A $ be an algebra over the ring $ R $ .
A left ideal $ I $ of $ A $ is a subalgebra of $ A $ such that $ axin I $ for all $ ain A $ , $ xin I $ .
Equivalently (if we want to elide the subalgebra condition), a left ideal $ I $ of $ A $ is a subset $ Isubset A $ such that for all $ x,yin I $ , $ ain A $ , and $ rin R $ , (i) $ x - yin I $ ; (ii) $ rxin I $ ; (iii) $ axin I $ .
Let $ V $ be a finite - dimensional vector space with $ text{dim}(V) = ell +1 $ .
Let $ mathfrak{sl}(V) $ be the set of endomorphisms of $ V $ with trace equal to zero.
Since $ text{Tr}(xy)= text{Tr}(yx) $ and $ text{Tr}(x+y) =text{Tr}(x)+text{Tr}(y) $ , $ mathfrak{sl}(V) $ is a subalgebra of the general linear algebra $ mathfrak{gl}(V) $ .
A neighborhood of a point $ x $ in a topological space $ X $ is a subset $ N $ of $ X $ containing $ x $ which contains an open set $ Usubseteq N $ that also contains $ x $ .
Let $ R $ be a ring.
A nonzero element $ rin R $ is a zero divisor if there exists nonzer $ sin R $ such that $ sr = 0 $ or $ rs = 0 $ .
A subspace $ A $ of a topological space $ X $ is compactly closed if the inverse image $ g^{ - 1}(A) $ is closed in $ K $ for any map $ g:Kto X $ from a compact Hausdorff space $ K $ into $ X $ .
Let $ L $ be a Lie algebra.
Its derived series is given by the derived algebras $ } $
The transition function of two charts $ (U,phi) $ and $ (V, psi) $ on a manifold $ M $ is the composition $ psi circ phi^{ - 1}: phi(Ucap V)to psi(Ucap V) $ .
Let $ G $ be a group.
The lower central series for $ G $ is defined as follows: Let $ G_0=G $ , let $ G_1 = G,G $ , and for each $ i geq 1 $ , define $ G_{i+1} = G,G_{i} $ where the bracket notation denotes a particular subgroup of the commutator subgroup: $ G,H = {ghg^{ - 1}h^{ - 1}mid gin G, hin H} $ .
Note that this series may or may not reach the trivial subgroup.
A norm $ |cdot | : Ato mathbb R $ on an algebra $ A $ is called multiplicative if $ |1_A| = 1 $ and for all $ a,bin A $ , $ |ab| leq |a|cdot |b| $ .
A topological space $ X $ is semi - locally simply connected if every point $ bin B $ has a neighborhood $ U $ such that the map $ pi_1(U,b)to pi_1(B,b) $ between fundamental groups is the trivial group homomorphism.
Let $ G $ be a finite group.
Consider the space of complex - valued class functions on $ G $ .
It has a Hermitian inner product given by $ langle f_1,f_2rangle_G = frac{1}{|G|}sum_{gin G} f_1(g)overline{f_2(g)} $ where $ |G| $ denotes the order of $ G $ .
The normalizer of a subspace $ U $ of a vector space $ V $ is given by $ N_V(U) = {xin Vmid x,Usubset U} $ .
Let $ (X,Sigma,mu) $ and $ (Y,T,nu) $ be measure spaces and let $ T: L_1(X,mu) to L_1(Y,nu) $ be an operator that maps measurable functions on $ X $ to measurable functions on $ Y $ .
Let $ 1leq p,qleq infty $ .
Then $ T $ is of type strong $ (p,q) $ if there exists a constant $ c > 0 $ such that $ ||Tf||_q leq c||f||_p $ .
This immediately implies that (i) $ T $ is a bounded operator.
(ii) Weak $ (p,infty) $ is equivalent to strong $ (p,infty) $ .
Let $ f:Ato B $ be a function.
A left inverse of $ f $ is a function $ g:Bto A $ such that $ gcirc f(a) = a $ for all $ a in A $ .
A right inverse of $ f $ is a function $ g:Bto A $ such that $ fcirc g(b) = b $ for all $ bin B $ .
An inverse of $ f $ is a function $ f^{ - 1} $ that is both a left and a right inverse of $ f $ .
A singular n - simplex is a map $ sigma: Delta^n to X $ from the standard n - simplex to a topological space $ X $ .
The opposite category $ mathcal C^{text{op}} $ of a category $ mathcal C $ is a category consisting of the same objects as $ mathcal C $ , but with morphisms $ mathcal C^text{op}(A,B) = mathcal C(B,A) $ .
Let $ f:S^nto S^n $ be a function.
The induced homomorphism $ f_*: tilde H_n(S^n)to tilde H_n(S^n) $ is a homomorphism from an infinite cyclic group to itself.
Thus it must be of the form $ f_*(alpha) = dalpha $ for some integer $ d $ depending only on $ f $ .
The integer $ d $ is the degree of $ f $ .
Let $ F $ be a D - shaped diagram in the category $ mathcal Dmathcal C $ .
The colimit of $ F $ , written $ text{colim}F $ , is an object of $ mathcal C $ together with a morphism of diagrams $ iota: Fto underline{text{colim} F} $ that is initial among all such morphisms.
That is, if $ eta:Fto underline A $ is a morphism of diagrams, then there is a unique map $ tilde eta:text{colim}(F)to A $ in $ mathcal C $ such that $ tilde etacirc iota = eta $ .
We can also express this as a diagram.
For each map $ d:Dto D' $ in $ mathcal D $ , the following diagram commutes:
Let $ text{Aut}(E)subset text{Cov}(E,E) $ in the category of covering spaces of some connected locally path - connected topological space $ B $ denote the group of automorphisms of the cover $ E $ .
A linear transformation $ T:Vto V $ is self - adjoint if $ T^*=T $ .
The inverse of an $ ntimes n $ matrix $ A $ is an $ ntimes n $ matrix $ A^{ - 1} $ such that the products $ AA^{ - 1} = A^{ - 1}A = I_n $ where $ I_n $ is the $ ntimes n $ identity matrix.
Let $ Msubsetmathbb R^n $ .
Then $ M $ is an embedded $ m $ - dimensional manifold if for all $ xin M $ , there exists a neighborhood $ U subsetmathbb R^n $ of $ x $ and a smooth function $ f:Utomathbb R^{n - m} $ such that $ Mcap U = f^{ - 1}(0) $ and $ Df(y):mathbb R^ntomathbb R^{n - m} $ is surjective for all $ y in U $ .
An equivalence relation $ sim $ on a set $ X $ is a binary relation that is reflexive, symmetric, and transitive.
That is, for all $ a,b,c in X $ , (i) $ asim a $ .
(ii) $ a sim b $ if and only if $ b sim a $ .
(iii) If $ a sim b $ and $ b sim c $ , then $ asim c $ .
A subset $ A $ of a topological space $ X $ has the Baire property if there exists an open set $ G $ such that the symmetric difference of $ A $ and $ G $ ( $ A triangle G $ ) is meager.
A field $ k $ is algebraically closed if every non - constant polynomial in $ kt $ has a root in $ k $ .
Let $ I $ be the unit interval.
A pair of compactly generated topological spaces $ (X,A) $ (where $ A $ is a subspace of $ X $ ) is called an NDR pair (stands for neighborhood deformation retract pair) if there exists a map $ u:Xto I $ such that the fiber $ u^{ - 1}(0) = A $ and if there exists a homotopy $ h:Xtimes I to X $ such that $ h_0 = text{id} $ , $ h(a,t) = a $ for all $ ain A $ and $ tin I $ , and $ h(x,q) in A $ if $ u(x)<1 $ .
Such a pair is a DR pair if $ u(x) < 1 $ for all $ xin X $ , and this implies that $ A $ is a deformation retract of $ X $ .
Given a group $ G $ and an action of $ G $ on two sets $ X $ and $ Y $ , then the collection of maps between $ X $ and $ Y $ (denoted $ text{Maps}X,Y $ ) is naturally a G - set in the following way: Given a function $ fin text{Maps}X,Y $ , consider the graph $ text{Graph}(f) = {(x,f(x))in Xtimes Y mid xin X} $ .
Let $ G $ act diagonally on $ Xtimes Y $ : that is, define $ g.(x,y) = (g.x,g.y) $ .
Then define $ g^*f $ by $ text{Graph}()g^*f = g(text{Graph}(f)) $ .
Explicitly, for $ gx' = x $ , we have $ g^*f(x) = g(f(x')) implies g^*f(x) = gf(g^{ - 1}x) $ .
We can see the map as the composite $ g^*f: Xoverset{g^{ - 1}}{to} X overset{f}{to} Y overset{g}{to} Y $ .
The general linear group $ GL(n) $ or $ GL(V) $ is the group of automorphisms of an $ n $ - dimensional vector space $ V $ .
Equivalently, it is the group of invertible $ ntimes n $ matrices.
A metric space is separable if it contains a countable, dense subset.
Let $ L $ be a formal language of propositional logic and let $ Gammasubseteq text{Form}(L) $ .
A model for $ Gamma $ is a truth function $ t:text{Form}(L) to {0,1} $ such that $ t(gamma) = 1 $ for all $ gammain Gamma $ .
For any $ phi in text{Form}(L) $ , we say that $ Gamma $ models $ phi $ if for every model $ t $ of $ Gamma $ , $ t(phi) = 1 $ .
The volume of a rectangle $ Q $ in $ mathbb R^n $ is the product of the lengths of the component intervals of $ Q $ : $ text{vol}(Q) = (b_1 - a_1)cdots(b_n - a_n) $ .
The symmetric differential basis is the differential basis consisting of balls centered around each $ x in mathbb R^n $ .
The equivalence classes with respect to the connection equivalence relation are called connected components of $ X $ .
Let $ (X,Sigma) $ be a measurable space with $ mu $ a signed measure.
The positive variation of $ mu $ is the function $ pi $ given by $ pi(A) = sup_{Ssubset A} mu(S) $ .
Let $ X $ be a topological space and let $ A $ be a subspace of $ X $ .
Define a chain complex whose elements are the quotients $ C_n(X,A) = C_n(X)/C_n(A) $ of elements of the chain complex on the groups of n - chains of $ X $ and $ A $ , respectively.
Note that chains in $ A $ are trivial in $ C_n(X,A) $ .
The boundary homomorphism $ partial:C_n(X)to C_{n - 1}(X) $ sends $ C_n(A) $ to $ C_{n - 1}(A) $ , so it induces a quotient boundary map $ partial: C_n(X,A)to C_{n - 1}(X,A) $ .
The requisite relation in which composition of consecutive boundary maps is zero is met because it is met before taking the quotient.
The homology groups of this chain complex are called the relative homology groups $ H_n(X,A) $ .
A partition $ P $ of a closed interval $ a,b $ is a finite collection of points in $ a,b $ such that $ a,b in P $ .
More generally, a partition $ P $ of a rectangle $ Q $ in $ mathbb R^n $ is a collection $ P = {P_i}_{i=1}^n $ of partitions of the component intervals $ a_i,b_i $ of $ Q $ .
The Stirling numbers of the second kind count the number of ways to partition a set of $ n $ distinct objects into $ k $ indistinct sets, leaving none empty.
They are denoted $ S(n,k) $ .
They also count the number of different equivalence relations with exactly $ k $ equivalence classes that can be defined on a set of size $ n $ .
The Choquet game is a topological game played
A topological $ m $ - dimensional manifold with boundary is a topological space $ M $ such that (i) for all $ p in M $ , there exists an open neighborhood $ U $ of $ p $ homeomorphic to an open set in the closed half - space $ H^m $ ; (ii) $ M $ is Hausdorff; (iii) $ M $ is second - countable.
Let $ X $ be a topological space.
The $ n $ th reduced homology group of $ X $ , denoted $ tilde H_n(X) $ , is the $ n $ th homology group of the chain complex $ cdots to C_2(X) to C_1(x) to C_0(X) to mathbb Zto 0 $ where the maps between the groups $ C_i $ of n - chains are the usual boundary homomorphisms and the map (call it $ varepsilon $ ) $ C_0(X)to mathbb Z $ is given by $ varepsilonleft(sum_i n_isigma_iright) = sum_i n_i $ .
Reduced homology groups are introduced so that a point has trivial homology group for all values of $ n $ including zero.
Let $ R $ be a ring with multiplicative identity $ 1 $ .
A left $ R $ - module $ M $ is an abelian group $ (M, +) $ and another operation $ cdot : Rtimes M to M $ such that fior all $ r,sin R $ and $ x,yin M $ , (i) $ rcdot (x+y) = rcdot x + rcdot y $ ; (ii) $ (r+s)dot x = rcdot x + scdot x $ ; (iii) $ (rs)cdot x = rcdot (scdot x) $ ; (iv) $ 1cdot x = x $ .
The operation $ cdot $ is called scalar multiplication .
Let $ X $ be a metric space and $ K subset X $ .
For $ x in X $ , the distance from $ x $ to $ K $ is given by the infimum $ text{dist}(x,K) = inf_{yin K}(rho(x, y)) $ .
A Borel function is a measurable function between Borel spaces.
A measure space $ (X,Sigma, mu) $ is bounded if $ mu(X) < infty $ .
Let $ lambda = {lambda_1,dots,lambda_k} $ be a partition.
Let $ alpha_i = {lambda_i, lambda_i + 1, dots, lambda_{i+1} - 1} $ so that $ {1,dots,n} = coprod_{i=1}^k alpha_i $ as disjoint subsets.
Corresponding to this decomposition is the Young subgroup of the symmetric group $ S_n $ given by $ S_{alpha_1} times cdots times S_{alpha_k} $ where $ S_{alpha_i} = {sigmain S_n mid sigma(j) = j text{ for all } jnotin alpha_i} $ .
Let $ Usubsetmathbb R^n $ be an open subset and let $ g: U tomathbb R^n $ be differentiable.
Then Jacobian determinant of $ g $ is the function $ Jg:Utomathbb R $ given by the absolute value of thedeterminant of the Jacobian matrix of $ g $ : $ Jg(x)= |text{det}(Dg(x))| $ .
The Euclidean inner product is an inner product on $ mathbb R^n $ given by $ langle x,yrangle = sum_{i=1}^n x_iy_i $ .
The notation $ langle x,yrangle $ notation is often written $ xcdot y $ instead.
A submodule $ N $ of a module over a ring $ M $ is a subset $ Nsubset M $ that forms a module over a ring under the same operations.
It must form a group under $ + $ .
A set is uncountable if it is not countable, that is if it has cardinal number greater than that of the natural numbers $ mathbb N $ .
A function $ f:Ato B $ is injective if for all $ a,b in A $ , $ aneq b $ implies that $ f(a)neq f(b) $ .
A binary relation $ R $ on sets $ X $ and $ Y $ is a subset of $ X times Y $ .
If $ (x,y) in R $ , then we say that " $ x $ is related to $ y $ ."
An associative algebra is an algebra $ A $ with an associative binary operation
An ideal $ I $ of a ring $ R $ is principal if it can be generated by a single element.
A lattice $ (L,lor,land) $ is distributed if it satisfies the following distributive laws for all $ a,b,cin L $ : - $ alor (bland c) = (alor b)land (alor c) $ ; - $ aland(blor c) = (aland b)lor (aland c) $ .
The orthogonal group $ text{O}(n) $ is the group of $ ntimes n $ orthogonal matrices.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
A subtree (a subgraph that happens to be a tree) of a graph $ X $ is maximal if it is contained in no strictly larger tree.
The transposition $ sigma_{ij} $ is a permutation in $ S_n $ such that for some $ i,j in {1,dots ,n} $ , $ sigma_{ij}(i) = j, sigma_{ij}(j) = i, text{ and } sigma_{ij}(k) = k $ for all $ kneq i,j $ .
Equivalently, a transposition is a 2 - cycle.
The fundamental groupoid $ Pi(X) $ of a topological space $ X $ is the category whose objects are the points of $ X $ and whose morphisms $ xto y $ are the equivalence classes of paths from $ x $ to $ y $ .
Note that this means that the set of endomorphisms of an object $ x $ in $ Pi(X) $ is exactly the fundamental group $ pi_1(X) $ .
The fundamental groupoid may be thought of as a functor from the category of topological spaces to the category of groupoids.
Two representations of a group $ G $ are equivalent if there exists an invertible intertwiner between them.
The special orthogonal group $ text{SO}(n) $ is the group of $ ntimes n $ orthogonal matrices with unit determinant.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
Let $ X $ be a topological space.
For any $ Asubset X $ , the exterior $ text{ext}(A) $ of $ A $ is the complement of the closure of $ A $ .
Let $ A $ be an algebra over the field $ mathbb k $ and let $ ain A $ .
The spectrum of $ a $ is $ text{spec}(a)= {lambdainmathbb kmid lambda - a text{ is not invertible}} $ .
The dual of a normed linear space $ X $ is the set of bounded linear functions $ f: Xto mathbb R $ .
This set is itself a normed linear space under the linear transformation norm.
Let $ hat{mathbb C} $ be the Riemann sphere.
The filled Julia set of a polynomial $ f $ is $ K(f) = {zinhat{mathbb C}mid exists T in mathbb R text{ s.t. }|f^{circ n}(z)| leq T forall n} $ .
Let $ L $ be a Lie algebra with subalgebra $ K $ .
If $ K $ is equal to its own normalizer of a vector space (i.e. $ K = N_L(K) $ ), then $ K $ is called self - normalizing .
A property is typical if it holds for every element of a residual set.
A Gale - Stewart game is a two - player game in which both players have perfect information, make an infinite sequence of moves, and there are no draws.
Let $ f $ be in the Schwartz space of the p - adic field $ mathbb Q_p $ .
That is, let $ fin C_c^infty(mathbb Q_p) $ .
Then the analogue to the Fourier transform of $ f $ is given by $ widehat{f}(u) = int_{mathbb Q_p}f(x)psi_u(x)text dx $ where $ psi_u $ is the multiple of the Basic Additive Character on $ mathbb Q_p $ and the integral is taken with respect to the Haar measure on $ mathbb Q_p $ .
A linear functional on a vector space $ V $ with underlying field $ F $ is a linear transformation from $ V $ to $ F $ when considered as a one - dimensional vector space over itself.
A matrix Lie group is a closed subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
Let $ G $ be a semigroup and let $ S $ be a subset of $ G $ .
The centralizer of $ S $ is $ C_G(S) = {gin Gmid gs=sg text{ for all } sin S} $ .
When $ G $ is a group, this may be written equivalently as $ C_G(S) = {gin Gmid gsg^{ - 1} =s text{ for all } sin S} $ .
Let $ sim $ be an equivalence relation on the set $ S $ .
The quotient of $ S $ by $ sim $ (denoted $ S/sim $ ) is the set of equivalence classes of elements of $ S $ with respect to $ sim $ .
Let $ T:Vto V $ be a linear transformation from the vector space $ (V,F) $ to itself.
Then $ lambda in F $ is an eigenvalue of $ T $ if there exists $ vin V $ such that $ T(v) = lambda v $ .
A topological space is contractible if it is homotopy equivalent to a point.
Let $ G $ be a group acting on a the set $ X $ .
A function $ f:Xto Y $ is $ G $ - invariant if for all $ gin G $ and all $ xin X $ , $ f(gx) = f(x) $ .
A Lie algebra $ L $ is solvable if some element $ L^{(n)} $ of its derived series of a Lie algebra is equal to 0.
The descending central series of ideals of a Lie algebra $ L $ is given by the derived algebras $ } $ Where $ L^{(1)} $ is the second element of the derived series of $ L $ .
An $ n $ - simplex is the smallest convex set in $ mathbb R^m $ containing $ n+1 $ points $ v_0,dots, v_n $ that do not lie in a hyperplane of dimension less than $ n $ .
The standard $ n $ - simplex is the set $ Delta^n = {(t_0,dots, t_n)inmathbb R^{n+1} mid sum_i t_i =1 text{ and } t_i geq 0 forall i} $ .
The order of the vertices $ v_i $ is part of the data of an $ n $ - simplex.
Writing an $ n $ - simplex as $ v_0,dots, v_n $ determines orientation of the edges $ v_i, v_j $ by their order.
Let $ R $ be a ring and let $ M $ , $ N $ and $ L $ be $ R $ - modules.
Then given homomorphisms $ f:Mto L $ and $ g:Nto L $ , there exists a unique homomorphism $ foplus g:Moplus Nto L $ from the direct sum $ Moplus N $ to $ L $ defined by $ foplus g(m,n) = f(m) + g(n) $ .
Let $ (X,*) $ be a magma with identity element $ e $ .
An element $ bin X $ is a right inverse element for $ ain X $ if $ a * b=e $ .
A nonempty family $ R $ of sets is a ring if it is closed under union and relative complement.
That is, for all $ A, B in R $ , - $ Acup B in R $ ; - $ Asetminus B in R $ .
Let $ G $ be a group and let $ gin G $ .
Define a homomorphism $ varphi_g: Gto G $ by $ varphi_g(x) = gxg^{ - 1} $ for all $ xin G $ .
These $ varphi_g $ are in fact automorphisms and moreover $ varphi_gcircvarphi_h = varphi_{gh} $ .
The $ varphi_g $ are called inner automorphisms of $ G $ .
Let $ W $ and $ V $ be vector spaces over the same ground field $ F $ .
A function $ T: V to W $ is a linear transformation if for all $ v,w in V $ and $ alpha,beta in F $ , $ T(alpha v+beta w) = alpha T(v) + beta T(w) $ .
This is the correct notion for a homomorphism of vector spaces.
If a linear transformation is bijective, then it is an isomorphism of vector spaces.
Let $ R $ be a ring.
An ideal $ I $ of $ R $ is a subset that is both a left ideal and a right ideal.
A surface is a two - dimensional manifold.
Let $ (X,Sigma,mu) $ and $ (Y,T,nu) $ be measure spaces and let $ T: L_1(X,mu) to L_1(Y,nu) $ be an operator that maps measurable functions on $ X $ to measurable functions on $ Y $ .
Let $ 1leq pleq infty $ and $ 1leq q < infty $ .
The operator $ T $ is of type weak $ (p,q) $ if there exists a constant $ c > 0 $ such that $ mu({x in Xmid |Tf(x)|>t})leqleft(frac{c||f||_p}{t}right)^q $ .
If $ q = infty $ , then $ T $ is weak $ (p,infty) $ if there exists a constant $ c > 0 $ such that $ ||Tf||_infty leq c||f||_p $ .
A graph $ X $ is locally finite if each vertex is a boundary point of only finitely many eges or if $ X $ is a locally compact topological space.
Let $ L $ be a formal language of propositional logic and let $ Gammasubseteq text{Form}(L) $ .
We say that $ Gamma $ is inconsistent if it is not consistent.
Let $ F $ be a field.
Then translation by en element $ bin F $ is a group homomorphism $ t_b:(F, +)to F $ given by $ t_b(x) = x+b $ and multiplication by a nonzero element $ ain F $ gives a group homomorphism $ m_a:Fto F $ given by $ m_a(x) = ax $ .
Considering $ bmapsto t_b $ and $ amapsto m_a $ gives rise to two more homomorphisms $ (F,+)to S_F $ and $ (F^times,times)to S_F $ respectively.
Thus the multiplicative and additive groups on $ F $ both act on $ F $ .
Combining the two actions, we get the affine transformations of $ F $ $ text{Aff}(F) = {t_bm_amid bin F, ain F^times} $
A measurable space is the pair $ (X,mathcal M) $ consisting of a set $ X $ and a σ - algebra $ mathcal M $ of subsets of $ X $ .
A subset $ E $ of $ X $ is measurable if it is an element of $ mathcal M $ .
An algebraic integer is a number that is the root of a monic polynomial over $ mathbb Z $ .
The algebraic integers form a ring.
Let $ (X,Sigma, mu) $ be a measure space.
For $ f in L_1(X,mu) $ , the Hardy - Littlewood maximal operator $ Mf $ is defined as follows: $ (Mf)(x)=sup_{r>0}frac{1}{mu(B(x,r))}int_{B(x,r)}|f|text dmu $ .
Let $ X $ be a topological space and let $ C_n(X) $ be the free group abelian group on the set of singular n - simplices in $ X $ .
Elements of $ C_n(X) $ are called singular $ n $ - chains and are written as finite formal sums $ sum_i n_isigma_i $ for $ n_iinmathbb Z $ and $ sigma_i:Delta^nto X $ .
A magma is a set $ X $ together with a binary operation $ * $ that is closed under $ * $ .
That is, $ a*b in X $ for all $ a,b in X $ .
Let $ Usubsetmathbb R^n $ be an open subset and let $ f:Utomathbb R $ be of class at least $ C^2 $ .
The Hessian matrix of $ f $ at the point $ a in U $ is the matrix $ H = }_{i,j=1}^n $ of the mixed second partial derivatives of $ f $ .
Let $ L $ be a Lie algebra with basis $ {x_1,dots,x_n} $ .
The structure constants $ a_{ij}^k $ are the constants in the expressions
$ x_ix_j = sum_{k=1}^n a_{ik}^k x_k $ .
Let $ mathfrak g $ be a Lie algebra and let $ (V,rho_V) $ be a representation of $ mathfrak g $ .
Then the dual representation of $ mathfrak g $ is the pair $ (V^*,rho_{V^*}) $ where $ V^* $ is the dual space of $ V $ and $ rho_{V^*} $ is given by $ rho_{V^*}(x) = - rho_V(x)^ * $ for all $ x in mathfrak g $ .
A unital subring of a ring $ R $ is a subset $ S $ of $ R $ that is itself a ring under the binary operations in $ R $ with the same additive and multiplicative identities.
Let $ mathfrak g $ be a Lie algebra over the field $ F $ with a basis $ {x_i} $ and a Lie bracket given by $ x_i,x_j = sum_k = c_{ij}^kx_k $ .
The universal enveloping algebra $ mathcal U(mathfrak g) $ is the associative algebra generated by the $ x_i $ with the relations $ x_ix_j - x_jx_i = sum_k c_{ij}^k x_k $ .
The $ c_{ij}^k $ are the structure constants of $ mathcal U(mathfrak g) $ .
The universal property for the universal enveloping algebra is as follows: Let $ mathfrak g $ be a Lie algebra over the field $ F $ and let $ A $ be an associative algebra over $ F $ .
By endowing $ A $ with the commutator as a Lie bracket, $ A $ becomes a Lie algebra.
Let $ phi: mathfrak g to A $ be a Lie algebra homomorphism.
Then the universal enveloping algebra $ mathcal U(mathfrak g) $ is the algebra such that the following diagram commutes: In a basis - free setting, let $ mathfrak g $ be a Lie algebra over the field $ F $ and let $ Tmathfrak g $ be the tensor algebra of $ mathfrak g $ .
Consider the ideal $ mathcal I = langle xotimes y - yotimes x - x,y mid x,y in mathfrak grangle $ of $ Tmathfrak g $ .
Then the universal enveloping algebra $ mathcal U(mathfrak g) $ is defined as the quotient $ Tmathfrak g/mathcal I $ .
The category of sets is the category whose objects are sets and whose morphisms are functions between sets.
A topological space $ X $ is totally disconnected if the connected components of $ X $ are exactly the singletons.
A (unital) ring is a set $ R $ together with two binary operations $ + $ and $ cdot $ such that (i) $ R $ is an abelian group under $ + $ ; (ii) $ R $ is a monoid under $ cdot $ ; (iii) $ cdot $ distributes over $ + $ : $ acdot (b+c)=(acdot b) + (acdot c) $ and $ (b+c)cdot a = (bcdot a) + (ccdot a) $ .
An elementary matrix is the result of applying a row operation to the identity matrix.
Let $ u,v $ be elements of the group $ G $ .
Define $ u,v = uvu^{ - 1}v^{ - 1} $ .
The subgroup of $ G $ generated by $ u,v $ for all pairs $ u,vin G $ is denoted $ G,G $ and is called the commutator subgroup of $ G $ .
A function $ f: Xto Y $ between metric spaces $ (X, rho) $ and $ (Y,sigma) $ is an isometry if for all $ a,b in X $ , $ sigma(f(a), f(b)) = rho(a,b) $ .
Two metric spaces are isometric if there exists an isometry between them.
The complex symplectic group $ text{Sp}(2n,mathbb C) $ is the group of $ 2ntimes 2n $ complex - valued matrices that preserve the form $ omega $ as in the definition of the real symplectic group.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
The image of a function $ f:Xto Y $ is the set $ text{im}(f) = {T(x) in Y mid x in X}subset Y $ .
Let $ (X,Sigma,mu) $ be a measure space.
A measurable subset $ A $ of $ X $ is outer regular if $ mu(A) = inf{mu(G) mid Gsupseteq A, Gtext{ open and measurable}} $ .
Let $ R $ be a commutative ring and let $ S $ be a subset of $ R $ that is closed under multiplication and contains the multiplicative identity.
Then $ S^{ - 1}R = left{leftfrac{r}{s}rightmid rin R,sin Sright} $ where equivalence of fractions is given by $ leftfrac{r _1}{s_1}right =leftfrac{r_2}{s_2}right $ if and only if there exists $ k in S $ such that $ kr_1s_2 = kr_2s_1 $ .
This is itself a ring under the usual operations we know and love for fractions of integers.
If $ S $ is exactly the nonzero elements of $ R $ , then this forms a field.
Let $ R $ be a ring and let $ M $ be a free $ R $ - module.
Then the cardinality of a basis for $ M $ is the rank of $ M $ .
We can also define the rank for modules that are not free: it is the maximum number of linearly independent elements of $ M $ .
Let $ A $ be an algebra over an algebraically closed field $ mathbb k $ and let $ M $ be a semisimple module over $ A $ of finite dimension over $ mathbb k $ .
The number of times the module $ V $ appears in a direct sum decomposition of $ M $ is denoted $ M:V $ and is called the multiplicity of $ V $ in $ M $ .
A short exact sequence of chain complexes is a diagram like the following: where the rows $ A $ , $ B $ , and $ C $ are chain complexes and the columns are exact sequences.
Let $ M $ and $ N $ be modules over the ring $ R $ .
A module homomorphism between $ M $ and $ N $ is a function $ f:Mto N $ such that for any $ x,y in M $ and $ rin R $ , (i) $ f(x+y) = f(x) + f(y) $ (ii) $ f(rx) = rf(x) $ where the operation in (2) is the scalar multiplication.
If a module homomorphism is bijective, then it is an isomorphism of modules.
Let $ mathcal P(X) $ be the power set of $ X $ .
An outer measure on $ X $ is a function $ mu: mathcal P(X) to 0,infty $ such that - $ mu(emptyset) = 0 $ - if $ A, B subset X $ with $ Asubset B $ , then $ mu(A) leqmu(B) $ - for arbitrary subsets $ B_i subset X $ for $ i in mathbb N $ , we have $ muleft(bigcuplimits_{iinmathbb N} B_i right)leq sumlimits_{iinmathbb N}mu(B_i) $ .
Let $ V $ be a vector space.
The set $ W subseteq V $ spans $ V $ if for all $ v in V $ , there exists $ n in mathbb N $ , a collection $ {alpha_i}_{i=1}^n subset mathbb R $ , and a collection $ {v_i}_{i=1}^n subset W $ such that $ v = sum_{i=1}^n alpha_i v_i $ .
Let $ X $ be a set.
The symmetric group $ S_X $ is the set of all bijections from $ X $ to itself.
When $ X $ is finite with cardinal number $ n $ , it is denoted $ S_n $ and consists of the set of permutations on $ n $ letters.
The action of a group $ G $ on the set $ X $ is transitive if for every pair $ x,y in X $ , there exists $ gin G $ such that $ gcdot x=y $ .
Let $ A $ be a Banach algebra with norm $ |cdot | $ .
Given a sequence $ {a_i}_{iin mathbb N} $ of elements of $ A $ , we say that the series $ sum_{i=1}^infty a_i $ is absolutely convergent if $ sum_{i=1}^infty |a_i| <infty, $ i.e. if the series $ {|a_i|}_{iinmathbb N} $ is convergent as real numbers.
This means that the partial sums $ sum_{i=1}^n a_i $ form a Cauchy sequence in $ A $ , which is complete, and therefore must have a limit in $ A $ .
We denote this limit by the infinite sum.
Let $ A $ be an elementary abelian p - group and let $ mathbf ain(a_1,dots,a_r)in A^r $ .
Let $ f_mathbf a:(mathbb Z/pmathbb Z)to A $ be the unique homomorphism satisfying $ f_mathbf a(1,0,dot,0) = a_1, dots, f_mathbf a(0,dots, 1) = a_r $ .
If the function $ f_mathbf a $ is injective, then the $ a_i $ are linearly independent .
Let $ R $ be an integral domain and let $ M $ be an $ R $ - module.
Then $ M $ is torsion - free if its torsion submodule is trivial.
Let $ (X,rho) $ be a metric space.
Then a function $ f:Xto X $ is a contraction mapping if there exists $ r in 0,1) $ such that for all $ x,y in x $ , $ rho(f(x),f(y)) leq rrho(x,y) $ .
The rank of a linear transformation $ T $ is the dimension of the image of $ T $ .
An at - most - countable collection $ {A_i}_{iin I} $ whose elements are called atomic formulae or propositional variables together with some subset of the logical connectors $ (land, lor, rightarrow, leftrightarrow, neg) $ and $ {top, bot} $ is called a language for propositional logic .
Name this whole agglomeration $ L $ .
Then $ text{Form}(L) $ is the smallest set containing $ {A_i}_{iin I} $ , $ top $ , and $ bot $ that is closed under the remaining logical operators, which at their most basic take two (one in the case of $ neg $ ) atomic variables and spit out a new formula.
We know that this set is nonempty because there exists a set $ D $ consisting of all finite strings with characters drawn from $ {A_i}_{iin I} $ , $ top $ , $ bot $ , and $ {land,lor, rightarrow, leftrightarrow, neg} $ .
Consider $ bigcap _{Sin F}S $ where $ F $ is a collection of sets that are subsets of $ D $ satisfying the closure property.
This is the smallest such set, but it does not tell us what the elements look like.
Consider a map $ LC: mathcal P(D) to mathcal P(D) $ given by $ LC(S)= Scup{(B)rightarrow (C), (B)leftrightarrow (C), (B)land (C), (B)lor (C), neg(B) mid B,Cin S} $ .
Let $ H = bigcup_{iin mathbb N} LC^i ({A_j}_{jin I} cup {bot,top}) $ .
Then $ H = text{Form}(L) $ : Let $ S $ be some candidate set for $ text{Form}(L) $ .
It suffices to show that $ Hsubseteq S $ .
To show this it suffices to show that $ LC^i ({A_j}_{jin I} cup {bot,top}) subseteq S $ for all $ i $ .
The base case $ i=0 $ follows from the definition we are looking to satisfy and the hypothesis that $ S $ satisfies it.
The inductive step follows immediately from the definition and its closure property.
Let $ X $ be a topological space.
A path in $ X $ is a continuous function from the unit interval $ I $ to $ X $ . $ p: Ito X $
The order of a group $ G $ is the cardinal number of the underlying set of $ G $ .
The order of a group does not distinguish between different infinite cardinals.
The kernel of a linear transformation $ T:Vto W $ is the set $ text{ker}(T)= {vin Vmid T(v) =0} $ .
It is a vector space.
Let $ R $ be a ring with ideals $ I $ and $ J $ .
Then the sum of $ I $ and $ J $ , defined by $ I+J= {i+j mid iin I,jin J} $ is itself an ideal.
Let $ V $ be a vector space.
The $ m $ - fold symmetric product of $ V $ is the quotient vector space of the $ m $ - fold tensor product $ V^{otimes m} $ by the subspace generated by the tensors $ v_1otimescdotsotimes v_m - v_{sigma(1)}otimes cdotsotimes v_{sigma(m)} $ for all $ v_i in V $ and all $ sigma in S_m $ (the set of permutations of $ {1,dots,m} $ ).
It is denoted $ text{Sym}^m(V) $ .
A linear transformation $ T:Vto V $ from a normed linear space $ V $ to itself is compact if the closure of the image of the closed unit ball $ B = {v in V mid ||v|| leq 1} $ is compact in the usual sense.
The quaternion algebra $ mathbb H $ is the four - dimensional associative algebra over $ mathbb R $ spanned by the elements $ 1,mathbf i,mathbf j,mathbf k $ where $ 1 $ is the identity element and the relations (i) $ mathbf i^2 = mathbf j^2 = mathbf k^2 = - 1 $ ; (ii) $ mathbf{ij} = mathbf k $ ; (iii) $ mathbf{ji} = - mathbf k $ ; (iv) $ mathbf{jk} = mathbf i $ ; (v) $ mathbf{kj} = - mathbf i $ ; (vi) $ mathbf{ki} = mathbf j $ ; (vii) $ mathbf{ik} = - mathbf j $ .
This can be considered as a sub(?) of the set of $ 2times 2 $ complex matrices $ M_2(mathbb C) $ by setting $ mathbf i = }, quad mathbf j = },quad mathbf k = } $ .
Then $ mathbb H $ is the space of real linear combinations of $ I_2 $ , $ mathbb i $ , $ mathbb j $ , and $ mathbb k $ .
Let $ X $ be a ∆ - complex.
An open simplex $ e_alpha^n $ of dimension $ n $ in $ X $ has a canonical map $ sigma_alpha: Delta^n to X $ that restricts to a homeomorphism from the interior of the standard n - simplex $ Delta^n $ onto $ e_alpha^n $ .
Let $ p $ be a prime number.
A Sylow $ p $ - subgroup of a group $ G $ is a maximal p - subgroup of $ G $ .
That is, it is a p - group that is a subgroup of $ G $ and that is not a proper subgroup of any other p - subgroup of $ G $ .
A field or algebra of subsets of a set $ X $ is a pair $ (X, mathcal F) $ consisting of the set $ X $ and a collection $ mathcal F $ of subset of $ X $ such that (i) $ mathcal F $ is closed under complementation: $ Xsetminus F in mathcal F $ for all $ F in mathcal F $ ; (ii) $ mathcal F $ contains the empty set.
Equivalently, if (1) holds, $ mathcal F $ contains $ X $ ; (iii) Any of the following equivalent (by induction and De Morgan's laws) conditions hold: - $ mathcal F $ is closed under binary union: $ Fcup G in mathcal F $ for all $ F, G in mathcal F $ ; - $ mathcal F $ is closed under binary intersection: $ Fcap G in mathcal F $ for all $ F,G in mathcal F $ ; - $ mathcal F $ is closed under finite unions; - $ mathcal F $ is closed under finite intersections.
An algebra is a ring of subsets of $ X $ containing the set $ X $ .
A topological game $ G(X, Y, mathcal S) $ is a game of perfect information played between two players on a topological space $ X $ where $ Y $ is a fixed subset of $ Y $ , and $ mathcal S $ is some family of subsets of $ X $ .
Let $ G $ be a matrix Lie group.
The identity component $ G_0 $ of $ G $ is the set of $ Ain G $ for which there exists a continuous path $ gamma:0,1to G $ such that $ gamma(0) = I $ and $ gamma(1) = A $ .
Because connectedness and path - connectedness are equivalent in this case, $ G_0 $ amounts to the connected component of $ G $ that contains the identity matrix $ I $ .
Let $ mathfrak g $ be a Lie algebra and let $ (V,rho_V) $ and $ (W,rho_W) $ be representations of $ mathfrak g $ .
Then the tensor product of these representations is the pair $ (Votimes W, rho_{Votimes W}) $ where $ rho_{Votimes W}(x) = rho_V(x) otimes text{Id} + text{Id}otimes rho_W(x) $ for all $ x in mathfrak g $ .
The projectivization of a vector space $ (V,F) $ is the quotient of $ V $ by the action of the multiplicative group of the base field $ F $ by scalar multiplication.
Let $ Lambda={mu = (mu_1,dots, mu_n)inmathbb Z^n mid mu_1geq cdotsgeq mu_n geq 0} subseteq mathbb Z^n_{geq 0} $ .
For each $ mu in Lambda $ , define $ nabla_mu in mathbb Zx_1,dots, x_n^text{sign} $ to be the antisymmetric polynomial under the sign representation of the symmetric group given by $ nabla_mu = sum_{sin S_n} text{sign}(s) x^{s(mu)} = detleft|} right| $ .
The second equality follows from the Leibniz formula for determinants.
Set $ rho in Lambda $ to $ rho = (n - 1,n - 2,dots, 1, 0) $ .
Up to sign, $ nabla_rho $ is the Vandermonde determinant.
Because $ mathbb Zx_1,dots, x_n^text{sign} $ is a rank - one free module over $ mathbb Zx_1,dots, x_n^{S_n} $ with generator $ Delta_n $ , it follows that $ nabla_rho $ is also a generator of this module.
Thus for every $ muin Lambda $ there exists a unique polynomial $ s_muinmathbb Zx_1,dots, x_n^{S_n} $ called the Schur polynomial of $ mu $ such that $ nabla_{mu + rho} = s_mu cdotnabla_rho $ .
Explicitly, $ s_mu = nabla_{mu + rho}/nabla_rho $ .
A chain complex is a sequence of homomorphisms of abelian groups $ cdots to C_{n+1} to C_n to C_{n - 1} to cdots to C_1to C_0 to 0 $ (where the arrow between $ C_{k} $ and $ C_{k - 1} $ is labelled $ partial_k $ ) such that the composition $ partial_{k+1}circpartial_k = 0 $ for all $ k $ .
A Banach algebra is an algebra equipped with a complete multiplicative norm.
Let $ G $ be a group.
The category of canonical orbits of $ G $ , denoted $ mathcal O(G) $ , is the category with objects the G - sets $ G/H $ for all subgroups $ H $ of $ G $ and morphisms the G - maps that exist between the objects. $ mathcal O(G) $ is isomorphic to the category $ mathcal G $ whose objects are the subgroups of $ G $ and whose morphisms are the distinct subconjugacy relations $ gamma^{ - 1}Hgamma subset K $ for $ gamma in G $ .
Proof G - equivariant homomorphisms between quotient G - sets Weyl group of stabilizer under transitive action isomorphic to G - automorphisms of G - set
Let $ G $ be a group.
A normal series is an ascending sequence of subgroups $ {e} = N_0subseteq N_1subseteq cdotssubseteq N_{k - 1} subseteq N_k = G $ such that $ N_i $ is normal in $ G $ for all $ 0leq ileq k $ .
Let $ f:Xto Y $ be a function.
The fiber of a point $ yin Y $ is the set of all points $ xin X $ such that $ f(x) = y $ .
That is, $ f^{ - 1}(y) = {xin Xmid f(x) = y} $ .
The fiber is the inverse image under $ f $ of the singleton $ {y} $ .
Let $ mathbb k $ be a field and consider the algebra of polynomials in $ n $ variables $ P_n =bigopluslimits_{dgeq 0} P_n^d $ .
Given a representation of a group $ G $ in $ mathbb k^n $ , we get an action of $ G $ on $ P_n $ by the induced action of group on maps: for $ fin P_n $ , $ g:fmapsto g^*f $ .
Then each homogeneous subspace $ P_n^d $ of $ P_n $ is a $ G $ - invariant subspace.
Moreover, the algebra of invariant polynomials, written $ P_n^G $ , is a graded subalgebra of $ P_n $ .
An element $ x $ of an integral domain is prime if the ideal generated by $ x $ is a prime ideal.
That is, for all $ a,b in R $ such that $ x $ divides $ ab $ , then either $ x $ divides $ a $ or $ x $ divides $ b $ .
% Options for packages loaded elsewhere PassOptionsToPackage{unicode}{hyperref} PassOptionsToPackage{hyphens}{url} % documentclass {article} usepackage{amsmath,amssymb} usepackage{lmodern} usepackage{iftex} ifPDFTeX usepackageT1{fontenc} usepackageutf8{inputenc} usepackage{textcomp} % provide euro and other symbols else %
if luatex or xetex usepackage{unicode - math} defaultfontfeatures{Scale=MatchLowercase} defaultfontfeaturesrmfamily{Ligatures=TeX,Scale=1} fi %
Use upquote if available, for straight quotes in verbatim environments IfFileExists{upquote.sty}{usepackage{upquote}}{} IfFileExists{microtype.sty}{% use microtype if available usepackage{microtype} UseMicrotypeSetprotrusion{basicmath} % disable protrusion for tt fonts }{} makeatletter @ifundefined{KOMAClassName}{% if non - KOMA class IfFileExists{parskip.sty}{% usepackage{parskip} }{% else setlength{parindent}{0pt} setlength{parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class KOMAoptions{parskip=half}} makeatother usepackage{xcolor} setlength{emergencystretch}{3em} % prevent overfull lines providecommand{tightlist}{% setlength{itemsep}{0pt}setlength{parskip}{0pt}} setcounter{secnumdepth}{ - maxdimen} % remove section numbering ifLuaTeX usepackage{selnolig} % disable illegal ligatures fi IfFileExists{bookmark.sty}{usepackage{bookmark}}{usepackage{hyperref}} IfFileExists{xurl.sty}{usepackage{xurl}}{} % add URL line breaks if available urlstyle{same} % disable monospaced font for URLs hypersetup{ pdftitle={∆ - complex}, hidelinks, pdfcreator={LaTeX via pandoc}} title{∆ - complex} author{} date{} }
A sequence $ {x_i}_{iinmathbb N} $ in a topological space $ X $ converges to $ x in X $ if for every open $ Usubset X $ such that $ x in U $ , there exists $ n_0 inmathbb N $ such that for every $ n geq n_0 $ , we have $ x_n in U $ .
In a general topological space, the limit of a sequence need not be unique.
A descending sequence $ {E_n}_{ninmathbb N} $ of nonempty subsets of a metric space $ X $ is contracting if the limit of the diameters goes to zero.
That is, $ lim_{nto infty} text{diam}(E_n) = 0 $ .
Let $ A $ and $ B $ be two algebras over the field or commutative ring $ K $ .
Then an algebra homomorphism between $ A $ and $ B $ is a map $ f:Ato B $ such that (i) $ f(kx) = kf(x) $ (ii) $ f(x+y) = f(x) + f(y) $ (iii) $ f(xy) = f(x)f(y) $ for all $ kin K $ and $ x,yin A $ .
If an algebra homomorphism is bijective, then it is an isomorphism of algebras.
A subset $ S $ of a topological space is perfect if it is closed and contains no isolated points.
Equivalently, $ S $ is perfect if it is equal to the set of its limit points.
The diameter of a nonempty subset $ E $ of a metric space $ (X,rho) $ is given by the supremum $ text{diam}(E) = sup{rho(x,y)mid x,y in E} $ .
Let $ V $ be a vector space and let $ T: Vto V $ be a linear transformation with corresponding matrix $ A $ .
The matrix coefficients of the matrix corresponding to the exterior product $ wedge^m(T) $ of $ T $ are called the minors of $ A $ .
Let $ (X,Sigma,mu) $ be a measure space.
The $ p $ - norm of a measurable function $ f to mathbb R $ is $ ||f||_p = left(int_X |f|^ptext dmuright)^{1/p} $ .
An elementary abelian $ p $ - group is an abelian group $ G $ in which every nontrivial element has order $ p $ where $ p $ is prime.
Let $ X $ be a locally compact Hausdorff topological space.
Fix an integral $ int_X $ and $ V $ a finite - dimensional vector space.
Choose a basis $ {v_i}_{i=1}^n $ for $ V $ so that any continuous function $ f:Xto V $ can be expressed as $ f(x) = sum_{i=1}^n f_(x)v_i text{ for all } xin X $ where the $ f_i $ are the components of $ f $ .
Define the vector - valued integral of $ f $ as $ int_X f(x) text{d} x = sum_{i=1}^n left(int_X f_i(x)text dx right)v_i $ .
We can say that it is given by the integrals of the coordinates.
A right ideal of a ring $ R $ is a subset $ Isubseteq R $ such that - $ (I,+) $ is a subgroup of $ (R,+) $ ; - $ I $ is closed under right multiplication in $ R $ (i.e. for all $ iin R $ and all $ rin R $ , $ ir in I $ ).
Let $ (V,F) $ be a vector space.
A function $ langlecdot,cdotrangle: Vtimes V to F $ is an inner product on $ V $ if (i) it is bilinear; (ii) it is commutative; (iii) $ langle v, vrangle geq 0 $ for all $ v in V $ and $ langle v,vrangle = 0 $ if and only if $ v=0 $ .
A function $ f: Xtomathbb R $ has compact support if the closure of its support $ S $ is compact as a subset of $ X $ .
Let $ X $ be a finite CW complex.
The Euler characteristic $ chi(X) $ is defined to be the alternating sum $ sum_n ( - 1)^nc_n $ where $ c_n $ is the number of $ n $ - cells of $ X $ .
Let $ R $ be a ring and let $ M $ be an $ R $ - module.
A composition series for $ M $ is a chain of submodules of $ M $ $ M = F_0 supset F_1 supset cdots supset F_{n - 1}supset F_n ={0} $ such that the quotient $ F_i/F_{i+1} $ is simple for all $ i $ .
The complex orthogonal group $ text{O}(n,mathbb C) $ is the set of $ ntimes n $ complex matrices whose transposes are equal to their inverses.
This definition mirrors the definition of the real orthogonal group, but instead we consider complex - valued matrices.
It is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
An algebraic number is a number that is a root of a nonzero polynomial in one variable over $ mathbb Q $ .
It is often denoted by $ overline{mathbb Q} $ .
The algebraic numbers form a field.
A topological space $ X $ is compactly generated if it a weak Hausdorff k - space.
Let $ mathcal E $ and $ mathcal B $ be small connected groupoids.
A covering $ p:mathcal Eto mathcal B $ is a functor that is surjective on objects and restricts to a bijection on the star for each object $ e $ of $ mathcal E $ .
That is, $ p(text{St}(e)) = text{St}(p(e)) $ .
For an object $ b $ of $ mathcal B $ , let $ F_b $ denote the set of objects of $ mathcal E $ such that $ p(e) = b $ .
That is, the fiber of $ b $ .
Then $ p^{ - 1}(text{St}(b)) $ is the disjoint union over $ ein F_b $ of $ text{St}(e) $ .
See covering space.
Let $ X $ be a ∆ - complex and let $ Delta_n(X) $ be the free abelian group on the open n - simplices of $ X $ .
Elements of $ Delta_n(X) $ are called $ n $ - chains and can be written as formal finite sums $ sum_alpha n_alpha e_alpha^n $ for $ n_alpha in mathbb Z $ .
Equivalently, this is sometimes written $ sum_alpha n_alphasigma_alpha $ where the $ sigma_alpha:Delta^nto X $ are the characteristic maps of the $ e_alpha^n $ .
This sum can be thought of as a finite collection/chain of n - simplices in $ X $ with integer multiplicities.
A non - Archimedean absolute value on a field $ F $ is an absolute value $ |cdot|:Fto mathbb R $ such that $ |x+y|leqmax{|x|,|y|} $ for all $ x,yin F $ .
This inequality is called the ultrametric inequality and it is stronger than the triangle inequality.
If a Lie algebra has no ideals except for $ {0} $ and itself, then it is simple .
A function $ f:a,bto mathbb R^n $ is absolutely continuous if for every $ varepsilon > 0 $ , there exists $ delta > 0 $ such that if $ {(x_i,y_i)}_{i=1}^k $ are pairwise disjoint subintervals of $ a,b $ with total length less than $ delta $ , then $ sum_{i=1}^k |f(y_i) - f(x_i)| < varepsilon $ .
Let $ A $ be a matrix and let $ A_i $ be the $ i $ th row of $ A $ .
A row operation on $ A $ is any of the following operations: (i) Switch two rows (ii) Multiply a row by a nonzero constant (iii) Add one row to another
The closed half - space $ H^n subsetmathbb R^n $ is $ H^n = {(x_1,dots,x_n)inmathbb R^nmid x_i in mathbb R, x_n geq 0} $ .
The center of a group $ G $ is $ Z(G) = {zin Gmid zg=gz text{ for all } gin G} $ .
A category $ mathcal C $ is a collection of objects together with a set $ mathcal C(A,B) $ of morphisms or maps between any two objects.
In particular, there is always for each object $ A $ in $ mathcal C $ an identity morphism $ text{id}_A inmathcal C(A,A) $ .
Finally, a category must obey the following composition law: $ circ: mathcal C(B,C)times mathcal C(A,B) to mathcal C(A,C) $ for all triples $ A $ , $ B $ , $ C $ of objects such that composition is associative, and identity morphisms are identities for composition.
That is, - $ hcirc(gcirc f) = (hcirc g)circ f $ ; - $ text{id}circ f = f $ ; - $ fcirc text{id} = f $ .
Let $ X $ be a set together with a binary operation $ * $ .
An element $ e in X $ is an identity element of $ X $ if it is both a left and right identity of $ X $ with respect to $ * $ .
Let $ X $ be a topological space.
The singular complex $ S(X) $ is the ∆ - complex with one n - simplex $ Delta_sigma^n $ for each singular n - simplex $ sigma:Delta^nto X $ with $ Delta_sigma^n $ attached in the obvious way to the $ (n - 1) $ - simplices that are the restrictions of $ sigma $ to the various $ (n - 1) $ - simplices in the boundary $ partialDelta^n $ .
If $ sigma in S_n $ is an element of the symmetric group on $ n $ letters, then it is the product of disjoint cycles of length $ n_1leqcdotsleq n_r $ (including the $ 1 $ - cycles!)
and moreover the sum of the $ n_i $ is equal to $ n $ .
The partition of $ n $ given by the $ n_i $ is called the cycle type of $ sigma $ .
Let $ X $ be a topological space.
The closure of the set $ A subset X $ is the smallest closed subset of $ X $ containing $ A $ , or the union of $ A $ and its limit points.
A monoid is a semigroup $ (X,*) $ that has an identity element under $ * $ .
A unit of a ring $ R $ is an element $ rin R $ that has a multiplicative inverse.
Let $ G $ be a group.
A subnormal series is an ascending sequence of subgroups $ {e} = N_0subseteq N_1subseteq cdotssubseteq N_{k - 1} subseteq N_k = G $ such that $ N_i $ is normal in $ N_{i+1} $ for all $ 0leq ileq k $ .
An oriented edge $ k:Ito X $ in a graph $ X $ where $ I $ is the unit interval is the traversal of an edge in either the forward or the backward direction.
Let $ mathfrak g $ be a Lie algebra with subalgebras $ mathfrak g_1 $ and $ mathfrak g_2 $ .
Then $ mathfrak g $ decomposes as the direct sum of Lie algebras $ mathfrak g_1 $ and $ g_2 $ if it is the direct sum of vector spaces $ mathfrak g_1oplus mathfrak g_2 $ and $ X_1,X_2=0 $ for all $ X_1inmathfrak g_1 $ and $ X_2in mathfrak g_2 $ .
Let $ S subsetmathbb R^n $ be rectifiable. $ S $ is symmetric with respect to the vector subspace $ x_k=0 $ if the function $ h:Stomathbb R^n $ given by $ h(x) = h(x_1,dots, - x_k, dots, x_n) $ has as its image $ S $ itself.
Let $ (X_1,Sigma_1,mu_1) $ and $ (X_2,Sigma_2,mu_2) $ be two measure spaces, and let $ Sigma_1otimes Sigma_2 $ be the σ - algebra generated by sets of the form $ B_1times B_2 $ for $ B_1in Sigma_1 $ , $ B_2in Sigma_2 $ .
A product measure $ mu_1times mu_2 $ is a measure on the measurable space $ (X_1times X_2, Sigma_1otimesSigma_2) $ such that $ (mu_1timesmu_2)(B_1times B_2) = mu_1(B_1)mu_2(B_2) $ for all $ B_1in Sigma_1 $ , $ B_2in Sigma_2 $ .
Note: When multiplying measures that are infinite, the product is zero if any factor is zero.
Note:
The product measure can *always* be constructed according to the Hahn - Kolomogorov theorem.
Let $ X $ be a topological space.
The subset $ A subset X $ is closed if its complement $ Xsetminus A $ is open.
Equivalently, if $ X $ is a metric space, then a subset $ A subset X $ is closed if and only if it contains all of its limit points.
Let $ S subsetmathbb R^n $ be a bounded set.
Then $ S $ is rectifiable if the constant function $ 1 $ is Riemann integrable over $ S $ .
A topological group is a topological space $ G $ that is also a group such that the binary operation on the group and the inverse map are both continuous.
That is, the two maps $ cdot:Gtimes Gto G text{ given by } (x,y)mapsto xy $ $ text{inv}:Gto G text{ given by } xmapsto x^{ - 1} $ are both continuous where $ Gtimes G $ is given the product topology.
Let $ X $ be a topological space and let $ A $ be a subspace of $ X $ .
Elements of the relative homology groups $ H_n(X,A) $ are called relative cycles .
They are n - chains $ alphain C_n(X) $ such that $ partial alpha in C_{n - 1}(A) $ .
The quotient of $ G $ by its commutator subgroup $ G,G $ (which is normal) is called the abelianization of $ G $ .
It is in fact abelian.
Let $ G $ be a group.
The upper central series for $ G $ is an ascending sequence of subgroups of $ G $ defined as follows: Let $ Z_0(G) = {e} $ and let $ Z_1(G) $ be the center $ Z(G) $ .
Then for each $ igeq 1 $ , define $ Z_{i+1}(G) $ to be a subgroup of $ G $ containing $ Z_i(G) $ such that the quotient $ Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G)) $ .
Note that because the center is a normal subgroup, these quotients are well - defined.
Also note that this series may or may not reach $ G $ .
An open cover $ {U} $ of a topological space $ B $ is numerable if for each $ U $ there exist continuous maps $ lambda_U:Bto I $ (where $ I $ is the unit interval) such that the inverse image $ lambda_U^{ - 1}((0,1) = U $ and if the cover is locally finite.
this seems to have a double word
Let $ (L,leq) $ be a lattice.
A complement of $ ain L $ is an element $ bin L $ such that the supremum $ sup(a,b) $ is a maximum element of $ L $ and the infimum $ inf(a,b) $ is a minimum element of $ L $ .
That is, if $ yleq sup(a,b) $ and $ inf(a,b)leq y $ for all $ yin L $ .
A set $ S $ of vectors in an inner product space is orthonormal if every element of $ S $ has norm $ 1 $ and each vector in the set is orthogonal to every other vector in the set.
Let $ G $ be a group with subgroup $ H $ .
Then the quotient $ G/H $ is the set of classes of cosets $ gH $ for $ gin G $ .
The quotient is itself a subgroup exactly when $ H $ is normal in $ G $ .
See the definition of the quotient when the subgroup is normal.
A function $ f:Ato mathbb R $ is locally monotonic at $ x in A $ if there exists $ r>0 $ such that whenever $ 0 < y - x < r $ , $ f(y)geq f(x) $ and whenever $ 0 < x - y < r $ , $ f(y)leq f(x) $ .
An ideal $ mathfrak p $ of a ring $ R $ is prime if it is proper and if $ a,bin R $ such that $ ab in I $ , then $ ain mathfrak p $ or $ bin mathfrak p $ .
The alternating group $ A_n $ is the subgroup of the symmetric group $ S_n $ given by the kernel of the sign homomorphism.
That is, it consists of all permutations whose sign is $ 1 $ .
The Lebesgue measure on $ mathbb R^n $ is constructed as follows: A set $ A subsetmathbb R^n $ is Lebesgue measurable if for every $ Ssubset mathbb R^n $ , we have the following relation in the Lebesgue outer measure $ lambda^* $ : $ lambda^*(S) = lambda^*(Scap A) + lambda^*(Ssetminus A) $ .
By Carathéodory's extension theorem, Lebesgue measurable sets $ A $ form a σ - algebra and the Lebesgue measure of any such set coincides with its Lebesgue outer measure.
Let $ (L,F) $ be a vector space with a binary operation $ Ltimes L to L $ denoted $ (x,y) mapsto xy $ .
This operation is called the bracket or commutator of $ x $ and $ y $ .
The vector space together with the operation is a Lie algebra over the field $ F $ if (i) The bracket operation is bilinear; (ii) $ xx = 0 $ for all $ x in L $ ; (iii) $ xyz + yzx+zxy =0 $ for all $ x,y,zin L $ .
The last axiom is called the Jacobi identity .
An ideal $ I $ of an algebra $ A $ is a subset $ Isubset A $ that is both a left algebra ideal and a right algebra ideal.
In the case that $ A $ is a commutative algebra, these notions coincide and every right ideal is also a left ideal is also a two - sided ideal.
As long as the ring $ R $ is not a rng, $ A $ will inherit the definition of ideal from its underlying ring structure.
Let $ R $ be a commutative ring and $ x $ a variable.
Then $ Rx $ is the polynomial ring of polynomials in $ x $ with coefficients in $ R $ .
We can also define the polynomial ring in multiple variables.
Let $ X = {x_1,dots,x_n} $ be a finite collection of variables.
Then $ RX = Rx_1,dots,x_n $ is defined as follows: To each function $ I:Xto mathbb N $ , associate a monomial $ x^I = x_1^{I(x_1)}+cdots + x_n^{I(x_n)} $ .
Then $ RX= left{ sum_{I:Xto mathbb N} a_IX^I mid a_Iin R, a_I = 0 text{ for all but finitely many } I right} $ .
Multiplication and addition are defined exactly how you would think.
A Lie group $ Gsubseteq text{GL}(n,mathbb C) $ is compact if it is compact in the usual sense when considered as a subset of $ M_n(mathbb C) cong mathbb R^{2n^2} $ .
A Δ - complex is a quotient vector space of a collection of disjoint simplices obtained by identifying certain faces by the canonical linear homeomorphisms described in the definition of barycentric coordinates.
Consider a collection of n - simplices $ Delta^n_alpha $ of various dimension (i.e. $ n $ is not fixed) and some sets $ mathcal F_i $ of faces of the $ Delta_alpha^n $ such that within each $ mathcal F_i $ , each face has the same dimension (i.e. value for $ n $ ).
We can take a quotient vector space of the disjoint union $ coprod_alpha Delta_alpha^n $ by identifying all of the faces in each $ mathcal F_i $ via the canonical linear homeomorphisms described in the definition of barycentric coordinates.
The generalized orthogonal group $ text{O}(n,k) $ is the group of $ (n+k)times (n+k) $ real - valued matrices that preserve the symmetric bilinear form $ x,y_{n,k} = x_1y_1 + cdots + x_ny_n - x_{n+1}y_{n+1} - cdots - x_{n+k}y_{n+k} $ .That is, $ A in text{O}(n,k) $ if and only if $ x,y_{n,k} = Ax,Ay_{n,k} $ for all $ x,y in mathbb R^n $ .
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
A metric space $ (X,rho) $ is complete if every Cauchy sequence is also convergent.
An atlas for a topological space $ M $ is a collection of charts on $ M $ that covers $ M $ .
Let $ mathcal C $ be a category.
When there exists a "suitable" relation of homotopy between morphisms in $ mathcal C $ , then the homotopy category $ hmathcal C $ , defined to be the category consisting of the same objects as $ mathcal C $ with with morphisms given by homotpy classes of morphisms in $ mathcal C $ .
Let $ R $ be a ring of subsets of $ X $ and let $ mu_0:0,infty $ .
Then $ mu_0 $ is a pre - measure if $ mu_0(emptyset) = 0 $ and for every sequence $ {A_n}_{ninmathbb N}subset R $ of pairwise disjoint subsets whose union is in $ R $ , $ mu_0left(bigcup_{ninmathbb N}A_nright) = sum_{ninmathbb N} mu_0(A_n) $ .
That is, $ mu_0 $ is σ - additive.
Let $ G $ be a group acting on the set $ X $ .
The stabilizer of a point $ xin X $ , denoted $ text{Stab}(x) $ , is the set of elements $ gin G $ such that $ gcdot x = x $ .
Let $ U subset mathbb R^n $ be open.
Let $ v in mathbb R^n $ .
The directional derivative of a function $ f:Utomathbb R^m $ in the direction of $ v $ at a point $ a in U $ is $ D_vf(a) = frac{partial f}{partial v}(a) = lim_{hto 0}frac{f(a+hv) - f(a)}{h} $ if this limit exists.
The characteristic polynomial of a linear transformation $ T $ is the polynomial in $ lambda $ given by the determinant of the linear transformation $ T - lambda I $ : $ p_T(lambda) = text{det}(T - lambda I) $ .
Let $ X $ be a metric space with $ A subset X $ .
Let $ mu $ be the Lebesgue measure.
A collection $ mathcal V $ of sets covering $ A $ is a Vitali covering if for each $ x in A $ and $ varepsilon > 0 $ , there exists $ U in mathcal V $ such that $ x in U $ , and the diameter of $ U $ is nonzero and less than $ varepsilon $ .
Let $ (X,Sigma, mu) $ be a measure space and let $ A subset X $ .
The measurable hull $ H $ of $ A $ is the measurable set $ Hsupseteq A $ such that
for any measurable set $ F $ satisfying $ Asubseteq Fsubseteq H $ , we have $ mu(F) = mu(H) $ .
Let $ X $ be a compact topological space and let $ 1_X $ be the constant function $ 1 $ on this space.
Then for a fixed Haar integral $ int_X $ , define the volume of $ X $ to be $ text{vol}(X) = int_X 1_X $ .
A Banach - Mazur game $ MB(X,Y, mathcal W) $ is a topological game played as follows: Let $ X $ be a topological space, $ Y $ a fixed subset of $ X $ , and $ mathcal W $ a family of subsets of $ X $ such that (i) each element of $ mathcal W $ has nonempty interior.
(ii) every nonempty subset of $ X $ contains an element of $ mathcal W $ .
Players 1 and 2 alternately choose elements of $ mathcal W $ to form a sequence $ W_1supset W_2supsetcdots $ .
Player 1 wins if and only if $ Ycap left(bigcap_{n<omega} W_nright)neq emptyset $ where $ omega $ is the first transfinite ordinal.
The generating set of a group $ G $ is a subset $ S $ of $ G $ such that every element of $ G $ can be expressed as a combination under the group operation of finitely many elements of $ S $ and their inverses.
A set $ {v_i}_{iin Lambda} $ in the vector space $ V $ is orthogonal if $ langle v_i, v_jrangle =0 $ for all $ ineq j $ .
A subgroup $ H $ of a group $ G $ is characteristic in $ G $ if every automorphism of $ G $ leaves $ H $ invariant.
Let $ G $ be a compact topological group acting continuously on a topological space $ X $ .
Let $ f $ be a continuous function $ Xto mathbb C $ .
An averaging of $ f $ is a continuous function $ text{Av}(f) $ on $ X $ given by $ text{Av}(f)(x) = frac{1}{text{vol}(G)} int_G f(gcdot x)text dx $ where $ text{vol}(G) $ is the volume of $ G $ and the integral is the Haar integral on $ G $ .
Note that for a finite group $ G $ , which is necessarily compact, we have simply $ text{Av}(f)(x) = frac{1}{|G|} sum_{gin G}f(gx) $ .
A group $ G $ is solvable if it admits a subnormal series whose quotients quotients $ N_{i+1}/N_i $ are all abelian.
Let $ R $ be a commutative ring with ideals $ I $ and $ J $ .
Then the product of $ I $ and $ J $ , defined by $ IJ= {i_1j_1+cdots + i_kj_j mid i_ellin I,j_ellin J, ell in mathbb N} $ is itself an ideal.
Let $ phi:Gto GL(V) $ be a representation of the group $ G $ .
A sub - representation is a representation $ psi:Gto GL(W) $ for $ W $ a subspace of $ V $ .
Let $ U subset mathbb R^n $ be open and let $ f:Utomathbb R^n $ be differentiable.
Then its total derivative $ Df(a) $ at a point $ ain U $ is given by the $ mtimes n $ matrix $ left frac{partial f_j}{partial x_i}(a)right_{i=1,j=1}^{n,m} $ where $ f = (f_1,dots, f_m) $ are the component functions of $ f $ .
This matrix is called the Jacobian matrix of $ f $ at $ a $ .
Let $ A $ be a compactly generated topological space.
A space under $ A $ (that is, an object of the category) is a map $ i:Ato X $ .
A morphism between spaces under $ A $ is a commutative diagram This is a homotopy category: a homotopy $ h $ between maps under $ A $ is a homotopy that at each "time" $ t $ is a map under $ A $ .
Then we write $ h:fsimeq f' text{ rel } A $ and $ h(i(a),t) = j(a) $ for all $ ain A $ and $ tin I $ .
The above notion of homotopy equivalence is called cofiber homotopy equivalence .
Two absolute values over the same field $ F $ , call them $ |cdot|_1 $ and $ |cdot|_2 $ , are equivalent if there exists a real number $ alpha>0 $ such that $ |x|_1^alpha = |x|_2 $ for all $ xin F $ .
The $ ntimes n $ identity matrix $ I_n $ is the $ ntimes n $ matrix whose entries $ (a_{ij}) $ are such that $ a_{ij}=1 $ when $ i=j $ and $ 0 $ otherwise.
That is, $ a_{ij} = delta_{ij} $ where $ delta_{ij} $ is the Kronecker delta.
A binary relation on a set $ X $ is transitive if for all $ a,b,cin X $ , if $ a $ is related to $ b $ and $ b $ is related to $ c $ , then $ a $ is related to $ c $ .
Let $ f:S^nto S^n $ be a continuous function such that for some $ yin S^n $ , the preimage $ f^{ - 1}(y) $ is finite.
Denote the elements of $ f^{ - 1}(y) $ by $ {x_i}_{i=1}^m $ .
Let $ {U_i}_{i=1}^m $ be disjoint neighborhoods of the $ x_i $ .
Then $ f $ maps each $ U_i $ into a neighborhood $ V $ of $ y $ .
Then $ f(U_i setminus {x_i}) subset Bsetminus {y} $ for all $ y $ and the following diagram of homology groups and relative homology groups commuptes: where all of the maps are "obvious:" $ k_i $ and $ p_i $ are induced by inclusion.
The two isomorphisms on the top of the diagram come from the Excision Theorem and the two on the bottom come from the long exact sequence of a pair.
The top two groups are then isomorphic to $ H_n(S^n) approxmathbb Z $ , so the top homomorphism is then multiplication by an integer $ d $ .
This integer is the local degree of $ f $ at $ x_i $ , written $ text{deg}(f)_{|x_i} $ .
Consider a chain complex whose abelian groups are the groups of n - chains of a ∆ - complex $ X $ .
The $ n $ th homology group $ text{Ker}(partial_{n})/text{Im}(partial_{n+1}) $ is denoted $ H_n^Delta(X) $ and is called the $ n $ th simplicial homology group of $ X $ .
Let $ f: Xto Y $ be a function between topological spaces $ X $ and $ Y $ .
Then $ f $ is continuous if and only if the preimage of an open set in $ Y $ is open in $ X $ .
A function $ f: Xto Y $ between metric spaces $ (X,rho) $ and $ (Y,sigma) $ is continuous at $ xin X $ if and only if for every $ varepsilon > 0 $ , there exists
$ delta> 0 $ such that $ sigma(f(x),f(x')) < varepsilon $ whenever $ rho(x,x') < delta $ .
A function $ f: Xto Y $ between metric spaces $ (X,rho) $ and $ (Y,sigma) $ is continuous at $ xin X $ if and only if for any sequence $ {x_n}_{ninmathbb N} $ in $ X $ converging to $ x $ , $ {f(x_n)}_{nin mathbb N} $ converges to $ f(x) $ .
These three definitions are all equivalent.
Let $ E $ and $ B $ be compactly generated topological spaces.
A surjective continuous function $ p:Eto B $ is a fibration if it satisfies the coveirng homotopy property .
That is, given the diagram where $ Y $ is any compactly generated topological space, $ f:Yto E $ is some continuous function, $ h:Ytimes Ito B $ is a homotopy, and $ i_0(y) = (y,0) $ , if $ hcirc i_0 = pcirc f $ , then there exists $ tilde h $ that makes the diagram commute.
The diagram can also be written as !
Screen Shot 2022 - 02 - 25 at 1(i)0(v)02 PM.png where $ X^I $ is the set of ( : continuous?
is this even what that is?) functions $ Ito B $ for $ I $ the unit interval and $ p_0(beta) = beta(0) $ for $ betain B^I $ .
Let $ (X,Sigma, mu) $ be a measure space.
The essential supremum $ alpha $ of a function $ f:Xto mathbb R $ is the smallest real number such that $ mu({xmid f(x) > alpha}) = 0 $ .
A subgraph $ A $ of a graph $ X $ is a graph $ Asubset X $ with $ A^0subset X^0 $ .
That is, $ A $ is a union of some of the vertices and edges of $ X $ .
A Hermitian inner product $ langlecdot,cdotrangle $ on a complex vector space $ V $ is a bilinear form on $ V $ that is antilinear in the second slot.
That is, (i) $ langle u+v,wrangle = langle u,wrangle +langle v,wrangle $ ; (ii) $ langle u,v+wrangle = langle u,vrangle +langle u,wrangle $ ; (iii) $ langle alpha u ,vrangle = alphalangle u,vrangle $ ; (iv) $ langle u ,alpha vrangle = overlinealphalangle u,vrangle $ ; (v) $ langle u,vrangle = overline{langle v,urangle} $ (vi) $ langle u,u rangle geq 0 $ (vii) $ langle u,u rangle = 0 $ if and only if $ u=0 $ .
Any subalgebra of a general linear algebra $ mathfrak{gl}(V) $ is called a linear Lie algebra .
A division algebra $ D $ is an algebra over the field $ F $ such that for any element $ a $ in $ D $ and any non - zero element $ b $ in $ D $ , there exists precisely one element $ xin D $ with $ a=bx $ and precisely one element $ y in D $ such that $ a=yb $ .
If $ D $ is an associative algebra, then we just require that $ D $ have an identity element under the binary operation and every nonzero element have a multiplicative inverse.
Let $ X $ be a ∆ - complex.
Then between elements of a chain complex, we define the boundary homomorphism $ partial_n: Delta_n(X) to Delta_{n - 1}(X) $ by specifying its values on generators: $ partial_n(sigma_alpha)= sum_i( - 1)^isigma_{alpha|v_0,dots,hat v_i,dots, v_n} $ .
Each restriction $ sigma_{alpha|v_0,dots,hat v_i,dots, v_n} $ is the characteristic map of an $ (n - 1) $ - simplex of $ X $ .
check to make sure this is the right link
A smooth topological $ m $ - dimensional manifold with boundary is a topological m - dimensional manifold with boundary $ M $ together with a collection $ mathcal A $ of charts on $ M $ such that (i) for all $ pin M $ , there exists a chart $ (U, phi)in mathcal A $ such that $ p in U $ ; (ii) for any two charts $ (U,phi) $ and $ (V, psi) in mathcal A $ , the transition function $ psicirc phi^{ - 1}:phi(Ucap V)topsi(Ucap V) $ is a smooth diffeomorphism.
(iii) the set $ mathcal A $ (called an atlas) is the maximal set of collections of charts satisfying (i) and (ii)
The Euler characteristic of a graph is the difference $ V - E $ between the number of vertices and the number of edges.
Let $ nu $ be a Borel measure on $ mathbb R^n $ (i.e. the underlying measure space is $ (mathbb R^n, mathcal B, nu) $ ).
The symmetric derivative of $ nu $ at $ x in mathbb R^n $ is $ (Dnu)(x) = lim_{rto 0} frac{nu(B(x,r))}{mu(B(x,r))} $ if this limit exists.
This is the derivative ( : link to 208 notions?) in the symmetric differential basis.
The automorphism group of a group $ G $ is the group consisting of automorphisms of $ G $ .
It is a subgroup of the symmetric group on the underlying set of $ G $ .
An edge path on a graph $ X $ is closed if it starts and ends at the same vertex.
It is therefore a loop.
Let $ f: Q tomathbb R $ be a bounded function on the rectangle $ Q $ in $ mathbb R^n $ .
Let $ a in Q $ .
For any $ delta > 0 $ , let $ A_delta(a) = {f(x) mid x in Q land ||x - a|| < delta} $ .
Set $ M_delta (f, a) = sup A_delta(a) $ and $ m_delta (f,a)= inf A_delta(a) $ .
We define the oscillation of $ f $ at $ a $ to be $ text{osc}(f,a) = inf(M_delta(f,a) - m_delta(f,a)) $ .
The $ p $ - adic integers is the subset of Qp defined by $ mathbb Z_p = {xin mathbb Q_pmid |x|_p leq 1} $ .
Let $ X $ be a set with binary operation $ * $ .
An element $ ein X $ is a right identity of $ X $ if $ a*x = a $ for all $ ain A $ .
Let $ G $ be a group and let $ X $ be a topological space.
Then $ X $ is a $ G $ - space if it is a G - set and the group action $ Gtimes Xto X $ is continuous.
The orbit space $ X/G $ is the set of orbits $ {Gxmid xin X} $ with topology given as the quotient of topological spaces.
Note that in the case of a discrete group $ G $ , the continuity condition means that each action by an element of $ G $ is a homeomorphism.
A natural transformation $ alpha:Fto G $ is a map between functors $ mathcal Cto mathcal D $ consisting of a morphism $ alpha_A: F(A)to G(A) $ for each object $ A $ of $ mathcal C $ such that the following diagram commutes:
The centralizer of a subset $ X $ of a Lie algebra $ L $ is $ C_L(X) = {xin L mid xX=0} $ .
all the Lie algebra stuff seems to mostly not appear on wikidata...
A group representation is irreducible if it has no nontrivial sub - representations.
Let $ f:mathbb Ctomathbb C $ , and write it as follows: $ f(z) = u(x,y) + iv(x,y) $ where $ zin mathbb C $ is such that $ z = x+iy $ and $ u,v:mathbb R^2to mathbb R $ .
The function $ f $ is analytic on the open set $ Usubsetmathbb C $ if it is complex differentiable at each point in $ U $ .
The unitary group $ text{U}(n) $ is the group of $ ntimes n $ unitary matrices together with the operation of matrix multiplication.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
Let $ V_1,dots,V_n, W $ be vector spaces.
A multilinear map is a map $ f: V_1 timescdotstimes V_n to W $ such that for all $ 1leq i leq n $ and $ v_j in V_j $ with $ j neq i $ , the map $ f(v_1,dots, v_{i - 1}, - , v_{i+1}, dots, v_n):V_i to W $ (which varies only in the $ i $ th component) given by $ V_i ni v mapsto f(v_1,dots, v_{i - 1}, v, v_{i+1}, dots, v_n) in W $ is a linear transformation.
Let $ (X,Sigma) $ and $ (Y, T) $ be measurable spaces.
A function $ f:Xto Y $ is measurable if the preimage under $ f $ of every set in $ T $ is in $ Sigma $ .
An automorphism is an isomorphism from an algebraic structure to itself
A topological space $ X $ is simply connected if it is path - connected and for every two continuous maps $ gamma_1:0,1to X $ and $ gamma_2:0,1to X $ there exists a homotopy $ F:0,1times0,1to X $ such that $ F(x,0)= gamma_1(x) $ and $ F(x,1) = gamma_2(x) $ .
A topological space $ X $ is simply connected if it is path - connected and has trivial fundamental group.
A topological space $ X $ is locally path - connected if for any $ xin X $ and any neighborhood $ U $ of $ x $ , there is a smaller neighborhood $ V $ of $ x $ , each of whose points can be connected to $ x $ by a path in $ U $ .
Let $ H $ and $ N $ be groups and let $ phi: Hto text{Aut}(N) $ be a group homomorphism from $ H $ to the automorphism group of $ N $ .
The semidirect product $ Hltimes N $ is a group on the Cartesian product $ Htimes N $ with operation $ * $ given by $ (h_1,n_1)*(h_2,n_2) = (h_1h_2, n_1phi(h_1)n_2) $ .
The identity element of $ Hltimes N $ is the ordered pair consisting of the identity elements of $ H $ and $ N $ , respectively.
The inverse with respect to $ * $ is given by $ (h,n)^{ - 1} = (h^{ - 1},phi(h^{ - 1})n^{ - 1}) $ .
A category is connected if any two of its objects can be connected by a chain of morphisms.
Note that $ Aleftarrow Bto C $ connects $ A $ to $ C $ even when there might not be a morphism $ Ato C $ or $ Cto A $ .
An edge path in a graph $ X $ is a finite composite of oriented edges $ k_n $ with $ k_{n+1}(0) = k_n(1) $ .
A projection is a linear transformation $ P:Vto W $ between vector spaces $ V $ and $ W $ such that $ P^2 = P $ .
That is, $ P $ is idempotent when considered as an element of the ring of linear transformations.
Equivalently, for $ V' $ a subspace of a vector space $ V $ , a linear transformation $ P:Vto V $ is a projection onto $ V' $ if (i) $ P_{|V'} = text{id}_V' $ .
That is, if the restriction of $ P $ to $ V' $ is equal to the identity function on $ V' $ , and (ii) $ text{im}(P) subseteq V' $ .
In particular the image is actually equal to $ V' $ , but the condition is sufficient.
Let $ U,Vsubsetmathbb R^n $ be open.
A class $ C^k $ diffeomorphism is a bijective $ C^k $ function $ g:Uto V $ such that the inverse $ g^{ - 1} $ is also of class $ C^k $ .
Let $ V $ be a module over the ring $ R $ .
A subset $ W subset V $ is linearly independent if for $ v_i in W $ and $ alpha_i in R $ , $ sum_{i=1}^n alpha_i v_i=0 $ only when $ alpha_i = 0 $ for all $ i $ .
This is a more general definition than the one for vector spaces.
The category of groupoids is the category whose objects are groupoids and whose morphisms are functors between groupoids.
If we have $ a $ ways of doing one thing and $ b $ ways of doing another thing but we cannot do both at the same time, then there are $ a+b $ ways to choose to do one thing.
Set theoretically, the sum of the sizes of a finite collection of pairwise disjoint sets is the size of the union of the sets.
That is, $ |S_1| + |S_2| + cdots + |S_n| = |S_1cup S_2cupcdotscup S_n| $ .
Let $ G $ be a Lie group with associated Lie algebra $ mathfrak g $ .
Then the adjoint map $ text{Ad}:G to text{GL}(mathfrak g) $ is given by $ text{Ad}_A(X) = AXA^{ - 1} $ .
Because the adjoint map is a Lie algebra homomorphism, it is a representation of $ mathfrak g $ , and therefore we may refer to $ text{ad} $ as the adjoint representation of $ G $ acting on $ mathfrak g $ when considered as a vector space.
A groupoid is a category in which all morphisms are isomorphisms.
A group may be considered a groupoid with a single object.
For any subset $ A subsetmathbb R^n $ , the Lebesgue outer measure is given by $ mu^*(A) = inf{sumlimits_{Rinmathcal C} $ vol $ (R) mid mathcal C $ is a countable collection of rectangles covering $ A} $ .
A covering space $ p:Eto B $ is universal if $ E $ is simply connected.
The barycentric subdivision of an n - simplex $ v_0,dots,v_n $ is the decomposition of $ v_0,dots,v_n $ into the n - simplices $ b,w_0,dots,w_{n - 1} $ where we have by induction $ w_0,dots, w_{n - 1} $ an $ (n - 1) $ - simplex in the barycentric subdivision of a face $ v_0,dots,hat v_i,dots,v_n $ .
The induction starts at $ n=0 $ where we define the barycentric subdivision of $ v_0 $ to be $ v_0 $ .
It's not too hard to see the cases $ n=1,2,3 $ .
The vertices of the barycentric subdivision of $ v_0,dots,v_n $ are thus the barycenters of all of the $ k $ - dimensional faces $ v_{i_0},dots, v_{i_k} $ of $ v_0,dots,v_n $ for $ 0leq kleq n $ .
The barycenter of this face has barycentric coordinates $ t_i = frac{1}{(k+1)} $ for $ i= i_0,dots, i_k $ and $ t_i=0 $ otherwise.
The n - simplices of the barycentric subdivision of the standard n - simplex $ Delta^n $ with all of their faces form a ∆ - complex and even a simplicial complex structure on $ Delta^n
Let $ G $ be a group.
Then $ G $ acts on itself by conjugation: $ gcdot x = gxg^{ - 1} $ .
The orbits of this action are called conjugacy classes of $ G $ .
Let $ Usubsetmathbb R^n $ be an open set.
A differential k - form on $ U $ is a $ C^r $ - form if it is of class $ C^r $ .
Let $ (V,F) $ be a vector space and let $ text{End}(V) $ be the set of linear transformations $ Vto V $ .
This is a ring with resepect to the usual product operation.
Define a new bracket operation by $ x,y = xy - yx $ .
Then $ text{End}(V) $ is a Lie algebra over $ F $ .
We call it the general linear algebra and denote it $ mathfrak{gl}(V) $ .
If $ V $ is of finite dimension, then $ text{dim}(text{End}(V)) = text{dim}(V)^2 $ .
The general linear algebra is a Lie algebra.
A class function on a group $ G $ is a function that is constant on the conjugacy classes of $ G $ .
Let $ (X,*) $ be a magma with identity element $ e $ .
An element $ ain X $ is a right inverse element for $ bin X $ if $ a * b=e $ .
Let $ V $ be a finite - dimensional vector space with $ text{dim}(V)=2ell $ with basis $ {v_1,dots,v_{2ell}} $ .
Define a nondegenerate skew - symmetric form $ f $ on $ V $ with the matrix $ s = } $ .
Note that the dimension being even is required for the existence of a nondegenerate bilinear form satisfying $ f(v,w) = - f(w,v) $ .
Then denote by $ mathfrak{sp}(V) $ the symplectic algebra consisting of all endomorphisms $ x $ of $ V $ such that $ f(x(v),w) = - f(v,x(w)) $ .
Let $ mathcal D $ be a small category and $ mathcal C $ any category.
A $ mathcal D $ - shaped diagram is a functor $ F:mathcal Dto mathcal C $ .
Morphisms $ Fto F' $ of $ mathcal D $ - shaped diagrams are natural transformations.
Then we get the category $ mathcal Dmathcal C $ of $ mathcal D $ - shaped objects in $ mathcal C $ .
An object $ C $ of $ mathcal C $ determines the constant diagram $ underline C $ that sends each object of $ mathcal D $ to $ C $ and each morphism of $ mathcal D $ to the identity morphism of $ C $ .
Let $ U subset mathbb R^n $ be open.
The partial derivative of a function $ f:Utomathbb R^m $ with respect to the $ i $ th coordinate $ x_i $ at $ a = (a_1,dots,a_n) in U $ is defined as $ frac{partial f}{partial x_i}(a) = lim_{hto 0} frac{f(a_1,dots,a_i+h,dots,a_n) - f(a)}{h} $ .
The special unitary group $ text{SU}(n) $ is the group of unitary matrices with unit determinant.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
Let $ G $ and $ H $ be groups.
The kernel of a group homomorphism $ f:Gto H $ is the set of all elements $ gin G $ such that $ f(g) = e_H $ , i.e. the elements of $ G $ that get mapped to the identity element in $ H $ .
Let $ (X,Sigma, mu) $ be a measure space.
Let $ f = sumlimits_{k=1}^n a_k chi_{A_k} $ be a simple function such that $ mu(A_k) < infty $ for $ a_k neq 0 $ .
The Lebesgue integral of the simple function $ f $ is defined as $ intleft(sum_{k=1}^n a_kchi_{A_k}right)dmu = sum_{k=1}^n a_k intchi_{A_k}dmu = sum_{k=1}^n a_kmu(A_k), $ consistent with the Lebesgue integral of characteristic functions.
If $ B $ is a measurable subset of $ X $ and $ g = sumlimits_{k=1}^n b_kchi_{B_k} $ is a measurable simple function on $ B $ , define $ int_B gdmu = intchi_Bgdmu = sum_{k=1}^nb_kmu(B_kcap B) $ .
An embedded m - dimensional manifold with boundary that admits an orientation is an oriented manifold .
A lower bound of a subset $ S $ of a poset $ (X,leq) $ is an element $ a $ of $ P $ such that for all $ xin S $ , $ aleq x $ .
The complex conjugate of a complex number $ z = x+iy $ is $ overline z = x - iy $ .
Let $ G $ be a group and $ N $ a normal subgroup of $ G $ .
Define an equivalence relation $ sim $ by setting $ gsim h $ if $ g,h in H $ .
Then the quotient of $ G $ by $ N $ (written $ G/N $ ) is the quotient by $ sim $ of $ N $ .
A manifold is a topological space that is second - countable, Hausdorff, and locally homeomorphic to $ mathbb R^n $ .
Let $ V $ be a vector space.
A subset $ W subseteq V $ is a vector subspace if for all $ alpha in F $ the ground field and $ u,v in W $ , $ alpha v in W $ and $ u+v in W $ .
Let $ E $ , $ E' $ , and $ B $ be connected and locally path - connected topological spaces.
Then $ g:Eto E' $ is a map of coverings $ p:Eto B $ and $ p':E'to B $ of $ B $ if the following diagram commutes:
Let $ mathfrak g $ and $ mathfrak h $ be Lie algebras.
The direct sum $ mathfrak g oplus mathfrak h $ of $ mathfrak g $ and $ mathfrak h $ is the vector space direct sum of $ mathfrak g $ and $ mathfrak h $ when considered as vector spaces endowed with the bracket $ (X_1,X_2,(Y_1,Y_2)=(X_1,Y_1,X_2,Y_2) $ .
The symmetric difference of two sets $ A $ and $ B $ is given by $ Atriangle B = (Asetminus B)cup(Bsetminus A) = (Acup B)setminus (Acap B) $ .
It is the set of elements that are in exactly one of $ A $ or $ B $ .
We define the Haar integral over a topological space $ X $ s follows: Any Borel measure $ mu $ on $ X $ gives a linear functional $ int:C_c(X) to mathbb C, fmapsto int_X(f) := int_X f(x)text dmu(x) $ where $ C_c(X) $ is the collection of continuous complex - valued functions on $ X $ with compact support and the integral is defined according to that measure.
The Haar integral is this $ mathbb C $ - linear function $ int:C_c(X) to mathbb C $ such that (i) for all $ fin C_c(X) $ , if $ f(x) geq 0 $ for all $ xin X $ then $ int_X f geq 0 $ .
Moreover, $ int_X f= 0 $ implies that $ f=0 $ .
(when $ f(x) $ is real) (ii) For any compact $ Ksubset X $ , there exists some $ c_Kgeq 0 $ such that for all continuous functions with support in $ K $ , $ left|int_X fright| leq c_Kcdot maxlimits_{xin K} |f(x)| $ .
If we want the integral to be invariant with respect to the action of a group, we choose the measure to be the Haar measure.
Also keep in mind that this is in fact a special case of the Lebesgue integral!
All of those properties of the integral still apply here with no extra work!
The Borel σ - algebra on a topological space $ X $ is the σ - algebra generated by the open sets in $ X $ .
Let $ M $ be a manifold with boundary.
The interior points of $ M $ are the points $ pin M $ for which there exists a chart $ (U,phi) $ such that $ phi(p) in H^m_+ $ , the interior of the closed half - space.
Let $ H $ be a subgroup of the group $ G $ .
The index of $ G $ is $ G:H = |G|/|H| $ .
This is always an integer by Lagrange's theorem.
The power set of a set $ X $ is the connection of all subsets of $ X $ , including $ X $ itself and the empty set.
It is denoted $ mathcal P(X) $ .
Let $ R $ be an integral domain.
Then $ a,bin R $ are associate if the ideals they generate are equal.
That is, if there exists a unit $ uin R $ such that $ ua=b $ .
The holomorph of a group $ G $ is in some sense a group that contains a copy of both $ G $ itself and the automorphism group of $ G $ .
Formally, $ text{Hol}(G) = Grtimes text{Aut}(G) $ where the homomorphism $ rho: text{Aut}(G)to text{Aut}(G) $ associated to the semidirect product is the identity function.
Thus the product of elements of $ text{Hol}(G) $ is as follows: $ (g,alpha)(h,beta) = (galpha(h),alphabeta) $ .
Equivalently, we can define $ text{Hol}(G) ={l_gcircsigma mid gin G, sigmaintext{Aut}(G)}subset S_G $ where $ l_g $ denotes left multiplication by $ g $ .
Let $ (X,leq) $ be a poset and let $ Ssubseteq X $ .
The supremum of this subset, if it exists, is an upper bound $ b $ of $ S $ in $ X $ such that for all upper bounds $ x $ of $ S $ in $ X $ , $ bleq x $ .
That is, the supremum is the least upper bound.
Let $ G $ be a group acting on a set $ S $ .
For a subgroup $ H $ of $ G $ , denote by $ NH $ the normalizer of $ H $ in $ G $ and let $ WH = NH/H $ .
The quotient $ WH $ is the Weyl group .
Let $ (X,*) $ be a magma with identity element $ e $ .
An element $ ain X $ is an inverse element for $ bin X $ if it is both a right and a left inverse element for $ b $ .
An attracting fixed point of a function $ f $ is a point $ x_0 $ in the domain of $ f $ for which there exists a neighborhood $ U $ of $ x_0 $ such that for all $ x in U $ , the iterated function sequence $ {f^{circ n}(x)}_{ninmathbb n} $ convergesto $ x_0 $ .
The volume of a rectifiable set $ S $ is $ text{vol}(S) = int_S 1 $ , or the integral of the constant function over $ S $ .
Let $ (X,Sigma) $ be a measurable space with $ mu $ a signed measure.
The negative variation of $ mu $ is the function $ nu $ given by $ nu(A) = pi(A) - mu(A) $ where $ pi $ is the positive variation of $ mu $ .
Let $ (X,Sigma,mu) $ be a measure space and let $ {f_n}_{ninmathbb N} $ be a sequence of functions $ f_n: Xto mathbb R $ .
The sequence converges in measure to $ f $ if for all $ varepsilon > 0 $ , the measure of the set $ mu({xin Xmid |f_n(x) - f(x)|>varepsilon}) $ converges to zero as $ n $ goes to infinity.
Let $ Asubset X $ be a subset of the topological space $ X $ .
The subspace topology on $ A $ is the topology in which the open sets are given by all sets $ A cap U $ where $ Usubset X $ is open in $ X $ .
The support of a function $ f: X to Y $ is the subset $ S subset X $ such that for all $ x in S $ , $ f(x) neq 0 $ .
For $ sigma $ a permutation in $ S_n $ , the sign of $ sigma $ denoted $ text{sgn}(sigma) $ is the determinant of the linear transformation $ P_sigma:mathbb R^ntomathbb R^n $ given by $ P_sigma(x_1,dots,x_n) =(x_{sigma(1)},dots, x_{sigma(n)}) $ .
Equivalently, let $ {x_i}_{i=1}^n $ be $ n $ distinct variables and let $ Delta $ be the polynomial $ Delta = prod_{1leq ileq j leq n}(x_i - x_j) $ .
For each $ sigmain S_n $ , let $ sigma $ act on $ Delta $ by permuting the variables as their indices.
That is, $ sigma(Delta)=prod_{1leq ileq jleq n} (x_{sigma(i)} - x_{sigma(j)}) $ .
Note that in general, $ Delta $ contains one factor of $ (x_i - x_j) $ for all $ i < j $ .
Because $ sigma $ is a bijection on the indices, $ sigma(Delta) $ contains either a factor of $ (x_i - x_j) $ or a factor of $ (x_j - x_i) $ but not both.
If the latter, we may write it as $ ( - 1)(x_i - x_j)) $ .
Collect the $ - 1 $ factors in the product and define $ epsilon(sigma) = } $ The function $ epsilon $ gives the sign of $ sigma $ .
A group action of a group $ G $ on a set $ X $ is a map $ Gtimes Ato A $ such that (i) $ g_1cdot(g_2cdot x)= (g_1g_2)cdot x $ for all $ g_1,g_2in G $ and for all $ xin X $ ; (ii) $ ecdot x = x $ for all $ xin X $ .
Equivalently, we can describe a group action in terms of the homomorphism $ rho:Gto S_X $ (to the symmetric group on $ X $ ) given by $ rho(g) = sigma_g $ where $ sigma_g in S_X $ is such that $ xmapsto gcdot x $ under $ sigma_g $ .
When thinking of the group $ G $ as a groupoid with a single object, then a left action of $ G $ is the same as a covariant functor $ Gto mathcal S $ from $ G $ to the category of sets while a right action is like a contravariant functor.
The determinant of a linear transformation $ T: Vto V $ (where $ V $ is finite - dimensional) is the unique antisymmetric $ n $ - variable multilinear map $ text{det} $ such that $ text{det}:(mathbb R^n)^n to mathbb R $ such that $ text{det}(e_1,dots,e_n)=1 $ .
It is defined for a the matrix $ A $ of $ T $ as $ text{det}(A) = text{det}(A_1,dots,A_n) $ where the $ A_i $ are the columns of $ A $ .
Let $ p $ be a prime and let $ (A,+) $ be an abelian group.
The $ p $ - primary component of $ A $ is $ A_{(p)} = {ain Amid exists ninmathbb N text{ st. }p^na= 0} $ .
Let $ G $ be a group with subgroup $ H $ and let $ G/H $ be the quotient of $ G $ by $ H $ .
Then $ G $ acts on $ H $ in a natural way: for all $ Z = aH in G/H $ , $ gcdot Z = gZ = (ga)H $ .
A CW complex is a topological space $ X $ constructed in the following way: (i) Consider a discrete set $ X^0 $ and call its points "0 - cells."
(ii) Form the $ n $ - skeleton $ X^n $ from $ X^{n - 1} $ by attaching $ n $ - cells $ e_alpha^n $ via maps $ varphi_alpha:S^{n - 1} to X^{n - 1} $ .
Thus $ X^n $ is the quotient vector space of the disjoint union $ X^{n - 1}coprod_alpha D_alpha^n $ of $ X^{n - 1} $ with a collection of disks $ D_alpha^n $ under the identification $ xsim varphi_alpha(x) $ for $ xin partial D_alpha^n $ .
As a set, $ X^n = X^{n - 1}coprod_alpha e_alpha^n $ where each $ e_alpha^n $ is an open $ n $ - disk.
(iii) Either stop this process at a finite value of $ n $ , setting $ X=X^n $ or continue indefinitely and endow $ X = bigcup_n X^n $ with the weak topology.
A matrix Lie group $ mathfrak g $ is complex if it is a complex subspace of $ M_n(mathbb C) $ .
That is, if $ iXin mathfrak g $ for all $ Xin mathfrak g $ .
Let $ X $ be a metric space.
Two metrics $ rho_1 $ , $ rho_2 $ are equivalent if there exists $ c>0 $ such that $ frac{1}{c}rho_1(x,y) leq rho_2(x,y) leq crho_1(x,y) $ .
A covering of groupoids $ p:mathcal Eto mathcal B $ is universal if and only if the automorphism group $ pi(mathcal B,b) $ of the object $ b $ acts freely on the fiber of $ b $ , denoted $ F_b $ .
Then $ F_b $ is isomorphic to $ pi(mathcal B, b) $ as a $ pi(mathcal B,b) $ - set.
A harmonic polynomial is a harmonic function that is also a polynomial.
Let $ G $ and $ G' $ be compact topological groups with respective Haar integrals $ int_G $ and $ int_{G'} $ .
For continuous functions $ f,f':G,G'to mathbb C $ , denote by $ fboxtimes f':Gtimes G to mathbb C $ the function such that $ (g,g')mapsto f(g)f'(g') $ .
From this and given continuous representations $ rho:Gto GL(V) $ and $ rho':Gto GL(V') $ , we can define the box product of representations into the tensor product of $ V $ and $ V' $ as $ rhoboxtimes rho' :Gtimes G to GL(Votimes V') $ given by $ (g,g')mapsto rho(g) otimes rho'(g') $ .
Let $ L $ be a finite - dimensional vector space.
Then consider $ L $ as a Lie algebra by setting $ xy=0 $ for all $ x,y in L $ .
Such a Lie algebra is called abelian .
Let $ G $ be a group and $ S $ some subset of $ G $ .
A group word in $ S $ is an expression of the form $ s_1^{ell_1}s_2^{ell_2}cdots s_n^{ell_n} $ where $ s_i in S $ and $ ell_i = pm 1 $ .
Let $ X $ be a metric space.
The completion of $ X $ is the complete metric space $ (tilde X, tilde rho) $ such that $ X $ is a dense subset of $ tilde X $ and $ rho(x,y) = tilderho(x,y) $ for all $ x,y in X $ as constructed here.
Let $ mathfrak g $ be a complex, semisimple Lie algebra and $ mathfrak h $ a Cartan subalgebra of $ mathfrak g $ .
Let $ (V,rho) $ be a representation of $ mathfrak g $ and let $ lambda $ be a linear functional on $ mathfrak h $ .
We can consider the vector space $ V $ as a $ mathfrak g $ - module by setting $ xcdot v = rho(x)v $ for all $ x in mathfrak g $ .
Then the weight space of $ V $ with weight $ lambda $ is the subspace $ V_lambda subset V $ given by $ V_lambda ={vin Vmid forall Hin mathfrak h, Hcdot v = lambda(H)c} $ .
Let $ G $ be a group and $ R $ a ring.
The group ring $ RG $ is the set of formal sums of elements in $ G $ with coefficients in $ R $ .
By convention, $ 1cdot g = g $ and $ 0cdot g $ terms are omitted from the sum.
Addition is defined by components and multiplication is defined by $ (ag_i)(bg_j) = abg_k $ where $ g_ig_j = g_k $ , and extended distributively.
Let $ (V,rho) $ be a representation of the group $ G $ .
A vector subspace $ W $ of $ V $ is $ G $ - invariant if for all $ win W $ and $ gin G $ , $ rho(g)w in W $ .
Let $ V $ and $ W $ be vector spaces.
An intertwiner between two representations $ phi:Gto GL(V) $ and $ psi:Gto GL(W) $ of the group $ G $ is a linear transformation $ T:Vto W $ such that $ Tcirc(phi(g)) = (psi(g))circ T $ for all $ g in G $ .
The category of topological spaces is the category whose objects are topological spaces and whose morphisms are continuous functions.
Let $ X $ be a topological space.
A set $ Asubset X $ is meager if it may be expressed as a countable union of nowhere dense sets
A complex group representation $ rho: Gto GL(V) $ is unitary if $ rho(g) $ is unitary for all $ gin G $ .
Explicitly, it is unitary if the Hermitian adjoint $ rho(g)^* = rho(g)^{ - 1} = rho(g^{ - 1}) $ for all $ gin G $ .
A topological space is connected if it cannot be expressed as the disjoint union of two nonempty open subsets.
Let $ W_1,W_2 subset mathbb R^n $ be open.
A function $ g: W_1to W_2 $ is a $ C^k $ - embedding if $ g $ is injective, $ g $ is of class $ C^k $ , and the total derivative $ Dg(x) $ is invertible for all $ x in W_1 $ .
A subset $ Hsubset G $ of the group $ G $ is a subgroup of $ G $ if it is itself a group under the operation inherited from $ G $ .
The subgroup axioms are - closure under $ * $ ; - the identity element of $ G $ is in $ H $ ; - for every $ hin H $ , the inverse element $ h^{ - 1} $ is also in $ H $ .
Let $ (V,rho) $ be a finite - dimensional representation of a group $ G $ where $ V $ is a vector space over the field $ F $ .
The character of this representation is the function $ chi_rho:Gto F $ given by $ chi_rho(g) = text{Tr}(rho(g)) $ .
Let $ G $ be a group acting on a set $ X $ .
The orbit of some $ x in X $ is the set of elements of $ X $ to which $ x $ can be moved by the elements of $ G $ .
It is denoted $ Gcdot x = {gcdot x mid g in G} $ .
The set of orbits of points of $ X $ creates a partition of $ X $ with associated equivalence relation given by $ xsim y $ if and only if there exists $ g in G $ with $ gcdot x =y $ , and the orbits are equivalence classes.
Equivalently, an orbit of $ x $ is the subset of $ X $ containing $ x $ on which the action of $ G $ is transitive.
Let $ R $ be an integral domain and let $ M $ be an $ R $ - module.
Then for any submodule $ N $ of $ M $ , the annihilator of $ N $ is defined as $ text{Ann}(N) = {rin Rmid rn=0text{ for all } nin N} $ .
An inner product space is a vector space $ V $ together with an inner product $ langlecdot,cdotrangle $ on $ V $ .
A category is complete if it "has all" limits.
Let $ V $ be an $ n $ - dimensional vector space.
A representation of a group $ G $ is a group homomorphism $ rho: G to GL(V) $ from $ G $ to the $ n $ - dimensional general linear group of invertible linear transformations of $ V $ .
Let $ X $ be a topological space.
The local homology groups of $ X $ at the point $ xin X $ are the relative homology groups $ H_n(X,Xsetminus {x}) $ .
A function $ f:Ato B $ is bijective if it is both injective and surjective.
A Lie algebra $ L $ is semisimple if its radical $ text{Rad}(L) = {0} $ .
A linear transformation $ phi:L to L' $ for Lie algebras $ L $ and $ L' $ is a homomorphism if it preserves the bracket operation, that is if $ phi(xy_L) = phi(x)phi(y)_{L'} $ .
A Lie algebra homomorphism is an isomorphism if it is bijective.
Assuming the Axiom of Choice, the cardinality of a set $ X $ is the least ordinal number $ alpha $ such that there exists a bijection between $ X $ and $ alpha $ .
There exists a well - ordering on the cardinal numbers under the relation $ |A| leq |B| $ if and only if there exists an injection from $ A $ to $ B $ .
Let $ p $ be a prime number.
A $ p $ - group is a group whose order is a power of $ p $ .
Equivalently, the order of every element in a $ p $ - group is a power of $ p $ .
A ring $ R $ is a Euclidean domain if there exists a norm $ N $ on $ R $ such that for all nonzero $ a,bin R $ , there exist $ q,rin R $ such that $ a = qb+r $ and either $ r=0 $ or $ N(r) < N(b) $ .
A ring ideal is proper if it is a proper subset of its parent ring.
A set $ X $ is countable if there exists a bijection between $ X $ and the set $ mathbb N $ of natural numbers. $ |mathbb N| $ is the cardinal number of the naturals.
Let $ f:Qtomathbb R $ be a bounded function defined on the rectangle $ Q $ in $ mathbb R^n $ .
The upper integral of $ f $ over $ Q $ is the infimum over all partitions of $ Q $ of the upper sum of $ Q $ : $ overline{int_Q}f = inf_P U(f,P) $
Let $ T: Vto V $ be a linear transformation from the vector space $ V $ to itself.
A vector subspace $ Wsubseteq V $ is an invariant subspace of $ T $ if the image of $ W $ is contained in $ W $ .
That is, $ T(W)subseteq W $ .
An integral domain $ R $ is a unique factorization domain if (i) Every nonzero, non - unit $ rin R $ can be written as a product of irreducibles; and (ii) Whenever $ p_1cdots p_n=q_1cdots q_m $ for $ p_i,q_j $ irreducible and $ m,ninmathbb N $ , then $ m=n $
and there is a relabeling/reordering of the $ q_j $ such that $ q_i $ is associate to $ p_i $ for all $ 1leq ileq n $ .
In particular, for every association class of irreducibles, we can choose a representative, and let $ P $ be the collection of such representatives so that every $ rin R $ can be written uniquely as $ r= uprod_{pin P} p^{e_p} $ for $ u $ a unit and all but finitely many of the $ e_p $ equal to zero.
Let $ R $ be a commutative ring and let $ a,b in R $ .
We say that $ a $ divides $ b $ if there exists $ rin R $ such that $ a = rb $ .
Equivalently, $ a $ divides $ b $ if the ideal generated by $ b $ is contained in the ideal generated by $ a $ .
Let $ f:mathbb Ctomathbb C $ , and write it as follows: $ f(z) = u(x,y) + iv(x,y) $ where $ zin mathbb C $ is such that $ z = x+iy $ and $ u,v:mathbb R^2to mathbb R $ .
Then $ f $ is complex differentiable at $ z_0in mathbb C $ with derivative $ f'(z_0) $ if the limit $ f'(z_0) = lim_{zto z_0} frac{f(z) - f(z_0)}{z - z_0} $ exists.
A complex $ ntimes n $ matrix $ U $ is unitary if its conjugate transpose $ U^* $ is equal to its inverse $ U^{ - 1} $ .
A category is cocomplete if it "has all" colimits.
The pair $ (U,phi) $ as in the definition of orientable defines an orientation on an embedded m - dimensional manifold with boundary $ M $ .
Two pairs $ (U,phi) $ and $ (V,psi) $ determine the same orientation if for every local parametrization $ f:Wto M $ where $ W subset H^m $ is open in $ H^m $ , there exists a positive function $ g:Wtomathbb R $ such that $ f^*(phi) = gf^*phi $ .
An orientation on $ M $ is an equivalence class if pairs that determine the same orientation on $ M $ .
Let $ (V,F) $ and $ (W,F) $ be vector spaces over the same field with respective inner products $ langlecdot,cdotrangle_V $ and $ langlecdot,cdotrangle_V $ .
The adjoint of a linear transformation $ T:Vto W $ is the linear transformation $ T^*: W to V $ such that $ langle T(v), wrangle_W = langle v, T^*(w)rangle_V $ for all $ v in V $ and $ w in W $ .
Fix $ {V_i}_{iin mathbb N} $ a (possibly infinite) sequence of finite - dimensional vector spaces over a field $ k $ .
Then we can define $ V = bigoplus_{igeq 0}V_i = {(v_{i_1},dots, v_{i_N}) mid Nin mathbb N, v_{i_k} in V_{i_k}} $ .
If we consider each $ V_i $ as a linearly independent subspace of $ V $ , we can consider the tuples as finite sums of vectors.
Such a $ V $ is a graded vector space .
A module over a ring $ M $ is simple if it has no nonzero proper submodules.
Let $ Usubsetmathbb R^n $ be an open set and let $ f:Utomathbb R^n $ be of class $ C^1 $ .
A point $ ain mathbb R^n $ is a critical value of $ f $ if there exists a critical point $ c in U $ such that $ f(c) =a $ .
A norm $ N $ on a ring $ R $ is a function $ N: Rto mathbb N $ (here $ mathbb N $ is assumed to include $ 0 $ ) such that $ N(0) = 0 $ .
Such a function is called positive if it is nonzero on every element except for the additive identity.
A covering space $ p:Eto B $ is regular if the image of the induced homomorphism|induced homomorphism $ p_*(pi_1(E,e)) $ is a normal subgroup of the fundamental group $ pi_1(B,b) $ .
Let $ R $ be a ring and consider the graded polynomial algebra
$ P = mathbb Rx_1,dots,x_d $ .
Then Vandermonde determinant is the product $ Delta_d = prod_{1leq i < j leq d}(x_j - x_i) $ .
It is equal to the determinant $ Delta_d = det} $ .
It is a homogeneous polynomial with integer coefficients of degree $ frac{d(d - 1)}{2} $ .
Let $ n > 0 $ .
As a generalization of the Vandermonde determinant, given any subset $ Jsubseteq {1,dots,n} $ define $ Delta(J) = prod_{{i,jin J mid i < j}}(x_i - j_i) $ .
Moreover, given a partition $ lambda = (lambda_1,dots, lambda_k) $ of $ n $ , let $ alpha_i = {lambda_i, lambda_i + 1, dots, lambda_{i+1} - 1} $ so that $ {1,dots,n} = coprod_{i=1}^k alpha_i $ as disjoint subsets.
Then we define $ Delta_lambda = prod_{i=1}^k Delta(alpha_i), $ which turns out to be a homogeneous polynomial of degree $ deg Delta_lambda = d_lambda = sum_{i=1}^k deg(Delta(alpha_i)) = sum_{i=1}^k frac{lambda_i(lambda_i +1)}{2} $ .
Moreover, if $ text{sign}_lambda $ is the restriction of the sign representation to the Young subgroup $ S_lambda $ , then $ Delta_lambda in(P^{d_lambda})^{text{sign}_lambda} $ .
That is, $ Delta_lambda $ is a degree - $ d_lambda $ invariant polynomials under the representation $ text{sign}_lambda $ .
Let $ X $ and $ Y $ be topological spaces.
A lift of a continuous map $ f:Yto X $ is a map $ tilde f:Yto tilde X $ such that $ ptilde f = f $ where $ p $ is the map associated to the covering space $ (p,tilde X) $ .
Let $ R $ be a ring, let $ M $ be an $ R $ - module, and let $ Nsubseteq N $ be a submodule of $ M $ .
Then the quotient module $ M/N $ is the set of cosets $ {m+Nmid min M} $ with addition defined by $ (m_1+N)+ (m_2+N) = (m_1+m_2) + N $ and scalar multiplication by $ r(m_1 + N) = (rm_1)+N $ .
The set $ Asubset mathbb R $ is an interval if for all $ a,b in A $ with $ a<b $ and $ xin mathbb R $ , $ a<x<b implies xin A $ .
Let $ Usubsetmathbb R^n $ be an open set.
A differential $ k $ - form on $ U $ is a map to the $ k $ th exterior product of the dual space of $ mathbb R^n $ : $ phi:Utowedge^k((mathbb R^n)^*) $ .
A $ 0 $ - form is just a function.
Wedging a $ 0 $ - form with something amounts to multiplication by the function.
Let $ Usubsetmathbb R^n $ be an open set.
Then we write $ mathcal A^k(U) $ for the vector space of k - forms on $ U $ .
Let $ R $ be a ring and let $ M $ be an $ R $ - module.
For $ A $ and $ B $ submodules of $ M $ , define the sum of $ A $ and $ B $ by $ A+B = {a+bmid ain A,bin B} $ .
It is itself a submodule of $ M $ , and in particular is the smallest one containing both $ A $ and $ B $ .
The Baire classes of continuous real - valued functions on a topological space $ X $ are defined as follows: - continuous functions are Baire - 0; - functions that are the pointwise limit of a sequence of continuous functions are Baire - 1; - for all $ alpha in mathbb N $ , functions that are the pointwise limit of a sequence functions of Baire class less than $ alpha $ are Baire - $ alpha $ .
Let $ R $ be a ring.
Two ideals $ A $ and $ B $ of $ R $ are comaximal if the sum $ A+B=R $ .
A Lie algebra $ L $ is nilpotent if some element $ L^n $ of the descending central series or the ascending central series is equal to $ {0} $ .
An abelian group $ (A,+) $ is divisible if for every $ din mathbb N $ and for every $ ain A $ , there exists $ a'in A $ such that $ da' = a $ .
A normed linear space is a vector space together with a norm.
Let $ E $ and $ B $ be compactly generated topological spaces.
A continuous map $ p:Eto B $ is a bundle if when restricted to the spaces $ U $ in a numerable open cover of $ B $ , there are homeomorphisms $ phi:Utimes Fto p^{ - 1}(U) $ such that $ picircphi = pi_1 $ for some fixed topological space $ F $ ; what is $ pi_1 $ ?
check if this is the right thing
Let $ G $ and $ H $ be Lie groups.
A map $ phi $ from $ G $ to $ H $ is an isomorphism of Lie groups if it is a homeomorphic group homomorphism.
Equivalently, $ Phi $ is an isomorphism of Lie groups if it is a bijective homomorphism of Lie groups with continuous inverse.
Let $ Usubsetmathbb R^n $ be an open subset and let $ phi $ be an n - form on $ U $ .
Then we can uniquely write $ phi $ as $ phi = g(x_1,dots,x_n)dx_1wedgecdotswedge dx_n $ for some ( what kind?) function $ g:Utomathbb R $ .
If $ phi $ has compact support, we can define the integral of the $ n $ - form to be $ int_U phi = int_U g, $ if it exists.
A category is small if both its collection of objects and its collection of morphisms are sets, not proper classes
The radical of a Lie algebra $ L $ is the maximal solvable ideal of $ L $ .
Let $ V $ be a vector space.
A subset $ W subset V $ is linearly dependent if it is not linearly independent.
Let $ G $ be a group and $ V $ a vector space.
The trivial representation is the representation $ rho: Gto text{GL}(V) $ such that for all $ gin G $ , $ rho(g) = text{id}_V $ .
Let $ S $ be a simple Lie algebra representation $ mathfrak g $ .
The isotypic component of type $ S $ of a representation $ V $ is the direct sum of subrepresentations of $ V $ isomorphic to $ S $ .
The strong differential basis is the differential basis consisting of rectangles containing $ x $ that are parallel to all axes of $ mathbb R^n $ .
A group $ G $ is cyclic if it is generated by a single element.
Let $ (X,Sigma) $ be a measurable space with measures $ mu $ and $ nu $ such that $ nu $ is absolutely continuous with respect to $ mu $ .
The Radon - Nikodym derivative $ frac{text dnu}{text dmu} $ is the function $ f in $ L $ _1(mu) $ such that $ nu(A) = int_A ftext dmu $ .
Let $ (X,Sigma) $ be a measurable space and let $ mu $ and $ nu $ both be measures on $ (X,Sigma) $ .
Then $ mu $ and $ nu $ are singular , written $ mu perp nu $ , if there exist disjoint sets $ A,Bsubset X $ such that $ mu equiv 0 $ outside of $ A $ and $ nu equiv 0 $ outside of $ B $ .
That is, $ mu(A') = 0 $ for all $ A'subset Xsetminus A $ and $ nu(B') =0 $ for all $ B'subset Xsetminus B $ .
Let $ mathcal B $ be a groupoid.
Then $ text{Cov}(mathcal B) $ denotes the category of coverings of $ mathcal B $ where the objects are coverings $ p:mathcal Eto mathcal B $ and the morphisms are maps of coverings of $ mathcal B $ .
When there is no ambiguity, we write $ text{Cov}(mathcal E,mathcal E') $ for the set of maps $ mathcal Eto mathcal E' $ .
Let $ f:Qtomathbb R $ be a bounded function defined on the rectangle $ Q $ in $ mathbb R^n $ .
The lower integral of $ f $ over $ Q $ is the supremum over all partitions of $ Q $ of the lower sum of $ Q $ : $ underline{int_Q}f = sup_P L(f,P) $
Let $ X $ be a topological space and let $ f,g:Ito X $ be two paths in $ X $ such that $ f(0) = g(0) = x $ and $ f(1) = g(1)=y $ .
That is, $ f $ and $ g $ are both paths *from* $ x $ *to* $ y $ in $ X $ .
Then $ f $ and $ g $ are said to be homotopically equivalent if there exists a homotopy $ h: Itimes I to X $ such that (i) $ h(s,0) = f(s) $ ; (ii) $ h(s,1) = g(s) $ ; (iii) $ h(0,t) = x $ ; (iv) $ h(1,t) = y $ .
Homotopy equivalence of paths forms an equivalence relation on paths in $ X $ .
Let $ P = mathbb Cx_1,dots, x_n $ be the graded polynomial algebra and let $ P^d $ be the subspace of degree - $ d $ homogeneous polynomials.
Let $ lambda $ be a partition of $ n $ .
Then define $ V(lambda)= mathbb Ctext{ - span}{s(Delta_lambda) mid sin S_n} $ .
That is, $ V(lambda) $ is the collection of $ mathbb C $ - linear combinations of images of the Vandermonde polynomial under the action of the symmetric group $ S_n $ on variables by permutation.
It is an $ S_n $ - invariant subspace of $ P^{d_lambda} $ , and moreover it defines a representation of $ S_n $ $ rho:S_nto GL(V(lambda)) $ called the Specht module associated to $ lambda $ .
It is a module via the correspondence between representations and modules over the group algebra.
An $ mtimes n $ (real or complex) matrix $ A $ is a collection of $ mn $ entries $ (a_{ij}) $ which are either real or complex numbers arranged in $ m $ rows and $ n $ columns as follows: $ } $ Two matrices of the same size may be added together entrywise.
A field is a set $ F $ together with two binary operations addition ( $ + $ ) and multiplication ( $ cdot $ ) such that - Addition and multiplication are both associative; - Addition and multiplication are both commutative; - There exist two distinct elements $ 0neq 1 $ in $ F $ such that $ 0 $ is an additive identity and $ 1 $ is a multiplicative identity; - Every element of $ F $ has an additive inverse; - Every nonzero element of $ F $ has a multiplicative inverse; - Multiplication distributes over addition as follows: $ acdot (b+c) = (acdot b) + (acdot c) $ for all $ a,b,cin F $ .
Equivalently, a field is a commutative division ring.
The elements of L $ _infty $ are called essentially bounded .
That is, their absolute values have finite essential supremum.
The barycenter of an n - simplex $ v_0,dots,v_n $ is the point $ b=sum_i t_iv_i $ whose barycentric coordinates $ t_i $ are all equal.
Specifically, $ t_i = frac{1}{(n+1)} $ for each $ i $ .
Let $ Dsubsetmathbb R^2 $ be connected and open and let $ f:Dto mathbb R $ be continuous.
Then let $ zinmathbb R^2 $ .
If for some $ varepsilon > 0 $ , the closure $ overline{B(z,varepsilon)} $ of the ball of radius $ varepsilon $ centered at $ z $ is contained within $ D $ (i.e. $ overline{B(z,varepsilon)}subset D $ ), then the circular mean value $ MV(f,z,varepsilon) $ is the average value of $ f $ on the boundary $ partial B(z,varepsilon) $ .
This is given by $ MV(f,z,varepsilon) =frac{1}{2pi varepsilon} int_{{winmathbb R^2 mid |w - z|=varepsilon}} f(w) |text dw| $ where the integral is taken with respect to arc length.
Moreover, $ MV(f,zvarepsilon) = frac{1}{2pi}int_0^{2pi} f(z+varepsilon e^{itheta})text dtheta $ where $ theta $ is the angle of the polar coordinate for $ z $ .
Let $ mathcal A $ be a collection of open sets in $ mathbb R^n $ and let $ A $ be their union.
A partition of unity subordinate to $ mathcal A $ is a countable collection of smooth functions $ {phi_i}_{iin I} $ such that $ phi_i:mathbb R^n to mathbb R $ such that (i) each $ phi_i geq 0 $ .
(ii) each $ phi_i $ has compact support contained in some $ U in mathcal A $ .
(iii) for all points $ a $ of $ A $ , there exists an open neighborhood $ V $ of $ a $ such that all but finitely many of the $ phi_i $ are zero on $ V $ .
(iv) for all points $ a $ in $ A $ , $ sumlimits_{iin I} phi_i(a) = 1 $ .
An open simplex is an n - simplex with all of its proper faces deleted.
A group homomorphism is a function $ f: Gto H $ between groups $ G $ and $ H $ is a function that preserves the group operation; that is $ f(gh)=f(g)f(g) $ for all $ g,h in G $ .
A group isomorphism is a homomorphism that is also bijective.
An ideal $ I $ of the ring $ R $ is maximal if for any ideal $ J $ such that $ Isubseteq J $ , either $ J=I $ or $ J=R $ .
A set $ B $ is a basis for the vector space $ V $ if it consists of linearly independent vectors that span $ V $ .
Let $ X $ be a topological space and let $ A $ be a subspace of $ X $ .
A relative cycle $ alpha $ is trivial in the relative homology groups $ H_n(X,A) $ if and only if it is a relative boundary , i.e. if $ alpha = partialbeta + gamma $ for some $ betain C_{n+1}(X) $ and $ gammain C_n(A) $ .
Let $ X $ be a topological space.
A set $ Asubset X $ is comeager if its complement is meager.
A semigroup is a magma $ (X,*) $ such that the binary operation $ * $ is associative.
A signed measure on the measurable space $ (X,Sigma) $ is a function $ mu: Sigma to mathbb R cup { - infty} text{ or } mathbb R cup {infty} $ such that $ mu(emptyset) =0 $ and $ mu $ is σ - additive.
Note that $ mu $ cannot take on both $ - infty $ and $ infty $ as values.
Let $ R $ be a ring and let $ M $ be an $ R $ - module.
We say that $ M $ is of finite length if it admits a composition series of finite length.
A homeomorphism between two topological spaces $ X $ and $ Y $ is a function $ f: Xto Y $ such that (i) $ f $ is continuous.
(ii) $ f $ is bijective.
(iii) $ f^{ - 1}:Yto X $ is continuous.
Two topological spaces $ X $ and $ Y $ are homeomorphic if there exists a homeomorphism between them.
There are two orthogonal algebras : one for even dimension one for odd dimension.
First let Let $ V $ be a finite - dimensional vector space with $ text{dim}(V) = 2ell +1 $ .
Let $ f $ be the nondegenerate, symmetric bilinear form on $ V $ with matrix $ s = } $ .
The orthogonal algebra $ mathfrak{o}(V) $ is defined as the set of all endomorphisms $ x $ of $ V $ satisfying $ f(x(v),w) = - f(v,x(w)) $ .
When $ text{dim}(V) = 2ell $ , $ f $ has the matrix $ s =} $ .
The rest of the construction is the same as for odd dimension.
Let $ G $ be a group and $ V $ a vector space.
The dimension of a group representation $ rho:Gto text{GL}(V) $ is just the usual dimension of $ V $ .
An outer derivation is any derivation that is not an inner derivation.
A representation $ phi $ of a Lie algebra $ L $ is a homomorphism to the general linear algebra ( $ phi:L to mathfrak{gl}(V) $ ) where $ V $ is a vector space.
Here we require the dimension of the vector space underlying $ L $ to be finite, but admit arbitrary dimension for $ V $ .
Let $ G $ be a group.
The subgroups $ H_1 $ and $ H_2 $ are conjugate if there exists $ ain G $ such that $ H_2= aHa^{ - 1} $ .
The character table of a group $ G $ is a table that displays the value taken by the characters of the irreducible representations of $ G $ on the different conjugacy classes of $ G $ .
Recall that characters are class functions
Let $ (X,Sigma) $ be a measurable space, and let $ {A_i}_{i=1}^n $ be a sequence of disjoint measurable sets, and let $ {a_i}_{i=1}^n $ be a sequence of real numbers.
A simple function is a function $ f:Xto mathbb R $ of the form $ f(x) = sum_{k=1}^n a_kchi_{A_k}(x) $ where $ chi_A $ is the characteristic function of $ A $ .
Let $ Msubsetmathbb R^n $ be an embedded m - dimensional manifold with boundary.
Then $ M $ is orientable if there exists an open set $ U subsetmathbb R^n $ containing $ M $ and an m - form $ phi $ on $ U $ such that for every local parametrization $ f:Wto M $ with $ W subset H^+ $ open in $ H^+ $ , the pullback $ f^*phi $ is an m - form on $ W $ which is nowhere zero.
Let $ mathfrak g $ be a Lie algebra.
Then the adjoint map $ text{ad}:mathfrak g to mathfrak{gl}(mathfrak g) $ is given by $ text{ad}_X(Y) = X,Y $ .
Because the adjoint map is a Lie algebra homomorphism, it is a representation of $ mathfrak g $ , and therefore we may refer to $ text{ad} $ as the adjoint representation of $ mathfrak g $ .
A game is determined if one player has a winning strategy.
An edge path is reduced if it is never the case that $ k_{n+1} $ is $ k_n $ with the opposite orientation.
Let $ F $ be a local field with a non - Archimedean absolute value.
Then the Schwartz space , or the collection of Schwartz - Bruhat functions , consists of the functions from $ F $ to $ mathbb C $ that are locally constant and have compact support.
It is often either denoted $ C_c^infty(F) $ or $ mathcal S(F) $ .
The quotient of a topological space $ X $ by a subspace $ Ysubeteq X $ is the quotient by the equivalence relation $ sim $ given by $ ysim y' $ if $ y $ and $ y'in Y $ .
The special linear group $ text{SL}(n,mathbb C) $ (or, over the reals, $ text{SL}(n,mathbb R) $ ) is the group of $ ntimes n $ complex (or real) valued matrices with determinant $ 1 $ together with the operation of matrix multiplication.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
A measure is regular if it is both inner regular and outer regular.
Let $ V $ be a vector space.
A subset $ Ksubset V $ is convex if for all $ x,y in K $ and $ t in 0,1 $ , the vector $ tx + (1 - t)y $ is in $ K $ .
Let $ f: Qto mathbb R $ be a bounded function defined on the rectangle $ Q $ and let $ P $ be a partition of a set of $ Q $ .
For each subrectangle $ R $ determined by $ P $ , let $ M_R(f) = suplimits_{xin R} f(x) $ .
The lower sum for $ f $ determined by $ P $ is $ U(f,P) = sum_{R} M_R(f)text{vol}(R) $ where $ text{vol} $ is the volume of the rectangle $ R $ .
Let $ G $ and $ H $ be Lie groups.
A map $ Phi $ from $ G $ to $ H $ is a Lie group homomorphism if it is a continuous group homomorphism.
A subset $ E $ of a metric space $ (X,rho) $ is bounded if it has finite diamter.
Let $ M subset mathbb R^n $ be an oriented manifold of dimension $ m $ .
Then the boundary $ partial M $ is an $ (m - 1) $ - dimensional oriented manifold (with the induced orientation).
Let $ phi Uto M $ be a local parametrization of $ M $ near $ p in partial M $ so that $ 0 in U $ and $ phi(0) = p $ , where $ U $ is open in the half space $ H^m_+ $ .
A tangent vector $ v in T_pM = Dphi(0)(mathbb R^m)subsetmathbb R^n $ is outward pointing if it is the image of a vector $ w in mathbb R^m $ which does *not* lie in $ H_+^m $ .
A game in normal form is a triplet $ G = (I, {S_i}_{iin I}, {h_i}_{iin I}) $ where (i) $ I $ is the set of _players_. $ I = {1,dots, n} $ for some $ n inmathbb N $ .
(ii) for each $ i in I $ , $ S_i $ is player $ i $ 's set of strategies. $ prodlimits_{iin I} S_i $ is the set of strategy profiles (iii) for each $ iin I $ , the function $ h_i: S_i to mathbb R $ is the payoff function for player $ i $ .
Let $ G $ be a group acting on a set $ A $ .
Then $ A $ is $ G $ - paradoxical if there exist disjoint subsets $ A_1,dots, A_n, B_1,dots, B_msubset A $ and group elements $ g_1,dots,g_n,h_1,dots,h_min G $ such that $ A=bigcup_{i=1}^n g_i(A_i) text{ and } A=bigcup_{i=1}^m h_i(B_i) $ .
A ring $ R $ is an integral domain if it is commutative and has no zero divisors.
A subspace $ I $ of a Lie algebra $ L $ is an ideal of $ L $ if $ xin L $ , $ yin I $ implies that $ xyin I $ .
Equivalently by antisymmetry of the bracket, $ yx in I $ .
Let $ (V,rho) $ be a representation of the group $ G $ .
The $ n $ th exterior power of $ V_1 $ and $ V_2 $ is a representation of $ G $ over the exterior power of a vector space $ wedge^nV $ under the linear transformation $ rho:Gto wedge^n(V) $ such that $ rho(g)(v_1wedge v_2wedge cdots wedge v_n) = (rho(g)v_1wedge rho(g)v_2wedgecdotswedgerho(g)v_n) $ .
Let $ A $ be an abelian group.
The torsion subgroup $ A_T $ of $ A $ is the subgroup consisting of all elements of $ A $ with finite order.
A truth function on a formal language of propositional logic $ L $ (using only the logical operators $ rightarrow $ and $ bot $ ) is a function $ f:text{Form}(L) to {0,1} $ such that $ t(bot) = 0 $ , $ t(top) = 1 $ , and $ t(Arightarrow B) = } $ for all $ A,B in text{Form}(L) $ .
Let $ R $ be a ring.
An $ R $ - module $ M $ is finitely generated if it is generated by a finite subset of itself.
A linear transformation that preserves the norm of vectors is called unitary .
Let $ U subset mathbb R^n $ be open.
The set $ U $ is the graph of a some function if for all $ x in U $ , there exists unique $ y in mathbb R^m $ such that $ (x,y) in A $ .
Equivalently, let $ f:Ato B $ be a set function.
The set $ Gamma = {(a,b) in Atimes B mid b=f(a)} $ is the graph of $ f $ .
An isomorphism of group representations is a morphism of group representations that happens to be bijective.
A binary operation $ cdot $ on the set $ A $ is associative if for all $ a,b,c in A $ , the following relationship holds: $ acdot (bcdot c) = (acdot b)cdot c $ .
Let $ G $ be a compact topological group and let $ dg $ be a Haar measure on $ G $ .
The $ L^2 $ - inner product on $ C(G) $ , the space of continuous functions $ Gto mathbb C $ , is a Hermitian inner product given by $ langle f_1,f_2rangle = frac{1}{text{vol}(G)}int_G f_1(g)overline{f_2(g)}text dg $ for any $ f_1,f_2in C(G) $ .
The integral is the Haar integral and the volume of $ G $ is the integral of the constant function $ 1 $ over $ G $ .
Let $ T:Vto V $ be a linear transformation from the $ n $ - dimensional vector space $ V $ to itself.
The trace of $ T $ is $ text{tr}(T) = sum_{i=1}^n langle T(e_i), e_irangle $ where $ {e_i}_{i=1}^n $ is an orthonormal basis for $ V $ .
Equivalently, if $ V^* $ is the dual ov $ V $ , then the trace of a linear transformation $ T: Vto V $ is the functional $ text{tr}:V^*otimes V to mathbb R $ from the tensor product of of $ V^* $ and $ V $ determined by $ phi otimes v mapsto phi(v) $ for $ phi in V^* $ and $ v in V $ .
A group $ G $ is nilpotent if its upper central series reaches $ G $ after some finite number of steps.
The number of steps is called the nilpotency class of $ G $ .
A function $ f:Xto Y $ between metric spaces $ (X,rho) $ and $ (Y, sigma) $ is uniformly continuous if for all $ x,y in X $ and all $ varepsilon > 0 $ , there exists
$ delta > 0 $ such that $ sigma(f(x), f(y)) < varepsilon $ whenever $ rho(x,y) < delta $ .
Let $ f:Xto Y $ be a continuous function between topological spaces $ X $ and $ Y $ .
Let $ C_n(X) $ and $ C_n(Y) $ be the groups of n - chains in the chain complex of $ X $ and $ Y $ , respectively.
Then we may define a homomorphism called a chain map $ f_sharp:C_n(X)to C_n(Y) $ by composing each singular n - simplex $ sigma:Delta^nto X $ with $ f $ to get a singular n - simplex $ f_sharp(sigma) = fsigma:Delta^nto Y $ and extending $ f_sharp $ linearly via $ f_sharpleft(sum_i n_isigma_iright) = sum_i n_if_sharp(sigma_i) = sum_i n_if_sharpsigma_i $ .
Let $ partial $ be a general boundary homomorphism between $ C_{k+1}(X) $ and $ C_{k}(X) $ and do a similar thing for $ Y $ .
Then the maps $ f_sharp:C_n(X)to C_n(Y) $ satisfy $ f_sharp partial = partial f_sharp $ and the following diagram commutes:
Let $ R $ be a ring, $ L $ an $ R $ - module, and $ {ell_i}_{iin I} $ a subset of $ L $ .
The submodule generated by the $ ell_i $ is the set $ left{sum_{i=1}^n a_iell_imid a_iin R, text{ all but finitely many } a_i=0right} $ .
Equivalently, it is the smallest submodule of $ L $ containing the $ ell_i $ .
A subnormal series for a group $ G $ is a composition series when all of the quotients (called composition factors ) $ N_{i+1}/N_i $ are simple.
Let $ X $ be a topological space.
The boundary $ partial A $ of a set $ A subset X $ is the intersection of the closure of $ A $ and the closure of the complement of $ A $ .
That is, $ partial A = overline A cap overline{(Xsetminus A)} $ .
A collection $ mathcal V $ of measurable subsets of $ mathbb R^n $ is a regular family if there exists $ C > 0 $ such that the diameter of each $ V in mathcal V $ satisfies the following inequality: $ text{diam}(V)^n leq Cmu(V) $ .
Any set $ X $ can be equipped with the discrete topology, namely the topology in which every set is both open and closed.
A topological space $ X $ is Hausdorff if for every pair of points $ x,yin X $ , there exist disjoint open sets $ U,Vsubset X $ such that $ xin U $ and $ yin V $ .
A measure $ mu $ on a measurable space $ (X,mathcal M) $ is a function $ mu:mathcal M to 0,infty $ valued on the extended real numbers for which $ mu(emptyset) = 0 $ for any countable, pairwise disjoint collection $ {E_i}_{iinmathbb N} $ of measurable sets, $ muleft(bigcup_{iinmathbb N} E_iright) = sum_{iinmathbb N} mu(E_i) $ .
A measure space is a measurable space $ (X,mathcal M) $ together with a measure $ mu $ on $ mathcal M $ .
Let $ G $ be a locally compact abelian topological group.
The Pontryagin dual of $ G $ , denoted $ widehat G $ , is the group $ G $ of continuous homomorphisms from $ G $ to the circle group $ mathbb T $ .
Let $ m< n $ .
Then an $ m $ - dimensional smooth manifold embedded in $ mathbb R^n $ with boundary is a subset $ M subsetmathbb R^n $ such that for all points $ p in M $ , there exists an open set $ U subset H^k $ the closed half - space and an open neighborhood $ V subset M $ of $ p $ and a map $ phi: U to V $ such that (i) $ phi $ is a homeomorphism (ii) $ phi $ is smooth in $ H^k $ ( : what is this exactly?)
(iii) the total derivative $ Dphi(x) $ has rank $ m $ for all $ x in U $ .
A cycle is an element $ sigma $ of the symmetric group $ S_X $ if the action on $ X $ of the subgroup generated by $ sigma $ has at most one orbit with more than one element.
A tree is a connected graph with no closed reduced edge paths.
Let $ G $ be a group acting on the set $ X $ .
This action is faithful if the associated homomorphism to the symmetric group $ S_X $ has a trivial kernel.
A sequence $ {x_n}_{ninmathbb N} $ in a metric space $ (X,rho) $ is Cauchy if for all $ varepsilon > 0 $ , there exists
$ n_0 > 0 $ such that $ rho(x_n, x_m) < varepsilon $ for all $ m,n geq n_0 $
A chart or coordinate patch on an $ m $ - dimensional manifold $ M $ is an open set $ Usubset M $ together with a homeomorphism $ phi: Uto mathbb R^m $ onto an open subset of the closed half - space $ H^m $ .
Let $ R $ be a ring and let $ M $ be an $ R $ - module.
Then $ M $ is Noetherian (or it satisfies the ascending chain condition on submodules ) if there are no infinite increasing chains of submodules.
That is, if whenever $ M_1subseteq M_2subseteq M_3subseteq cdots $ is an increasing chain of submodules of $ M $ , then there is a positive integer $ m $ such that for all $ kgeq m $ , $ M_k=M_m $ .
Let $ Usubsetmathbb R^n $ and $ Vsubsetmathbb R^m $ be open sets and let $ f:Uto V $ be differentiable.
Let $ phi $ be a k - form on $ V $ .
Then we define the pullback of $ phi $ along $ f $ to be the k - form on $ U $ $ f^*(phi):Uto wedge^k((mathbb R^n)^*) $ such that $ f^*(phi)(x) = (wedge^k(Df(x)^*))(phi(f(x)) $ .
The order of an element $ g $ of the group $ G $ is the smallest integer $ n geq 1 $ such that $ g^n = e $ , where $ e $ is the group's identity element Equivalently, this is the cardinal number of the cyclic subgroup of $ G $ generated by $ g $ alone, if it exists.
Let $ P = (P_1,dots,P_n) $ and $ P'=(P_1', dots, P_n') $ be partitions of the rectangle $ Q $ in $ mathbb R^n $ .
Then $ P' $ is a refinement of $ O $ if $ P_i subset P_i' $ for all $ i $ .
The diamteter of an n - simplex $ v_0,dots,v_n $ is the maximum distance (in the ambient Euclidean metric inner product induces metric) between any of its vertices.
This is equivalent to the diameter of the n - simplex when considered as a set because the distance between two points $ v $ and $ sum_i t_iv_i $ of $ v_0,dots,v_n $ satisfies $ } $
A normed linear space $ (X, ||cdot||) $ is Banach if it is complete under the induced metric.
Let $ V $ be a vector space.
The $ m $ - fold exterior power of $ V $ is the quotient vector space of the $ m $ - fold tensor product of $ V $ by the subspace generated by the pure tensors $ v_1otimescdotsotimes v_m $ for $ v_i in V $ such that $ v_j = v_k $ for some $ jneq k $ .
It is denoted $ wedge^m(V) $ and the elements are written $ v_1wedgecdotswedge v_m $ .
A topological space $ X $ is sequentially compact if every sequence in $ X $ has a convergent subsequence.
Let $ sim $ be an equivalence relation on $ X $ .
The equivalence class of $ a in X $ , denoted $ a $ , is the set $ {x in X mid a sim X} $ .
Let $ G $ be a group.
A maximal subgroup of $ G $ is proper a subgroup of $ G $ that is not contained in any other proper subgroup of $ G $ .
A homogeneous relation on a set $ X $ is a binary relation where the two sets being related are both $ X $ .
Given two vector spaces $ V_1 $ and $ V_2 $ , $ V_1 otimes V_2 $ is called the tensor product of $ V_1 $ and $ V_2 $ and it is defined as follows: Let $ F(V_1 times V_2) $ be the free vector space generated by $ V_1 times V_2 $ , and let $ K subset F(V_1 times V_2) $ be the subspace of $ F(V_1 times V_2) $ generated by the elements $ } $ for all $ v_1,v_1'in V_1 $ , $ v_2, v_2' in V_2 $ , and $ lambda in mathbb R $ .
The tensor product $ V_1 otimes V_2 $ is the quotient vector space $ V_1 otimes V_2 = F(V_1 times V_2)/K $ .
Let $ A $ be an elementary abelian p - group and let $ mathbf ain(a_1,dots,a_r)in A^r $ .
Then the $ a_i $ form a basis for $ A $ if they generate $ A $ and if they are linearly independent.
A graded algebra over the field $ k $ is a graded vector space $ A = bigoplus_{igeq 0} A_i $ equipped with an algebra structure such that (i) multiplication is compatible with the grading: i.e. $ A_iA_isubseteq A_{i+j} $ for all $ i,jgeq 0 $ .
(ii) The identity element on $ A $ , written $ 1_A $ , is in $ A_0 $ .
The subspaces $ A_i $ of $ A $ are called homogeneous components of $ A $ , and an element $ ain A $ is homogeneous of degree $ d $ if $ ain A_d $ .
By simple linear algebra, any element $ ain A $ can be written *uniquely* as a sum $ sum_{igeq 0} a^{(i)} $ where $ a^{(i)} $ is homogeneous of degree $ i $ and $ a^{(i)} = 0 $ for all but finitely many $ i $ .
A lattice $ (L,leq) $ is said to be complemented if it has a maximum and a minimum element and such that each element of $ L $ admits a lattice complement.
Let $ X $ be a topological space.
A deformation retraction of $ X $ onto a subspace $ A $ is a family of continuous maps $ f_t: Xto X $ for $ tin 0,1 $ such that $ f_0 = mathbb 1 $ (the identity), the image $ f_1(X) = A $ , and $ {f_t}_{|A} = mathbb 1 $ for all $ t $ and such that the associated map $ X times 0,1 to X $ given by $ (x,t)mapsto f_t(x) $ is also continuous.
Alternatively, a subspace $ A $ is a deformation retract of $ X $ if there exists a homotopy $ h:Xtimes Ito X $ such that $ h(x,0) = x $ , $ h(a,t) = a $ , and $ h(x,1)in A $ for all $ ain A $ and $ tin T $ .
Such a homotopy is a deformation retraction of $ X $ onto $ A $ .
A ring $ R $ is Noetherian if it is Noetherian as a module over itself.
That is, if there are no infinite increasing chains of left ideals of $ R $ .
A function $ f: a,bto mathbb R $ is of bounded variation if for every partition of a set $ P ={a= x_0 < <x_1 < cdots < x_n = b} $ , we have $ sup_P sum_{i=1}^n |f(x_i) - f(x_{i - 1})| <infty $ .
Consider the circle $ S^1 $ as the unit interval quotient by its boundary: $ S^1= I/partial I $ .
Denote its basepooint by $ 1 $ .
Then the reduced suspension of a based topological space $ X $ is the smash product $ Sigma X = Xwedge S^1 $ .
Explicitly, $ Sigma X = (Xtimes S^1)/(({*}times S^1)cup (Xtimes{1})) $ .
The usual suspension is done without identifying the line $ {*}times I $ through the basepoint of $ X $ to a point.
A topological space $ X $ is weak Hausdorff if for every map $ g:Kto X $ from a compact Hausdorff space $ K $ is closed in $ X $ .
Note that $ g(K) $ is itself Hausdorff, ( : why?)
so $ g(K) $ is a compact subspace of $ X $ .
An element $ x $ of an integral domain is irreducible if it is not a unit and if whenever $ x=ab $ for some $ a,bin R $ , then either $ a $ or $ b $ is a unit.
Let $ B $ be a compactly generated topological space.
A space over $ A $ (that is, an object of the category) is a map $ p:Eto B $ .
A morphism between spaces under $ A $ is a commutative diagram This is a homotopy category: a homotopy $ h $ between maps over $ B $ is a homotopy that at each "time" $ t $ is a map over $ B $ .
The above notion of homotopy equivalence is called fiber homotopy equivalence .
It is dual to cofiber homotopy equivalence.
Let $ X $ be a CW complex.
The cellular homology groups are the homology groups of the cellular chain complex of $ X $ .
They are denoted $ H_n^{text{CW}}(X) $ .
Consider the polynomial ring $ P_n =mathbb Cx_1,dots,x_n $ .
Let $ partial_i = frac{partial}{partial x_i} $ be the partial derivative with respect to the variable
$ x_i $ .
Note that it is a linear transformation from $ P_n $ to itself.
Let $ D_n $ be the subalgebra of the algebra of $ mathbb C $ - linear maps $ P_nto P_n $ generated by the $ partial_i $ .
Since partial derivatives commute with each other, the algebra $ D_n $ is commutative.
Elements of $ D_n $ take the form $ u = sum_{k_1,dots,k_n geq 0} c_{k_1,dots,k_n} frac{partial^{k_1+cdots + k_n}}{partial x_1^{k_1}cdots partial x_n^{k_n}} $ for $ c_{k_1,dots,k_n}inmathbb C $ .
This is the algebra of polynomial differential operators in $ n $ variables with constant coefficients .
The $ mathbb C $ - span of the $ partial_i $ is a vector subspace of $ D_n $ given by the directional derivatives along $ vin mathbb C^n $
Let $ A $ be an algebra.
Then its ideals are generated in the same way that ring ideals are generated by the inheritance following from the fact that algebras are rings.
An element $ x $ of a ring is idempotent if $ x^2 = x $ .
Let $ X $ be a topological space.
A set $ Asubset X $ is nonmeager if it is not meager.
Let $ mathfrak g $ be a Lie algebra.
A semisimple representation of $ mathfrak g $ is a linear representation of $ mathfrak g $ that may be written as the direct sum of irreducible representations.
Let $ G $ be a group.
The Frattini subgroup of $ G $ , denoted $ Phi(G) $ , is theintersection of all of the maximal subgroups of $ G $ .
If there are $ a $ ways of doing one thing and $ b $ ways of doing another thing, then there are $ ab $ ways of doing both things.
This can be extended inductively.
Set theoretically, this is the definition of the product of cardinal numbers: $ |S_1|cdot|S_2|cdots|S_n| = |S_1times S_2timescdots S_n| $ where $ |cdot| $ is the cardinal number of the sets $ S_i $ and $ times $ is the Cartesian product of sets.
The sets need not be finite nor does the number of sets.
Let $ (X,Sigma) $ be a measurable space with $ mu $ a signed measure.
The total variation of $ mu $ is the function $ tau $ given by $ tau(A) = pi(A) + nu(A) $ where $ pi $ is the positive variation of $ mu $ and $ nu $ is the negative variation of $ mu $ .
Let $ A $ be an algebra over the ring $ R $ .
A right ideal $ I $ of $ A $ is a subalgebra of $ A $ such that $ xain I $ for all $ ain A $ , $ xin I $ .
Equivalently (if we want to elide the subalgebra condition), a right ideal $ I $ of $ A $ is a subset $ Isubset A $ such that for all $ x,yin I $ , $ ain A $ , and $ rin R $ , (i) $ x - yin I $ ; (ii) $ rxin I $ ; (iii) $ xain I $ .
Let $ G $ be a semigroup and let $ S $ be a subset of $ G $ .
The normalizer of $ S $ is $ C_G(S) = {gin Gmid gS=Sg} $ .
When $ G $ is a group, this may be written equivalently as $ C_G(S) = {gin Gmid gSg^{ - 1}} $ .
The real symplectic group $ text{Sp}(2n,mathbb R) $ is the group of real $ 2ntimes 2n $ matrices that preserve the skew - symmetric bilinear form $ omega(x,y) = langle x,Omega yrangle $ where $ langlecdot,cdotrangle $ is the Euclidean inner product and $ Omega = } $ for $ I_n $ the $ ntimes n $ identity matrix.
That is, $ Ain text{Sp}(2n, mathbb R) $ if and only if $ omega(Ax,Ay) = omega(x,y) $ for all $ x,y in mathbb R^n $ .
Equivalently, $ Ain text{Sp}(2n, mathbb R) $ if and only if $ - Omega A^T Omega = A^{ - 1} $ where , as usual, $ A^T $ is the transpose of $ A $ and $ A^{ - 1} $ is its inverse.
This group is a matrix Lie group because it is a subgroup of the general linear group $ text{GL}(n,mathbb C) $ .
Let $ V $ and $ W $ be vector spaces.
If $ phi: Gto GL(V) $ and $ psi:Gto GL(W) $ are two different representations of the group $ G $ , the direct sum $ phi oplus psi:G to GL(Voplus W) $ is also a representation of $ G $ where $ Voplus W $ is given by the usual Cartesian product of vector spaces.
$ I $ denotes the unit interval $ 0,1 $ .
Let $ X $ and $ Y $ be topological spaces.
A homotopy is a family of continuous functions $ f_t:Xto Y $ for $ tin I $ such that the associated map $ F:Xtimes Ito Y $ given by $ F(x,t) = f_t(x) $ is continuous.
Two maps $ f_0,f_1:Xto Y $ are homotopic if there exists a homotopy $ f_t $ connecting them.
Equivalently, let $ X $ and $ Y $ be topological spaces.
A homotopy between two continuous functions $ p,q:Xto Y $ is a continuous function $ h: Xtimes Ito Y $ such that $ h(x,0) = p(x) $ and $ h(x,1) = q(x) $ .
A subgroup $ H $ of the group $ G $ is normal if for any $ g in H $ , the left and right cosets of $ H $ are equal.
That is, if $ gH = Hg $ for all $ g in G $ .
This is denoted $ Htriangleleft G $ .
A topological space $ X $ is path - connected if for every pair of points $ x,y in X $ , there exists a continuous function $ gamma:0,1to X $ such that $ gamma(0)=x $ and $ gamma(1)=y $ .
A division ring is a ring $ R $ in which every nonzero element is a unit.
A topological space $ X $ is compact if every open cover of $ X $ has a finite subcover.
Let $ phi $ be a differentiable k - form on the open set $ Usubset mathbb R^n $ given by $ phi = sum_{i_1<dots<i_k} phi_{i_1,dots,i_k}(x_1,dots,x_n)dx_{i_1}wedgecdotswedge dx_{i_k}in mathcal A^k(U) $ .
Define the differential of $ phi $ by $ dphi = sum_{i_1<dots<i_k} dphi_{i_1,dots,i_k}wedge dx_{i_1}wedgecdotswedge dx_{i_k} in mathcal A^{k+1}(U) $ where the $ phi_j $ are ( : what??).
That is, $ d $ is the linear transformation between the vector space of k - and k+1 - forms $ d: mathcal A^k(U) to mathcal A^{k+1}(U) $ given by $ g(x)dx_{i_1} wedgecdotswedge dx_{i_k} mapsto dgwedge x_{i_1}wedgecdotswedge dx_{i_k} $ .
: what is d for the functions??????
Let $ C $ be a category.
A morphism $ fin text{Hom}_C(A,B) $ is an epimorphism if for all objects $ Z $ of $ C $ and all morphisms $ alpha,betain text{Hom}_C(B,Z) $ , $ alpha circ f= betacirc fimplies alpha = beta $ .
A function $ f: mathbb R^n to mathbb R $ is Lebesgue measurable if for every open $ G in mathbb R $ , the preimage $ f^{ - 1}(G) $ is Lebesgue measurable.
The derived series of a group $ G $ is the descending series of consecutive derived subgroups of $ G $ : $ G = G^{(0)} supseteq G^{(1)} = G,G supseteq cdots supseteq G^{(i)}supseteq cdots $ where $ G^{(i+1)} = G^{(i)},G^{(i)} $ .
Note that this series may not reach the trivial subgroup.
The center of an algebra $ A $ over the ring $ R $ is the set $ Z(A) = {zin A mid za=az text{ for all } ain A} $ .
Let $ f: Qto mathbb R $ be a bounded function defined on the rectangle $ Q $ and let $ P $ be a partition of a set of $ Q $ .
For each subrectangle $ R $ determined by $ P $ , let $ m_R(f) = inflimits_{xin R} f(x) $ .
The lower sum for $ f $ determined by $ P $ is $ L(f,P) = sum_{R} m_R(f)text{vol}(R) $ where $ text{vol} $ is the volume of the rectangle $ R $ .
Let $ X $ be a topological space.
A set $ Asubset X $ is nowhere dense in $ X $ if for every nonempty open subset $ Gsubset X $ , there exists a nonempty open subset $ Hsubset G $ such that $ Acap H=emptyset $ .
Equivalently, a set $ Asubset X $ id nowhere dense in $ X $ if its closure has empty interior.
A covering of groupoids $ p:mathcal Eto mathcal B $ is regular if $ p(mathcal E,e)) $ is a normal subgroup of $ pi(mathcal B,b) $ where $ pi(mathcal B,b) $ denotes the automorphism group of $ b $ .
Let $ R $ be a ring and $ I $ an ideal in $ R $ .
Define an equivalence relation $ sim $ on $ R $ by $ asim b $ when $ a - b in I $ .
The equivalence classes are given by $ a = a+ I = {a+r mid rin I} $ .
The set of all such equivalence classes is denoted $ R/I $ , or the quotient of $ R $ by $ I $ .
The operations addition ( $ (a+I)+ (b+I) = (a+b)+I $ ) and multiplication ( $ (a+I)(b+I) = (ab)+I $ ) are well - defined.
Moreover, this itself is a ring .
The quotient comes with a surjective homomorphism $ Rto R/I $ given by $ rmapsto r+I $ .
Let $ U, V subsetmathbb R^n $ be open and let $ f:Uto V $ be a $ C^k $ diffeomorphism.
Then $ f $ is orientation - preserving if the determinant of its total derivative is always positive:
$ text{det}(Df(x)) > 0 $ for all $ x in U $ .
The nullity of a linear transformation $ T $ is the dimension of the kernel of $ T $ .
A continuous map ( : the inclusion? is $ A $ a subspace?) $ i:Ato X $ between compactly generated topological spaces $ A $ and $ X $ is a cofibration if it satisfies the homotopy extension property .
That is, if $ hcirc i_0=fcirc i $ in the diagram below, then there exists $ tilde h $ that makes the diagram commute.
Note that $ i_0(x) = (x,0) $ .
The map $ tilde h $ s not required to be unique.
The logarithm of an $ ntimes n $ matrix $ A $ is given by $ ln(A) = sum_{m=1}^infty ( - 1)^{m+1}frac{(A - I)^m}{m} $ whenever the sequence converges.
Let $ X $ be a Hausdorff topological space and let $ (X, Sigma, mu) $ be a measure space where $ Sigma $ is the Borel σ - algebra on $ X $ .
The measure $ mu $ is a Radon measure if it is inner regular and locally finite.
Let $ Dsubsetmathbb R^2 $ be connected and open and let $ f:Dto mathbb R $ be of class $ C^2 $ .
Then the Laplacian of $ f $ is given by the sums of the second partial derivatives $ Delta f(z) = f_{xx}(z) + f_{yy}(z) $ .
Let $ Qsubsetmathbb R^n $ be a rectangle.
For any partition of a set $ P $ of $ Q $ , the mesh of $ P $ is the maximum length of the component intervals of $ P $ .
Let $ M $ be an embedded m - dimensional manifold.
The tangent space of $ M $ at $ x $ , denoted $ T_xM $ , is the vector space given by the kernel of the total derivative of the function given in the definition of the kind of manifold.
In light of the functorial definition of group actions, we can think of a covariant functor $ T:mathcal Bto mathcal S $ from a small groupoid to the category of sets as a generalization of a group action.
For each object $ b $ of $ mathcal B $ , $ T $ restricts to an action of the group $ pi(mathcal B,b) $ ^(i)
Then $ T $ is transitive if this group action is transitive for all objects $ b $ .
Note that if $ mathcal B $ is connected, if the restriction of $ T $ on $ pi(mathcal B,b) $ is transitive for one object $ b $ then it is transitive for all objects $ b $ .
^1: Note that $ pi(mathcal B,b) $ denotes the automorphism group of $ b $ as in the definition of star of an object.
A metric space is a set $ X $ together with a function $ rho: Xtimes X to mathbb R $ such that for all $ x,y,z in X $ , the following properties are satisfied: (i)
$ rho(x,y) = 0 $ if and only if $ x=y $ .
(ii) $ rho(x,y) = rho(y,x) $ .
(iii) $ rho(x,z) leq rho(x,y) + rho(y,z) $ .
Property 3 is called the triangle inequality and the function $ rho $ is called a metric .
The metric generates a topology on $ X $ where the open sets $ G $ are those for which there exists
$ varepsilon > 0 $ for all $ x in G $ such that $ {yin Xmid rho(x,y) < varepsilon}subset G $ .
Let $ Usubsetmathbb R^n $ be an open subset and let $ f:Utomathbb R $ be differentiable on $ U $ .
The point $ a in U $ is a critical point of $ f $ if $ Df(a) =0 $ .
Let $ 1leq p < infty $ and let $ (X,Sigma, mu) $ be a measure space.
Let $ F $ be the set of functions $ f: Xto mathbb R $ such that $ ||f||_p = left(int_X|f|^ptext dmuright)^{1/p} < infty $ where $ ||f||_p $ is the p - norm.
Equivalently, $ F $ is the set of functions such that the $ p $ th power of their absolute value is integrable.
Let $ N = {f in F mid f = 0 text{almost everywhere}} $ .
Because the integral of a positive function is zero if and only if it is zero almost everywhere, we have that $ N $ is the kernel of the p - norm.
The normed linear space given by the quotient vector space of $ F $ by $ N $ is $ L_p(X,mu) $ .
For $ p = infty $ , consider the set $ G $ of functions $ f:X to mathbb R $ for which there exists $ c inmathbb R $ such that $ mu({|f| > c}) = 0 $ and take the quotient vector space of this set by the equivalence relation $ sim $ on $ G $ given by $ fsim g $ if $ f(x) = g(x) $ almost everywhere on $ X $ .
This space with the essential supremum of $ f $ as the norm is $ L_infty(X,mu) $ .
Let $ G $ be a group acting on the set $ X $ .
A point $ xin X $ is a fixed point for or an invariant of the $ G $ - action if $ gcdot x=x $ for every $ gin G $ .
The set of these points is denoted $ X^G $ .
Let $ (X,Sigma, mu) $ be a measure space.
The measure $ mu $ is σ - finite if it satisfies one of the following equivalent criteria: - $ X $ can be covered with at most countably many (pairwise disjoint
, this is not required) measurable sets of finite measure; - $ X $ can be covered with a monotone sequence of measurable sets of finite measure; - There exists a positive measurable function $ f $ whose integral is finite.
A topological field is a field $ F $ that is a topological ring and also whose inverse maps $ Ftimes F^timesto F $ for both binary operations are also continuous.
Let $ P $ be a partition of a set of the rectangle $ Q $ in $ mathbb R^n $ .
The subrectangles of $ Q $ given by $ P $ are the products of the subintervals of the component intervals given by each $ P_i $ as in the definition of partition of a set of a rectangle.
Let $ mathcal C $ be a category and $ x $ an object of $ mathcal C $ .
The category $ x/mathcal C $ of objects under $ x $ has objects given by the morphisms $ f:xto y $ in $ mathcal C $ .
For objects $ f:xto y $ and $ g:xto z $ in $ x/mathcal C $ , the morphisms $ gamma:fto g $ are the morphisms $ gamma yto z $ in $ mathcal C $ such that $ gammacirc f = g:xto z $ .
Composition and the identity maps are given by composition and the identity maps in $ mathcal C $ .
When $ mathcal C $ is a groupoid, $ gamma = gcirc f^{ - 1} $ and the objects of $ x/mathcal C $ therefore determine the category.
The ring of Gaussian integers is exactly $ mathbb Zi = {a+bimid a,binmathbb Z, i^2= - 1} $ .
It is a Euclidean domain with norm $ N(a+bi) = a^2+b^2 = (a+bi)(a - bi) $ .
This norm is multiplicative.
Let $ U subset mathbb R^n $ be open.
A function $ f:mathbb U to mathbb R^n $ may be written as an $ m $ - tuple of functions $ f_i:Uto mathbb R $ so that $ f=(f_1,dots,f_m) $ .
Let $ (X,Sigma, mu) $ and $ (Y, T, nu) $ be measure spaces.
A function $ f:Xto Y $ is Lusin or Luzin if the image of every set of $ mu $ - measure zero has zero $ nu $ - measure.
Two elements of a commutative ring are coprime if their greatest common divisor is $ 1 $ .
Specifically, if it is the additive identity element of the ring.
Let $ mathbb Q_p $ be the field of p - adic numbers under the $ p $ - adic absolute value.
Define $ mathcal P = {xinmathbb Q_pmid |x|_p< 1} $ .
We call this curly $ mathcal P $ because it has no other name that I have been able to find.
It is a subset of the p - adic integers, and in particular it is an open ball in the $ p $ - adic absolute value topology which means that it is also closed.
We can also define the "powers" of $ mathcal P $ by $ mathcal P^n = {xinmathbb Q_pmid |x_p|<p^{ - k}} $ .
find a better name for this!!!
Let $ (V,rho) $ be a representation of the group $ G $ .
The $ n $ th symmetric power of $ V_1 $ and $ V_2 $ is a representation of $ G $ over the symmetric power of vector spaces $ text{Sym}^n(V) $ under the linear transformation $ rho:Gto text{Sym}^n(V) $ such that $ rho(g)(v_1odot v_2odot cdots odot v_n) = (rho(g)v_1odot rho(g)v_2odotcdotsodotrho(g)v_n) $ .
Let $ U subset mathbb R^n $ be open.
A function $ f: Uto mathbb R^m $ is differentiable at $ a in U $ if there exists a linear transformation $ A:mathbb R^n to mathbb R^m $ such that $ lim_{hto 0} frac{f(a+h) - f(a) - A(h)}{||h||} = 0 $ .
The transformation $ A $ is denoted $ Df(a) $ and is called the total derivative of $ f $ at $ a $ .
Equivalently, let $ V $ and $ V $ ' be vector spaces with norms $ |cdot| $ and $ |cdot|' $ , respectively.
A function $ f:Uto V' $ from $ U $ an open subset of $ V $ is differentiable at $ xin U $ if there exists a linear transformation $ d_x f :Vto V' $ called the differential of $ f $ at $ x $ and a small neighborhood $ U_xsubseteq U $ of $ x $ such that for all $ vin U $ , we have that $ f(x+v) = f(x) + d_x(f) + o(v) $ where $ o: U_xto V' $ is a map such that $ lim_{|v|to 0} frac{|o(v)|'}{|v|} = 0 $ .
Let $ A $ be a graded algebra.
A graded ideal of $ A $ is an left algebra ideal that is also a graded vector subspace.
It is also called a homogeneous ideal of $ A $ .
An $ ntimes n $ Hermitian complex matrix $ M $ is positive definite if $ langle x, Mxrangle > 0 $ for all $ xneq 0 $ in $ mathbb C^n $ where $ langlecdot,cdotrangle $ is the standard inner product on $ mathbb C^n $ If we allow the dot product to be zero, then $ M $ is postive semidefinite .
The Mandelbrot set $ mathcal M subset mathbb C $ is the set of $ c in mathbb C $ such that the filled Julia set $ K(f_c) $ is connected, where $ f_c:mathbb C to mathbb C $ is the polynomial $ f_c(z) = z^2+c $ .
Define an equivalence relation $ sim $ on a vector space $ V $ with subspace $ W $ by $ v_1 sim v_2 text{ if } v_1 - v_2 in W $ .
Then the quotient of $ V $ by $ W $ is the set of equivalence classes of $ sim $ in $ V $ .
Let $ G $ be a magma with binary operation $ cdot $ and sub - structure $ H $ .
A left coset of the $ H $ for the element $ xin G $ is the set $ xcdot H = {xcdot hmid h in H} $ .
Let $ Usubsetmathbb R^n $ be an open subset and $ f:Uto mathbb R $ be differentiable on $ U $ .
The gradient of $ f $ at $ a in U $ is the vector of the partial derivatives of $ f $ : $ nabla f(a) = left(frac{partial f}{partial x_1}(a),cdots, frac{partial f}{partial x_n}(a) right) $ .
Let $ G,H $ be groups, let $ R $ be a ring, and consider the trivial homomorphism $ f:Gto H $ .
This induces a ring homomorphism on the group ring given by $ f_*:RGto Rtext{ such that } sum_{gin G} a_gg mapsto sum_{gin G}a_g $ for $ a_g in R $ .
The kernel of this map is called the augmentation ideal .
We have immediately that it is an ideal because the kernel of a ring homomorphism is an ideal.
The Cantor set $ mathcal C $ is constructed from the unit interval as follows: Let $ C_n = frac{C_{n - 1}}{3} cup left(frac{2}{3} +frac{C_{n - 1}}{3}right) $ with $ C_0 = 0,1 $ .
Then $ mathcal C = bigcap_{ninmathbb N} C_n $ .
A set $ Esubset mathbb R $ is called a Cantor set if it is compact, has empty interior, and has no isolated points.
An integral domain $ R $ is a principal ideal domain if every ideal of $ R $ is principal.
Let $ A $ be a graded algebra.
A graded subalgebra of $ A $ is a subalgebra of $ A $ that is also a graded vector subspace of $ A $ as a graded vector space.
A bounded function $ f:Qtomathbb R $ defined on the rectangle $ Q $ in $ mathbb R^n $ is integrable if the lower integral of $ f $ is equal to the upper integral of $ f $ .
The integral of $ f $ over $ Q $ is defined to be $ int_Q f = underline{int_Q} f = overline{int_Q} f $ .
Alternatively, let $ f: Sto mathbb R $ be a bounded function from the bounded set $ S $ to $ mathbb R $ , and let $ f_S $ be the extension of $ S $ in the usual sense to $ mathbb R $ .
Choose $ Q $ a rectangle containing $ S $ .
The integral of this function over $ S $ is $ int_S f = int_Q f_S, $ if it exists.
The skeleton $ text{sk}mathcal C $ of a category $ mathcal C $ is a subcategory with one object from each isomorphism class of objects of $ mathcal C $ such that the morphisms between two objects in $ text{sk}mathcal C $ are all of the morphisms between these objects in $ mathcal C $ .
Let $ p:Xto Y $ be a continuous map between topological spaces.
Define $ p_*:pi_1(X,x) to pi_1(Y,p(x)) $ to be the map between the fundamental groups of $ X $ at $ x $ and $ Y $ at $ p(x) $ given by $ p_*f = pcirc f $ .
This map is a homomorphism as $ p_*fcdot g = pcirc (fcdot g) = p_*fp_*g $ .
A game is something which may be represented as a normal - form game or an extended - form game.
A representation of a topological group $ G $ (written $ rho:Gto GL(V) $ for $ GL(V) $ the general linear group of some vector space $ V $ ) is called continuous if it is of finite dimension and if $ rho $ is continuous in the usual sense as a function between topological spaces.
Let $ X $ be a topological space and let $ x in X $ .
Denote by $ pi_1(X,x) $ the set of equivalence classes of loops at $ x $ under homotopy equivalence of paths.
This is called the fundamental group of $ X $ at $ x $ .
The limit of a D - shaped diagram is given by reversing the arrows in the definition of the colimit of the same D - shaped diagram: Then $ lim F $ is an object of $ mathcal C $ together with a morphism of diagrams $ pi:underline{lim F} to F $ that is terminal among all such morphisms.
That is, if $ varepsilon:underline Ato F $ is a morphism of diagrams, then there is a unique morphism $ tilde varepsilon: Ato lim F $ in $ mathcal C $ such that $ picirc tildevarepsilon = varepsilon $ .
The following diagram commutes:
The ideal generated by a subset $ T $ of a ring $ R $ is the smallest ideal containing $ T $ .
Equivalently, $ (T) = bigcaplimits_{Isupseteq T} I $ where $ I $ varies over all of the ideals in $ R $ that contain $ T $ .
Equivalently, if $ T = {t_i}_{i=1}^n $ , then $ (T) = left{sumlimits_{i=1}^n r_it_iright} $ , the set of all formal sums of elements of $ T $ with coefficients in $ R $ .
Let $ X $ and $ Y $ be topological spaces.
The mapping cylinder $ M_f $ of a continuous map $ f:Xto Y $ is the quotient vector space of the disjoint union $ (Xtimes 0,1)amalg Y $ given by identifying $ (x,1) in Xtimes 0,1 $ with $ f(x) in Y $ .
Equivalently, let $ X $ and $ A $ be compactly generated topological spaces.
Then in the test diagram for the homotopy extenson property let $ Y $ be the mapping cylinder $ Xcup_i (Atimes I) $ be the pushout of $ i $ and $ i_0 $ .
By the universal property for pushouts, if there exists a map $ r $ such that the following diagram commutes: the maps $ f $ , $ h $ from the *original* test diagram induces a map $ Mi to Y $ (where $ Y $ is the same $ Y $ from that original diagram) whose composite with $ r $ gives a homotopy $ tilde h $ that makes the test diagram commute.
Let $ x $ be a unit of the p - adic field $ mathbb Q_p $ .
That is, let $ xinmathbb Q_p^times $ .
Then $ x $ has a unique p - adic expansion $ x= sum_{n=text{val}(x)}^infty a_np^n $ .
Denote by $ lambda(x) $ the sum $ lambda(x) = sum_{n=text{val}(x)}^{ - 1}a_np^n $ and call it the tail of $ x $ .
Then we define the basic additive character $ psi:mathbb Q_pto mathbb Tsubseteq mathbb C^times $ (the middle set being the circle group) by $ psi(x) = e^{2pi i lambda(x)} $ .
Auxiliary definition We can also define a "scalar multiple" of the character by defining, for each $ uinmathbb Q_p $ , a new function $ psi_u:mathbb Q_pto mathbb T $ by $ xmapsto psi(ux) $ .
Proof that it is a character First, we show that $ psi_u $ is a group homomorphism: $ psi_u(x+y) = psi(u(x+y)) = psi(ux+uy) = psi(ux)psi(uy) = psi_u(x)psi_u(y) $ from the fact that $ psi $ is itself a group homomorphism.
Finally, $ psi_u $ is continuous because the composition of continuous functions is continuous and $ psi $ is the composition of $ psi $ , which is itself continuous by virtue of it being a multiplicative character, and because $ mathbb Q_p $ is a topological field, the map given by "multiplication by $ u $ " is also continuous.
Let $ n $ be a positive integer.
A partition of $ n $ is a non - decreasing sequence of positive integers whose sum is $ n $ .
The σ - algebra generated by a collection of subsets is the σ - algebra containing all of the subsets such that any σ - algebra containing the sets also contains it.
The adjoint map of a Lie algebra $ mathfrak g $ is the map $ text{ad}: mathfrak g to text{End}(mathfrak g) $ that sends $ X $ to $ text{ad}_X $ in the set of endomorphisms of $ mathfrak g $ .
A binary relation on the set $ X $ is symmetric if for all $ a,b in X $ , $ a $ is related to $ b $ if and only if $ b $ is related to $ a $ .
Let $ A $ be an algebra over the field $ mathbb k $ and let $ ain A $ .
Then the subalgebra $ A_a $ generated by $ a $ is the image of the algebra homomorphism from the polynomials $ mathbb kt $ to $ A $ given by $ fmapsto f(a) $ .
The kernel of this map is an ideal of $ mathbb kt $ consisting of polynomials such that $ f(a) = 0 $ .
Denote this ideal by $ I_a $ and note that it is a principal ideal because $ mathbb kx $ is a principal ideal domain.
The element $ ain A $ is algebraic if there exists a nonzero polynomial $ finmathbb kt $ such that $ f(a)=0 $ , i.e. if $ I_aneq 0 $ .
Then because $ I_a $ must be principal, there exists a unique monic polynomial $ p_a $ that generates $ I_a $ and this is called the minimal polynomial of $ a $ .
A left ideal of a ring $ R $ is a subset $ Isubseteq R $ such that - $ (I,+) $ is a subgroup of $ (R,+) $ ; - $ I $ is closed under left multiplication in $ R $ (i.e. for all $ iin R $ and all $ rin R $ , $ ri in I $ ).
A point $ x $ is an isolated point of a subset $ S $ of a topological space $ X $ if it is an element of $ S $ but there exists a neighborhood of $ x $ containing no other points of $ S $ .
The real projective space $ mathbb RP^n $ of dimension $ n $ is the quotient of $ mathbb R^{n+1} $ by the equivalence relation $ sim $ given by $ xsim y $ if and only if $ x= - y $ .
This amounts to considering $ mathbb RP^n $ as the set of lines through the origin in $ mathbb R^{n+1} $ , or the unit sphere $ S^n $ where we identify antipodal points $ u $ and $ - u $ .
A $ sigma $ - algebra on the set $ X $ is a subset $ mathcal M subset mathcal P(X) $ of the power set of $ X $ satisfying the following properties: (i) $ X in mathcal M $ ; (ii) $ mathcal M $ is closed under complementation, i.e. if $ A in mathcal M $ , then $ Xsetminus A in mathcal M $ ; (iii) $ mathcal M $ is closed under countable unions, i.e. if $ A_i in mathcal M $ for $ i in mathbb N $ , then $ bigcuplimits_{iinmathbb N} A_i in mathcal M $ .
A rng is a set $ R $ together with two binary operations $ + $ and $ cdot $ such that (i) $ R $ is an abelian group under $ + $ ; (ii) $ R $ is a semigroup under $ cdot $ ; (iii) $ cdot $ distributes over $ + $ : $ acdot (b+c)=(acdot b) + (acdot c) $ and $ (b+c)cdot a = (bcdot a) + (ccdot a) $ .
Note that this is essentially the definition of ring without a multiplicative identity.
Let $ X $ be a topological space.
A set $ A subset X $ is dense in $ X $ if every point in $ x $ is either in $ A $ or is a limit point of $ A $ .
Equivalently, $ A $ is dense in $ X $ if its closure is all of $ X $ .
Let $ H_n $ be the $ n $ th homology group of a chain complex.
Elements of $ H_n $ are cosets of $ text{Im}(partial_{n+1}) $ and are called homology classes .
Representatives of the same homology class are homologous .
Let $ V $ be a vector space and let $ T:Vto V $ be a linear transformation.
For integer $ m $ , there exists a multilinear transformation $ wedge^m(T): wedge^m(V) to wedge^m(V) $ from the $ m $ - fold exterior product to itself given by $ v_1wedgecdotswedge v_m mapsto T(v_1)wedgecdotswedge T(v_m) $ . : show is well - defined
A Hilbert space $ (H, langlecdot,cdotrangle) $ is an inner product space that is complete as a metric space with metric induced by the norm induced by the inner product.
Let $ X $ be a CW complex.
Then by These three results, we can write parts of the long exact sequences of the pairs $ (X^{n+1}, X^n) $ , $ (X^n, X^{n - 1}) $ , and $ (X^{n - 1}, X^{n - 2}) $ of the $ n $ - skeleta of $ X $ can be written in the commutative diagram of the homology groups and relative homology groups where $ d_{n+1} $ and $ d_n $ are the compositions $ j_npartial_{n+1} $ and $ j_{n - 1}partial_n $ , which in turn are "relativizations" of the boundary maps $ partial_n $ and $ partial_{n+1} $ .
The composition $ d_nd_{n+1} $ is zero because the composition of consecutive boundary homomorphisms is 0.
The horizontal part of this diagram is therefore a chain complex called the cellular chain complex of $ X $ .
Because $ H_n(X^n, X^{n - 1}) $ is free with basis in bijection with $ n $ - cells of $ X $ , we can think of its elements as linear combinations of $ n $ - cells of $ X $ .
Let $ X $ be a topological space.
For any $ Asubset X $ , the interior $ text{int}(A) $ of $ A $ is the largest open set contained in $ A $ .
Let $ Usubsetmathbb H^n $ be open in the subspace topology on $ H^n $ considered as a subset of $ mathbb R^n $ .
Then a function $ f:Utomathbb R^m $ is differentiable if there is an open set $ V subsetmathbb R^n $ such that $ U = Vcap H^n $ onto which we may extend $ f $ to a differentiable function $ f_V:Vtomathbb R^n $ .
A topological group $ G $ is locally compact if the underlying topological space is both locally compact and Hausdorff.
Let $ (X,Sigma,mu) $ be a measure space.
A property $ P $ holds almost everywhere if there exists a set $ N in Sigma $ such that $ mu(N) = 0 $ and $ P $ is true for all $ x in Xsetminus N $ .
A binary relation $ R $ on a set $ X $ (i.e. a subset of $ X times X $ ) is reflexive if it relates every $ x in X $ to itself.
That is, for all $ x in X $ , $ (x,x) in R $ .
Let $ S subsetmathbb R^n $ be rectifiable with nonzero volume.
The centroid of $ S $ is the point $ c(S) in mathbb R^n $ whose $ k $ th coordinate is given by the integral $ c_k(S) = frac{1}{text{vol}(S)} int_S pi_k, $ where $ pi_k:mathbb R^n tomathbb R $ is such that $ (x_1,dots,x_n)mapsto x_k $ .
Consider a chain complex with abelian groups $ C_i $ and homomorphisms $ partial_i $ .
Because $ partial_{k+1}circpartial_k = 0 $ for all $ k $ , the image of $ partial_{k=1} $ is a subset of the kernel of $ partial_k $ : $ text{Im}(partial_{k+1})subset text{Ker}(partial_k) $ .
The $ n $ th homology group of the chain complex is the quotient group $ H_n = text{Ker}(partial_n)/text{Im}(partial_{n+1}) $ .
Elements of $ text{Ker}(partial_n) $ are called cycles and elements of $ text{Im}(partial_{n+1}) $ are called boundaries .
A prime number is a prime element of $ mathbb N $ .
Equivalently, it is a natural number that is not divisible by any number other than $ 1 $ and itself.
A partially ordered set is a set $ X $ together with a homogeneous relation $ leq $ such that for all $ x,y,zin X $ , (i) $ leq $ is reflexive, i.e. $ xleq x $ ; (ii) $ leq $ is antisymmetric, i.e. if $ xleq y $ and $ yleq x $ then $ x=y $ ; (iii) $ leq $ is transitive, i.e. if $ xleq y $ and $ yleq z $ then $ xleq z $ .
Let $ (X,Sigma,mu) $ be a measure space.
A set $ Asubseteq mathbb X $ is d - open or open in the density topology (or in the Denjoy topology) if (i) $ A $ is measurable (ii) The measure density $ d(A,x) = 1 $ for every $ x in A $ .
An $ ntimes n $ real matrix $ A $ is orthogonal if its column vectors (equivalently, its row vectors) are orthonormal.
That is, the product of $ A $ with its transpose $ A^T $ is equal to the identity matrix $ I_n $ : $ AA^T = A^TA = I_n $ Equivalently, the transpose of an orthogonal matrix $ A $ is equal to its inverse: $ A^T = A^{ - 1} $ .
Let $ R $ be a ring.
An $ R $ - module $ M $ is free over a subset $ Ssubseteq M $ if $ S $ is linearly independent and generates $ M $ .
We say that $ S $ forms a basis for $ M $ .
A pushout is a colimit of a $ cdot leftarrow cdot to cdot $ shaped diagram.
That is, in a given category $ mathcal C $ , the pushout of the morphisms $ f:Zto X $ and $ g:Zto Y $ in $ mathcal C $ consists of an object $ P $ of $ mathcal C $ and two morphisms $ i_1:Xto P $ and $ i_2:Yto P $ such that the diagram commutes and such that $ (P, i_1,i_2) $ is universal with respect to the diagram.
Again, that is, for any other triple $ (Q, j_1,j_2) $ in $ mathcal C $ such that the following diagram commutes, there exists a unique $ u:Pto Q $ so that the diagram commutes. !
Screen Shot 2022 - 02 - 17 at (iii)5(vi)27 PM.png
A set $ X $ is second - countable if there exists a countable collection $ mathcal A $ of open sets in $ X $ such that every open set is a union of sets of $ A $ .
Let $ V $ be a vector space.
The dimension of $ V $ is the cardinal number of any basis for $ V $ .
A subset $ A subsetmathbb R^n $ has measure zero if for every $ varepsilon > 0 $ , there exists a countable covering of $ A $ by rectangles $ {Q_i}_{iinmathbb N} $ such that the sum of the volumes of the $ Q_i $ is less than $ varepsilon $ , i.e. that $ sum_{iinmathbb N} text{vol}(Q_i) < varepsilon $ .
The orthogonal complement of $ W $ a subspace of the vector space $ V $ is the set $ W^perp = {v in V mid forall w in W, text{ } langle v, wrangle = 0} $ .
This is itself a vector subspace:
A group $ G $ is simple if it contains no normal subgroups other than the subgroup containing the identity and itself.
Let $ mathfrak g $ be a Lie algebra over the field $ F $ .
A $ mathfrak g $ - module over a ring is a pair $ (V,cdot) $ of a vector space $ V $ over $ F $ and $ cdot:mathfrak g times V to V $ a map such that (i) $ (lambda x + mu y)cdot v = lambda (xcdot v) + mu(ycdot v) $ (ii) $ xcdot (lambda v + mu w) = lambda (xcdot v) + mu(ycdot v) $ (iii) $ x,ycdot v = xcdot (ycdot v) - ycdot (xcdot v) $ for all $ x,y in mathfrak g $ , $ v,win V $ , and $ lambda,mu in F $ .
The Lorentz group is the generalized orthogonal group $ text{O}(3,1) $ (or, alternatively, $ text{O}(1,3) $ depending on the metric signature convention being used).
Let $ R $ be a commutative ring and let $ a,bin R $ .
An element $ din R $ is a greatest common divisor of $ a $ and $ b $ if $ d $ divides $ a $ and $ d $ divides $ b $ and for every $ d'in R $ such that $ d' $ divides $ a $ and $ d' $ divides $ b $ , we have that $ d' $ divides $ d $ .
Equivalently, $ d $ is a greatest common divisor of $ a $ and $ b $ if the ideal generated by $ d $ is the smallest principal ideal that contains both $ a $ and $ b $ .
This ideal is unique.
Let $ V $ be an $ n $ - dimensional vector space and let $ T:Vto V $ be a linear transformation.
The adugate of $ T $ is $ text{adj}(T):Vto V $ the unique linear transformation such that for all $ v in V $ and all $ w in wedge^{n - 1}(V) $ the $ n - 1 $ - fold exterior product of $ V $ , $ text{adj}(T)(v) wedge w = v wedge (wedge^{n - 1}(T)(w)) in wedge^n(V) $ .
Let $ T:Vto V $ be a linear transformation from the vector space $ (V,F) $ to itself.
Then $ v in V $ is an eigenvector of $ T $ if there exists $ lambdain F $ such that $ T(v) = lambda v $ .
Let $ G $ be a subgroup of the general linear group $ GL_n(mathbb C) $ acting naturally on $ mathbb C^n $ .
A polynomial $ fin mathbb Cx_1,dots,x_n $ is $ G $ - harmonic if it is annihilated by all $ G $ - invariant differential operators without constant term.
That is, $ f $ is $ G $ - harmonic if $ u(f) = 0 $ for all $ u in D^G_{>0} $ .
Let $ (X,Sigma,mu) $ be a measure space.
A measurable subset $ A $ of $ X $ is outer regular if $ mu(A) = sup{mu(F) mid Fsubseteq A, Ftext{ compact and measurable}} $ .
An endomorphism is a morphism from an object to itself.
A group $ G $ is abelian if its operation is commutative.
A covariant functor $ F:mathcal Cto mathcal D $ is a map between categories $ mathcal C $ and $ mathcal D $ .
It must assign to each object $ A $ of $ mathcal C $ an object $ F(A) $ an object of $ mathcal D $ and to each morphism $ f:Ato B $ of $ mathcal C $ a morphism $ F(f):f(A) to f(B) $ of $ mathcal D $ such that - $ F(text{id}_A)) = text{id}_{F(A)} $ ; - $ F(gcirc f) = F(g)circ F(f) $ .
A contravariant functor "reverses the direction of arrows;" that is $ F $ sends $ f:Ato B $ to $ F(f): f(B) to f(A) $ and $ F(gcirc f) = F(f) circ F(g) $ .
Let $ G $ be a finitely generated abelian group.
By the Fundamental Theorem of Finitely Generated Abelian Groups, $ G $ may be uniquely written as a direct sum of cyclic groups.
The rank of $ G $ is the number of $ mathbb Z $ summands in this expression.
For $ xin L $ a Lie algebra, $ x $ is ad - nilpotent if $ text{ad}(x) $ is a nilpotent endomorphism.
Let $ X $ be a normed linear space with dual space $ X^* $ .
A sequence $ {x_i}_{iinmathbb N} $ in $ X $ converges weakly to $ x in X $ if $ limlimits_{itoinfty} f(x_i)=f(x) $ for all $ fin X^* $ .
Let $ mathbb k $ be some field.
Let $ G $ be a group and let $ rho:Gto GL(V) $ be a representation of $ G $ in the $ mathbb k $ - vector space $ V $ .
Let $ V^* = text{Hom}_mathbb k(V,mathbb k) $ be the dual space of $ V $ .
By the Riesz representation theorem, there is a canonical pairing $ langle cdot,cdot rangle:V^*times V tomathbb k $ that lets us write linear functionals $ phi:vmapsto phi(v) $ as $ langle phi,vrangle $ .
The $ G $ - action $ vmapsto rho(g)v $ on $ V $ induces the action on the space of functions on $ V $ , of which $ V^* $ is an invariant subspace.
Explicitly, $ gin G $ acts on $ phi in V^* $ by $ vmapsto phi((rho(g))^{ - 1}(v)) = langle phi, (rho(g))^{ - 1}vrangle = langle ((rho(g))^{ - 1})^intercal, vrangle = langle (rho(g^{ - 1})^intercal, vrangle $ where $ (rho(g^{ - 1}))^intercal $ is the adjoint of the operator $ rho(g):Vto V $ .
This action on $ V^* $ gives a group homomorphism $ rho^vee:Gto GL(V^*) $ where $ rho^vee(g) = (rho(g^{ - 1}))^intercal $ that is called the contragradient representation of the original representation $ rho $ .
A based topological space is a topological space $ X $ together with a point $ x_0in X $ , denoted $ (X, x_0) $ .
A covering space of a topological space $ X $ is another topological space $ tilde X $ together with a continuous map $ p:tilde Xto X $ such that there exists an open cover $ {U_alpha} $ of of $ X $ such that for all $ alpha $ , the inverse image $ p^{ - 1}(U_alpha) $ is a disjoint union of open sets in $ tilde X $ , each of which is mapped homeomorphically onto $ U_alpha $ .
Note that $ p^{ - 1}(U_alpha) $ does not need to be nonempty.
Thus $ p $ need not be surjective.
Alternatively, let $ E $ and $ B $ be locally path - connected topological spaces.
A continuous function $ p:Eto B $ is a covering space (of $ B $ ?) if it is surjective and if each point $ pin B $ has an open neighborhood $ V $ such that each connected component of the inverse image $ p^{ - 1}(V) $ is open in $ E $ and is mapped homeomorphically onto $ V $ by $ p $ .
A path - connected open subset $ V $ with this property is a fundamental neighborhood of $ B $ .
The space $ E $ is called the total space and the space $ B $ is called the base space .
Write the fiber of the point $ b $ as $ F_b = p^{ - 1}(b) $ .
Let $ p:mathcal Eto mathcal B $ be a covering of groupoids.
Then the fiber translation functor $ T=T(p): mathcal Bto mathcal S $ , where $ mathcal S $ is the category of sets.
For an object $ b $ of $ mathcal B $ , define $ T(b) = F_b $ , the fiber of $ b $ .
For a morphism $ f:bto b' $ of $ mathcal B $ , define T(f):F_bto F_{b'} $ by $ T(f)(e) = e' $ where $ e' $ is the target of the unique $ g $ in the star $ text{St}(e) $ such that $ p(g)=f $ .
A graph $ X $ is a topological space obtained from a discrete set $ X^0 $ of points called vertices and a discrete set $ J $ of functions $ j:S^0to X^0 $ as the quotient space of the disjoint union $ X^0/coprod (Jtimes I) $ where $ I $ is the unit interval obtained by identifying $ (j,0) $ with $ j(0) $ and $ (j,1) $ with $ j(1) $ .
The images of the intervals $ jtimes I $ are called edges .
A graph can also be though of as a 1 - dimensional CW complex whose $ 0 $ - cells are the vertices and whose $ 1 $ - cells are the edges and where the endpoints of each edge are identified with the vertices attached to it.
A group $ G $ is free if there exists a generating set of a group $ X $ of $ G $ such that every nonempty reduced group word in $ X $ defines a nontrivial element of $ G $ .
The ordinary generating function of a sequence $ {a_k}_{kinmathbb N} $ is $ A(x) = sum_{k=0}^infty a_kx^k $
A monic polynomial is a polynomial whose leading coefficient is the unit of the ring over which it is defined.
Let $ (X,Sigma) $ be a measurable space and let $ mu $ and $ nu $ be two measures on $ (X,Sigma) $ . $ mu $ is absolutely continuous with respect to $ nu $ (written $ mu ll nu) $ if $ mu(A) =0 $ for every $ Ain Sigma $ for which $ nu(A)=0 $ .
Let $ X $ be a set with binary operation $ * $ .
An element $ ein X $ is a left identity of $ X $ if $ x* a = a $ for all $ ain A $ .
A norm on a vector space $ V $ is a real - valued functional $ ||cdot|| $ such that for all $ v,w in V $ and $ alpha in mathbb R $ , the following properties hold: (i) $ ||v+w|| leq ||v|| + ||w|| $ ; (ii) $ ||alpha v|| = |alpha| cdot ||v|| $ ; (iii) $ ||v|| geq 0 $ and $ ||v|| = 0 $ if and only if $ v=0 $ .
hAn orthonormal sequence of vectors $ {v_i}_{iinmathbb N} $ in a Hilbert space $ H $ is an orthonormal (Hamel) basis for $ H $ if $ h = sum_{iinmathbb N} langle v_i, hrangle v_i $ for all $ h in H $ .
A ring homomorphism is a map $ f:Sto R $ between rings $ S $ and $ R $ such that - $ f:(S,+) to (R,+) $ is a group homomorphism; - $ f(s_1s_2)= f(s_1)f(s_2) $ for all $ s_1,s_2in S $ ; - $ f(1) = 1 $ .
A ring isomorphism is a homomorphism that happens to also be a bijection.
Let $ V = bigoplus_{igeq0} V_i $ be a graded vector space.
A graded subspace is a vector subspace of $ V $ of the form $ W = bigoplus_{igeq 0}W_i $ for $ W_i subseteq V_i $ .
Clearly, this is only the case when $ W= bigoplus_{igeq 0} (Wcap V_i) $ .
A topological ring is a ring $ R $ together with a topology on the underlying set where the addition map $ +:Rtimes Rto R $ (given by $ +(a,b) = a+b $ for all $ a,bin R $ ) and the multiplication map $ cdot:Rtimes Rto R $ (given by $ cdot(a,b)= ab $ for all $ a,bin R $ ) are both continuous.
Let $ L $ be a Lie algebra and $ Isubset L $ an ideal.
Then the quotient $ L / I $ is constructed as follows: - The underlying vector space is the usual quotient vector space $ L/I $ when $ L $ and $ I $ are considered as vector spaces. - Lie multiplication is given by $ x+I,y+I = xy+I $ .
Let $ (V_1,rho_1) $ and $ (V_2,rho_2)) $ be representations of the group $ G $ .
The tensor product of $ V_1 $ and $ V_2 $ is a representation of $ G $ over the tensor product of vector spaces $ V_1otimes V_2 $ under the linear transformation $ rho_1otimes rho_2 = rho:Gto V_1otimes V_2 $ such that $ rho(g)(v_1otimes v_2) = (rho_1(g)v_1)otimes (rho_2(g)v_2) $ .
The measure density of a set $ A subsetmathbb R^n $ at a point $ x in mathbb R^n $ is given by $ lim_{rto 0} frac{mu(Acap B(x,r))}{mu(B(x,r))} $ where $ mu $ is some measure on $ mathbb R^n $ .
Definition 1 (second category sets)
A topological space is a Baire space if every nonempty open set in $ X $ is nonmeager.
Definition 2 (intersection of open dense sets)
A topological space is a Baire space if every countable intersection of open dense sets is also dense.
A derivation of the algebra over a field $ U $ is a linear transformation $ delta $ satisfying the product rule: $ delta(ab) = adelta(b) + delta(a) b $ .
The group algebra of a group $ G $ is the group ring of $ G $ over $ mathbb C $ is denoted $ mathbb CG $ and is the set of maps $ f:Gto mathbb C $ with finite support.
This forms a module over a ring over $ mathbb C $ , and because $ mathbb C $ is a field, it is a vector space over $ mathbb C $ .
The module scalar product of $ alphain mathbb C $ and $ fin mathbb CG $ is the map $ xmapsto alphacdot f(x) $ .
The module group sum of mappings $ f $ and $ g $ in $ mathbb CG $ is the map $ xmapsto f(x)+g(x) $ .
To make this whole thing into a ring, we define the product of $ f,gin mathbb CG $ by the map $ xmapsto sum_{uv=x} f(u)g(v) = sum_{uin G} f(u)g(u^{ - 1}x) $ .
We often write elements of $ mathbb CG $ as formal linear combinations of elements in $ G $ with coefficients in $ mathbb C $ , which looks like $ sum_{gin G} f_gg $ .
Equivalently, let $ mathbb k $ be a field and let $ G $ be a group.
The group algebra $ mathbb k G $ is a $ mathbb k $ - vector space with basis $ {1_g}_{gin G} $ and multiplication table given by $ 1_gcdot 1_h = 1_{gh} $ for all $ g,hin G $ .
The elements of $ mathbb kG $ are linear combinations $ sum_{gin G} c_g1_g $ for $ c_g inmathbb k $ zero for all but finitely many $ gin G $ .
A category is discrete if its morphisms are only the identity morphisms.
A differential basis is a collection $ mathcal B $ of pairs $ (B,x) $ where $ Bsubsetmathbb R^n $ and $ xin mathbb R^n $ such that for each $ xin mathbb R^n $ , there exists sequence $ {(B_i, x)}_{iinmathbb N}subset mathcal B $ such that $ B_i to {x} $ .
That is, the diameter $ text{diam}(B_icup{x}) to 0 $ .
A right coset of the subgroup $ H $ of the group $ G $ for the element $ xin G $ is the set $ Hx = {hxmid h in H} $ .