Algebras Revisited Hopf Algebra Axioms Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras Hopf Algebras are Everywhere Thomas Kerler May 7, 2007 Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Hopf Algebra Axioms Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras 1 Algebras Revisited Conventional Definition Examples ideals Categorical Definition 2 Hopf Algebra Axioms Commutative Algebra Coalgebras Bialgebras Hopf Algebras 3 Bicommutative Hopf Algebras Simple Set Example Reduced Coproduct Structure of Bicommutative Hopf Algebras Hopf Algebras of Graphs Graded Commutative Hopf algebras Cohomology of Lie groups 4 Semi-Commutative Hopf Algebras Department of Lie algebras Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Associativity & Unit Diagrams Definition An algera A over a field k is a vector space over k with a mutliplication m : A × A −→ A, (x, y) 7→ x ·y such that the following properties hold: 1 Distributivity ∀a, b, c ∈ A, ∀λ ∈ k :(λa + b)·c = λa·c + b·c 2 Associativity ∀x, y, z ∈ A :(x ·y)·z = x ·(y ·z) 3 Unit ∃e ∈ A : ∀x ∈ A : e·x = x = x ·e Remark clearly Algebras can also be defined over a commutative ring B (rather than a field k). Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Examples: Finite Dimensional, Semisimple Algebras Example n×n-Matrices 2 Matn(k) : Spanned by Eij Relations: Eij ·Ekl = δjk Eil . dim (Matn(k)) = n Example Group Algebras k[G] (finite group G) Elements: λ= λg g (λg ∈ k) dim (k[G])) = |G| gX∈G λ· µ = λg g µhh = λg µhgh = ( λkh−1 µh) k h∈G gX∈G hX∈G gX,h∈G kX∈G X Definition Sums of Algebras For two algebras A and B their direct sum A ⊕ B , is the sum of the underlying vector spaces equipped with component wise mulitplication. (a1, b1)·(a2, b2) = (a1 ·a2, b1 ·b2) e⊕ = (eA, eB ) Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Other Examples Example Upper (Block) Triangular Matrices ∗ ∗ ∗ n Matrices of form 0 ∗ ∗ Preserving Filtration V0 ⊂ V1 ⊂ ... ⊂ Vm = k 0 0 ∗ Example Algebras of infinte Groups & Monoids k[M] with M any (in/finite) monoid. k[G] (with |G| = ) −1 k[Z] = k[t, t ] Laurent-Polynomials∞ k[N] = k[X] Polynomials k{X1,..., Xn} Free algebras with generators X1,..., Xn Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Ideals and Quotients Definition Ideals An (2-sided) ideal of A is a k-subspace I ⊂ A such that I·A ⊆ I and A·I ⊆ I Lemma Quotients A For an ideal I ⊂ A the quotient A = I inherits conical algebra structure. Example Polynomial For p(X) ∈ k[X] have (principal) ideal I = p(X)·k[X] k[X] Denote A = p(X) . dim(A) = deg(p(X)) Representations: ρ : f (X) 7→ f (A) , with A ∈ Matk (k) s/th µA(X) p(X). Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition More Examples: Ideals and Quotients Example Augmentation Ideal [ Define augmentation ideal I ⊂ k[G] by I = (g − 1)k[G], or g∈G I = ker() where : k[G] → k : g 7→ 1 . A Consider A = I2 0 H A =∼ e ⊕ where H = G ⊗ k 0 0 [G,G] k Example Borel Algebra I ⊂ A = C{H, E} defined by I = A(H ·E − E ·H − γE)A Language: I ideal generated by relation HE − EH = γE. A i j 0 A = I has basis {H E : i, j ∈ N }. Department of A has no simple representations with 1 < dim < . Mathematics Thomas Kerler Hopf Algebras are Everywhere∞ Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Tensor Products Recall Distributivity Relation: m(λx + y, z) = λm(x, z) + m(y, z) Definition Tensor Product Given two k-vector spaces V and W , the tensor product V ⊗ W is the k-vector space spanned by pairs [v, w] modulo the k-vector space spanned by expressions [λx + y, z] − λ[x, z] − [y, z] Remark Product Properties For v ∈ V and w ∈ W denote v ⊗ w ∈ V ⊗ W the class of [v, w]. If {vi } basis of V and {wj } basis of W , then {vi ⊗ wj } basis of V ⊗ W . Hence dim(V ⊗ W ) = dim(V ) dim(W ). For three k-vector spaces U, V , and W , there is a canonical isomorphism =∼ α :(U ⊗ V ) ⊗ W −→ U ⊗ (V ⊗ W ) induced by (u ⊗ v) ⊗ w 7→ u ⊗ (vDepartment⊗ w) of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Distributivity & Associativity Lemma Bilinearity A map f : V × W → U is bilinear/distributiveif and only if it factors as f f : V × W → V ⊗ W −→ U where f is a linear map. Hence we can circumvent distributivty by using m : A ⊗ A → A Metamorphosis of Associativity (x ·y)·z = x ·(y ·z) m(m(x, y), z) = m(x, m(y, z)) m(m(x ⊗ y) ⊗ z) = m(x ⊗ m(y ⊗ z)) m((m ⊗ id)((x ⊗ y) ⊗ z)) = m((id ⊗ m)(x ⊗ (y ⊗ z))) m ◦ (m ⊗ id) = m ◦ (id ⊗ m) ◦ α Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Associativity & Unit Diagrams α (A ⊗ A) ⊗ A A ⊗ (A ⊗ A) id ⊗ m m ⊗ id A ⊗ A A ⊗ A m m A id Unit as map e : k → A : 1 7→ e A A Rephrasing Unit Axiom x ·e = x =∼ m m(x ⊗ e) = x id ⊗ e m((id ⊗ e)(x ⊗ 1)) = id(x) A ⊗ k A ⊗ A ∼ Department of m ◦ (id ⊗ e)◦ = = id Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Categorical Algebras Definition Algebras in Tensor Categories Let C be a tensor category, with unit object 1. An object A ∈ obj(C) is called an algebra in C if there are morphisms e : 1 → A and m : A ⊗ A → A such that the above diagrams commute. If C is category of k-vector spaces and linear maps this is equivalent to the conventional definition. Here a very different example: Definition Category of 1+1-dim Cobordisms F 1 Objects are 1-dim compact manifolds, that is, unions Sk = k S of circles. Morphisms between Sk and Sl are 2-dim oriented surfaces Σ with ∂Σ = −Sk t Sl . Composition by sewing surfaces. Tensor product by disjoint union. Department of 1 = ∅ Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition 1+1-dim Bordisms Generic example of morphism Σ : S3 → S2 Theorem Circle Algebra The circle A = S1 is an algebra with multiplication m : A ⊗ A → A given by unit e : 1 → A given by Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Conventional Definition Hopf Algebra Axioms Examples Bicommutative Hopf Algebras ideals Semi-Commutative Hopf Algebras Categorical Definition Sketch of Proof Associativity = Unit Axiom = Remark More Properties The 1+1-dim cobordism category is a symmetric category. A is a commutative algebra: We have m ◦ c = m A is a Frobenius algebra: Non-degenerate trace given by opposite capDepartment of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras Commutative and Function Algebras Example Function Algebras Let X be some space and B commutative ring over k with unit. Let C(X, B) space of functions f : X → B C(X, B) is k-algebra under point wise multiplication. Commutative Algebras essentially all look like some algebra of functions. A precise statement for k = C: Theorem [“Classical” Gel’fand-Naimark Theorem] Suppose A is a commutative C∗-algebra. Then there is a compact Hausdorff space X such that A is (isometrically) isomor- phic to C0(X, C). See also in algebraic geometry, algebra of coordinate functions on variety give by zeros of polynomialsvs commutative algebra for which polynomials are relationsDepartment of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras Functoriality & Monoids Lemma Functoriality for Function Spaces Any map φ : X → Y induces homomorphism of algebras φ∗ : C(Y ) → C(X) by φ∗(f )(x) = f (φ(x)). There is a canonical inclusion of algebras χ : C(X, k) ⊗ C(Y , k) → C(X × Y , k) defined by χ(f ⊗ g)(x, y) = f (x)g(y) which is an isomorphism for finite spaces Remark Structure Induced by Monoids Suppose X is a finite, associative monoid with multiplication µ : X × X → X. Induces dual map µ∗ : C(X, k) → C(X, k) ⊗ C(X, k). If X also has a unit 1 ∈ X then define evaluation map ev1 : C(X, k) → k : f 7→ f (1). Department of Mathematics Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras Coproducts & Counits Lemma Coalgebra Structure for Functions ∗ Denote ∆ = µ , = ev1, and A = C(X, k) as above.