Hopf Algebras Are Everywhere

Algebras Revisited Hopf Algebra Axioms Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras

Hopf Algebras are Everywhere

Thomas Kerler

May 7, 2007

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).

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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

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

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

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

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

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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

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

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

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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) isomorphic 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. Then these maps make the diagrams below commute. id ∆ A A A A ⊗ A ∆ A ⊗ A =∼ ∆ id ⊗ ∆ id ⊗ ∆ ⊗ id A ⊗ k A ⊗ A α (A ⊗ A) ⊗ A A ⊗ (A ⊗ A)

Deﬁnition Coalgebra A coalgebra over k is a k-vector A space with maps ∆ : A → A ⊗ A (coproduct) and : A → k (counit) such that the above diagrams commute. 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

Examples of Coalgebras

Example Duals For every algebra A its dual A∗ is a colagebra- vice-versa.

Particularly, for ﬁnite monoid X have C(X, k) =∼ (k[X])∗

Example Cobordisms As before - but read from right to left.

Deﬁnition Coideal For k-coalgebra A a coideal is a k-subspace J ⊂ A such that ∆(A) ⊆ A ⊗ I ⊕ I ⊗ A

Remark Historical Lie Group Cohomology Instead of C(X, k) considered H∗(G, C) with G a Lie group. ∗ Department of H is graded commutative algebra Mathematics Kunneth¨ Theorem ensures coproduct. Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras

Comensurability of Products

Lemma Cohomomorphy for Functions For X a ﬁnite monoid, view C(X) as a commutative algebra. The map ∆ = µ∗ : C(X) → C(X × X) =∼ C(X) ⊗ C(X) is a homomorphism.

Proof: µ∗(φ∗ψ∗)(g, h) = (φ∗ψ∗)(gh) = (φ∗(gh))(ψ∗(gh)) = (µ∗(φ∗)(g, h))(µ∗(ψ∗)(g, h)) = ((µ∗(φ∗))·(µ∗(ψ∗))(g, h)

Lemma Algebras for Tensors The natural algebra structure on A ⊗ A is given by m(2) = (m ⊗ m) ◦ (id ⊗ τ ⊗ id):(A ⊗ A) ⊗ (A ⊗ A) −→ (A ⊗ A) where τ(x ⊗ y) = y ⊗ x and all α supressed

Proof: (a ⊗ b)·(c ⊗ d) = m ((a ⊗ b) ⊗ (c ⊗ d) = (m ⊗ m)((id ⊗ τ⊗)(a ⊗ b ⊗ c ⊗ d)) = 2 Department of (m ⊗ m)(a ⊗ c ⊗ b ⊗ d) = (m(a ⊗ c)) ⊗ (m(b ⊗ d)) = ac ⊗ bd 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

Compatibility Axiom

Thus ∆ is an algebra homomorphism if: m2 ◦ (∆ ⊗ ∆) = ∆ ◦ m As commutative diagram:

m ∆ A ⊗ A A A ⊗ A

∆ ⊗ ∆ m ⊗ m

id ⊗ τ ⊗ id A ⊗ A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ A Deﬁnition Bialgebras A bialgebra is a k-vector space A which carries both an algebra and coalgebra structure such that the following hold: 1 The above diagram commutes. 2 ∆ ◦ e = e ⊗ e 3 ◦ m = ⊗ Department of Mathematics 4 ◦ e = 1 Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras

Examples of Bialgebras

Example Self Dual For a bialgebra A also A∗ is also a bialgebra. Particularly, for monoid X, since C(X, k) is bialgebra also k[X] bialgebra. Coproduct on k[X] is ∆(x) = x ⊗ x for x ∈ X (“group like”) Group like elements are precisely X ⊂ k[X].

Proof 3rd Claim: Verify (φ∗ ·ψ∗)(x) = φ∗ ⊗ ψ∗(∆(x)). Remark NO GO

1 For n > 1 have that Matn(k)(⊕k) does not extend to bialgebra. 2 The product and coproduct for cobordisms are no compatible:

genus(∆ ◦ m) = 0 but genus(m2 ◦ (∆ ⊗ ∆)) = 1

Example Borel Algebra As above let A be the algebra generated by H and E with relation HE −EH = γE.

Then ∆(H) = H ⊗e+e⊗H and ∆(E) = E ⊗e+e⊗E yields a bialgebraDepartment of Mathematics structure on A .

Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras

Primitive Element

Deﬁnition Primitives For a bialgebra A deﬁne P(A) = {x ∈ A : ∆(x) = x ⊗ e + e ⊗ x}.

Lemma Lie Type P(A) is closed under the bracket [x, y] = xy − yx. Particularly, P(A) ⊂ A is a Lie algebras

Proof: Verify ∆([x, y]) = [x, y] ⊗ e + e ⊗ [x, y] for x, y ∈ P(A)

Lemma p-Modular Algebra Let char(k) = p , and assume x is a primitive generator of k[x]. p k[x] Then x k[x] is a bi-ideal so that A = xp is a bialgebra. p Proof: Compute ∆(x p) = (∆(x))p= (x ⊗ e + e ⊗ x)p = x p−j ⊗ x j j Department of j Mathematics X

Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Commutative Algebra Hopf Algebra Axioms Coalgebras Bicommutative Hopf Algebras Bialgebras Semi-Commutative Hopf Algebras Hopf Algebras

Antipode

For a group G consider “inversion map” S : k[G] → k[G] deﬁned by S(g) = g−1. Express gS(g) = e ! ⇔ m(g ⊗ S(g)) = e ⇔ m((id ⊗ S)(g ⊗ g)) = e ⇔ m((id ⊗ S)(∆(g))) = e

More generally, by linearity m ◦ (id ⊗ S) ◦ (∆)( λg g) = ( λg )e g g X X Note that for x = λg g have (x) = λg g g Hence above readsX m ◦ (id ⊗ S) ◦ (∆)(Xx) = (x)e

id ⊗ S A ⊗ A A ⊗ A ∆ m A A e

k 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

Hom-Bialgebras

Lemma Algebra of Morphisms Let B be an algebra and C is a coalgebra. For f , g ∈ Hom(B, C) deﬁne ∆ f ⊗g m f ? g : C −→ C ⊗ C −→ A ⊗ A −→ A Then ? deﬁnes an associative product on Hom(B, C) with unit I = e.

Remark Antipode The antipode axiom is equivalent to the following identity on Hom(A, A)

S ? idA = idA ? S = I

Remark Bialgebra of Endomorphisms Hom(A, A) equipped with ? as above and m f ∆ δ(f ): A ⊗ A −→ A −→ A −→ A ⊗ A deﬁnes a bialgebra. 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

Hopf Algebra Deﬁnition

Deﬁnition Hopf Algebras A Hopf algebra is a bialgebra A with a map S : A → A (antipode) which satiﬁes the commutative diagram (+reverse) above.

Remark Restrictions If g is group like then S(g) = g−1 is 2-sided inverse. k[G] has Hopf algebras structure only if G is a group. S(x) = −x for primitive x. S is always anti-homomorphism and anti-cohomomorphism.

Deﬁnition Hopf Ideal For a Hopf algebra A a Hopf ideal is a k-subspace I ⊂ A which is, both, an ideal and a coideal Department of and satiﬁes S(I) = I 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

Basic Hopf Algebra Notions

Remark Duality A If I ⊂ A is a Hopf ideal then A = I is also a Hopf algebra. For any Hopf algebra A its dual A∗ is also a Hopf algebra.

Note that “x and y” commute means x ·y = y ·x ⇔ m(x ⊗ y) = m(y ⊗ x) = m(τ(x ⊗ y)).

Deﬁnition Commutativity An algebra is called commutative if m = m ◦ τ

Deﬁnition Cocommutativity A coalgebra is called cocommutative if ∆ = τ ◦ ∆ Department of Mathematics

Thomas Kerler Hopf Algebras are Everywhere Simple Set Example Algebras Revisited Reduced Coproduct Hopf Algebra Axioms Structure of Bicommutative Hopf Algebras Bicommutative Hopf Algebras Hopf Algebras of Graphs Semi-Commutative Hopf Algebras Graded Commutative Hopf algebras Cohomology of Lie groups

Deﬁnition Set Algebra Let S be the k-span of all (inequivalent) ﬁnite sets [Q], equipped with the following structures: Product given by [Q]·[R] = [Q t R]. Coproduct given by ∆(Q) = [U] ⊗ [Q\U]. U⊆Q X Antipode given by S([Q]) = (−1)|Q|[Q]

Units: e = ∅ and ([S]) = δS,∅.

Lemma Set Hopf Algebra The above make S into a commutative and cocommutative Hopf algebra.

Remark Set Hopf Algebra Primitive elements are multiples of [{∗}], since ∆([{∗}]) = [∅] ⊗ [{∗}] + [{∗}] ⊗ [∅]. Department of Any set is a product of [{∗}]. Mathematics

Lemma Sets and Polynomials The Hopf algebra S is equivalent to polynomial algebra k[X] with primitive generator, that is, ∆(X) = X ⊗ 1 + 1 ⊗ X.

Remark Sets to Polynomials Isomorphism give by Q 7→ X |Q| . k ∆(X k ) = (X ⊗ 1 + 1 ⊗ X)k = X j ⊗ X k−j j j X

Deﬁnition Reduced Coproduct Denote ∆ : A → A ⊗ A : x 7→ ∆(x) = ∆(x) − x ⊗ 1 − 1 ⊗ x

Observe, if X is primitive then ∆(X) = 0 Department of ∆(X 2) = 2X ⊗ X Mathematics ∆(X 3) = 3X 2 ⊗ X + 3X ⊗ X 2 ... etc Thomas Kerler Hopf Algebras are Everywhere Simple Set Example Algebras Revisited Reduced Coproduct Hopf Algebra Axioms Structure of Bicommutative Hopf Algebras Bicommutative Hopf Algebras Hopf Algebras of Graphs Semi-Commutative Hopf Algebras Graded Commutative Hopf algebras Cohomology of Lie groups Disassemble with Reduced Coproduct

Deﬁnition Iterated Reduced Coproduct

(n) Deﬁne iteratively ∆ : A → A⊗(n+1) = A ⊗ A ⊗ ... ⊗ A : as follows: (1) ∆ = ∆ n+1 (n+1) (n) | {z } ∆ = (∆ ⊗ idA⊗n ) ◦ ∆

Lemma Iteration of Primitives

For commutative Hopf algebra A suppose Z1, Z2,..., ZN ∈ A are primitive: (N) ∆ (Z1Z2 ... ZN ) = 0 (N−1) ∆ (Z Z ... Z ) = Z ⊗ ... ⊗ Z 1 2 N π∈SN π(1) π(N) P Remark

(N) (N−1) If X ∈ A primitive ∆ (X N ) = 0 and ∆ (X N ) = (N!)X ⊗ ... ⊗ X. Department of Mathematics Thus for char(k) = 0, can conclude 1, X, X 2, X 3, . . . are linearly independent. Thomas Kerler Hopf Algebras are Everywhere Simple Set Example Algebras Revisited Reduced Coproduct Hopf Algebra Axioms Structure of Bicommutative Hopf Algebras Bicommutative Hopf Algebras Hopf Algebras of Graphs Semi-Commutative Hopf Algebras Graded Commutative Hopf algebras Cohomology of Lie groups

Lemma Polynomial Ring of Primitives Suppose A is a commutative Hopf algebra over k with char(k) = 0 Denote by P(A) the primitive elements.

Then k[P(A)] → A is a Hopf monomorphism

Deﬁnition Graded Hopf Algebras M A graded Hopf algebra is of the form A = An such that n>0

Ak ·Al ⊆ Ak+l n ∆(An) = j=0 Aj ⊗ An−j A0 = ke P (An) = 0 for n > 0. Department of Mathematics

Theorem Even Hopf Theorem Suppose A is a graded, commutative and cocommutative Hopf algebra over a ﬁeld k with char(k) = 0. Then A is freely generated by its primitive elements That is, A is the polynomial algebra A = k[P(A)] = S(P(A))

For a non-trivial illustration consider the following graph theory example:

Graph Notation Denote G the set of ﬁnite graphs. Let G be the k-span of all graphs G . For G ∈ G denote by V (G) the set of vertices. For U ⊂ V (G) denote (G|U) the “restriction” of G to the subset U. That is, the set of vertices is U, and edges in (G|U) are the edges in G that Department of run between vertices in U. Mathematics

Deﬁnition Graph Algebra The space G is a graded, bicommutative Hopf algebra with Grading by deg(G) = |V (G)|. Product [G]·[H] = [G t H] by disjoint union. Coproduct, by ∆([G]) = [G|U] ⊗ [G|V (G) − U] U⊆XV (G) Unit by e = [∅] and counit by (G) = δG,∅ . Antipode deﬁned by induction on the number of edges in each degree. *)

*) Assume [G] is connected. Then the antipode axiom can be written as S([G]) + [(G|U) t (G|V (G) − U)] + [G] ∅6=UX$V (G) and hence S([G]) = −[G] + (Expression in Graphs with less edges) For example, S([•]) = −[•] S([• •]) = −[• •] + 2[•]·S([•]) = −[• •] − 2[• t •] Department of Mathematics

By Structure Theorem this is a polynomial Hopf algebra in primitives

Find P[G]:

∆[•] = [•] ⊗ [∅] + [∅] ⊗ [•] = [•] ⊗ e + e ⊗ [•] ⇒ [•] ∈ P

∆([• •]) = [• •] ⊗ e + 2[•] ⊗ [•] + e ⊗ [• •]

∆([• t •]) = [• t •] ⊗ e + 2[•] ⊗ [•] + e ⊗ [• t •]

Cancel mixed term with difference: so that Q2 := [• •] − [• t •] ∈ P

Similarly, obtain:

∆([• • •]) = [• • •]⊗e + 2[• •]⊗[•] + [•t•]⊗[•] + 2[•]⊗[• •] + [•]⊗[•t•] + e⊗[• • •]

Department of Q3 = [• • •] − 2[• • t•] + [• t • t •]∈ P Mathematics

Theorem Structure Graph Algebra

The graph Hopf algebra G is isomorphic to a free polynomial algebra k[{QG}] in independent variables QG , which are in one-to-one correspondence with connected graphs.

Remark Applications Similar Graph algebras are used to encode knots in 3-space and their invariants.

Equivalence under changes in projections are expressed in a Hopf ideal IIHX . Enumeration of (ﬁnite type) knot invariants reduces to counting dimensions of quotient algebra. Structure Theorem allows simpliﬁcation - only dimensions of primitive spaces have to be counted.

Department of Mathematics

Deﬁnition Graded (Co)Commutative Suppose A is a graded Hopf algebra, and τ : A ⊗ A → A is given by

k·l τ(x ⊗ y) = (−1) y ⊗ x for x ∈ Ak and y ∈ Al . A is called graded commutative if m = m ◦ τ. (That is, x ·y = (−1)k·l y ·x) A is called graded cocommutative if τ ◦ ∆ = ∆

Divide primitives P(A) = P+(A) ⊕ P−(A) into

P+(A) = primitives of even degree, and P−(A) = primitives of odd degree

Theorem Hopf Structure Theorem Suppose A is a graded-bicommutative Hopf algebra over a ﬁeld k with char(k) = 0.

Then A is freely generated by its primitive elements Department of That is, A = S(P+(A)) ⊗ Λ(P−(A)) Mathematics

Note, that we can have dim(A) < only if P+(A) = 0 .

“Cohomology” H∗(G, R) of a compact∞ Lie group(that is, a group that is a diffble manifold) is a ﬁnite dimensional, graded-bicommutative Hopf algebra

Theorem Hopf Structure Theorem H∗(G, C) is a free exterior alegbra generated by elements of odd degree.

Example Lie groups of dim=4 There are only two poissble cohomology rings for a Lie group of dimension=4: H∗(G, C) has four generator of degree=1. H∗(G, C) has one generator of degree=1, and one generator of degree=3.

Department of These are realized by U(1)4 and U(1) × SU(3) respectively. Mathematics

Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Hopf Algebra Axioms Lie algebras Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras

Universal Enveloping Algebras

Deﬁnition Lie Algebra A Lie algebra g is a k-space with a (bi)linear bracket operation g ⊗ g −→ g : X ⊗ Y 7→ [X, Y ]

which staiﬁes the following: 1 Skew: [X, Y ] = −[Y , X] 2 Jacobi: [X, [Y , Z]] + [Y , [Z, X]] + [Z, [X, Y ]] = 0

Every Lie algebra and it bracket is realized in the by an associative algebra in which the bracket is a commutator. The “most free” such algebra is the following:

Deﬁnition Universal Enveloping Algebra The universal enveloping algebra U(g) of a Lie algebra g is the associative algebra freely(distributively) generated by elements in g modulo the relations XY − YX = [X, Y ]. Department of Mathematics

Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Hopf Algebra Axioms Lie algebras Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras

Lemma Universal Enveloping Hopf Algebra U(g) has a unique cocommutative Hopf algebra structure for which the generators g ⊂ U(g) are primitive.

Theorem [Friedrich] If is U(g) is as above then P(U(g)) = g

This is the essential tool to proof the following

Theorem [Poincare-Birkhoff-Witt]´

If {X1,..., XN } is a basis of g,, then the ordered monomials

n1 n2 nN {X1 ·X ...·XN : nj > 0} are a basis of U(g)

Department of Mathematics

Thomas Kerler Hopf Algebras are Everywhere Algebras Revisited Hopf Algebra Axioms Lie algebras Bicommutative Hopf Algebras Semi-Commutative Hopf Algebras

Using similar arguments as for the commutatice case we ﬁnd that primitive elements behave like free generators:

Theorem [Universal Sub Algebra] For a Hopf algebra over k with char(k) = 0 we have a monomorphism: U(P(A)) −→ A which eventually gives us the following structure theorem for cocommutative Hopf algebras

Theorem [Structure Theorem via Lie Algebras] Suppose A is a cocommutative, irreducible Hopf algebra over k with char(k) = 0

Then the above is an isomorphism- i.e., A itself is universal enveloping algebra. Department of Mathematics

Thomas Kerler Hopf Algebras are Everywhere