Hopf Algebras, definitions and examples March 26, 2002 1 Introduction This presentation, as the title indicates, is designed to be an introduction to Hopf algebras, covering basic definitions and some examples. Most of the material is from [Mon91] and [Kas95]. Only a couple of the simpler ”proofs” are mine. These are generally the type of proof that were omitted from the source document, being considered trivial or obvious. In general, such things are not so obvious to me, so to clarify the material for myself, I wrote out the explanations. If time permits I would like to also talk about how Hopf algebras are related to what I am studying. 2 Algebras and Coalgebras Before stating the definition of a Hopf algebra, we should first go over a few preliminary definitions. First recall the definition of an algebra over a field k. Definition 2.1 An algebra over a field k is a vector space, A, together with two linear maps, a multiplication µ : A N A → A, and a unit map η : k → A such that the following diagrams commute: µ ⊗ id A ⊗ A ⊗ AA- ⊗ A id ⊗ µ µ ? ? - A ⊗ AAµ 1 and η ⊗ id id ⊗ η k ⊗ AA- ⊗ A A ⊗ k @ @ @ µ @ @@R ? © A where the lower left and right maps are simply scalar multiplication. Note: the second diagram implies 1A = u(1k) Examples of algebras are the polynomial algebras and the matrix algebras. Definition 2.2 A coalgebra is a vector space C together with two linear maps, comultiplication ∆ : C → C ⊗ C and counit ε : C → k, such that the following two diagrams commute. CC∆ - ⊗ C ∆ ∆ ⊗ id ? ? C ⊗ CC- ⊗ C ⊗ C id ⊗ ∆ and 1⊗ ⊗1 k ⊗ CC - C ⊗ k @I@ @ ε ⊗ id @ ∆ id ⊗ ε @ @ ? C ⊗ C Definition 2.3 If C and D are coalgebras with respective comultiplication maps ∆C and ∆D, and respective counit maps εC and εD then 2 1. A map f : C → D is a coalgebra morphism if ∆D ◦f = (f ⊗f)◦∆C i.e. the following diagram commutes. f CD- ∆C ∆D ? ? C ⊗ CD- ⊗ D f ⊗ f 2. A subspace I ⊆ C is a coideal if ∆I ⊆ I ⊗ C + C ⊗ I and ε(I) = 0 Example 2.1 (A(X)) As an example of a coalgebra, consider the polyno- mials of four variables over the field C C = C[x11, x12, x21, x22] We can define the algebra morphisms ∆ and ε on the generators of C by 2 X ∆(xij) = xik ⊗ xkj k=1 ε(xij) = δij For later use, we will define det = x11x22 − x12x21 ∈ C. Also, this algebra will be referred to as A(X) where X is the set of all 2 × 2 matrices with complex coefficients. This algebra is also referred to as the algebra of regular functions of X. Example 2.2 (Divided Powers) If we let C = C[t] be the polynomials of one variable over C, then we can define a comultiplication and counit by X ∆(tn) = tp ⊗ tq p+q=n n ε(t ) = δn0 We can demonstrate the coassociativity (for n = 2) with the following calcu- lations: (id ⊗ ∆) ◦ ∆(t2) = (id ⊗ ∆)(t2 ⊗ 1 + t ⊗ t + 1 ⊗ t2) = t2 ⊗ 1 ⊗ 1 + t ⊗ 1 ⊗ t + t ⊗ t ⊗ 1 +1 ⊗ t2 ⊗ 1 + 1 ⊗ t ⊗ t + 1 ⊗ 1 ⊗ t2 = (∆ ⊗ id)(t2 ⊗ 1 + t ⊗ t + 1 ⊗ t2) = (∆ ⊗ id) ◦ ∆(t2) 3 n n To show that ε(t ) = δn0 defines a counit map we check that (id⊗ε)◦∆(t ) = tn ⊗ 1 and that (ε ⊗ id) ◦ ∆(tn) = 1 ⊗ tn ! X (id ⊗ ε) ◦ ∆(tn) = (id ⊗ ε) tp ⊗ tq p+q=n = tn ⊗ 1 Similarly, (ε ⊗ id) ◦ ∆(tn) = 1 ⊗ tn. Remark 2.1 (Construction of an Algebra from a Coalgebra) Given an coalgebra C over a field k, we may consider the dual space C∗ = Hom(C, k). We see here that there is a natural way of dualizing the coalgebra structure of C into an algebra structure on C∗. Associated with the coalgebra structure of C we may define the following multiplication and unit maps on C∗ µ(f ⊗ g)(c) = (f ⊗ g) ◦ ∆(c) η(α)(c) = αε(c) f, g ∈ C∗, c ∈ C, α ∈ k 3 Bialgebras and Hopf Algebras We see in example 2.1 that we defined a coalgebra structure on top of an existing algebra. This leads us to the following definition which combines both ideas. Definition 3.1 Given a space B, B is a bialgebra if (B, ∆, ε) is a coalge- bra, (B, µ, η) is an algebra and either of the following equivalent conditions is true: 1. ∆ and ε are algebra morphisms 2. µ and η are coalgebra morphisms This bialgebra structure is often denoted by (B, µ, η, ∆, ε) Proposition 3.1 The conditions in the previous definition are indeed equiv- alent. (Here we will simply show that ∆ and µ are compatible, leaving it to the interested reader to verify the conditions for the unit and counit maps). 4 Proof: Here, we must make use of the algebra and coalgebra structures of B ⊗ B, which are defined respectively by the maps µB⊗B and ∆B⊗B [Kas95]: µB⊗B = µB ⊗ µB ◦ (id ⊗ τ ⊗ id) ∆B⊗B = (id ⊗ τ ⊗ id) ◦ (∆B ⊗ ∆B) Where τ : B ⊗ B → B ⊗ B defined by τ(a ⊗ b) = b ⊗ a. Now, if µB is a coalgebra morphism we have ∆B ◦ µB = (µB ⊗ µB) ◦ ∆B⊗B = (µB ⊗ µB) ◦ [(id ◦ τ ◦ id) ◦ (∆B ⊗ ∆B)] and if ∆B is an algebra morphism then ∆B ◦ µB = µB⊗B ◦ (∆B ⊗ ∆B) = [(µB ⊗ µB) ◦ (id ◦ τ ◦ id)] ◦ (∆B ⊗ ∆B) which are equivalent because the composition of maps is associative. In Example 2.1, ∆ (and ε) were defined to be algebra morphisms, thus the defined structure is automatically a bialgebra. The following are a few more examples of bialgebras: Example 3.1 If we let G be a group then B = CG, the associated group algebra, becomes a bialgebra with the following defined maps ∆(g) = g ⊗ g, ∀g ∈ G ε(g) = 1, ∀g ∈ G Definition 3.2 If C is a any coalgebra, then for c ∈ C, we say that c is grouplike if ∆(c) = c ⊗ c and if ε(c) = 1. The set of all grouplike elements of a coalgebra is denoted G(C). An interesting point to mention here is that if B is a group algebra, then G(B) = G, the original group. [Mon91]. Also, it can be shown through direct calculation that det = x11x22 − x12x21 from the bialgebra of Example 5 2.1 is grouplike. As follows: ∆(det) = ∆(x11x22 − x12x21) = ∆(x11)∆(x22) − ∆(x12)∆(x21) = (x11 ⊗ x11 + x12 ⊗ x21)(x21 ⊗ x12 + x22 ⊗ x22) −(x11 ⊗ x12 + x12 ⊗ x22)(x21 ⊗ x11 + x22 ⊗ x21) = (x11x22 − x12x21) ⊗ (x11x22 − x12x21) = det ⊗ det Example 3.2 (U(sl(2))) Consider the universal enveloping algebra of sl(2), U(sl(2)). One can think of U(sl(2)) as the polynomial algebra of three gen- erators e, f, and h, with the added relations [x, y] = H, [h, x] = 2x, [h, y] = −2y i j k Also, note that the set {x y h |i, j, k ∈ Z+} is a basis of U(sl(2)) as a result of the Poincar´e-Birkhoff-Witt theorem [Kas95]. If we define the comultiplication and counit maps on U(sl(2)) in the following manner, then it has a bialgebra structure. ∆(x) = x ⊗ 1 + 1 ⊗ x, ε(x) = 0, ∀x ∈ sl(2) Definition 3.3 Given an algebra (A, µ, η), a coalgebra (C, ∆, ε) and two lin- ear maps f, g : C → A then the convolution of f and g is the linear map f ? g : C → A defined by f ? g(c) = µ ◦ (f ⊗ g) ◦ ∆(c), c ∈ C Definition 3.4 Let (H, µ, η, ∆, ε) be a bialgebra. An endomorphism S of H is called an antipode for the bialgebra H if idH ?S = S ? idH = η ◦ ε A Hopf algebra is a bialgebra with an antipode. Example 3.3 The (group) bialgebra B, of Example 3.1 is a Hopf algebra with antipode S defined by S(g) = g−1 6 We can show that S is an antipode by id ? S(g) = µ ◦ (id ⊗ S) ◦ ∆(g) = µ ◦ (id ⊗ S)(g ⊗ g) = µ(g ⊗ g−1) = 1B = η ◦ ε(g) Proposition 3.2 Given a Hopf algebra (H, µ, η, ∆, ε), with antipode S, then for any grouplike element g ∈ H, S(g) = g−1. Proof. Using the definition of the antipode S, we have id ? S(g) = η ◦ ε(g) µ ◦ (id ⊗ S) ◦ ∆(g) = 1H µ(g ⊗ S(g)) = 1H g · S(g) = 1G Similarly, S(g) · g = 1 Example 3.4 An example of a bialgebra that is not a Hopf algebra, refer back to Example 2.1, A(X). If it were a Hopf algebra (i.e. if it had an antipode S) then S(det) = det−1, because det is grouplike. However, det is not invertible, so A(X) is not a Hopf algebra. However, we can use this bialgebra to construct the next example of a Hopf algebra. Example 3.5 Consider the polynomial algebra C[x11, x12, x21, x22][1/det]. This is obtained by simply adjoining the inverse of det to the previous bialgebra. Other notations found in the literature for describing the same algebra are −1 C[x11, x12, x21, x22, det ] and C[x11, x12, x21, x22][t]/(t(det) − 1). Now we are free to define the map S in the following manner [Kas95]. −1 −1 S(x11) = det x22,S(x12) = −det x12 −1 −1 S(x22) = det x11,S(x21) = −det x21 This is often referred to as the algebra of regular functions on the group GL(2, C), and is denoted as A(GL(2, C)).