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Hopf algebra∗ mhale† 2013-03-21 14:59:11 A Hopf algebra is a bialgebra A over a field K with a K-linear map S : A → A, called the antipode, such that m ◦ (S ⊗ id) ◦ ∆ = η ◦ ε = m ◦ (id ⊗ S) ◦ ∆, (1) where m : A ⊗ A → A is the multiplication map m(a ⊗ b) = ab and η : K → A is the unit map η(k) = k1I. In terms of a commutative diagram: k A SSSS kkk SSS SS∆S SSS SS) ∆ kkkk A⊗A S⊗id k kkk ukkk ε K A⊗A id⊗S η A⊗A S A⊗A k SSS kkk SSS kkk SSS k k m k m SSS SS) ukkkkk A The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group scheme to its affine 1-space is a commutative Hopf algebra, with coalgebra structure given by dualising the group structure of the affine group scheme. Further, a commutative Hopf algebra is a cogroup object in the category of commutative algebras. ∗ hHopfAlgebrai created: h2013-03-21i by: hmhalei version: h33524i Privacy setting: h1i hDefinitioni h16W30i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Example 1 (Algebra of functions on a finite group) Let A = C(G) be the algebra of complex-valued functions on a finite group G and identify C(G × G) with A ⊗ A. Then, A is a Hopf algebra with comultiplication (∆(f ))(x, y) = f (xy), counit ε(f ) = f (e), and antipode (S(f ))(x) = f (x−1 ). Example 2 (Group algebra of a finite group) Let A = CG be the complex group algebra of a finite group G. Then, A is a Hopf algebra with comultiplication ∆(g) = g ⊗ g, counit ε(g) = 1, and antipode S(g) = g −1 . The above two examples are dual to one another. Define a bilinear form C(G) ⊗ CG → C by hf, xi = f (x). Then, hf g, xi = hf ⊗ g, ∆(x)i, h1, xi = ε(x), h∆(f ), x ⊗ yi = hf, xyi, ε(f ) = hf, ei, hS(f ), xi = hf, S(x)i, hf ∗ , xi = hf, S(x)∗ i. Example 3 (Polynomial functions on a Lie group) Let A = Poly(G) be the algebra of complex-valued polynomial functions on a complex Lie group G and identify Poly(G × G) with A ⊗ A. Then, A is a Hopf algebra with comultiplication (∆(f ))(x, y) = f (xy), counit ε(f ) = f (e), and antipode (S(f ))(x) = f (x−1 ). Example 4 (Universal enveloping algebra of a Lie algebra) Let A = U(g) be the universal enveloping algebra of a complex Lie algebra g. Then, A is a Hopf algebra with comultiplication ∆(X) = X ⊗ 1 + 1 ⊗ X, counit ε(X) = 0, and antipode S(X) = −X. The above two examples are dual to one another (if g is the Lie algebra of . . G). Define a bilinear form Poly(G) ⊗ U(g) → C by t. t=0 f (exp(tX)) 2