Hopf algebra

A Hopf algebra emerges by a proper redefinition of the antilinear characteristics of TFD. Consider g = giti = 1,2,3,.. be an associative algebra defined on the field of the complex numbers and let g be equipped with a Lie algebra structure specified by giOgj = C gk, where 0 is the Lie product and Cfj are the structure constants (we are assuming the rule of sum over repeated indeces). Now we take g first realized by C = Ai,i = 1,2,3,.. such that the commutator [Ai,Aj is the Lie product of elements Ai,Aj G C. Consider tp and (p two representations of C, such that ip (A) (linear operators defined on a representation vector space As a consequence,... 

T 1— The set of Hopf algebra structures on O such that the associated... 

Theorem. Affine group schemes over k correspond to Hopf algebras over k. 

Thinking of the usual axioms for groups, we can see that A is the most important part of a Hopf algebra structure on an algebra A. For suppose we have a representable functor G and a map A A A <8> A giving a composition law on the G(li). If they happen to be groups, the unit and inverses are... 

Prove the following Hopf algebra facts by interpreting them as statements about group functors ... 

A homomorphism of affine group schemes is a natural map G -+ H for which each G(R) - H(R) is a homomorphism. We have already seen the example det GL - Gm. The Yoneda lemma shows as expected that such maps correspond to Hopf algebra homomorphisms. But since any map between groups preserving multiplication also preserves units and inverses, we need to check only that A is preserved. An algebra homomorphism between Hopf algebras which preserves A must automatically preserve S and e. 

The name diagonalizable will be justified in (4.6). But we can already distinguish these groups Hopf-algebraically over fields. We first need the following result, which in group language states the independence of characters. 

Lemma. If A is a Hopf algebra over a field k, the group-like elements in A are linearly independent. 

Proof. The last sentence is obvious. To show that A° is indeed a co-commutative Hopf algebra is nothing but a collection of verifications, of which we give samples done by different methods. 

Looking back to the previous section, we find the duals of the finite constant groups are precisely the finite diagonalizable groups the dual algebra of kr is kpT]. In general this would not be one of our Hopf algebras, since it is not commutative. But when T is commutative we can write it as a product of various Z/nZ and compute that the dual of Z/nZ is ft,. 

Let / be an ideal in a Hopf algebra A. Work out the conditions necessary for A/l to represent a closed subgroup which is normal. 

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