Invariants of the adjoint coaction and Yetter-Drinfeld categories

Let H be a Hopf algebra over a field k. We study O(H), the subalgebra of in variants of H under the adjoint coaction, and prove that it is closely rela ted to questions about the antipode and the integral. It may differ from C( H), the subalgebra of cocommutative elements of H. In fact, we prove that i f H is unimodular then C(H)= O(H) is equivalent to assuming that the antipo de is an involution. We prove that if H is a semisimple Hopf algebra over a n algebraically closed field then O(H*) is a symmetric Frobenius algebra co ntaining the left integral of H*. This enables us to prove that if H is als o cosemisimple then C(H*), C(H) are all separable algebras. It has been rec ently shown by Etingof and Gelaki (On finite-dimensional semisimple and cos emisimple Hopf algebras in positive characteristic, preprint) that in this situation S-2 = id and hence O(H)= C(H). In characteristic 0 semisimple Hop f algebras are cosemisimple and O(HC) and C(H*) coincide land equal the so- called "character ring''). In positive characteristic O(H) not equal C(H) i n some cases, and O(H) may be a more natural object. For example, quasitria ngular Hopf algebras are endowed with an algebra homomorphism between O(H*) and the center of H. We show that if this homomorphism is a monomorphism t hen H is factorizable (a notion connected to computing invariants of 3-mani folds). We prove that if (H,R) is factorizable and semisimple then it is co semisimple and so C(H*) and C(H) are separable algebras. We apply these res ults to the associated Yetter-Drinfeld category. (C) 2001 Elsevier Science B.V. All rights reserved.