Shenglin Zhu

Frobenius and separable functors for generalized module categories and nonlinear equations

Part I: Entwined modules and Doi-Koppinen Hopf modules.- 1. Generalities.- 2. Doi-Koppinen Hopf modules and entwined modules.- 3. Frobenius and separable functors for entwined modules.- 4. Applications.- Part II: Nonlinear equations.- 5. Yetter-Drinfeld modules and the quantum Yang-Baxter equation.- 6. Hopf modules and the pentagon equation.- 7. Long dimodules and the Long equation.- 8. The Frobenius-Separability equation.- References.- Index.

The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R : H ⊗ L → L ⊗ H and W : L ⊗ H → H ⊗ L. We introduce a product K = LW ⋈RH and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dăscălescu and Grünenfelder appear as special cases.

Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties

We study the following question: when is the right adjoint of the forgetful functor from the category of (H,A,C)-Doi-Hopf modules to the category of A-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that C ⊗ A and the smash product A#C∗ are isomorphic as (A,A#C∗)-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case A = H, and this leads to the notion of k-FrobeniusH-module coalgebra. In the special case of Yetter-Drinfel′d modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if H is finite dimensional and unimodular. 0. Introduction Let H be a Hopf algebra with bijective antipode over a commutative ring k. Let A be a (left) H-comodule algebra, and C a (right) H-module coalgebra. Doi [6] introduced the notion of unified Hopf module (we will call such a module a DoiHopf module). This is a k-module, which is at once a right A-module and a left C-comodule, satisfying the compatibility relation (1.7). Several module structures appear as special cases; let us mention Sweedler’s Hopf modules [20], Takeuchi’s relative Hopf modules [21], graded modules, and modules graded by a G-set. It was proved recently [2] that Yetter-Drinfel′d modules are also a special case of Doi-Hopf modules. In [3], induction functors between categories of Doi-Hopf modules are discussed. It turns out that many pairs of adjoint functors are special cases, for example the functor forgetting action or coaction, extension and restriction of scalars and coscalars. In this paper, we focus attention on the functor F from the category of Doi-Hopf modules to the category of right A-modules forgetting the C-coaction. This functor has a right adjoint G = C ⊗ •. A natural question that arises is the following: when is G also a left adjoint of F? In fact, we can view this problem as a Frobenius type problem. It can be shown (cf. [13, Theorem 3.15]) that a ring extension R → S is Frobenius (of the first type) if and only if the induction functor (which is a left adjoint of the restriction Received by the editors May 9, 1995. 1991 Mathematics Subject Classification. Primary 16W30.

Determinants, integrality and Noether’s theorem for quantum commutative algebras

The aim of this paper is to generalize Noether’s theorem for finite groups acting on commutative algebras, to finite-dimensional triangular Hopf algebras acting on quantum commutative algebras. In the process we construct a non-commutative determinant function which yields an analogue of the Cayley-Hamilton theorem for certain endomorphisms.

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 invariants of H under the adjoint coaction, and prove that it is closely related 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 if H is unimodular then C(H)=O(H) is equivalent to assuming that the antipode is an involution. We prove that if H is a semisimple Hopf algebra over an algebraically closed field then O(H∗) is a symmetric Frobenius algebra containing the left integral of H∗. This enables us to prove that if H is also cosemisimple then C(H∗),C(H) are all separable algebras. It has been recently shown by Etingof and Gelaki (On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, preprint) that in this situation S2=id and hence O(H)=C(H). In characteristic 0 semisimple Hopf algebras are cosemisimple and O(H∗) and C(H∗) coincide (and equal the so-called “character ring”). In positive characteristic O(H)≠C(H) in some cases, and O(H) may be a more natural object. For example, quasitriangular 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 then H is factorizable (a notion connected to computing invariants of 3-manifolds). We prove that if (H,R) is factorizable and semisimple then it is cosemisimple and so C(H∗) and C(H) are separable algebras. We apply these results to the associated Yetter–Drinfeld category.

Finite dimensional Nichols algebras over certain p-groups

ABSTRACT We investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p3, p4 and p5, and list all the pointed Hopf algebras of order p6, whose group of grouplikes is non-abelian.

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End A′(V ⊗ H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.

Faithfully Flat Hopf Bi-Galois Extensions

This article is devoted to faithfully flat Hopf bi-Galois extensions defined by Fischman, Montgomery, and Schneider. Let H be a Hopf algebra with bijective antipode. Given a faithfully flat right H-Galois extension A/R and a right H-comodule subalgebra C ⊂ A such that A is faithfully flat over C, we provide necessary and sufficient conditions for the existence of a Hopf algebra W so that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra. As a consequence, we prove that if R = k, there is a Hopf algebra W such that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra if and only if C is an H-submodule of A with respect to the Miyashita–Ulbrich action.