G. Militaru

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.

Crossed modules and doi-hopf modules

We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product.

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.

Classifying Bicrossed Products of Hopf Algebras

Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object

FROBENIUS FUNCTORS OF THE SECOND KIND

{\mathcal H}^{2} (A, H)

Extending structures for Lie algebras

. Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A ⋈ H associated to all possible matched pairs of Hopf algebras

Extending Structures I: The Level of Groups

(A, H, \triangleleft, \triangleright)

The Structure of Frobenius Algebras and Separable Algebras

that can be defined between A and H. In the construction of

Unified products for Leibniz algebras. Applications

{\mathcal H}^{2} (A, H)