Florin Panaite

Yetter-Drinfeld modules for Hom-bialgebras

The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category

General twisting of algebras

_H^H{\mathcal YD}

Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications

of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively coquasitriangular) the first (respectively second) quasi-braided pre-tensor category

Yetter-Drinfeld Categories for Quasi-Hopf Algebras

_H^H{\mathcal YD}

Some bialgebroids constructed by Kadison and Connes-Moscovici are isomorphic ∗

contains, as a quasi-braided pre-tensor subcategory, the category of modules (respectively comodules) with bijective structure maps over H.

DEFORMATION COHOMOLOGY FOR YETTER-DRINFEL'D MODULES AND HOPF (BI)MODUL ES

Abstract We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra ( A , μ , u ) in a monoidal category, as a morphism T : A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that ( A , μ ○ T , u ) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevichs braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.

A structure theorem for quasi-Hopf comodule algebras

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.

HOPF BIMODULES ARE MODULES OVER A DIAGONAL CROSSED PRODUCT ALGEBRA

ABSTRACT We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H 0, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. Communicated by M. Takeuchi.

L-R-Smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Equivalent Crossed Products and Cross Product Bialgebras

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present name is taken from [10]). If A is a finite dimensional Hopf algebra then the category of left-right Yetter-Drinfel’d modules is equivalent to the category of left modules over D(A), the Drinfel’d double of A (see [6], [9]), even as braided tensor categories, and also to the category of Hopf bimodules over A (see [1], [11], [12], [17]). An important class of examples occurs as follows: if M is a finite dimensional vector space and R ∈ End(M ⊗ M) is a solution to the quantum Yang-Baxter equation, then the so-called “FRT construction” (see for instance [2]) associates to R a certain bialgebra A(R), and M becomes a left-right Yetter-Drinfel’d module over A(R) (see [5], [9]). In this paper we introduce a cohomology theory for left-right Yetter-Drinfel’d modules. If A is a bialgebra and M,N are left-right Yetter-Drinfel’d modules over A, we construct a double complex {Y (M,N)} whose total cohomology is the desired cohomology H(M,N). For M = N = k, this cohomology