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Certain quantization problems are equivalent to the construction of morphisms from "quantum" to "classical" props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof…
We apply categorical machinery to the problem of defining anti-Yetter-Drinfeld modules for quasi-Hopf algebras. While a definition of Yetter-Drinfeld modules in this setting, extracted from their categorical interpretation as the center of…
We investigate the relation between Connes-Kreimer Hopf algebra approach to renomalization and deformation quantization. Both approaches rely on factorization, the correspondence being established at the level of Wiener-Hopf algebras, and…
A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classical r-matrix satisfying the modified Yang-Baxter…
In this paper we outline an approach to calculus over quasitriangular Hopf algebras. We study differential operators in the framework of monoidal categories equipped with a braiding or symmetry. To be more concrete, we choose as an example…
We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid…
A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to…
If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category…
We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra $H_0$ in the category of…
Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…
Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive…
In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with…
We study a functor from anti-Yetter Drinfeld modules to contramodules in the case of a Hopf algebra $H$. Some byproducts of this investigation are the establishment of sufficient conditions for this functor to be an equivalence,…
For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel'd module category ${}_{A}\mathcal{YD}^{A}$ is equivalent to the centre $Z({}_{A}\mathcal{M})$ of the unital left $A$-module category ${}_{A}\mathcal{M}$. Then we introduce the…
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular,…
We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules…
We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra $H$ are isomorphic. We prove also that the category $\yd^{\rm fd}$ of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute…
We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one to one correspondence between anti-Yetter-Drinfeld modules, which serve as coefficients for the Hopf…
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an…
For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an…