| Résume | Following Drinfeld, one can think about Drinfeld-Jimbo's quantum universal enveloping algebras of a simple Lie algebra $\mathfrak{g}$ in more geometrical terms: these Hopf algebras are twist equivalent to the classical universal enveloping algebras with a non-trivial $\mathfrak{g}$-invariant coassociator defined via monodromies of solutions to the Knizhnik-Zamolodchikov equation on 3 points. In other words, we deal with the representation category of $U \mathfrak{g}$ equipped with a non-trivial associator, called Drinfeld's associator. I am interested in a similar deformation problem for Lusztig's small quantum groups at roots of unity, and more generally, in deformations of associators in tensor categories. As it is often in algebra, infinitesimal deformations are controlled by Hochschild type complexes, called in this case Davydov-Yetter complex. We have recently reformulated the corresponding deformation cohomologies in terms of relative Ext groups of the Drinfeld center. In this talk, I will show how to use the relative homological algebra in a rather explicit study of deformations of tensor categories in Hopf algebra theory, in particular for Taft algebras and Lusztig's small quantum groups for sl(2). In the latter case we discover new non-trivial deformations (joint with M. Faitg and Ch. Schweigert). If time allows I will also discuss a relation of these new deformations to the classification of graded extensions of tensor categories. |
