Universal deformation formulas and braided module algebras

Abstract

We study formal deformations of a crossed product S(V )# f G, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras generated by automorphisms and skew derivations. We show that they are non-trivial in the characteristic free context, even if G is infinite, by showing that their infinitesimals are not coboundaries. For this we construct a new complex which computes the Hochschild cohomology of S(V )# f G.

PBW deformations of Koszul algebras over a nonsemisimple ring

“…Theorem 4.6.1, for the notions see Sect. 4.6), generalizing the corresponding results in [7,20,21,24].…”

Deformations of quantum symmetric algebras extended by groups

“…The question that naturally arises is what about skew derivations. The answer is that sometimes skew derivations do lead to a deformation via a Hopf algebra action, as shown in [12] and [20] for the case of S(V )#G. The work presented here generalizes these results to the quantum version, that is to the case of S q (V )#G. This is particularly relevant since explicit examples of deformations of S q (V )#G have proven to be difficult to find. Moreover, it turns out that some of the deformations constructed in this work are not graded in the sense of Braverman and Gaitsgory [2] (see Remark 3.10).…”

“…Remark 4.7. This result was obtained for the case of S(V )#G in [12]. Thus, Theorem 4.5 is a generalization of [12, Theorem 3.5] to S q (V )#G with α being the identity map on S q (V )#G, s the twist map, χ α (g) = 1 and χ ς (g) = χ 1 (g −1 ) for all g ∈ G.…”

“…Let us present some generalizations of the results obtained in [12] and [20] to the case of S q (V )#G. We provide the necessary and sufficient conditions for S q (V )#G to have the structure of an H q -module algebra under some assumptions. As a consequence, by applying (2.3), we obtain more explicit examples of deformations.…”

Deforming group actions on Koszul algebras

“…We will look more closely at the Hochschild 2-cocycle condition (3.4) and the obstructions (3.5) and (3.6) in the case that A is a group twisted Koszul algebra S ⋊ G. A convenient resolution for this purpose was introduced by Guccione, Guccione, and Valqui [8]. We now recall from [13] a modified version of this construction.…”