Universal deformation formulas and braided module algebras

TL;DR

This paper investigates formal deformations of crossed product algebras using universal deformation formulas, demonstrating their nontriviality through Hochschild cohomology computations in a characteristic-free setting.

Contribution

It introduces a new complex for Hochschild cohomology of crossed products and proves the nontriviality of deformations from universal formulas in characteristic-free contexts.

Findings

Deformations are nontrivial even for infinite groups.

Constructed a new complex for Hochschild cohomology.

Showed infinitesimals are not coboundaries.

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 nontrivial 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 $S(V)#_f G$.