Deformations of Quantum Symmetric Algebras Extended by Groups

TL;DR

This paper introduces a novel deformation of quantum symmetric algebras extended by groups, utilizing a Hopf algebra action, resulting in nontrivial deformations with new relations involving original vector space elements.

Contribution

It presents a new method to deform smash product algebras using a specific Hopf algebra, including the first example with relations involving original vector space elements.

Findings

Constructed a deformation of smash product algebra with Hopf algebra action.

Identified the first known deformation involving original vector space elements.

Used Hochschild cohomology to prove the deformation's nontriviality.

Abstract

We discuss a general construction of a deformation of a smash product algebra coming from an action of a particular Hopf algebra. This Hopf algebra is generated by skew-primitive and group-like elements, and depends on a complex parameter. The smash product algebra is defined on the quantum symmetric algebra of a finite-dimensional vector space and a group. In particular, an application of this result has enabled us to find a deformation of such a smash product algebra which is, to the best of our knowledge, the first known example of a deformation in which the new relations in the deformed algebra involve elements of the original vector space. Using Hochschild cohomology, we show that the resulting deformations are nontrivial by giving the precise characterization of the infinitesimal.