Multiparameter quantum groups, bosonizations and cocycle deformations

TL;DR

This paper explores multiparameter quantum groups as pointed Hopf algebras, demonstrating they depend on fewer parameters under certain conditions, and relates this to known cocycle deformation results.

Contribution

It shows that multiparameter quantum groups can be simplified to depend on a single parameter per Dynkin diagram component, up to cocycle deformation.

Findings

Dependence on one parameter per Dynkin component under certain assumptions

Connection to known cocycle deformation results

Representation of quantum groups as pointed Hopf algebras

Abstract

We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We show that under some assumptions, these Hopf algebras depend only on one parameter on each connected component of the Dynkin diagram, up to a cocycle deformation. In particular, we obtain in this way a known result of Hu, Pei and Rosso.