Pointed Hopf Algebras with Triangular Decomposition -- A Characterization of Multiparameter Quantum Groups

TL;DR

This paper characterizes a class of pointed Hopf algebras with triangular decomposition, generalizing multiparameter quantum groups and introducing asymmetric braided Drinfeld doubles, linking them to Lie algebra structures.

Contribution

It introduces a new framework for classifying multiparameter quantum groups via weakly separable Hopf algebras and asymmetric braided Drinfeld doubles, extending prior results.

Findings

Classified weakly separable Hopf algebras of triangular type.

Established connection between these Hopf algebras and multiparameter quantum groups.

Revealed Lie algebra structures generated by sl2 subalgebras.

Abstract

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2.