Small quantum groups associated to Belavin-Drinfeld triples

TL;DR

This paper constructs new finite-dimensional quantum groups from Belavin-Drinfeld triples on simple Lie algebras of types A, D, E, and analyzes their algebraic structures and representations.

Contribution

It introduces a method to generate new small quantum groups via Drinfeld twists derived from Belavin-Drinfeld triples, expanding the class of known quantum groups.

Findings

New finite-dimensional factorizable Hopf algebras constructed

Identification of grouplike elements and Drinfeld elements

Description of irreducible representations of the dual

Abstract

For a simple Lie algebra L of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of L produces a collection of Drinfeld twists for Lusztig's small quantum group u_q(L). These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e. new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in L associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of u_q(L) which express a known algebraic group action on its category of representations, and pose a subsequent question regarding the classification of all twists.