Small quantum groups associated to Belavin-Drinfeld triples

Abstract:

For a simple Lie algebra $\mathsf {g}$ of type $A$, $D$, $E$ we show that any Belavin-Drinfeld triple on the Dynkin diagram of $\mathsf {g}$ produces a collection of Drinfeld twists for Lusztig’s small quantum group $u_q(\mathsf {g})$. 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 $\mathsf {g}$ associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of $u_q(\mathsf {g})$ which express a known algebraic group action on its category of representations and pose a subsequent question regarding the classification of all twists.