Factorizable $R$-Matrices for Small Quantum Groups

TL;DR

This paper classifies factorizable R-matrices for small quantum groups at roots of unity, producing new and known modular tensor categories, especially in cases with common divisors between the order of q and root lengths.

Contribution

It determines all solutions to Lusztig's ansatz that yield non-degenerate braidings for small quantum groups, including cases with common divisors, and analyzes transparent objects in degenerate cases.

Findings

Classified all factorizable R-matrices for small quantum groups.

Constructed new series of non-semisimple modular tensor categories.

Identified transparent objects in degenerate cases.

Abstract

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig to endow these categories with the structure of a braided tensor category. In this article we determine all solutions to this ansatz that lead to a non-degenerate braiding. Particularly interesting are cases where the order of $q$ has common divisors with root lengths. In this way we produce familiar and unfamiliar series of (non-semisimple) modular tensor categories. In the degenerate cases we determine the group of so-called transparent objects for further use.