### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 076, 25 pages      arXiv:1612.07960      https://doi.org/10.3842/SIGMA.2017.076

### Factorizable $R$-Matrices for Small Quantum Groups

Simon Lentner and Tobias Ohrmann
Fachbereich Mathematik, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Received January 16, 2017, in final form September 15, 2017; Published online September 25, 2017

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.

Key words: factorizable; $R$-matrix; quantum group; modular tensor category; transparent object.

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