TY - JOUR

T1 - Factorizable R-matrices for small quantum groups

AU - Ohrmann, Tobias

AU - Lentner, Simon

N1 - Funding information: Both authors thank Christoph Schweigert for helpful discussions and support. They also thank the referees, who gave a relevant contribution to improve the article with their comments. The first author was supported by the DAAD P.R.I.M.E program funded by the German BMBF and the EU Marie Curie Actions as well as the Graduiertenkolleg RTG 1670 at the University of Hamburg. The second author was supported by the Collaborative Research Center SFB 676 at the University of Hamburg.

PY - 2017

Y1 - 2017

N2 - Representations of small quantum groups uq(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.

AB - Representations of small quantum groups uq(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.

KW - quantum algebra

KW - Transparent object

KW - Quantum group

KW - Factorizable

KW - Modular tensor category

KW - R-matrix

UR - http://www.scopus.com/inward/record.url?scp=85030223011&partnerID=8YFLogxK

U2 - https://doi.org/10.3842/SIGMA.2017.076

DO - https://doi.org/10.3842/SIGMA.2017.076

M3 - Article

VL - 13

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

IS - 076

M1 - 076

ER -