Finitedimensional quantum groups of type Super A and nonsemisimple modular categories
Abstract
We construct a series of finitedimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a nonsemisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finitedimensional modules over these quantum groups provide examples of nonsemisimple modular categories. In the ranktwo case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a fourdimensional simple module of the ranktwo quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials and are related to a specialization of the LinksGould invariant.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.10685
 arXiv:
 arXiv:2301.10685
 Bibcode:
 2023arXiv230110685L
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Category Theory;
 Mathematics  Representation Theory;
 18M15;
 18M20;
 17B37;
 57K14;
 57K16
 EPrint:
 61 pages. Comments welcome. New connections to the LinksGould invariant