Finite-dimensional quantum groups of type Super A and non-semisimple modular categories

TL;DR

This paper constructs and classifies finite-dimensional quantum groups of type Super A at even roots of unity, demonstrating their role in forming non-semisimple modular categories and computing related knot invariants.

Contribution

It introduces a new family of quantum groups from Nichols algebras of type Super A, classifies their ribbon structures, and explores their module categories and link invariants.

Findings

Existence of ribbon structures depends on even rank and all odd simple roots.

Categories of modules form non-semisimple modular categories.

Link invariants distinguish knots beyond Jones and HOMFLYPT polynomials.

Abstract

We construct a series of finite-dimensional 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 non-semisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials and are related to a specialization of the Links-Gould invariant.