Modular data of non-semisimple modular categories

Abstract

We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification, and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen-Westreich modular data, which is obtained from the Lyubashenko-Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen-Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_{q}sl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{SL}(2,\mathbb{Z})$-representation on their centers, Cohen-Westreich modular data, and the congruence kernel theorem's validity.

50 pages. Minor changes to fix typos. Updated Conjecture 5.3.4 to include an explicit level