Slightly Degenerate Categories and ℤ-Modular Data

Abstract

Given a slightly degenerate braided pivotal fusion category $\mathscr{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories. Finally, we give an interpretation in this framework of the Bonnafé–Rouquier categorification of the $\mathbb{Z}$-modular datum associated to nontrivial family of the cyclic complex reflection group.

A Soergel-like category for complex reflection groups of rank one

“…One can cite for instance the 'Spetses' program [4], which provides a sort of generalization of unipotent characters of reductive groups to non-existing reductive groups attached to complex reflection groups. In this framework, some categorification results were obtained for cyclic groups in [1] (building up on [6,7]) and later extended in [14,15]. Note that the ring considered in [1,Theorem 5.5] is related to the ring A W studied in the present paper, as observed in Remark 5.4.…”

An asymptotic cell category for cyclic groups

“…This can be understood in terms of super-categories, as explained recently by Lacabanne [La1]. We have…”

“…The root of unity −ζ appearing in this formula has been interpreted in terms of super-categories by Lacabanne [La1]: it is due to the fact that our category is not spherical. Finally, note that (6.7)…”

“…The constructions of this paper have recently been extended by Lacabanne [La1], [La2] to some families of the complex reflection groups G (d , 1, n). His construction involves super-categories instead of triangulated categories, as suggested by Etingof.…”

Fourier matrices for $G(d,1,n)$ from quantum general linear groups

“…the Drinfeld double of the Taft algebra, which is a finite dimensional version of the quantum enveloping algebra of the standard Borel of sl 2 . In [14], the author explained how to reinterpret the category of Bonnafé and Rouquier into the framework of slightly degenerate categories. This framework turned out to be well adapted for the problem of categorifying modular data: the modular data of some families of the complex reflection group G(d, 1, n) arise from the representation of the Drinfeld double of the quantum enveloping algebra of the standard Borel of sl m [15].…”