Modular Categories Associated to Unipotent Groups

TL;DR

This paper constructs modular categories from unipotent algebraic groups over fields of positive characteristic and characterizes them as exactly those in a specific class C_p^{\u2212} or C_p^{+}.

Contribution

It establishes a precise correspondence between modular categories derived from unipotent groups and the class C_p^{\u2212} or C_p^{+}, providing a classification framework.

Findings

Modular categories from unipotent groups are exactly those in class C_p^{b} or C_p^{+}.

The construction uses minimal idempotents in the equivariant derived category.

The classification links algebraic group theory with modular tensor categories.

Abstract

Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under conjugation action) \bar{Q_l}-complexes on G. We can associate to G and e a modular category M_{G,e}. In this article, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class C_p^{\pm}.