Unitarizablity of premodular categories

TL;DR

This paper investigates when premodular categories derived from quantum group representations at roots of unity can be made unitary, introducing Grothendieck unitarizability and providing new results for specific Lie types.

Contribution

It introduces the concept of Grothendieck unitarizability and extends unitarizability results to quantum groups of types F4, G2, B, and C.

Findings

New unitarizability results for F4 and G2 quantum groups

Improved unitarizability results for types B and C

Introduction of Grothendieck unitarizability as a generalization

Abstract

We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce \emph{Grothendieck unitarizability} as a natural generalization of unitarizability to any class of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types $F_4$ and $G_2$, and improve the known results for Lie types $B$ and $C$.