Quantum $G_{2}$ categories have property (T)

TL;DR

This paper proves that the categories of finite-dimensional unitary representations of quantum G2 groups exhibit property (T), indicating strong rigidity properties in their structure.

Contribution

It establishes that quantum G2 categories for positive q have property (T), a significant rigidity property not previously confirmed for these quantum groups.

Findings

Quantum G2 categories have property (T) for positive q.

The result applies to finite-dimensional unitary representations.

It extends understanding of rigidity in quantum group categories.

Abstract

We show that the rigid C*-tensor categories of finite dimensional type 1 unitary representations of the quantum groups $U_{q}(\mathfrak{g}_{2})$ corresponding to the exceptional Lie group $G_2$ for positive $q\ne 1$ have property (T).