Tube representations and twisting of graded categories

TL;DR

This paper investigates how the tube algebra of a graded monoidal category can be deformed using group 3-cocycles, revealing a twisting structure that relates the original and deformed algebras.

Contribution

It introduces a method to deform the tube algebra via 3-cocycles, connecting the deformation to a 2-cocycle twist of the original algebra.

Findings

Tube algebra deformation corresponds to a 2-cocycle twist.

The deformation is governed by a torus-valued 3-cocycle.

The structure uses a Fell bundle over the action groupoid.

Abstract

We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $\Gamma$, a torus-valued 3-cocycle on $\Gamma$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $\Gamma$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.