A tensor product for representations of the Cuntz algebra and of the R. Thompson groups

TL;DR

This paper introduces a tensor product for a broad class of representations of the Richard Thompson groups and the Cuntz algebra, establishing a tensor category structure and computing fusion rules.

Contribution

It defines a new tensor product for representations of Thompson groups and the Cuntz algebra, and demonstrates the tensor category structure with explicit fusion rule computations.

Findings

Established a tensor category structure for certain representations

Introduced a tensor product for representations of Thompson groups and the Cuntz algebra

Performed explicit calculations of fusion rules

Abstract

The authors continue a series of articles studying certain unitary representations of the Richard Thompson groups $F,T,V$ called Pythagorean. They all extend to the Cuntz algebra $\mathcal{O}$ and conversely all representations of $\mathcal{O}$ are of this form. Via this approach we introduce a tensor product for a large class of representations of $F,T,V,\mathcal{O}$. We prove that a sub-category forms a tensor category and perform a number of explicit computations of fusion rules.