Irreducible Pythagorean representations of R. Thompson's groups and of the Cuntz algebra

TL;DR

This paper introduces the Pythagorean dimension to classify certain representations of the Cuntz algebra and Thompson's groups, revealing a rich geometric structure and extending previous results in the field.

Contribution

It provides a complete, functorial classification of Pythagorean representations using finite-dimensional linear algebra and describes their moduli space as a real manifold.

Findings

Classified all Pythagorean representations for each natural number d

Identified the moduli space as a real manifold of dimension 2d^2+1

Connected the new framework to existing literature and extended previous results

Abstract

We introduce the Pythagorean dimension: a natural number (or infinity) for all representations of the Cuntz algebra and certain unitary representations of the Richard Thompson groups called Pythagorean. For each natural number d we completely classify (in a functorial manner) all such representations using finite dimensional linear algebra. Their irreducible classes form a real manifold of dimension $2d^2+1$ playing the role of a moduli space. Apart from a finite disjoint union of circles, each point of the manifold corresponds to an irreducible unitary representation of Thompson's group F (which extends to the other Thompson groups and the Cuntz algebra) that is not monomial. The remaining circles provide monomial representations which we previously fully described and classified. We translate in our language a large number of previous results in the literature. We explain how our techniques extend them.