Classifying representations of the Thompson groups and the Cuntz algebra

Dilshan Wijesena – 13 March 2024

Recently Vaughan Jones developed a powerful tool to diagrammatically construct actions of groups permitting a calculus of fractions. One particular application leads to a rich family of infinite-dimensional representations of the celebrated Thompson group F and the Cuntz algebra O2. In this talk, we will reveal a surprisingly strong rigidity theorem between subcategories of Rep(F) and Rep(O2). We will also see how to reduce very difficult questions concerning irreducibility and equivalence of infinite-dimensional representations into problems in finite-dimensional linear algebra. This will allow to show that the irreducible classes of dimension d form a moduli space of real manifold dimension 2d2+1. Finally, we introduce the first known tensor product for a large class of representations of the Cuntz algebra. This tensor product has many properties reminiscent of the classical tensor product, and endows a tensor category structure on a subcategory of Rep(O2) which consists of infinite-dimensional representations.