Limits of traces of Temperley-Lieb algebras

TL;DR

This paper classifies positive extremal traces on the infinite Temperley-Lieb algebra, extending results to roots of unity, and constructs associated Hilbert space structures, advancing understanding of algebra representations at roots of unity.

Contribution

It extends the classification of extremal traces to non-semisimple cases at roots of unity and constructs Hilbert space structures for these algebras.

Findings

Classification of positive extremal traces at roots of unity

Construction of Hilbert space structures for the algebra

Extension of previous semisimple case results

Abstract

We review the classification of positive extremal traces on the generic infinite Temperley-Lieb algebra, and then extend the classification to the non-semisimple root of unity case. As a result, we obtain Hilbert space structures on the full infinite Temperley-Lieb algebra at roots of unity.