PT-invariance and representations of the Temperley-Lieb algebra on the unit circle

TL;DR

This paper explores PT-invariance in self-adjoint representations of the Temperley-Lieb algebra on the unit circle, emphasizing graphical calculus and potential generalizations.

Contribution

It introduces a conjecture on PT-invariant representations and discusses their graphical calculus formulation, highlighting possible extensions.

Findings

PT-invariance may be crucial for generalizing representations

Graphical calculus effectively illustrates algebraic structures

Specific examples demonstrate the conjecture's applicability

Abstract

We present in detail a recent conjecture on self-adjoint representations of the Temperley-Lieb algebra for particular values on the unit circle. The formulation in terms of graphical calculus is emphasized and discussed for several examples. The role of PT (parity and time reversal) invariance is highlighted as it might prove important for generalising the construction to other cases.