Image of the braid groups inside the finite Temperley-Lieb algebras

TL;DR

This paper characterizes the image of braid groups within finite Temperley-Lieb algebras over finite fields and establishes conditions for their representations to be unitary, advancing understanding of algebraic structures in quantum topology.

Contribution

It determines the braid group images in finite Temperley-Lieb algebras and proves unitarity of Hecke algebra representations under specific conditions.

Findings

Braid groups' images are explicitly characterized in finite Temperley-Lieb algebras.

Hecke algebra representations are shown to be unitary under certain parameter conditions.

Results apply to semisimple cases with large enough quantum parameters.

Abstract

We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras over a finite field are unitary for the action of the braid groups.