Local representations of the loop braid group

TL;DR

This paper explores how the loop braid group can be represented through extensions of braid group representations and investigates finite-dimensional quotient algebras as a generalization of classical algebraic paradigms.

Contribution

It introduces a framework for representing the loop braid group and proposes a generalization of the braid/Hecke/Temperley-Lieb algebra paradigm.

Findings

New representations of the loop braid group are constructed.

A generalization of classical algebraic structures to the loop braid context is proposed.

Potential applications in topological quantum computing are suggested.

Abstract

We study representations of the loop braid group $LB_n$ from the perspective of extending representations of the braid group $B_n$. We also pursue a generalization of the braid/Hecke/Temperlely-Lieb paradigm---uniform finite dimensional quotient algebras of the loop braid group algebras.