Quantum symmetry algebras of spin systems related to Temperley-Lieb R-matrices

TL;DR

This paper constructs a quantum symmetry algebra related to Temperley-Lieb R-matrices acting on spin systems, revealing its structure as a quasi-triangular Hopf algebra with implications for open spin chain symmetries.

Contribution

It introduces a reducible representation of the Temperley-Lieb algebra and identifies its centraliser as a quantum algebra with a representation ring akin to sl_2.

Findings

Constructed a reducible Temperley-Lieb algebra representation.

Identified the centraliser as a quantum algebra U_q.

Linked the algebra's representation ring to sl_2.

Abstract

A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation ring equivalent to the representation ring of the sl_2 Lie algebra. This algebra U_q is the symmetry algebra of the corresponding open spin chain.