Spin chains as modules over the affine Temperley-Lieb algebra

TL;DR

This paper explicitly describes the structure of representations of the affine Temperley-Lieb algebra acting on spin chains, revealing their composition factors and differences from cellular modules, with implications for understanding algebraic actions in quantum spin systems.

Contribution

It provides a detailed analysis of the structure of spin chain representations over the affine Temperley-Lieb algebra, including new maps and morphisms that clarify their composition factors and differences from cellular modules.

Findings

Representation structures are explicitly characterized.

Differences in Loewy diagram arrow directions are identified.

New intertwining maps between algebra actions are introduced.

Abstract

The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$ is an infinite-dimensional algebra parametrized by a number $\beta \in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on $(\mathbb{C}^2)^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in\mathbb C^\times$. The structure of these representations, which were first introduced by Pasquier and Saleur in their study of spin chains, is here made explicit. They share their composition factors with the cellular $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-modules of Graham and Lehrer, but differ from the latter representations by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin as well as new maps that intertwine various $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$-actions on the XXZ chain and generalize applications studied by Deguchi $\textit{et al}$ and after by Morin-Duchesne and Saint-Aubin.