Quantum Loop Modules and Quantum Spin Chains

TL;DR

This paper constructs quantum affine algebra modules, analyzes their irreducibility, develops crystal bases, and explores their application in diagonalizing quantum spin chains, revealing key differences from traditional modules.

Contribution

It introduces level-0 modules of quantum affine algebra as $q$-deformed loop modules, providing criteria for irreducibility and constructing crystal bases, with implications for quantum spin chain analysis.

Findings

Criteria for module irreducibility established

Crystal bases constructed for specific modules

Differences from highest weight modules identified

Abstract

We construct level-0 modules of the quantum affine algebra $\Uq$, as the $q$-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the crystal base for some of these modules and find significant differences from the case of highest weight modules. We also consider the role of loop modules in the recent scheme for diagonalising certain quantum spin chains using their $\Uq$ symmetry.