Standard Modules, Induction and the Temperley-Lieb Algebra

TL;DR

This paper reviews the properties of the Temperley-Lieb algebra, focusing on standard modules, their reducibility at roots of unity, and the structure of indecomposable modules, with applications to physical models.

Contribution

It provides a detailed analysis of the structure of standard modules and their radicals, especially at roots of unity, and explicitly computes principal indecomposable modules.

Findings

Standard modules are reducible when q is a root of unity.

The radicals of standard modules are irreducible.

The space of homomorphisms between standard modules is fully characterized.

Abstract

The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $\beta = q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the structure of the standard modules. Finally, a particular central element $F_n$ of $TL_n$ is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case $\beta = 0$, that plays an important role in physical models, is studied systematically.