Restriction and induction of indecomposable modules over the Temperley-Lieb algebras

TL;DR

This paper classifies all indecomposable modules over Temperley-Lieb algebras and their dilute variants, detailing their structure, and explores how these modules behave under restriction and induction.

Contribution

It provides a complete classification of indecomposable modules over Temperley-Lieb algebras using homological and Auslander-Reiten theory methods.

Findings

Explicit descriptions of indecomposable modules including projective covers and injective hulls.

Complete classification of indecomposable modules up to isomorphism.

Two proofs of module classification: homological and Auslander-Reiten theory.

Abstract

Both the original Temperley-Lieb algebras $\mathsf{TL}_{n}$ and their dilute counterparts $\mathsf{dTL}_{n}$ form families of filtered algebras: $\mathsf{TL}_{n}\subset \mathsf{TL}_{n+1}$ and $\mathsf{dTL}_{n}\subset\mathsf{dTL}_{n+1}$, for all $n\geq 0$. For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over $\mathsf{TL}_{n}$ (or $\mathsf{dTL}_{n}$) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its socle and head, and its extension groups with irreducible modules. These data are also used to prove the completeness of the list of indecomposable modules, up to isomorphism. In fact, two completeness proofs are given, the first is based on elementary homological methods and the second uses Auslander-Reiten theory. The latter proof offers a detailed example of this algebraic tool that may be of independent interest.