# On the Representation theory of the Infinite Temperley-Lieb algebra

## TL;DR

This paper explores the representation theory of the infinite Temperley-Lieb algebra, classifying finite and infinite-dimensional representations, and proposing new constructions and conjectures related to its structure.

## Contribution

It provides a complete classification of finite-dimensional representations and introduces new infinite link state representations and their properties.

## Key findings

- Classified all finite-dimensional representations.
- Introduced infinite link state representations and analyzed their irreducibility.
- Proposed a generalization of Schur-Weyl duality.

## Abstract

We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for $TL_{n}$ that generalizes to give extensions of $TL_{\infty}$ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.