Search for a basis of the Temperley-Lieb algebra, using rewriting systems

TL;DR

This paper develops an algorithmic approach using rewriting theory and category theory to find bases for the Temperley-Lieb algebra and its oriented generalization, enhancing understanding and computational methods.

Contribution

It introduces a basis algorithm for the Temperley-Lieb algebra via rewriting systems and extends it to an oriented generalization using category theory.

Findings

Algorithm successfully computes bases for the algebra

Category theory simplifies basis derivation for the generalized algebra

Provides a new framework for algebraic basis computation

Abstract

We begin by defining Temperley-Lieb algebra, in two different ways: as a presented algebra or as a diagrammatic algebra. Next, we look for a basis algorithmically, using rewriting theory. Finally, we introduce a generalization of the Temperley-Lieb algebra, which is an oriented version of the previous one. This pushes us to employ a more efficient tool, category theory, to use rewriting to easily obtain a basis for the algebra.