The center of the affine nilTemperley-Lieb algebra

TL;DR

This paper characterizes the center of the affine nilTemperley-Lieb algebra using a grading and fermionic representations, providing a basis and showing finite generation over its center, with an embedding result.

Contribution

It introduces a basis for the algebra and demonstrates its finite generation over the center, along with an embedding between algebras with different numbers of generators.

Findings

Explicit description of the algebra's center

Construction of a basis via a normal form

Embedding of N-generator algebra into (N+1)-generator algebra

Abstract

We give a description of the center of the affine nilTemperley-Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley-Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb algebra on N + 1 generators.