The nil Temperley--Lieb algebra of type affine C

TL;DR

This paper introduces a new affine type C nil Temperley--Lieb algebra, explores its structure, representations, and connections to statistical physics models, providing a detailed classification of certain finite-dimensional modules.

Contribution

It defines a novel affine type C nil Temperley--Lieb algebra, describes its faithful particle-based representation, and classifies finite-dimensional indecomposable modules where a key element is invertible.

Findings

The algebra $T(n)$ is a quotient of a Kac--Moody algebra's positive part.

It has a faithful particle configuration representation similar to TASEP.

The algebra's center is generated by a polynomial in element $Q$, and localizing at $Q$ yields a matrix ring.

Abstract

We introduce a type affine $C$ analogue of the nil Temperley--Lieb algebra, in terms of generators and relations. We show that this algebra $T(n)$, which is a quotient of the positive part of a Kac--Moody algebra of type $D_{n+1}^{(2)}$, has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of $T(n)$ consists of polynomials in a certain element $Q$, and $T(n)$ is a free module of finite rank over its centre. We show how to localize $T(n)$ by adjoining an inverse of $Q$, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although $T(n)$ has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of $T(n)$ in which $Q$ acts invertibly.