The nil Temperley–Lieb algebra of type affine C

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 Dn+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.