The representation theory of cyclotomic Temperley-Lieb algebras

Abstract

Abstract.A class of associative algebras called cyclotomic Temperley-Lieb algebras is introduced in terms of generators and relations. They are closely related to the group algebras of complex reflection groups on the one hand and generalizations of the usual Temperley-Lieb algebras on the other hand. It is shown that the cyclotomic Temperley-Lieb algebras can be defined by means of labelled Temperley-Lieb diagrams and are cellular in the sense of Graham and Lehrer. One thus obtains not only a description of the irreducible representations, but also a criterion for their quasi-heredity in the sense of Cline, Parshall and Scott. The branching rule for cell modules and the determinants of Gram matrices for certain cell modules are calculated, they can be expressed in terms of generalized Tchebychev polynomials, which therefore play an important role for semisimplicity.

Markov traces on cyclotomic Temperley–Lieb algebras

“…In this section, we recall the notion of a cyclotomic Temperley-Lieb algebra in [6] and state the main result in Theorem 2.9.…”

“…Suppose x 1 , x 2 , · · · , x n are indeterminates over R. Let A n = (a ij ) be the n-by-n matrix with a ii = x i , a ij = 1 for |i − j| = 1 and a ij = 0 in the remaining case. Following [6], we call…”

“…In [6], Xi and the author introduced a class of finite-dimensional algebras called cyclotomic Temperley-Lieb algebras of type G(m, 1, n). When m = 1, it turns out to be the usual Temperley-Lieb algebra.…”

“…When m = 1, it turns out to be the usual Temperley-Lieb algebra. Using generalized Tchebychev polynomials in [6], Xi, Yu and the author gave a criterion for the semisimplicity of a cyclotomic Temperley-Lieb algebra in [7]. When m = 1, such a result can be found in [2].…”

“…In section 2, we collect some of results on cyclotomic Temperley-Lieb algebras in [6] and [7]. We also state Theorem 2.9, the main result of this note.…”