Representations of Graph Temperley-Lieb Algebras

Abstract

We define graph Temperley-Lieb algebras, which are certain generalisations of Temperley-Lieb algebras including those appropriate for building the transfer matrices for lattice models in arbitrary dimensions. We construct various representations for these algebras, which are parameterised by a scalar Q. In particular we give the generically irreducible representations associated with statistical mechanical models. These representations are well defined for all Q, whereas the representations coming directly from physical models are typically only defined for certain values of Q. We also give representations with bases derived from the partitions of n distinguishable objects (n is the number of nodes in the graph). We show how to compute the dimensions of these epresentations by a diagrammatic technique.