UC Santa Barbara

In this paper, we give an overview of the classical Temperley-Lieb algebra, reviewing some of the basic results about the algebra. Then we will discuss the construction of the generalized Temperley-Lieb algebras from Coxeter groups, and the connection between fully commutative elements of the Coxeter group and the reduced elements of the corresponding Temperley-Lieb algebra. Finally, we will look at the diagrammatic representations of some finite-dimensional Temperley-Lieb algebras, specifically those of type B, D and E, and describe their bases.