  In this paper, we describe the irreducible representations and give a
dimension formula for the Framisation of the Temperley-Lieb algebra. We then
prove that the Framisation of the Temperley-Lieb algebra is isomorphic to a
direct sum of matrix algebras over tensor products of classical Temperley-Lieb
algebras. This allows us to construct a basis for it. We also study in a
similar way the Complex Reflection Temperley-Lieb algebra.
