Representation theory and an isomorphism theorem for the Framisation of the Temperley-Lieb algebra

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.