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

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