The Temperley-Lieb algebras have been used in a wide variety of applications, from topological quantum field theory to logic and computation. In this paper, we show the equivalence of various definitions of the\ud Temperley-Lieb algebras, and compute some of their properties. We then\ud include a detailed discussion of the Jones basic construction, with as many proofs as possible. The results presented in this discussion are finally used to derive the irreducible representations of the Temperley-Lieb algebras
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