Percolation, conformai invariance, critical phenomena, conformai quantum field theory. A first version of part of the material of this paper was presented by the first author as part of the AMS Colloquium lectures in Baltimore in January 1992. The third author was supported in part by NSERC Canada and the Fonds FCAR pour l'aide et le soutien à la recherche (Québec).

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The basic properties of the Temperley-Lieb algebra TL n with parameter β = q + q −1 , q ∈ C \ {0}, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterise their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and TL n is nonsemisimple. This happens only when q is a root of unity. Use of restriction and induction allows for a finer description of the structure of the standard modules. Finally, a particular central element F n ∈ TL n is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case β = 0, that plays an important role in physical models, is studied systematically.the Andrews-Baxter-Forrester models [4]. That Temperley-Lieb algebras play a fundamental role in our modern understanding of phase transitions cannot be overstated. These algebras were subsequently rediscovered by Jones [5] who used them to define what is now known as the Jones polynomial in knot theory. The Temperley-Lieb algebras are also intimately connected with the representation theory of the symmetric groups through their realisation as natural quotients of the (type A) Hecke algebras (see [6] for example).As is usual in physical applications, it is the representation theory of the Temperley-Lieb algebra TL n which is the main focus of attention. Indeed, Temperley and Lieb's original contribution takes place in a 2 n -dimensional representation commonly used by physicists for the study of spin chains. Such representations still form an active direction of research in mathematical physics. There are, in addition, somewhat smaller representations that are perhaps more natural to consider including, in particular, those which we shall refer to in what follows as link representations. From these, one obtains quotients that have come to be described as being standard. It is well-known that these standard representations are irreducible for almost all values of the parameter β and that for such β , the (finite-dimensional) representations of TL n are completely reducible. However, it is a curious fact that the β which are thereby excluded consist, to a large degree, of the parameter values that are of most interest to physicists.This failure of complete reducibility is also well-known and there have been significant efforts by physicists and mathematicians to understand the representation theory for these exceptional values of β . There is a continuing interest in this quest due to the current reinvigoration of the study of logarithmic conformal field theories. To ...

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