The representation theory of affine Temperley-Lieb algebras. (English) Zbl 0964.20002

Graham, J. J.; Lehrer, G. I.

Enseign. Math., II. Sér. 44, No. 3-4, 173-218 (1998).

The authors introduce a category whose morphisms are certain diagrams. Using a result of Fan and Green, they show how the endomorphisms in this diagram category are extended versions of the affine analogue of the well-known Temperley-Lieb algebra. Functors on the category of diagrams are given which give rise to the so-called cell modules, or Weyl modules, which come equiped with bilinear forms. By analyzing these forms, the authors determine all the homomorphisms between the cell modules, which allows the decomposition matrices and the dimensions of all the irreducible modules to be determined. The paper also gives explicit formulae for the discriminants of these forms which lead to sharp criteria for semisimplicity and extensive results on the modular representation theory. The results of this paper also have applications to the representation theory of closely related algebras such as the affine Hecke algebra (at roots of unity), the ordinary Temperley-Lieb algebra and Jones’ annular algebra.

Reviewer: Richard M.Green (Lancaster)

MSC:

20C08	Hecke algebras and their representations
16G30	Representations of orders, lattices, algebras over commutative rings
20C20	Modular representations and characters

Keywords:

Temperley-Lieb algebras; categories of diagrams; cell modules; Weyl modules; bilinear forms; decomposition matrices; irreducible modules; semisimplicity; modular representations; affine Hecke algebras; Jones’ annular algebras