The representation theory of affine Temperley-Lieb algebras
type: Item
instance of: scholarly article
title: The representation theory of affine Temperley-Lieb algebras
zbMATH Open document ID: 0964.20002
published in: L'Enseignement Mathématique. 2e Série
publication date: 2001-01-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1558732
zbMATH Keywords: Temperley-Lieb algebras
zbMATH Keywords: categories of diagrams
zbMATH Keywords: cell modules
zbMATH Keywords: Weyl modules
zbMATH Keywords: bilinear forms
zbMATH Keywords: decomposition matrices
zbMATH Keywords: irreducible modules
zbMATH Keywords: semisimplicity
zbMATH Keywords: modular representations
zbMATH Keywords: affine Hecke algebras
zbMATH Keywords: Jones' annular algebras
author: John J. Graham
author: Gus I. Lehrer
reviewed by: R. M. Green
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1558732