The representation theory of the Temperley-Lieb algebras
type: Item
instance of: scholarly article
title: The representation theory of the Temperley-Lieb algebras
zbMATH Open document ID: 0840.16008
DOI: 10.1007/BF02572380
published in: Mathematische Zeitschrift
publication date: 1996-02-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174788
review text: It is known (Temperley, Lieb, 1971) that the single bond transfer matrices \(u_i\) for the ice model and for the Potts model both satisfy the relations defining the so-called Temperley-Lieb algebra: \(u^2_i=\delta u_i\), \(u_iu_{i\pm 1}u_i=u_i\), \(u_iu_j=u_ju_i\) (\(|i-j|>0\)) where \(\delta\) is a parameter that depends on the model. The main objective of the present paper is to determine all solutions of the Temperley-Lieb relations for matrices with complex entries. As a main result, there is given a finite list of such solutions with the property that any solution can be obtained from this list using two operations: direct sums (decomposable solutions) and change of basis (equivalent solutions). The paper is organized as follows: 1. Diagrams (Yetter, 1989). 2. Paranthesis diagrams (Jones, 1983). 3. Temperley-Lieb algebras (Kaufmann, 1987). 4. Representations. 5. Gram matrices (James, Murphy, 1979; Kazhdan, Lusztig, 1979; Garsia, McLarnan, 1988). 6. Quasi-hereditary algebras (Scott, 1987). 7. Kazhdan-Lusztig representations. 8. The block structure (Jones, 1983). 9. Nonsemisimple case. 10. Indecomposables and resolutions.
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 82B23
Mathematics Subject Classification ID: 17B81
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 16G99
zbMATH DE Number: 795321
zbMATH Keywords: decomposable solutions
zbMATH Keywords: equivalent solutions
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: bond transfer matrices
zbMATH Keywords: ice model
zbMATH Keywords: Potts model
zbMATH Keywords: relations
zbMATH Keywords: Temperley-Lieb algebras
zbMATH Keywords: Temperley-Lieb relations
zbMATH Keywords: matrices
zbMATH Keywords: Kazhdan-Lusztig representations
author: Bruce W. Westbury
reviewed by: A. A. Bogush
MaRDI profile type: Publication
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OpenAlex ID: W2315567303
description: scientific article; zbMATH DE number 795321