The two-boundary Temperley–Lieb algebra

2007/03/31 by Jan de Gier, Alexander Nichols · 4 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Tensor decomposition and applications #cond-mat.stat-mech #math-ph #math.MP #math.RT

paper · pdf · doi:10.1016/j.jalgebra.2008.10.023

published as Journal of Algebra 321 (2009), 1132-1167 · 45 pages Latex, 21 eps figures, revised version

arxiv created 2008/10/27 · openalex publication_date 2008/12/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algebra to show that all irreducible representations lie within a finite dimensional diagrammatic quotient. These representations are fully characterised by an additional parameter related to the action of the centre. For generic values of this parameter there is a unique representation of dimension 2^N and we show that it is isomorphic to a tensor space representation. We construct a basis in which the Gram matrix is diagonal and use this to discuss the irreducibility of this representation.

Citations

• Boundary conditions for integrable quantum systems
• A Guide to Stochastic Löwner Evolution and Its Applications

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