The principal indecomposable modules of the dilute Temperley-Lieb algebra

Abstract

The Temperley-Lieb algebra \tln(β) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the concatenation of diagrams. The dilute Temperley-Lieb algebra \dtl n(β) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \tl n, the dilute \dtl n depends on a parameter β\in\mathbb C, often given as β=q+q^{-1} for some q\in\mathbb C^\times. In statistical physics, the algebra plays a central role in the study of dilute loop models. The paper is devoted to the construction of its principal indecomposable modules. Basic definitions and properties are first given: the dimension of \dtl n, its break up into even and odd subalgebras and its filtration through n+1 ideals. The standard modules \U{n,k} are then introduced and their behaviour under restriction and induction is described. A bilinear form, the Gram product, is used to identify their (unique) maximal submodule \dr{n,k} which is then shown to be irreducible or trivial. It is then noted that \dtl n is a cellular algebra. This fact allows for the identification of complete sets of non-isomorphic irreducible modules and projective indecomposable ones. The structure of \dtl n as a left module over itself is then given for all values of the parameter q, that is, for both q generic and a root of unity.

45 pages, rewrote section 5 to use cellular algebras

References in corpus (3)

• Lattice fusion rules and logarithmic operator product expansions
• The Jordan Structure of Two Dimensional Loop Models
• On Representations of Affine Temperley--Lieb Algebras

Cited by in corpus (11)

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• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• Critical loop models are exactly solvable
• Fusion rules for the Temperley-Lieb algebra and its dilute generalisation
• Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models
• The representation theory of seam algebras
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Fusion and monodromy in the Temperley-Lieb category
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip