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- W3098738110 abstract "The Temperley-Lieb algebra tln(beta) 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(beta) 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 betainmathbb C, often given as beta=q+q^{-1} for some qinmathbb 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." @default.
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- W3098738110 date "2014-11-01" @default.
- W3098738110 modified "2023-10-03" @default.
- W3098738110 title "The principal indecomposable modules of the dilute Temperley-Lieb algebra" @default.
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- W3098738110 doi "https://doi.org/10.1063/1.4901546" @default.
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