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The principal indecomposable modules of the dilute Temperley-Lieb algebra

### Abstract

The Temperley-Lieb algebra
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
has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like
, the dilute
depends on a parameter
, often given as β = q + q
^{−1} for some
. 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
, its break up into even and odd subalgebras and its filtration through n + 1 ideals. The standard modules
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
which is then shown to be irreducible or trivial. It is then noted that
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
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.

© 2014 AIP Publishing LLC

Received 01 August 2014
Accepted 30 October 2014
Published online 25 November 2014

Acknowledgments:
The authors would like to thank David Ridout for useful discussions and the referee whose constructive suggestions brought us to use cellular algebras as the main tool in Sec.
V
and make it easier to understand. J.B. holds a scholarship from Fonds de recherche Nature et Technologies (Québec). Y.S.A. holds a grant from the Canadian Natural Sciences and Engineering Research Council. This support is gratefully acknowledged.

Article outline:

I. INTRODUCTION
II. BASIC PROPERTIES OF THE DILUTE ALGEBRA
A. Definition of
B. The dimension of
III. LEFT (AND RIGHT) -MODULES
A. The link modules and
B. The standard modules
C. The dimension of
D. The restriction of
E. The induction of
IV. THE GRAM PRODUCT
A. The bilinear form ⟨*, *⟩_{
n, k
}
B. The structure of the radical
C. Symmetric pairs of standard modules
D. Restriction and induction of irreducible modules
V. THE STRUCTURE OF AT A ROOT OF UNITY
A. The dilute Temperley-Lieb algebra as a cellular algebra
B. The indecomposable modules and the structure of for *q* a root of unity
C. Induction of
VI. CONCLUSION

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