Standard Modules, Induction and the Temperley-Lieb Algebra

Abstract

The basic properties of the Temperley-Lieb algebra $T{L}_n$ with parameter $\beta =q+q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $T{L}_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the structure of the standard modules. Finally, a particular central element $F_n$ of $T{L}_n$ is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case $\beta =0$, that plays an important role in physical models, is studied systematically.

47 pages, 4 figures, many diagrams; v2: 70 pages, reset with new class as per journal requirements; v3: 78 pages, added in Sec. 2 proof that abstract TL is isomorphic to diagram TL and rewrote part of Sec. 8 to avoid Lemma 8.1 (which was wrong), results unchanged; v4: 51 pages, reset in amsart, to appear in ATMP vol. 18

References in corpus (4)

• Associative-algebraic approach to logarithmic conformal field theories
• From Percolation to Logarithmic Conformal Field Theory
• Lattice fusion rules and logarithmic operator product expansions
• Continuum limit and symmetries of the periodic gl(1|1) spin chain

Cited by in corpus (15)

• Logarithmic Conformal Field Theory: a Lattice Approach
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Jordan cells of periodic loop models
• Integrability and conformal data of the dimer model
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra
• On the computation of fusion over the affine Temperley-Lieb algebra
• Dimer representations of the Temperley-Lieb algebra
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• Topological actions of Temperley-Lieb algebras and Representation Stability
• The idempotents of the TL_n-modules \otimes^nC^2 in terms of elements of U_qsl_2
• Deforming algebras with anti-involution via twisted associativity