[Paper Review] Standard Modules, Induction and the Temperley-Lieb Algebra

David Ridout, Yvan Saint-Aubin|arXiv (Cornell University)|Apr 20, 2012

Algebraic structures and combinatorial models10 references21 citations

TL;DR

This paper provides a comprehensive, pedagogical analysis of the Temperley-Lieb algebra $τ\mathsf{TL}_n$ with parameter $β = q + q^{-1}$, focusing on standard (cell) modules, their structure via bilinear forms and induction, and the non-semisimple case when $q$ is a root of unity. The key contribution is a proof that the radicals of standard modules are irreducible and a complete determination of homomorphisms between standard modules using a central element $F_n$ with non-diagonalizable action.

ABSTRACT

The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $β= q + q^{-1}$, 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 $TL_n$ 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 $TL_n$ 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 $β= 0$, that plays an important role in physical models, is studied systematically.

Motivation & Objective

• To provide a pedagogical review of the Temperley-Lieb algebra $τ\mathsf{TL}_n$ and its standard (cell) modules, especially in the non-semisimple case.
• To characterize the structure of standard modules using a natural bilinear form and its radical, identifying when modules are reducible (when $q$ is a root of unity).
• To use restriction and induction functors to refine the structure of standard modules and determine homomorphisms between them.
• To study the action of a central element $F_n$ to prove that the radicals of standard modules are irreducible and to compute principal indecomposable modules explicitly.
• To systematically treat the physically important case $β = 0$, which is excluded in prior works.

Proposed method

• Define standard modules as cell modules and equip them with a natural $τ\mathsf{TL}_n$-invariant bilinear form to analyze their maximal submodules via the radical of the form.
• Use Frobenius reciprocity and induction/restriction functors between $τ\mathsf{TL}_n$ and $τ\mathsf{TL}_{n-1}$ to relate module structures across different $n$.
• Introduce a central element $F_n \in \mathsf{TL}_n$ whose non-diagonalizable action on certain indecomposable modules is key to proving irreducibility of module radicals.
• Construct principal indecomposable modules explicitly as direct summands of induced standard modules, using the decomposition of induced modules via the bilinear form and induction theory.
• Apply recursion and combinatorial techniques inspired by James and Murphy to compute determinants of bilinear forms, extending Westbury’s work with corrected recursion.
• Use Bratteli diagrams and block decomposition to analyze the representation theory in the non-semisimple regime, particularly when $q$ is a root of unity.

Experimental results

Research questions

• RQ1 When is the standard module of $τ\mathsf{TL}_n$ reducible, and how can this be detected via the bilinear form?
• RQ2 How do induction and restriction functors refine the structure of standard modules in the non-semisimple case?
• RQ3 What is the structure of the homomorphisms between standard modules, and how can they be completely determined?
• RQ4 Why is the case $β = 0$ physically significant, and how is it treated systematically in this work?
• RQ5 Is the radical of a standard module irreducible, and what structural role does the central element $F_n$ play in proving this?

Key findings

• The standard modules of $τ\mathsf{TL}_n$ are reducible if and only if $q$ is a root of unity, which occurs precisely when the bilinear form on the module has a non-trivial radical.
• The action of the central element $F_n$ on certain indecomposable modules is non-diagonalizable, a key technical tool in proving that the radicals of standard modules are irreducible.
• The space of homomorphisms between standard modules is completely determined, and this determination relies on the non-diagonalizability of $F_n$ and Frobenius reciprocity.
• The radicals of standard modules are irreducible, a result proven via the non-diagonalizable action of $F_n$ and the structure of induced modules.
• Principal indecomposable modules are explicitly constructed as direct summands of induced standard modules, providing a concrete realization in terms of standard modules and their inductions.
• The case $β = 0$ is systematically analyzed, filling a gap left in earlier works that excluded this physically relevant parameter value.