[Paper Review] Fusion and braiding in finite and affine Temperley-Lieb categories

Azat M. Gainutdinov, Hubert Saleur|arXiv (Cornell University)|Jan 1, 2016

Algebraic structures and combinatorial models18 references18 citations

TL;DR

This paper constructs infinite arc-tower categories from finite and affine Temperley–Lieb algebras via idempotent subalgebras, establishing direct-limit categories endowed with braided monoidal structures. For finite TL algebras at roots of unity, it reveals new non-rigid braided monoidal categories linked to Virasoro representations, while for affine TL algebras, it introduces a stable N-graded fusion product and proves associativity and braiding, suggesting deep connections to logarithmic conformal field theories in the continuum limit.

ABSTRACT

Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group. We study asymptotic `$N o\infty$' representation theory of these quotients (parametrized by $q\in\mathbb{C}^{ imes}$) from a perspective of braided monoidal categories. Using certain idempotent subalgebras in the finite and affine algebras, we construct infinite `arc' towers of the diagram algebras and the corresponding direct system of representation categories, with terms labeled by $N\in\mathbb{N}$. The corresponding direct-limit category is our main object of studies. For the case of the finite TL algebras, we prove that the direct-limit category is abelian and highest-weight at any $q$ and endowed with braided monoidal structure. The most interesting result is when $q$ is a root of unity where the representation theory is non-semisimple. The resulting braided monoidal categories we obtain at different roots of unity are new and interestingly they are not rigid. We observe then a fundamental relation of these categories to a certain representation category of the Virasoro algebra and give a conjecture on the existence of a braided monoidal equivalence between the categories. This should have powerful applications to the study of the `continuum' limit of critical statistical mechanics systems based on the TL algebra. We also introduce a novel class of embeddings for the affine Temperley-Lieb algebras and related new concept of fusion or bilinear $\mathbb{N}$-graded tensor product of modules for these algebras. We prove that the fusion rules are stable with the index $N$ of the tower and prove that the corresponding direct-limit category is endowed with an associative tensor product. We also study the braiding properties of this affine TL fusion.

Motivation & Objective

• To develop an asymptotic representation theory for finite and affine Temperley–Lieb algebras as N → ∞ via direct systems of representation categories.
• To construct infinite arc-tower categories using idempotent subalgebras, forming direct-limit categories with braided monoidal structure.
• To study the representation theory of finite TL algebras at roots of unity, revealing non-semisimple, non-rigid braided monoidal categories.
• To define and analyze a novel N-graded fusion product (bilinear tensor product) for affine TL algebras, proving stability and associativity.
• To conjecture a braided monoidal equivalence between the resulting categories and certain Virasoro representation categories, relevant to continuum limits of critical lattice models.

Proposed method

• Constructs infinite arc-tower diagrams from finite and affine Temperley–Lieb algebras using specific idempotent subalgebras.
• Defines a direct system of representation categories labeled by N ∈ ℕ, leading to a direct-limit category for each algebra type.
• Introduces a tensor product via induction functors FN: TLN-mod → TLN+2-mod, preserving standard and projective modules.
• Establishes associators and braidings satisfying the pentagon and hexagon identities, proving the direct-limit category is a braided monoidal category.
• Introduces a new fusion product for affine TL algebras, defined via induced modules from affine Hecke algebras, with stability under N.
• Uses matrix representations of affine Hecke generators to compute fusion rules and verify that ideals from the algebraic relations determine the fusion outcomes.

Experimental results

Research questions

• RQ1 What is the structure of the direct-limit category of finite Temperley–Lieb algebras as N → ∞, and is it braided monoidal?
• RQ2 How does the representation theory of finite TL algebras behave at roots of unity, and what new categories arise?
• RQ3 Can a stable, N-graded fusion product be defined for affine TL algebras, and is it associative?
• RQ4 What is the relationship between the resulting braided monoidal categories and representations of the Virasoro algebra?
• RQ5 Does the fusion of standard modules in the affine TL case match diagrammatic fusion rules, and under what conditions is the fusion non-zero?

Key findings

• The direct-limit category of finite Temperley–Lieb algebras is abelian and highest-weight at any q ∈ ℂ×, and carries a braided monoidal structure.
• At roots of unity, the resulting braided monoidal categories are non-semisimple and non-rigid, representing new mathematical structures.
• A conjectured braided monoidal equivalence is proposed between the direct-limit category at roots of unity and a certain category of Virasoro algebra representations.
• For affine TL algebras, a novel N-graded fusion product is defined, and its fusion rules are stable with respect to N, ensuring consistency in the inductive limit.
• The fusion product is shown to be associative, and the corresponding direct-limit category admits an associative tensor product.
• Explicit computation using affine Hecke algebra representations confirms that the fusion of standard modules is non-zero only for specific values of spectral parameters, matching diagrammatic fusion rules from the main text.