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Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors

Cameron Howat, Robert Laugwitz, Martin Ray

TL;DR

The paper develops a coloured Temperley–Lieb framework by labeling regions with a semisimple commutative algebra and introducing two-variable Chebyshev polynomials to control semisimplicity. It constructs coloured Jones–Wenzl projectors to produce simple objects indexed by colour sequences, and proves a precise semisimplicity criterion: TL$(k^ extell,oldsymbol{\omega})$ is semisimple iff $U_n(oldsymbol{\omega}_{ij},oldsymbol{\omega}_{ji}) eq 0$ for all $i,j$ and $n\ge 1$. It then derives Clebsch–Gordan-type tensor product decompositions for simple objects, and shows Gram determinants of trace pairings factor as products of the same two-variable Chebyshev polynomials, yielding a coloured interpretation of meander determinants and resolving KL*Conjecture 23 in this setting. The results provide a complete description of semisimplicity, simple-object structure, and tensor products for coloured TL categories, with explicit connections to meanders and potential applications in knot theory and representation theory of coloured TL algebras.

Abstract

Generalised Temperley-Lieb categories with regions labelled by elements of a commutative algebra were introduced by M. Khovanov and the second author in [Pure Appl. Math. Q. 19 (2023), no. 5]. We consider the case where the regions are labelled by colours, corresponding to a complete set of orthogonal idempotents of a semisimple commutative algebra. We determine when these generalised Temperley-Lieb categories are semisimple and find the direct sum decompositions of tensor products of simple objects. As the main tool we use two-variable versions of Chebychev polynomials and coloured Jones-Wenzl projectors. As a consequence, we prove a conjecture of M. Khovanov and the second author on Gram determinants and non-degeneracy of trace pairings for the associated Temperley-Lieb algebras with coloured regions.

Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors

TL;DR

The paper develops a coloured Temperley–Lieb framework by labeling regions with a semisimple commutative algebra and introducing two-variable Chebyshev polynomials to control semisimplicity. It constructs coloured Jones–Wenzl projectors to produce simple objects indexed by colour sequences, and proves a precise semisimplicity criterion: TL is semisimple iff for all and . It then derives Clebsch–Gordan-type tensor product decompositions for simple objects, and shows Gram determinants of trace pairings factor as products of the same two-variable Chebyshev polynomials, yielding a coloured interpretation of meander determinants and resolving KL*Conjecture 23 in this setting. The results provide a complete description of semisimplicity, simple-object structure, and tensor products for coloured TL categories, with explicit connections to meanders and potential applications in knot theory and representation theory of coloured TL algebras.

Abstract

Generalised Temperley-Lieb categories with regions labelled by elements of a commutative algebra were introduced by M. Khovanov and the second author in [Pure Appl. Math. Q. 19 (2023), no. 5]. We consider the case where the regions are labelled by colours, corresponding to a complete set of orthogonal idempotents of a semisimple commutative algebra. We determine when these generalised Temperley-Lieb categories are semisimple and find the direct sum decompositions of tensor products of simple objects. As the main tool we use two-variable versions of Chebychev polynomials and coloured Jones-Wenzl projectors. As a consequence, we prove a conjecture of M. Khovanov and the second author on Gram determinants and non-degeneracy of trace pairings for the associated Temperley-Lieb algebras with coloured regions.
Paper Structure (17 sections, 21 theorems, 47 equations, 1 figure)

This paper contains 17 sections, 21 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Krull--Schmidt categories
  4. Quotients and filtrations
  5. Semisimple categories
  6. Temperley--Lieb categories with labelled regions
  7. Definition of labelled Temperley--Lieb categories
  8. The case of a semisimple algebra of labels
  9. Filtrations on coloured Temperely--Lieb categories
  10. Monoidal structure
  11. Semisimplicity of Temperley--Lieb categories with coloured regions
  12. Classical Jones--Wenzl projectors
  13. Definition of the coloured Jones--Wenzl projectors
  14. Semisimplicity
  15. Gram determinants and meanders
  16. ...and 2 more sections

Key Result

Proposition 2.1

[proposition]prop:FLP1 Let $\mathcal{C}$ be enriched over $\Bbbk$-vector spaces and hom-finite. Then $\mathcal{C}$ is Krull--Schmidt if and only if it is additive and idempotent complete. In particular, $\mathrm{Kar}(\mathcal{C})$ is Krull--Schmidt.

Figures (1)

  • Figure 1: A meander on the left, and the same meander with coloured regions on the right.

Theorems & Definitions (58)

  • Proposition 2.1: FLP*Lemma 4.1
  • Lemma 2.2: FLP*Lemma 4.3
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 48 more