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arXiv · 2510.20613

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

Cameron Howat·Robert Laugwitz·Martin Ray
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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.

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BibTeXRIS

Cameron Howat, Robert Laugwitz, Martin Ray. 2025-10-23. Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors. https://doi.org/10.1112/blms.70363

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