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

TL;DR

This paper studies coloured Temperley-Lieb categories with Jones-Wenzl projectors, establishing conditions for semisimplicity, decomposing tensor products, and proving a conjecture on Gram determinants and trace pairings.

Contribution

It extends Temperley-Lieb categories to coloured regions, determines their semisimplicity, and proves a conjecture on Gram determinants and trace non-degeneracy.

Findings

Categories are semisimple under certain conditions.

Explicit decompositions of tensor products of simple objects.

Confirmed a conjecture on Gram determinants and trace pairings.

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.