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

Abstract Generalised Temperley–Lieb (TL) 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 TL 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 (JW) 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 TL algebras with coloured regions.