2025/10/23 by Howat, Cameron, Laugwitz, Robert, Ray, Martin
#18M30 #Category Theory (math.CT) #FOS: Mathematics #Primary 18M05 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Secondary 16L60
paper · doi:10.48550/arxiv.2510.20613
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.