Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914515856130048 |
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| author | Howat, Cameron Laugwitz, Robert Ray, Martin |
| author_facet | Howat, Cameron Laugwitz, Robert Ray, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors Howat, Cameron Laugwitz, Robert Ray, Martin Quantum Algebra Category Theory Representation Theory Primary 18M05, Secondary 16L60, 18M30 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. |
| title | Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors |
| topic | Quantum Algebra Category Theory Representation Theory Primary 18M05, Secondary 16L60, 18M30 |
| url | https://arxiv.org/abs/2510.20613 |