Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor
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arXiv
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| Format: | Preprint |
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2025
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| author | Ikhlef, Yacine Morin-Duchesne, Alexi |
| author_facet | Ikhlef, Yacine Morin-Duchesne, Alexi |
| contents | The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states.
We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor Ikhlef, Yacine Morin-Duchesne, Alexi Mathematical Physics Statistical Mechanics High Energy Physics - Theory Representation Theory The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative. |
| title | Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor |
| topic | Mathematical Physics Statistical Mechanics High Energy Physics - Theory Representation Theory |
| url | https://arxiv.org/abs/2509.11756 |