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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2308.11966 |
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| _version_ | 1866909113957482496 |
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| author | Nurcombe, Madeline |
| author_facet | Nurcombe, Madeline |
| contents | We introduce the ghost algebra, a two-boundary generalisation of the Temperley-Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang-Baxter equations, given existing solutions to the Yang-Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang-Baxter integrable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11966 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The ghost algebra and the dilute ghost algebra Nurcombe, Madeline Mathematical Physics High Energy Physics - Theory We introduce the ghost algebra, a two-boundary generalisation of the Temperley-Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang-Baxter equations, given existing solutions to the Yang-Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang-Baxter integrable. |
| title | The ghost algebra and the dilute ghost algebra |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2308.11966 |