Frobenius algebra objects in Temperley-Lieb categories at roots of unity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914171553054720 |
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| author | Grant, Joseph Pugh, Mathew |
| author_facet | Grant, Joseph Pugh, Mathew |
| contents | We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object, and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order.
Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley-Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frobenius algebra objects in Temperley-Lieb categories at roots of unity Grant, Joseph Pugh, Mathew Category Theory Quantum Algebra Representation Theory 18M05 (Primary) 18M20, 18D25, 16L60 (Secondary) We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object, and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley-Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices. |
| title | Frobenius algebra objects in Temperley-Lieb categories at roots of unity |
| topic | Category Theory Quantum Algebra Representation Theory 18M05 (Primary) 18M20, 18D25, 16L60 (Secondary) |
| url | https://arxiv.org/abs/2409.12920 |