Computing the center of a fusion category
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909890791866368 |
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| author | Mäurer, Fabian Thiel, Ulrich |
| author_facet | Mäurer, Fabian Thiel, Ulrich |
| contents | We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,γ)$ where $X$ is an object and $γ$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the center of a fusion category Mäurer, Fabian Thiel, Ulrich Representation Theory Mathematical Physics Category Theory Quantum Algebra We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,γ)$ where $X$ is an object and $γ$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper). |
| title | Computing the center of a fusion category |
| topic | Representation Theory Mathematical Physics Category Theory Quantum Algebra |
| url | https://arxiv.org/abs/2406.13438 |