Flatness of $α$-induced bi-unitary connections and commutativity of Frobenius algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908473115017216 |
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| author | Kawahigashi, Yasuyuki |
| author_facet | Kawahigashi, Yasuyuki |
| contents | The tensor functor called $α$-induction produces a new unitary fusion category from a Frobenius algebra, or a $Q$-system, in a braided unitary fusion category. A bi-unitary connection, which is a finite family of complex number subject to some axioms, realizes an object in any unitary fusion category. It also gives a characterization of a finite-dimensional nondegenerate commuting square in subfactor theory of Jones and realizes a certain $4$-tensor appearing in recent studies of $2$-dimensional topological order. We study $α$-induction for bi-unitary connections, and show that flatness of the resulting $α$-induced bi-unitary connections implies commutativity of the original Frobenius algebra. This gives a converse of our previous result and answers a question raised by R. Longo. We furthermore give finer correspondence between the flat parts of the $α$-induced bi-unitary connections and the commutative Frobenius subalgebras studied by Böckenhauer-Evans. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flatness of $α$-induced bi-unitary connections and commutativity of Frobenius algebras Kawahigashi, Yasuyuki Quantum Algebra High Energy Physics - Theory Mathematical Physics Operator Algebras 46L37 18M15 18M20 81T40 The tensor functor called $α$-induction produces a new unitary fusion category from a Frobenius algebra, or a $Q$-system, in a braided unitary fusion category. A bi-unitary connection, which is a finite family of complex number subject to some axioms, realizes an object in any unitary fusion category. It also gives a characterization of a finite-dimensional nondegenerate commuting square in subfactor theory of Jones and realizes a certain $4$-tensor appearing in recent studies of $2$-dimensional topological order. We study $α$-induction for bi-unitary connections, and show that flatness of the resulting $α$-induced bi-unitary connections implies commutativity of the original Frobenius algebra. This gives a converse of our previous result and answers a question raised by R. Longo. We furthermore give finer correspondence between the flat parts of the $α$-induced bi-unitary connections and the commutative Frobenius subalgebras studied by Böckenhauer-Evans. |
| title | Flatness of $α$-induced bi-unitary connections and commutativity of Frobenius algebras |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Operator Algebras 46L37 18M15 18M20 81T40 |
| url | https://arxiv.org/abs/2408.05501 |