$α$-induction for bi-unitary connections
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909281837645824 |
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| author | Kawahigashi, Yasuyuki |
| author_facet | Kawahigashi, Yasuyuki |
| contents | The tensor functor called $α$-induction arises from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of $N$ to $M$ arising from a subfactor $N\subset M$ of finite index and finite depth giving a braided fusion category of endomorpshisms of $N$. It is also understood in terms of Ocneanu's graphical calculus. We study this $α$-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares and gives certain 4-tensors appearing in recent studies of 2-dimensional topological order. We show that the resulting $α$-induced bi-unitary connections are flat if we have a commutative Frobenius algebra, or a local Q-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05577 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $α$-induction for bi-unitary connections Kawahigashi, Yasuyuki Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Operator Algebras 46L37 18M15 18M20 81T40 The tensor functor called $α$-induction arises from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of $N$ to $M$ arising from a subfactor $N\subset M$ of finite index and finite depth giving a braided fusion category of endomorpshisms of $N$. It is also understood in terms of Ocneanu's graphical calculus. We study this $α$-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares and gives certain 4-tensors appearing in recent studies of 2-dimensional topological order. We show that the resulting $α$-induced bi-unitary connections are flat if we have a commutative Frobenius algebra, or a local Q-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented. |
| title | $α$-induction for bi-unitary connections |
| topic | Quantum Algebra Strongly Correlated Electrons High Energy Physics - Theory Mathematical Physics Operator Algebras 46L37 18M15 18M20 81T40 |
| url | https://arxiv.org/abs/2302.05577 |