Invertible bimodule categories and generalized Schur orthogonality
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916113387880448 |
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| author | Bridgeman, Jacob C. Lootens, Laurens Verstraete, Frank |
| author_facet | Bridgeman, Jacob C. Lootens, Laurens Verstraete, Frank |
| contents | The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. We discuss applications to generalized symmetries, including a generalized Wigner-Eckart theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01947 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Invertible bimodule categories and generalized Schur orthogonality Bridgeman, Jacob C. Lootens, Laurens Verstraete, Frank Quantum Algebra Mathematical Physics Category Theory Quantum Physics The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. We discuss applications to generalized symmetries, including a generalized Wigner-Eckart theorem. |
| title | Invertible bimodule categories and generalized Schur orthogonality |
| topic | Quantum Algebra Mathematical Physics Category Theory Quantum Physics |
| url | https://arxiv.org/abs/2211.01947 |