Distortion for multifactor bimodules and representations of multifusion categories
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| Format: | Preprint |
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2020
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| author | Bischoff, Marcel Charlesworth, Ian Evington, Samuel Giorgetti, Luca Penneys, David |
| author_facet | Bischoff, Marcel Charlesworth, Ian Evington, Samuel Giorgetti, Luca Penneys, David |
| contents | We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results.
We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions.
Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_01067 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Distortion for multifactor bimodules and representations of multifusion categories Bischoff, Marcel Charlesworth, Ian Evington, Samuel Giorgetti, Luca Penneys, David Operator Algebras Category Theory Quantum Algebra 46L37, 18M20 Primary, 18M30, 18N10 Secondary We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions. |
| title | Distortion for multifactor bimodules and representations of multifusion categories |
| topic | Operator Algebras Category Theory Quantum Algebra 46L37, 18M20 Primary, 18M30, 18N10 Secondary |
| url | https://arxiv.org/abs/2010.01067 |