Invertible Bimodule Categories and Generalized Schur Orthogonality

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 Morit...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-09, Vol.402 (3), p.2691-2714
Main Authors: Bridgeman, Jacob C., Lootens, Laurens, Verstraete, Frank
Format: Article
Language:English
Subjects:
Algorithms
Classical and Quantum Gravitation
Complex Systems
Equivalence
Mathematical and Computational Physics
Mathematical Physics
Modules
Orthogonality
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Tensors
Theoretical
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Summary: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. Ultimately, the condition arises from Schur orthogonality relations on the characters of the annular algebra associated to a module category. 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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-023-04781-y