Invertible bimodule categories and generalized Schur orthogonality

• URL: http://arxiv.org/abs/2211.01947v1
• Date: Thu, 3 Nov 2022 16:34:01 GMT
• Title: Invertible bimodule categories and generalized Schur orthogonality
• Authors: Jacob C. Bridgeman, Laurens Lootens, Frank Verstraete
• Abstract summary: We use a generalization to weak Hopf algebras to determine whether a given bimodule category is invertible. We show that our condition for invertibility is equivalent to the notion of MPO-injectivity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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