Invertible bimodule categories and generalized Schur orthogonality

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.

14+6 pages, comments welcome

References in corpus (4)

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• Bulk-to-boundary anyon fusion from microscopic models
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• From gauging to duality in one-dimensional quantum lattice models
• Les Houches Lecture Notes on Tensor Networks