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.

arXiv comment14+6 pages, comments welcome

References in corpus (20)

• Fault-tolerant quantum computation by anyons
• String-net condensation: A physical mechanism for topological phases
• Classifying quantum phases using Matrix Product States and PEPS
• Models for gapped boundaries and domain walls
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• TFT construction of RCFT correlators I: Partition functions
• Duality and defects in rational conformal field theory
• Anyons and matrix product operator algebras
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• The many faces of Ocneanu cells
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Characterizing Topological Order with Matrix Product Operators
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Computing data for Levin-Wen with defects
• Fusing Binary Interface Defects in Topological Phases: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ case
• Computing associators of endomorphism fusion categories
• On the Classification of Pointed Fusion Categories up to weak Morita Equivalence
• Classification of Pointed Fusion Categories of dimension 8 up to weak Morita Equivalence
• Computing Defects Associated to Bounded Domain Wall Structures: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case

Cited by in corpus (10)

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• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• 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