Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories

2025/06/19 by Thibault D. Décoppet, Décoppet, Thibault D. · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2506.16241

Abstract

Fix a finite symmetric tensor category E over an algebraically closed field. We derive an E-enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. In order to do so, we consider, associated to any E-enriched finite braided tensor category A satisfying a mild technical assumption, a Hopf algebra \mathbbF_E/A in A. This is a generalization of Lyubashenko's universal Hopf algebra \mathbbF_A in A. In fact, we show that there is a short exact sequence \mathbbF_E→\mathbbF_A→\mathbbF_E/A of Hopf algebras in A, and that the canonical pairing on \mathbbF_A descends to a pairing ω_E/A on \mathbbF_E/A. We prove that A is E-non-degenerate, i.e. its symmetric center is exactly E, if and only if the pairing ω_E/A is non-degenerate. We then use the above characterization to show that an E-enriched finite braided tensor category is invertible in the Morita 4-category of E-enriched pre-tensor categories if and only if it is E-non-degenerate. As an application of our relative invertibility criterion, we extend the full dualizability result of Brochier-Jordan-Snyder by showing that a finite braided tensor category is fully dualizable as an object of the Morita 4-category of braided pre-tensor categories if its symmetric center is separable.

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Cited by

• Representation theory of non-factorizable ribbon Hopf algebras

Related

• Non-degeneracy conditions for braided finite tensor categories
• Duality Theorem and Drinfeld Double in Braided Tensor Categories
• Invertible Fusion Categories
• Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories
• Duality Theorems for Infinite Braided Hopf Algebras