An analogue of Radford's S4 formula for finite tensor categories

By Pavel Etingof, D. Nikshych, V. Ostrik. 2004

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We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ:V**→D⊗**V⊗D−1. This provides a categorical generalization of Radford's S 4 formula for finite-dimensional Hopf algebras (presented in 1976), which was proved by Nikshych (2002) for weak Hopf algebras and by Hausser and Nill (1999) for quasi-Hopf algebras, and was conjectured in general by Etingof and Ostrik (2003). When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of Radford (1992). We also show that a factorizable braided tensor category is unimodular (i.e., D=1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories defined by Etingof and Ostrik (2003).

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/38d3d6c2ec597ea5ddc61ed4a3a10a361603792a

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Citations in the corpus

Tensor products of finitely cocomplete and abelian categories (2012, background): “In recent years Deligne’s tensor product of abelian categories has encountered new applications apart from the original [8], especially in works on tensor categories and their actions on categories, in relation to Hopf algebra theory [11, 9, 10], but also in derived geometry [22], reconstruction the”
Non-degeneracy conditions for braided finite tensor categories (2016, background): “On the other hand, Etingof, Nikshych and Ostrik [ENO04] introduced a certain functor G : C ⊠ C → Z(C) defined in terms of the braiding (see §3.2), and then defined that C is factorizable if G is an equivalence.”
On braided fusion categories I (2009, background): “According to [15,16], in any fusion category C there is a canonical isomorphism g between the tensor functors idC and the fourth duality functor X → X∗∗∗∗, X ∈ C.”