An Analogue of Radford’s S-formula for Finite Tensor Categories
نویسنده
چکیده
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. This provides a categorical generalization of Radford’s S formula for finite dimensional Hopf algebras [R1], which was proved in [N] for weak Hopf algebras, in [HN] for quasi-Hopf algebras, and conjectured in general in [EO]. When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of [R2]. 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 in [EO].
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