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

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).