An analogue of Radford's S^4 formula for finite tensor categories
Abstract
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 ?^{**} and D tensor ^{**}? tensor D^{-1}. This provides a categorical generalization of D. Radford's S^4-formula for finite dimensional Hopf algebras and its generalizations for weak Hopf algebras and for quasi-Hopf algebras, and conjectured in general in \cite{EO}. When C is braided, we establish a connection between the above isomorphism and the Drinfeld isomorphism of C. 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.
Keywords
Cite
@article{arxiv.math/0404504,
title = {An analogue of Radford's S^4 formula for finite tensor categories},
author = {Pavel Etingof and Dmitri Nikshych and Viktor Ostrik},
journal= {arXiv preprint arXiv:math/0404504},
year = {2009}
}
Comments
14 pages, AMS-Latex