An analogue of Radford's S-4 formula for finite tensor categories

Academic Article

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.
  • Authors

  • Etingof, P
  • Nikshych, Dmitri
  • Ostrik, V
  • Status

    Publication Date

  • 2004
  • Has Subject Area

    Keywords

  • math.CT
  • math.QA
  • Start Page

  • 2915
  • End Page

  • 2933
  • Issue

  • 54