When is a Cleft Extension H-Azumaya?

Abstract

We show which H op-cleft extensions \( k\# _{\sigma } H^{{op}} \) of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map \( \theta _{\sigma } :H \to H* \) defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).