When is a cleft extension Azumaya?

Abstract

We show which $H^{op}$-cleft extensions $k#_σH^{op}$ of the base field $k$ for a dual quasitriangular Hopf algebra $(H, r)$ are $H$-Azumaya. The result is given in terms of bijectivity of a map $θ_σ\colon H\to H^*$ defined in terms of $r$ and the 2-cocycle $σ$, generalizing a well-known result for the commutative and cocommutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).