When is a cleft extension Azumaya?

We show which $H^{op}$-cleft extensions k#_\sigma H^{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 ${\theta }_{\sigma }\text{\hspace{0.17em}⁣}:H\to H^{\ast }$ defined in terms of $r$ and the 2-cocycle $\sigma$, 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).