Galois Objects over Generalized Drinfeld Doubles, with an Application to uq( 2)

Ralf Günther has determined all the cleft extensions over the finite quotient Hopf algebra uq( 2) of the quantized universal enveloping algebra of 2 at a root of unity [R. Günther, Ph.D. thesis, Universität München, 1999]. His techniques (applications of the diamond lemma) are similar to those used by A. Masuoka [Comm. Algebra22 (1994), 4537–4559] for the two-generator Taft algebras. In the present paper we give another proof of a special case of Güntherʹs classification, namely, the case of (cleft) Galois extensions of the base field. The idea is that uq( 2) is the quotient of the Drinfeld double of a Taft algebra by a normal Hopf subalgebra. We use techniques that allow us to calculate all Galois objects of such a composed Hopf algebra.