Abstract :
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.
