 
Summary: Advances in Mathematics 218 (2008) 14531495
www.elsevier.com/locate/aim
Polynomial identities and noncommutative versal
torsors
Eli Aljadeff a
, Christian Kassel b,
a Department of Mathematics, Technion Israel Institute of Technology, 32000 Haifa, Israel
b Institut de Recherche Mathématique Avancée, CNRS Université Louis Pasteur,
7 rue René Descartes, 67084 Strasbourg, France
Received 26 September 2007; accepted 16 March 2008
Available online 21 April 2008
Communicated by Michel Van den Bergh
Abstract
To any cleft Hopf Galois object, i.e., any algebra H obtained from a Hopf algebra H by twisting its
multiplication with a twococycle , we attach two "universal algebras" A
H and U
H . The algebra A
H is
obtained by twisting the multiplication of H with the most general twococycle formally cohomologous
to . The cocycle takes values in the field of rational functions on H. By construction, A
