Cohomological Aspects of Hopf Algebra Liftings

A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a ‘non-abelian’ cohomology theory, which gives a method for an explicit description of cocycles relevant to the lifting process. 0. Introduction The Nichols algebra B(V ) of a crossed kG-module V is a connected braided Hopf algebra. In terms of generators and relations it can be described via a certain pushout diagram K(V ) κ −−−−→ R(V )