Polynomial identities and universal constructions for Hopf algebras

Cocycle deformations of Hopf algebras are noncommutative analogues (in the sense of Connes's noncommutative geometry) of principal fiber bundles. I will show how to extend the classical theory of polynomial identities to these objects and how to use these extended identities to construct versal families of deformations of quantum principal fiber bundles (joint work with Eli Aljadeff).