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Hopf cocycle deformations and invariant theory. (English) Zbl 1436.16042

Let \(H\) be a finite dimensional Hopf algebra over an algebraically closed field \(k\) of characteristic zero. In this paper the author studies the classification problem of equivalence classes of cocycle deformations of \(H\). The main result of the paper states that the set of all such equivalence classes has a natural structure of an affine algebraic variety and the elements of its coordinate ring can be thus thought of as a complete set of scalar invariants of Hopf 2-cocycles on \(H\). This variety is presented as a quotient of an affine variety by the action of some reductive group, by using methods of geometric invariant theory. As an application, an explicit presentation by generators and relations of its coordinate algebra is given. As a consequence of the main result, an application to the study of the fields of definition of a given cocycle deformation is obtained.
The main construction of the paper is shown to generalize the universal coefficients theorem of group cohomology. Other families of Hopf algebras are also discussed, like dual group algebras and bosonisations of Nichols algebras.

MSC:

16T05 Hopf algebras and their applications
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