Hopf cocycle deformations and invariant theory

TL;DR

This paper characterizes all cocycle deformations of finite dimensional Hopf algebras as an affine variety using geometric invariant theory, providing classifications and examples across various Hopf algebra families.

Contribution

It introduces a geometric approach to classify cocycle deformations of Hopf algebras and applies it to specific cases like group algebras and Nichols algebras.

Findings

Classified cocycle deformations of dual pointed Hopf algebras of symmetric group.

Identified examples with only rational invariants not definable over rationals.

Connected deformation classification to geometric invariant theory.

Abstract

For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the Universal Coefficients Theorem in the case of a group algebra, and we also give examples from other families of Hopf algebras, including dual group algebras and Bosonizations of Nichols algebras. In particular, we use the methods developed here to classify the cocycle deformations of a dual pointed Hopf algebra associated to the symmetric group on three letters. We also give an example of a cocycle deformation over a dual group algebra, which has only rational invariants, but which is not definable over the rational field. This differs from the case of group algebras, in which every two-cocycle is equivalent to one which is definable by its invariants.