Normal Hopf subalgebras in cocycle deformations of finite groups

TL;DR

This paper provides an explicit criterion for when certain subalgebras remain normal after cocycle deformation of the Hopf algebra of functions on a finite group.

Contribution

It establishes a necessary and sufficient condition for the normality of Hopf subalgebras in cocycle deformations of finite group function algebras.

Findings

Derived explicit normality condition for Hopf subalgebras after deformation

Connected normality in deformed Hopf algebras to group homomorphisms

Enhanced understanding of subalgebra structure in cocycle deformations

Abstract

Let $G$ be a finite group and let $\pi: G \to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^{\sigma}$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $\sigma$. The (normal) Hopf subalgebra $k^{G'} \subseteq k^G$ corresponds to a Hopf subalgebra $L' \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$.