Gauge deformations for Hopf algebras with the dual Chevalley property

TL;DR

This paper proves that Hopf algebras with a finite-dimensional coradical can be transformed via gauge transformations into a bosonization form involving a connected dual quasi-bialgebra, revealing their structural decomposition.

Contribution

It establishes a gauge transformation that decomposes such Hopf algebras into a bosonization of a connected dual quasi-bialgebra and a finite-dimensional sub-Hopf algebra.

Findings

Existence of a gauge transformation $ abla$ transforming $A$ into a bosonization form.

Structural decomposition of Hopf algebras with the dual Chevalley property.

Identification of the dual quasi-bialgebra $Q$ in the decomposition.

Abstract

Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation $\zeta $ on $A$ such that $A^{\zeta}\cong Q#H$ where $A^\zeta$ is the dual quasi-bialgebra obtained from $A$ by twisting its multiplication by $\zeta$, $Q$ is a connected dual quasi-bialgebra in $^H_H\mathcal{YD}$ and $Q #H $ is a dual quasi-bialgebra called the bosonization of $Q$ by $H$.