Cocycle deformations for Hopf algebras with a coalgebra projection

TL;DR

This paper investigates how cocycle deformations can be applied to Hopf algebras with a coalgebra projection, providing explicit constructions and examples involving Nichols algebras and Radford biproducts.

Contribution

It introduces a method to deform such Hopf algebras via cocycles, extending the understanding of their structure and classifications.

Findings

Deformation of $A$ via $H$-bilinear cocycles results in isomorphic twisted algebras.

Explicit cocycle constructions for liftings of Nichols algebras are provided.

The approach generalizes the Radford biproduct deformation theory.

Abstract

Let $H$ be a Hopf algebra over a field $K$ of characteristic $0$ and let $A$ be a bialgebra or Hopf algebra such that $H$ is isomorphic to a sub-Hopf algebra of $A$ and there is an $H$-bilinear coalgebra projection $\pi$ from $A$ to $H$ which splits the inclusion. Then $A \cong R \#_\xi H$ where $R$ is the pre-bialgebra of coinvariants. In this paper we study the deformations of $A$ by an $H$-bilinear cocycle. If $\gamma$ is a cocycle for $A$, then $\gamma$ can be restricted to a cocycle $\gamma_R$ for $R$, and $A^\gamma \cong R^{\gamma_R} \#_{\xi_\gamma} H$. As examples, we consider liftings of $\mathcal{B}(V) \# K[\Gamma]$ where $\Gamma$ is a finite abelian group, $V$ is a quantum plane and $\mathcal{B}(V)$ is its Nichols algebra, and explicitly construct the cocycle which twists the Radford biproduct into the lifting.