Unified products and split extensions of Hopf algebras

TL;DR

This paper characterizes the structure of unified products of Hopf algebras using split morphisms and normality, providing new conditions for their isomorphism to Radford biproducts and methods for their construction.

Contribution

It offers an equivalent description of unified products via split morphisms, establishes criteria for their isomorphism to Radford biproducts, and introduces a general construction method from non-associative bialgebras.

Findings

Unified product characterization via split morphisms.

Conditions for isomorphism to Radford biproduct.

A new construction method for unified products.

Abstract

The unified product was defined in \cite{am3} related to the restricted extending structure problem for Hopf algebras: a Hopf algebra $E$ factorizes through a Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1\in H$ if and only if $E$ is isomorphic to a unified product $A \ltimes H$. Using the concept of normality of a morphism of coalgebras in the sense of Andruskiewitsch and Devoto we prove an equivalent description for the unified product from the point of view of split morphisms of Hopf algebras. A Hopf algebra $E$ is isomorphic to a unified product $A \ltimes H$ if and only if there exists a morphism of Hopf algebras $i: A \rightarrow E$ which has a retraction $\pi: E \to A$ that is a normal left $A$-module coalgebra morphism. A necessary and sufficient condition for the canonical morphism $i : A \to A\ltimes H$ to be a split monomorphism of bialgebras is proved, i.e. a condition for the unified product $A\ltimes H$ to be isomorphic to a Radford biproduct $L \ast A$, for some bialgebra $L$ in the category $_{A}^{A}{\mathcal YD}$ of Yetter-Drinfel'd modules. As a consequence, we present a general method to construct unified products arising from an unitary not necessarily associative bialgebra $H$ that is a right $A$-module coalgebra and a unitary coalgebra map $\gamma : H \to A$ satisfying four compatibility conditions. Such an example is worked out in detail for a group $G$, a pointed right $G$-set $(X, \cdot, \lhd)$ and a map $\gamma : G \to X$.