Unified products and split extensions of Hopf algebras

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 $A⋉H$. 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 $A⋉H$ if and only if there exists a morphism of Hopf algebras $i:A\to E$ 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 $i:A\to A⋉H$ to be a split monomorphism of bialgebras is proved, i.e. a condition for the unified product $A⋉H$ to be isomorphic to a Radford biproduct $L\ast A$, for some bialgebra $L$ in the category ${}_A^A\mathcal{Y}D$ 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 $(X,\cdot ,⊲)$ and a map $\gamma :G\to X$.

arXiv comment16 pages, to appear in AMS Contemporary Math