Unified products and split extensions of Hopf algebras

The unified product was defined in Agore and Militaru (2011) 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 E 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 proportional to H if and only if there exists a morphism of Hopf algebras i : A -> E which has a retraction A that is a normal left A-module coalgebra morphism. A necessary and sufficient condition for the canonical morphism : A A v H to be a split monomorphism of bialgebras is proved, i.e. a condition for the unified product A proportional to H to be isomorphic to a Radford biproduct L * A, for some bialgebra L in the category A(A)gamma 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 A satisfying four compatibility conditions. Such an example is worked out in detail for a group G, a pointed right G-set (X, ., (sic)) and a map gamma : G -> X.