Volume 585, 2013
Unified products and split extensions of Hopf algebras
A. L. Agore and G. Militaru
Abstract. 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 ∈ 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 → E which has a retraction
π : E → A that is a normal left A-module coalgebra morphism. A necessary
and suﬃcient condition for the canonical morphism i : A → 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 ∗ A, for some bialgebra L in
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 γ : H → A satisfying four compatibility conditions.
Such an example is worked out in detail for a group G, a pointed right G-set
(X, ·, ) and a map γ : G → X.
A morphism i : C → D in a category C is a split monomorphism if there exists
p : D → C a morphism in C such that p ◦ i = IdC. Fundamental constructions
like the semidirect product of groups or Lie algebras, the smash product of Hopf
algebras, Radford’s biproduct, etc can be viewed as tools to answer the following
problem: describe, when is possible, split monomorphisms in a given category C.
The basic example is the following: a group E is isomorphic to a semidirect product
G A of groups if and only if there exists i : A → E a split monomorphism of
groups. In this case E
= G A, where G = Ker(p), for a splitting morphism
p : E → A. The generalization of this elementary result at the level of Hopf
algebras was done in two steps. The first one was made by Molnar in [8, Theorem
4.1]: if i : A → E is a split monomorphism of Hopf algebras having a splitting map
p : E → A which is a normal morphism of Hopf algebras then E is isomorphic as
2010 Mathematics Subject Classification. Primary 16T10; Secondary 16T05, 16S40.
Key words and phrases. Split extensions of Hopf algebras, (bi)crossed products, biproducts.
The first author is Aspirant fellow of FWO-Vlaanderen; the first named author was also
partially supported by CNCS grant TE 45, contract no. 88/02.08.2010. This research is part of
the CNCS - UEFISCDI grant no. 88/5.10.2011 ’Hopf algebras and related topics’.
c 2013 American Mathematical Society