UNIFIED PRODUCTS AND SPLIT EXTENSIONS OF HOPF ALGEBRAS 3
has a retraction π : E A that is a normal left A-module coalgebra morphism
(Theorem 2.3). The proof of Theorem 2.3 is different from the one in [10] and
is based on the factorization theorem [1, Theorem 2.7]. As a bonus, a simplified
and more transparent version of [8, Theorem 4.1] is obtained in Corollary 2.5: if
π : E A is a normal split epimorphism of Hopf algebras, then E

= A#H,
where A#H is the right version of the smash product of bialgebras for some right
A-module bialgebra H.
The next aim of the paper is to make the connection between the unified prod-
uct and Radford’s biproduct: for a Hopf algebra A, Proposition 2.6 gives necessary
and sufficient conditions for iA : A A H to be a split monomorphism of bial-
gebras. In this case the unified product A H is isomorphic as a bialgebra to
a biproduct L A, and the structure of L as a bialgebra in the category AYD
A
of
Yetter-Drinfel’d modules is explicitly described. Furthermore, in this context a new
equivalent description of the unified product A H, as well as of the associated
biproduct L A, is given in Proposition 2.7: both of them are isomorphic to a new
product A H, which is a deformation of the smash product of bialgebras A#H
using a coalgebra map γ : H A. Finally, Theorem 2.8 gives a general method for
constructing unified products, as well as biproducts arising from a right A-module
coalgebra (H, ) and a unitary coalgebra map γ : H A. Example 2.9 gives an
explicit example of such a product starting with a group G, a pointed right G-set
(X, ·, ) and a map γ : G X satisfying two compatibility conditions.
1. Preliminaries
Throughout this paper, k will be a field. Unless specified otherwise, all modules,
algebras, coalgebras, bialgebras, tensor products, homomorphisms and so on are
over k. For a coalgebra C, we use Sweedler’s Σ-notation: Δ(c) = c(1) c(2),
(I Δ)Δ(c) = c(1) c(2) c(3), etc (summation understood). We also use the
Sweedler notation for left C-comodules: ρ(m) = m
−1
m
0
, for any m M
if (M, ρ)
CM
is a left C-comodule. Let A be a bialgebra and H an algebra and
a coalgebra. A k-linear map f : H H A will be denoted by f(g, h) = f(g h);
f is the trivial map if f(g, h) = εH (g)εH (h)1A, for all g, h H. Similarly, the k-
linear maps : H A H, : H A A are the trivial actions if ha = εA(a)h
and respectively h a = εH (h)a, for all a A and h H.
For a Hopf algebra A we denote by
AM
A
the category of left-left A-Hopf modules:
the objects are triples (M, ·,ρ), where (M, ·) AM is a left A-module, (M, ρ)
AM
is a left A-comodule such that
ρ(a · m) = a(1)m
−1
a(2) · m
0
for all a A and m M. For (M, ·,ρ)
AM
A
we denote by M
co(A)
= {m
M | ρ(m) = 1A m} the subspace of coinvariants. The fundamental theorem for
Hopf modules states that for any A-Hopf module M the canonical map
ϕ : A M
co(A)
M, ϕ(a m) := a · m
for all a A and m M is bijective with the inverse given by
ϕ−1
: M A M
co(A), ϕ−1(m)
:= m
−2
S(m
−1
) · m
0
for all m M.
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