Let
f
\mathfrak {f}
,
g
\mathfrak {g}
be finite-dimensional Lie algebras over a field of characteristic zero. Regard
f
\mathfrak {f}
and
g
∗
\mathfrak {g} ^*
, the dual Lie coalgebra of
g
\mathfrak {g}
, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair
(
f
,
g
∗
)
(\mathfrak {f} , \mathfrak {g} ^*)
of Lie bialgebras is given, which has structure maps
⇀
,
ρ
\rightharpoonup , \rho
. Then it induces a matched pair
(
U
f
,
U
g
∘
,
⇀
′
,
ρ
′
)
(U\mathfrak {f}, U\mathfrak {g}^{\circ },\rightharpoonup ’, \rho ’)
of Hopf algebras, where
U
f
U\mathfrak {f}
is the universal envelope of
f
\mathfrak {f}
and
U
g
∘
U\mathfrak {g}^{\circ }
is the Hopf dual of
U
g
U\mathfrak {g}
. We show that the group
O
p
e
x
t
(
U
f
,
U
g
∘
)
\mathrm {Opext} (U\mathfrak {f},U\mathfrak {g}^{\circ })
of cleft Hopf algebra extensions associated with
(
U
f
,
U
g
∘
,
⇀
′
,
ρ
′
)
(U\mathfrak {f}, U\mathfrak {g} ^{\circ }, \rightharpoonup ’, \rho ’ )
is naturally isomorphic to the group
Opext
(
f
,
g
∗
)
\operatorname {Opext}(\mathfrak {f},\mathfrak {g} ^*)
of Lie bialgebra extensions associated with
(
f
,
g
∗
,
⇀
,
ρ
)
(\mathfrak {f}, \mathfrak {g}^*, \rightharpoonup , \rho )
. An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If
g
=
[
g
,
g
]
\mathfrak {g} =[\mathfrak {g} , \mathfrak {g}]
, there follows a bijection between the set
E
x
t
(
U
f
,
U
g
∘
)
\mathrm {Ext}(U\mathfrak {f} , U\mathfrak {g}^{\circ })
of all cleft Hopf algebra extensions of
U
f
U\mathfrak {f}
by
U
g
∘
U\mathfrak {g}^{\circ }
and the set
E
x
t
(
f
,
g
∗
)
\mathrm {Ext}(\mathfrak {f}, \mathfrak {g}^*)
of all Lie bialgebra extensions of
f
\mathfrak {f}
by
g
∗
\mathfrak {g} ^*
.