In mathematics, a **weak Lie algebra bundle**

ξ
=
(
ξ
,
p
,
X
,
θ
)
is a vector bundle
ξ
over a base space *X* together with a morphism

θ
:
ξ
⊗
ξ
→
ξ
which induces a Lie algebra structure on each fibre
ξ
x
.

A **Lie algebra bundle**
ξ
=
(
ξ
,
p
,
X
)
is a vector bundle in which each fibre is a Lie algebra and for every *x* in *X*, there is an open set
U
containing *x*, a Lie algebra *L* and a homeomorphism

ϕ
:
U
×
L
→
p
−
1
(
U
)
such that

ϕ
x
:
x
×
L
→
p
−
1
(
x
)
is a Lie algebra isomorphism.

Any Lie algebra bundle is a weak Lie algebra bundle, but the converse need not be true in general.

As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space
s
o
(
3
)
×
R
over the real line
R
. Let [.,.] denote the Lie bracket of
s
o
(
3
)
and deform it by the real parameter as:

[
X
,
Y
]
x
=
x
⋅
[
X
,
Y
]
for
X
,
Y
∈
s
o
(
3
)
and
x
∈
R
.

Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. However globally the total space might fail to be Hausdorff.