In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones. Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".
More precisely, a Lie algebroid is a triple
(
E
,
[
⋅
,
⋅
]
,
ρ
)
consisting of a vector bundle
E
over a manifold
M
, together with a Lie bracket
[
⋅
,
⋅
]
on its module of sections
Γ
(
E
)
and a morphism of vector bundles
ρ
:
E
→
T
M
called the anchor. Here
T
M
is the tangent bundle of
M
. The anchor and the bracket are to satisfy the Leibniz rule:
[
X
,
f
Y
]
=
ρ
(
X
)
f
⋅
Y
+
f
[
X
,
Y
]
where
X
,
Y
∈
Γ
(
E
)
,
f
∈
C
∞
(
M
)
and
ρ
(
X
)
f
is the derivative of
f
along the vector field
ρ
(
X
)
. It follows that
ρ
(
[
X
,
Y
]
)
=
[
ρ
(
X
)
,
ρ
(
Y
)
]
for all
X
,
Y
∈
Γ
(
E
)
.
Every Lie algebra is a Lie algebroid over the one point manifold.
The tangent bundle
T
M
of a manifold
M
is a Lie algebroid for the Lie bracket of vector fields and the identity of
T
M
as an anchor.
Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
To every Lie groupoid is associated a Lie algebroid, generalizing how a Lie algebra is associated to a Lie group (see also below). For example, the Lie algebroid
T
M
comes from the pair groupoid whose objects are
M
, with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible, but every Lie algebroid gives a stacky Lie groupoid.
Given the action of a Lie algebra g on a manifold M, the set of g -invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
The Atiyah algebroid of a principal G-bundle P over a manifold M is a Lie algebroid with short exact sequence:
0
→
P
×
G
g
→
T
P
/
G
→
ρ
T
M
→
0.
The space of sections of the Atiyah algebroid is the Lie algebra of
G-invariant vector fields on
P.
A Poisson Lie algebroid is associated to a Poisson manifold by taking E to be the cotangent bundle. The anchor map is given by the Poisson bivector. This can be seen in a Lie bialgebroid.
To describe the construction let us fix some notation. G is the space of morphisms of the Lie groupoid, M the space of objects,
e
:
M
→
G
the units and
t
:
G
→
M
the target map.
T
t
G
=
⋃
p
∈
M
T
(
t
−
1
(
p
)
)
⊂
T
G
the t-fiber tangent space. The Lie algebroid is now the vector bundle
A
:=
e
∗
T
t
G
. This inherits a bracket from G, because we can identify the M-sections into A with left-invariant vector fields on G. The anchor map then is obtained as the derivation of the source map
T
s
:
e
∗
T
t
G
→
T
M
. Further these sections act on the smooth functions of M by identifying these with left-invariant functions on G.
As a more explicit example consider the Lie algebroid associated to the pair groupoid
G
:=
M
×
M
. The target map is
t
:
G
→
M
:
(
p
,
q
)
↦
p
and the units
e
:
M
→
G
:
p
↦
(
p
,
p
)
. The t-fibers are
p
×
M
and therefore
T
t
G
=
⋃
p
∈
M
p
×
T
M
⊂
T
M
×
T
M
. So the Lie algebroid is the vector bundle
A
:=
e
∗
T
t
G
=
⋃
p
∈
M
T
p
M
=
T
M
. The extension of sections X into A to left-invariant vector fields on G is simply
X
~
(
p
,
q
)
=
0
⊕
X
(
q
)
and the extension of a smooth function f from M to a left-invariant function on G is
f
~
(
p
,
q
)
=
f
(
q
)
. Therefore, the bracket on A is just the Lie bracket of tangent vector fields and the anchor map is just the identity.
Of course you could do an analog construction with the source map and right-invariant vector fields/ functions. However you get an isomorphic Lie algebroid, with the explicit isomorphism
i
∗
, where
i
:
G
→
G
is the inverse map.