Koszul complexes and Chevalley's theorems for Lie algebroids
We use Koszul complexes and Chevalley-type theorems to calculate the cohomology $\mathbf{H} \left( A\right) $ of a transitive Lie algebroid $A$ under some assumptions on the isotropy Lie algebras.
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552778138648576 |
|---|---|
| author | Kubarski, J. Kubarski, J. |
| author_facet | Kubarski, J. Kubarski, J. |
| author_institution_txt_mv | [
{
"author": "J. Kubarski",
"institution": "Institute of Mathematics Lodz University of Technology"
}
] |
| author_sort | Kubarski, J. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | We use Koszul complexes and Chevalley-type theorems to calculate the cohomology $\mathbf{H} \left( A\right) $ of a transitive Lie algebroid $A$ under some assumptions on the isotropy Lie algebras. |
| first_indexed | 2026-08-04T01:04:50Z |
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| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 97–117
Jan Kubarski
(Institute of Mathematics
Lodz University of Technology, Poland)
Koszul complexes and Chevalley’s
theorems for Lie algebroids
We use Koszul complexes and Chevalley-type theorems to calculate the
cohomology H (A) of a transitive Lie algebroid A under some assumptions
on the isotropy Lie algebras.
1 Introduction
How can we calculate the cohomology H (A) of a transitive Lie al-
gebroid A with the Atiyah sequence 0 −→ ggg ↪→A #A−→ TM −→ 0 ?
This is one of the fundamental questions for the topology of Lie alge-
broids [I-K-V], [M]. A classical method is to use spectral sequences. We
can use the Leray spectral sequence for the Čech–de Rham complex of
transitive Lie algebroids [K-M-1] as well as the Hochschild-Serre spectral
sequence for the pair of Lie algebras (Secggg,SecA) and the observation
that the vector bundle of the cohomology H (ggg) of the isotropy Lie alge-
bras, H (ggg)|x = H
(
ggg|x
)
, is flat, and that Ej,i2 = Hj
∇
(
M ; Hi (ggg)
)
where
∇ is the flat covariant derivative in Hi (ggg) [K-M-2], [K-M-3].
In this paper we propose an adaptation of the method of Koszul com-
plexes and Chevalley-type theorems [G-H-V, Vol. III] to the calculation
of H (A). Originally the method is based on the operation of a reduc-
tive Lie algebra in a graded differential algebra admitting an algebraic
connection. A fundamental theorem of Chevalley gives a homomorphism
from the corresponding Koszul complex which induces an isomorphism of
cohomology. Classically, this isomorphism is applied to the cohomology
c© Jan Kubarski, 2013
98 Koszul complexes and Chevalley’s theorems for Lie algebroids
of principal fibre bundles. Namely: the Chevalley theorem (for pfb’s)
says that under some assumptions, the cohomology of the total space
H (P ) of a pfb P depends uniquely on the cohomology of the base man-
ifold M and the characteristic classes (the Chern-Weil homomorphism
hP :
(∨
g∗
)
IG
−→ H (M)). It turns out that this assertion has a coun-
terpart for Lie algebroids, but in this context we cannot use the standard
operation of a Lie algebra directly. We propose some modification of this
method.
2 Lie algebroid of a principal fibre bundle,
Lie functor
2.1 Examples of Lie algebroids
2.1.1 Lie algebroid of a Lie group
The Lie algebroid of a Lie group G (the infinitesimal object of a Lie
group G) is simply its Lie algebra g = TeG = TG/G (for example,
through the right action of G on TG we obtain the "right Lie algebra of
a Lie group").
2.1.2 Lie algebroid of a principal fibre bundle
The vector space A (P ) := TP/G of cosets of the right action of
G on TP (introduced by M. Atiyah in 1955) is an infinitesimal object
of a principal fibre bundle P (M,G). It has two extra structures: a
Lie bracket in the space of global cross-sections Sec A (P ) and a linear
homomorphism #A(P ) : A (P ) −→ TM called the anchor. The Lie
bracket in Sec A (P ) is introduced via the isomorphism Sec (A (P )) ∼=
XR (P ) where XR (P ) is the space of right invariant vector fields on P with
the usual Lie bracket. The anchor is defined by #A(P ) : A (P ) −→ TM,
[v] 7−→ π∗ (v) where π : P →M is the projection of P. The anchor #A(P )
is bracket-preserving: #A(P ) ([[ξ1, ξ2]]) =
[
#A(P ) (ξ1) ,#A(P ) (ξ2)
]
, and
the Leibniz formula holds: [[ξ1, f · ξ2]] = f · [[ξ1, ξ2]] +
(
#A(P ) (ξ1)
)
(f) · ξ2.
The Lie algebroid of the trivial principal fibre bundle P = M×G is equal
to
A (P ) = TP/G = T (M ×G) /G = TM × (TG/G) = TM × g,
Jan Kubarski 99
with the bracket [[(X,σ) , (Y, η)]] = ([X,Y ] , X (η)− Y (σ) + [σ, η]) for
X,Y ∈ X (M) , σ, η ∈ C∞ (M, g) , and the anchor #TM×g = pr1 :
TM × g→ TM.
2.2 Pradines’ definition of a Lie algebroid
Generalizing the structure
(
A (P ) , [[·, ·]],#A(P )
)
for a pfb P (M,G)
J. Pradines gives the definition of a Lie algebroid [P]:
Definition 1. A Lie algebroid on a manifold M is a triple (A, [[·, ·]],#A)
where A is a vector bundle on M , (SecA, [[·, ·]]) is an R-Lie algebra, #A :
A→ TM is a linear homomorphism of vector bundles and the following
Leibniz condition is satisfied:
[[ξ, f · η]] = f · [[ξ, η]] + γL (ξ) (f) · η, f ∈ C∞ (M) , ξ, η ∈ SecA.
The anchor is bracket-preserving, #A◦[[ξ, η]] = [#A ◦ ξ,#A ◦ η].
The image of the anchor, Im #A ⊂ TM , is an integrable non-constant-
rank (in general) distribution whose leaves form a Stefan foliation of M .
If the anchor #A is of constant rank then the Lie algebroid A is called
regular and Im #A forms a regular foliation on M . The Lie algebroid
is called transitive if #A is an epimorphism. A transitive Lie algebroid
is called integrable if it is isomorphic to the Lie algebroid of a principal
fibre bundle.
We deal here only with transitive Lie algebroids.
For a transitive Lie algebroid A we have the Atiyah sequence
0 −→ ggg ↪→A #A−→ TM −→ 0.
The vector bundle ggg is a Lie algebra bundle, called the adjoint of A; in
particular, all the isotropy Lie algebras ggg|x are isomorphic.
Example 2. (1) A single Lie algebra g is a Lie algebroid over a one–point
set and with the zero anchor.
(2) The tangent bundle TM of a manifold M is a Lie algebroid on M
with idTM as anchor and with the usual Lie bracket of vector fields.
(3) Trivial Lie algebroid: TM × g with the projection pr1 as anchor
and with the bracket given by
[[ (X,σ) , (Y, η) ]] = ([X,Y ] , X (η)− Y (σ) + [σ, η]) ,
100 Koszul complexes and Chevalley’s theorems for Lie algebroids
X,Y ∈ X (M) , σ, η ∈ C∞ (M ; g), is a transitive Lie algebroid, called
trivial. (Each transitive Lie algebroid L over a contractible manifold is
isomorphic to the trivial one).
(4) The Lie algebroid A(P ) = TP/G of a G-principal fibre bundle
P = P (M,G).
(5) The Lie algebroid A (f) of a vector bundle f: With a vector bundle
f we associate a transitive Lie algebroid A (f) (isomorphic to the Lie
algebroid of the principal fibre bundle of all frames of f, A (f) = A (L f))
whose space of global cross-sections Sec A (f) is equal to the space of all
covariant differential operators for f. The Lie algebra bundle adjoint to
A (f) is equal to End (f) , so the Atiyah sequence reads
0 −→ End (f) −→ A (f) −→ TM → 0.
Example 3 (Other examples). (6) The Lie algebroid A(M,F) of a
transversally complete foliation (M,F) of a connected Hausdorff para-
compact manifold M , in particular:
(6’) The Lie algebroid A(G;H) of a nonclosed Lie subgroup H of G:
It is the Lie algebroid of the TC-foliation FG,H = {aH; a ∈ G} of left
cosets of a nonclosed Lie subgroup H in a Lie group G. These include
nonintegrable Lie algebroids.
(7) Poisson manifolds yield nontransitive Lie algebroids.
Definition 4. By a homomorphism of Lie algebroids F :
(A, [[·, ·]],#A) −→ (A′, [[·, ·]],#A′) on a manifold M we mean a lin-
ear homomorphism F : A → A′ of vector bundles commuting with the
anchors:
A
F−→ A′
↓ #A ↓ #A′
TM = TM
and such that F is a homomorphism of the Lie algebras of global cross-
sections:
F ([[ξ1, ξ2]]) = [[Fξ1, F ξ2]], ξi ∈ SecA.
A homomorphism F : A −→ B of transitive Lie algebroids induces a
linear homomorphism of the adjoint Lie algebra bundles F+ : ggg −→ ggg′
and for any x ∈M, F+
x : ggg|x −→ ggg′|x is a homomorphism of Lie algebras.
Jan Kubarski 101
We obtain in this way a homomorphism of Atiyah sequences,
0 0
↓ ↓
ggg
F+
−→ ggg′
↓ ↓
A
F−→ A′
↓ ↓
TM = TM
↓ ↓
0 0
2.3 Lie functor
To have a Lie functor for pfb’s we need to define a homomorphism
of Lie algebroids induced by a homomorphism of pfb’s. Let P and P ′
be two pfbs with structural Lie groups G and G′, respectively. Assume
that µ : G −→ G′ is a homomorphism of Lie groups, and F : P −→ P ′
a µ-homomorphism of pfbs, i.e. F (z · a) = F (z) · a′. Then the linear
homomorphism (the Lie algebroid differential of F )
F∗ : A (P ) −→ A (P ′) , [vz] 7−→ [dF∗z (vz)] ,
is a homomorphism of the induced Lie algebroids.
2.4 Cohomology of a Lie algebroid
To a Lie algebroid A we associate the cohomology algebra H (A) de-
fined via the DG-algebra of A-differential forms (with real coefficients)
(Ω (A) , dA) , where
Ω (A) = Sec
∧
A∗,
dA : Ω∗ (A) −→ Ω∗+1 (A)
(dAz) (ξ0, ..., ξk) =
k∑
j=0
(−1)
j
(#A ◦ ξj) (z (ξ0, ...̂..., ξk))
+
∑
i<j
(−1)
i+j
z ([[ξi, ξj ]], ξ0, ...̂ı...̂..., ξk) ,
102 Koszul complexes and Chevalley’s theorems for Lie algebroids
z ∈ Ωk (A), ξi ∈ SecA. The exterior derivative dA induces the cohomol-
ogy algebra
H (A) = H (Ω (A) , dA) .
Why the differential dA must be given by the above formula? It is easy
to obtain this formula starting with the Lie algebroid of a Lie groupoid
Φ = (Φ, (α, β) , ·.M) on a manifold M , with source α and target β and
partial multiplication · . Let i : M → Φ be the embedding of M onto the
submanifold of units, i (x) = ux, of this Lie groupoid. Then
A (Φ) = i∗ (TαΦ)
where TαΦ is the subbundle of α-vertical vectors. We see that for any
x ∈ M, the submanifold Φx = α−1 (x) of all elements starting at x (i.e.
having x as source) forms a Φxx-pfb (Φxx is the Lie isotropy group at x,
Φxx = {h ∈ Φ : αh = βh = x}) with the projection βx : Φx −→ M. We
have A (Φ)|x = Tux (Φx), the tangent space to the total space Φx at the
unit x. For all pfb’s Φx we can consider standard differential operators,
like the exterior derivative of usual differential forms (or Lie derivative
and substitution operator), and pass to the units ux and "glue". By this
procedure we obtain just dA.
Example 5. (1) If A = A (P ) = TP/G for a G-principal fibre bundle
P −→M then
Ω (A) ∼= ΩR (P ) ↪→ Ω (P ) ,
ΩR (P ) are G-right invariant differential forms on P and
H (A) ∼= H
(
ΩR (P )
) i−→ HdR (P ) .
The homomorphism i is an isomorphism if G is compact and connected.
(2) If A = A (M ;F) −→ W is the Lie algebroid of a TC-foliation F
on M (W is the so called basic manifold of the foliation F), then [K3,
Th. 6.2]
Ω (A) ∼= Ωb (M ;F) ,
Ωb (M ;F) is the algebra of F-basic differential forms, therefore H (A) ∼=
Hb (M ;F) is the algebra of basic cohomology.
Below, we will propose a calculation of H (A) using the old technique
of Koszul complexes and the so-called Chevalley theorems known for prin-
cipal fibre bundles with structural Lie groups with reductive Lie algebras.
Jan Kubarski 103
These Chevalley theorems (for pfb’s) say that under some assumptions,
the cohomology of the total space H (P ) of a pfb P depends uniquely
on the cohomology of the base manifold M and the characteristic classes
(the Chern-Weil homomorphism hP :
(∨
g∗
)
IG
−→ H (M)). It turns
out that this assertion has a counterpart for Lie algebroids.
3 Koszul complexes and Chevalley’s theorem
in the framework of Lie algebroids
3.1 Representations of Lie algebroids and invariant
cross-sections
Consider an arbitrary transitive Lie algebroid A on a manifoldM with
the Atiyah sequence 0 −→ ggg −→ A
#A−→ TM −→ 0 and a vector bundle f
on M.
Definition 6. By a representation of A on f we mean a homomorphism
of Lie algebroids
T : A −→ A (f) .
Look at the induced homomorphism of Atiyah sequences:
0 0
↓ ↓
ggg
T+
−→ End (f)
↓ ↓
A
T−→ A (f)
↓ ↓
TM = TM
↓ ↓
0 0
At each point x we get a representation of the isotropy Lie algebra ggg|x
on the vector space f|x,
T+
x : ggg|x −→ End
(
f|x
)
.
For a cross-section ξ ∈ SecA its image Tξ ∈ Sec A (f) determines a
covariant differential operator
LTξ : Sec f −→ Sec f.
104 Koszul complexes and Chevalley’s theorems for Lie algebroids
Example 7. (The representation of a Lie algebroid induced by a repre-
sentation of a pfb) Let G be any Lie group and µ : G −→ GL (V ) be
any representation of G on a vector space V . If F : P −→ L f is a µ-
homomorphism of pfb’s (called a µ-representation of P on f) then its Lie
algebroid’s differential F∗ : A (P ) −→ A (f) is a representation of A (P )
on f.
Definition 8. A cross-section ν ∈ Sec f is called T -invariant (or T -
parallel) if it belongs to the kernel of LTξ for each ξ, i.e.
LTξ (ν) = 0 for all ξ ∈ SecA.
The space of all T -invariant cross-sections is denoted by (Sec f)IT . If
ν ∈ Sec f is invariant then its value νx at x is invariant with respect to
T+
|x : ggg|x → End
(
f|x
)
, i.e.
νx ∈
(
f|x
)
I
T
+
x
.
One can prove that for each transitive Lie algebroid A and each rep-
resentation T : A −→ A (f) the following theorem holds.
Theorem 9. If ν1 and ν2 are T -invariant cross-sections of f and they
are equal at some point x0 ∈ M, ν1 (x0) = ν2 (x0), then they are equal
globally, ν1 = ν2 (M is assumed to be connected), see [M], [K2].
Therefore, the evaluation map
(Sec f)IT −→
(
f|x
)
I
T
+
x
, ν 7−→ ν (x) ,
is a monomorphism. Denote its image by(
f̃|x
)
I
T
+
x
;
it contains all invariant vectors u ∈
(
f|x
)
I
T
+
x
which can be extended to
globally defined invariant cross-sections, i.e.
(Sec f)IT
∼=
(
f̃|x
)
I
T
+
x
⊂
(
f|x
)
I
T
+
x
.
Moreover, each invariant vector u ∈
(
f|x
)
Io(T+
x ) can be extended to a
locally defined (on some neighbourhood of x) invariant cross-section of
the vector bundle f.
Jan Kubarski 105
There is a wider class of Lie algebroids (integrable and nonintegrable)
and representations where each invariant vector u ∈
(
f|x
)
I
T
+
x
can be
extended to globally defined invariant cross-sections.
Theorem 10 ([K1]). Let P be a connected G-principal fibre bundle (G
can be disconnected) and let F : P −→ L f be any µ-representation of P
on f where µ : G −→ GL (V ) is a representation of G on V. Denote by
µ∗ : g −→End (V ) the differential of µ (it is a representation of the Lie
algebra g of G on V ). Then for the induced representation F∗ : A (P ) −→
A (f) of the Lie algebroid A (P ) on f we have
(Sec f)IF∗
∼= VI(µ) ⊂ VI(µ∗)
∼=
(
f|x
)
I
F
+
∗x
.
If additionally G is connected then each invariant vector v ∈
(
f|x
)
I
F
+
∗x
(with respect to the representation F+
∗|x) can be extended to a globally
defined F∗-invariant cross-section of f and
(Sec f)IF∗
∼= VI(µ) = VI(µ∗)
∼=
(
f|x
)
I
F
+
∗x
.
If G is not connected then there may be invariant vectors which some-
times extend to global cross-sections and sometimes not (the Pfaffian is
a typical example).
A representation T : A −→ A (f) extends to representations on the
associated vector bundles such as the dual bundle f∗, the exterior and
symmetric powers
∧
f∗,
∨l
f∗ and their tensor products
∧
f∗ ⊗
∨l
f∗.
3.2 Weil algebra for Lie algebroids [K1]
A fundamental example of a representation is the adjoint representa-
tion of A on the adjoint Lie algebra bundle ggg defined by
adA : A −→ A (ggg) ,
adA (ξ) : Secggg −→ Secggg, ν 7−→ [[ξ, ν]].
Clearly the induced representation at an arbitrary point x,
(
ad+
A
)
|x,
is the adjoint representation of the Lie algebra ggg|x,(
ad+
A
)
|x = adggg|x : ggg|x −→ End
(
ggg|x
)
.
106 Koszul complexes and Chevalley’s theorems for Lie algebroids
The adjoint representation adA induces representations on the associated
vector bundles
∧
ggg∗,
∨l
ggg∗ (the skew symmetric and symmetric powers
of the dual bundle ggg∗) and on
(Wggg)
k,2l
:=
∧
ggg∗ ⊗
∨l
ggg∗,
denoted also by adA. Put
(Wggg)
k,2l
= Sec (Wggg)
k,2l
,
Wggg=
⊕
k,l
Sec (Wggg)
k,2l
.
For a point x ∈M we take the anticommutative (bi)graded tensor prod-
uct of anticommutative graded algebras, i.e. the Weil algebra of the space
ggg|x,
Wggg|x =
∧
ggg∗|x
⊗∨
ggg∗|x,
Wggg|x =
⊕
k.l
(
Wggg|x
)k,2l
,
(
Wggg|x
)k,2l
=
∧k
ggg∗|x
⊗∨l
ggg∗|x.
The module Wggg is a bigraded algebra with multiplication defined
pointwise, called the Weil algebra of the Lie algebroid A.
In the space Wggg|x =
∧
ggg∗|x
⊗∨
ggg∗|x (as for an arbitrary Lie alge-
bra) there exist three standard operators: the substitution operator, the
differential, and the adjoint representation, here denoted by
(ιx)ν , δWx , (θx)ν , ν ∈ ggg|x.
It is easy to see that the adjoint representation θk,2lx : ggg|x →
End
(
Wggg|x
)k,2l
is induced by the adjoint representation adA of the Lie
algebroid A on (Wggg)
k,2l
=
∧k
ggg∗
⊗∨l
ggg∗, k, l ≥ 0, at a point x.
We have:
(a) (ιν)x is an antiderivation of degree −1 defined by
(ιν)x (Φ⊗ Γ) = (ιν)x Φ⊗ Γ,
Φ ∈
∧
ggg∗|x, Γ ∈
∨
ggg∗|x,
Jan Kubarski 107
(b) δWx
is an antiderivation of degree +1 defined by
δWx
(h∗ ⊗ 1) = 1⊗ h∗ + δggg|xh
∗ ⊗ 1,
where h∗ ∈ ggg∗|x, δggg|x is the Chevalley-Eilenberg differential
δWx
(1⊗ h∗) ∈ (Wggg)
1,2
= ggg∗ ⊗ ggg∗,
such that
(ιν)x (δWx
(1⊗ h∗)) = (θx)ν h
∗.
The operators (ιx)ν , δWx
, (θx)ν , x ∈ M, together give operators on
smooth cross-sections
ιν , δW , θν :Wggg −→Wggg, ν ∈ Secggg.
The cross-section Θ ∈ Wggg is called horizontal if ινΘ = 0 for all
ν ∈ Secggg. Denote by
(Wggg)ι
the space of horizontal elements.
Lemma 11. The space (Wggg)ι of horizontal elements is a subalgebra of
the Weil algebra Wggg and contains only symmetric tensors:
(Wggg)ι =
⊕
l
Sec
∨l
ggg∗.
Denote the space of global cross-sections of the vector bundle
(Wggg)
k,2l
=
∧k
ggg∗
⊗∨l
ggg∗
invariant with respect to the adjoint representation of A on (Wggg)
k,2l (for
brevity) by
(Wggg)
k,2l
Io ⊂ (Wggg)
k,2l
and put
(Wggg)Io =
⊕
k,l
(Wggg)
k,2l
Io ⊂ Wggg.
Proposition 12. (Wggg)Io is a subalgebra of the Weil algebra Wggg. De-
note by (Wggg)Io,ι the subalgebra of invariant and horizontal elements of
the Weil algebra Wggg. The operator δW : Wggg −→ Wggg maps invariant
elements of Wggg into invariant ones defining an antiderivation
δW,Io : (Wggg)Io −→ (Wggg)Io ,
and
δW,Io | (Wggg)Io,ι = 0.
108 Koszul complexes and Chevalley’s theorems for Lie algebroids
3.3 Connections and the Chern-Weil homomorphism
of Lie algebroids
Definition 13. By a connection in a (transitive) Lie algebroid A we
mean a splitting ∇ : TM −→ A of the Atiyah sequence,
0 −→ ggg −→ A −→
←−
∇
TM → 0.
If A = A (P ) is the Lie algebroid of a G-principal fibre bundle
P (M,G) then connections in A (P ) correspond 1-1 to usual connections
in P.
Fix an arbitrary connection ∇ in A and consider:
a) the connection form ω : A −→ ggg, i.e. the 1-form on A with values
in ggg (ω|ggg = Id and kerω = Im∇),
ω ∈ Ω1 (A;ggg) ,
b) the curvature form of ∇,
Ω ∈ Ω2 (A;ggg) ,
defined by
Ω (ξ1, ξ2) = ω[[Hξ1, Hξ2]], ξ1, ξ2 ∈ SecA,
where H = Id− ω : A −→ A is the horizontal projection,
c) the identification Ω (A) = Ω
(
M ;
∧
ggg∗
)
.
For each point x ∈ M the mappings ω|x : A|x −→ ggg|x and Ω|x :∧2
A|x −→ ggg|x determine linear mappings
χω,x : ggg∗|x −→ A∗|x ⊂
∧
A∗|x, h∗ 7−→ h∗ ◦ ω|x,
and
χΩ,x : ggg∗|x −→
∧2
A∗|x ⊂
∧
A∗|x, h∗ 7−→ h∗ ◦ Ω|x.
By the universal properties of the exterior algebra
∧
ggg∗|x and the sym-
metric algebra
∨
ggg∗|x we obtain the existence and uniqueness of homo-
morphisms of algebras of degree 0, extending the above ones,
χ∧ω,x :
∧
ggg∗|x −→
∧
A∗|x,
χ∨Ω,x :
∨
ggg∗|x −→
∧ev
A∗|x
Jan Kubarski 109
(such that 1 7−→ 1). The above morphisms define a homomorphism of
algebras
χW,x : Wggg|x =
∧
ggg∗|x
⊗∨
ggg∗|x −→
∧
A∗|x,
χW,x (Φx ⊗ Γx) = χ∧ω,x (Φx) ∧ χ∨Ω,x (Γx) .
Passing to smooth cross-sections we obtain homomorphisms of algebras
χ∧ω : Sec
∧
ggg∗ −→ Ω (A) ,
χ∨Ω :
⊕l
Sec
∨l
ggg∗ −→ Ωev (A) ,
and
χW :Wggg→Ω (A)
χW (Φ⊗ Γ) = χ∧ω (Φ) ∧ χ∨Ω (Γ) .
Following [G-H-V, Vol. III, p. 341], χW is called the classifying homo-
morphism corresponding to the connection ∇.
One can prove that for Γ ∈ Sec
∨l
ggg∗,
χ∨Ω (Γ) =
1
k!
〈Γ,Ω ∨ · · · ∨ Ω︸ ︷︷ ︸
l times
〉
(the notation Ω∨· · ·∨Ω comes from [G-H-V, Vol. II], it is the usual skew
multiplication of differential forms whose values are multiplied according
to the multilinear symmetric mapping ∨ : ggg × · · · × ggg −→
∨l
ggg).
Theorem 14. (a) The classifying homomorphism χW commutes with
the substitution operators ιν , ν ∈ Secggg:
ιν (χWΘ) = χW (ινΘ) .
(b) The homomorphism χW,Io : (Wggg)Io −→ Ω (A), the restriction of
χW to the invariant elements, commutes with the differentials δW,Io and
dA:
dA (χW,IoΘ) = χW,Io (δW,IoΘ) .
As a simple consequence we obtain the Chern-Weil homomor-
phism of the Lie algebroid A. Consider the restriction χW,Io,ι of
χW : Wggg −→ Ω (A) to the horizontal invariant elements. Since
110 Koszul complexes and Chevalley’s theorems for Lie algebroids
δW,Io | (Wggg)Io,ι = 0 we see that all differential forms in ImχW,Io,ι are
closed and horizontal:
χW,Io,ι : (Wggg)Io,ι → Zι (A) ;
on the other hand, (Wggg)ι =
⊕
l
Sec
∨l
ggg∗, therefore
(Wggg)Io,ι =
⊕
l
(
Sec
∨l
ggg∗
)
Io
and Ω (M)
f∼= Ωι (A) (via the anchor f (Ψ)x (v1, ..., vk) =
(Ψ)x (#v1, ...,#vk)) and
χW,Io,ι : (Wggg)Io,ι −→ Zι (A)
‖ ‖
hA :
⊕
l
(
Sec
∨l
ggg∗
)
Io
−→ Z (M) −→ H (M)
3.4 Koszul complexes and Chevalley’s theorem in the
framework of Lie algebroids
We now apply the technique of Koszul complexes and Chevalley’s
theorem [G-H-V, Vol. III] to Lie algebroids. We recall that the adjoint
representation adA of A on (Wggg)
k,2l
=
∧k
ggg∗
⊗∨l
ggg∗ determines at each
point x the adjoint representation θk,2lx : ggg|x −→ End
(
Wggg|x
)k,2l
, which
together determine the representation on the Weil algebra θx : ggg|x −→
End
(
Wggg|x
)
. Denote by
(
W̃ggg|x
)
Iθx
the subspace of
(
Wggg|x
)
Iθx
consist-
ing of all vectors whose homogeneous parts can be extended to globally
defined cross-sections of (Wggg)
k,2l
=
∧k
ggg∗
⊗∨l
ggg∗ invariant with respect
to the adjoint representation of the Lie algebroid A,
(Wggg)Io
∼=
(
W̃ggg|x
)
Iθx
⊂
(
Wggg|x
)
Iθx
.
We assume the following (rather strong) assumptions:
(A1) the isotropy Lie algebras ggg|x are reductive,
Jan Kubarski 111
(A2) each homogeneous invariant element
Θx ∈
(
Wggg|x
)k,2l
Iθx
=
(∧k
ggg∗|x
⊗∨l
ggg∗|x
)
Iθx
can be extended to a globally defined invariant cross-section of the
vector bundle
∧k
ggg∗
⊗∨l
ggg∗, i.e.
(Wggg)Io
∼=
(
W̃ggg|x
)
Iθx
=
(
Wggg|x
)
Iθx
.
(In particular, the cohomology vector bundle H (ggg) , H (ggg)x =
H
(
ggg|x
)
, is trivial).
Now we return to an arbitrarily chosen connection ∇ in the Lie alge-
broid A, 0 −→ ggg −→ A −→
←−
∇
TM −→ 0, and take χW : Wggg −→ Ω (A),
the classifying homomorphism corresponding to the connection ∇, and
its restriction to the invariant elements,
χW,Io : (Wggg)Io
∼=
(
W̃ggg|x
)
Iθx
=
(
Wggg|x
)
Iθx
−→ Ω (A) .
Now we use the assumed reductivity of the isotropy Lie algebras ggg|x. Let
Px ⊂
(∧
ggg∗|x
)
Iθx
be the graded primitive subspace. We recall that homogeneous primitive
elements have odd degree (which implies that Φ ∧ Φ = 0 when Φ ∈ Px),
therefore the inclusion Px ⊂
(∧
ggg∗|x
)
Iθx
extends to a homomorphism of
algebras
κx :
∧
Px −→
(∧
ggg∗|x
)
Iθx
.
The Hopf-Samelson theorem [G-H-V, Vol. III, 5.18, Theorem III] says
that if ggg|x is reductive then κx is an isomorphism of graded algebras.
Further
τx : Px −→
(∨+
ggg∗|x
)
Iθx
denotes a fixed transgression in
(
Wggg|x
)
Iθx
, i.e. a linear mapping such
that
112 Koszul complexes and Chevalley’s theorems for Lie algebroids
(1) τx is homogeneous of degree +1, τx : P 2r−1
x −→
(∨
ggg∗|x
)2r
Iθx
=(∨r
ggg∗|x
)
Iθx
,
(2) for each Φ ∈ Px there exists Ω ∈W+
(
ggg|x
)
Iθx
such that
δWx
Ω = 1⊗ τxΦ and Ω− Φ⊗ 1 ∈
(∧
ggg∗|x ⊗
∨j≥1
ggg∗|x
)
Iθx
.
It turns out that we can demand that Ω depends linearly on Φ, and Φ
and Ω are of the same degree, i.e. that there exists a linear mapping
αx : Px −→W+
(
ggg|x
)
Iθx
,
homogeneous of degree 0, such that
(*) δWx
(αxΦ) = 1⊗ τx (Φ) ,
(**) αx (Φ)− Φ⊗ 1 ∈
(∧
ggg∗|x ⊗
∨j≥1
ggg∗|x
)
Iθx
.
In the following we fix such a mapping αx. Now we can define a Koszul
complex for the Lie algebroid. To this end we recall the homomorphism
χW,Io,ι :=
(∨
ggg∗|x
)
Iθx
∼=
⊕
l
(
Sec
∨l
ggg∗
)
Io
−→ Zι (A) ∼= Z (M)
(Z (M) = closed differential forms on M),
(after passing to cohomology, this yields the Chern-Weil homomorphism
of A). Composing it with the transgression τx : Px −→
(∨+
ggg∗|x
)
Iθx
we
obtain
τA : Px −→
(∨
ggg∗|x
)
Iθx
−→ Z (M) ⊂ Ω (M) .
Definition 15. In the skew tensor product of the graded algebras
Ω (M)⊗
(∧
ggg∗|x
)
Iθx
= Ω (M)⊗
∧
Px
we introduce the operator
∇A : Ω (M)⊗
∧
Px → Ω (M)⊗
∧
Px
uniquely determined by the conditions:
Jan Kubarski 113
(1) ∇A (z ⊗ 1) = d (z)⊗ 1, /d the de Rham differential
(2) ∇A (z ⊗ (Φ0 ∧ ... ∧ Φp)) = dz ⊗ (Φ0 ∧ ... ∧ Φp) +
(−1)
q
∑p
i=0
(−1)
i
τA (Φi) ∧ z ⊗
(
Φ0 ∧ ...̂i... ∧ Φp
)
, z ∈ Ωq (M) ,
Φi ∈ Px. In particular ∇A (z ⊗ Φ) = dz ⊗ Φ + (−1)
q
τA (Φ) ∧ z ⊗ 1 and
∇A (1⊗ Φ) = τA (Φ)⊗ 1.
Lemma 16. The operator ∇A is an antiderivation of square 0, homoge-
neous of degree +1.
Definition 17. The pair
(
Ω (M)⊗
∧
Px,∇A
)
is called the Koszul com-
plex of the Lie algebroid A.
We see that the Koszul complex for a Lie algebroid depends only on
the base manifold and the Chern-Weil homomorphism of A.
Now we define a Chevalley homomorphism. Take the restriction of the
classifying homomorphism χW :Wggg −→ Ω (A) to the invariant tensors,
χW,Io : (Wggg)Io
∼=
(
W̃ggg|x
)
Iθx
=
(
Wggg|x
)
Iθx
−→ Ω (A) .
Composing it with the mapping αx : Px →W+
(
ggg|x
)
Iθx
⊂
(
Wggg|x
)
Iθx
,
Px
αx−→W+
(
ggg|x
)
Iθx
⊂
(
Wggg|x
)
Iθx
∼= (Wggg)Io
χW,Io−→ Ω (A, )
we obtain a linear mapping homogeneous of degree 0,
ϑA : Px −→ Ω (A) .
Hence, since Ω (A) is anticommutative and P kx = 0 for even k, ϑA extends
to a homomorphism of graded algebras
ϑ∧A :
∧
Px −→ Ω (A) .
Finally, we extend ϑ∧A :
∧
Px −→ Ω (A) to a homomorphism of graded
algebras
ϑA : Ω (M)⊗
∧
Px −→ Ω (A)
by setting
ϑA (z ⊗ Φ) = #∗A (z) ∧ ϑ∧A (Φ)
(#∗A (z) is the pull back, via the anchor #A, of the differential form
z ∈ Ω (M) to a horizontal one on the Lie algebroid A).
114 Koszul complexes and Chevalley’s theorems for Lie algebroids
Definition 18. The homomorphism ϑA : Ω (M) ⊗
∧
Px −→ Ω (A) is
called the Chevalley homomorphism of A associated with the connection
∇ and the linear map αx.
Theorem 19 (The fundamental theorem). (A) The Chevalley homo-
morphism ϑA is a homomorphism of graded differential algebras
ϑA :
(
Ω (M)⊗
∧
Px,∇A
)
−→ (Ω (A) , dA) .
(B) Under the assumptions (A1) and (A2), i.e. that the isotropy Lie
algebras ggg|x are reductive, and
(
W̃ggg|x
)
Iθx
=
(
Wggg|x
)
Iθx
, the induced
homomorphism in cohomology
ϑ#
A : H
(
Ω (M)⊗
∧
Px,∇A
)
−→ H (A)
is an isomorphism of graded algebras.
Proof. (A) It is sufficient to check the equality dA ◦ ϑA = ϑA ◦ ∇A on
simple tensors z ⊗ 1 and 1⊗ Φ (Φ ∈ Px) only. We have
dA ◦ ϑA (z ⊗ 1) = dA (#∗Az) = #∗A (dz) = ϑA (dz ⊗ 1) = ϑA ◦ ∇A (z ⊗ 1) ,
and
dA ◦ ϑA (1⊗ Φ) = dA (ϑ∧A (Φ)) = dA (χW,Io (αx (Φ)))
Th (14)
= χW,Io (δW,Io (αx (Φ))) = χW,Io (δWx
(αx (Φ)))
(*)
= χW,Io (1⊗ τx (Φ)) = χW,Io (τx (Φ)) = #∗A (τAΦ)
= ϑA (τAΦ⊗ 1) = ϑA ◦ ∇A (1⊗ Φ) .
(B) The proof is analogous to that in the classical case for principal
fibre bundles [G-H-V, Vol. III, 9.3-4, p. 359]: we use some spectral
sequences and the comparison theorem for the first terms (the mapping
induced on the first terms is an isomorphism).
Step 1. Filtrations: For a given Lie algebroid A with the Atiyah
sequence 0 −→ ggg −→ A −→ TM −→ 0 we consider the pair of real
(infinite dimensional) Lie algebras (Γ (A) ,Γ (ggg)) of global cross-sections
of A and ggg.
Jan Kubarski 115
Following [H-S], [K-M-2], we introduce the Hochschild-Serre filtration
in Ω (A)j in Ω (A) as follows:
Ω (A)j =
{[
Ω (A) for j ≤ 0,⊕
k≥j Ω (A)
k
j for j > 0.
]
where Ω (A)
k
j consists of all those k-differential forms z ∈ Ωk (A) for
which
z (ξ1, ..., ξk) = 0
whenever k − j + 1 of the arguments ξi ∈ Γ (A) belong to Γ (ggg) . In this
way we obtain a graded filtered differential space and its spectral sequence(
Ej,iA,s, dA,s
)
.
Analogously, following [G-H-V, Vol. III] we introduce in the space
Ω (M)⊗
∧
Px the filtration(
Ω (M)⊗
∧
Px
)
j
=
⊕
k≥j
Ω (M)
k ⊗
∧
Px.
We obtain a graded filtered differential space and its spectral sequence(
Ej,is , ds
)
.
Step 2. We show that the Chevalley homomorphism ϑA is filtration
preserving. Firstly we notice that ϑA (z ⊗ 1) = #∗A (z) and ϑA (1⊗ Φ)−
χW,Io (Φ⊗ 1) ∈ Ω (A)1 . The first statement is obvious. To prove the
second, it is sufficient to consider the case Φ ∈ Px. According to (**)
above it follows that
ϑA (1⊗ Φ)− χW,Io (Φ⊗ 1) = χW,Io (αxΦ)− χW,Io (Φ⊗ 1) (1)
= χW,Io (αxΦ− Φ⊗ 1) ∈ Ω (A)1 .
By definition, ϑA
[
Ω (M)
k ⊗ 1
]
⊂ Ω (A)k . Since
(
Ω (M)⊗
∧
Px
)
j
is the
ideal generated by
⊕
k≥j Ω (M)
k ⊗ 1, and since Ω (A)j is an ideal, this
implies that ϑA preserves filtrations.
Step 3. We show that the mapping of the first terms of the spectral
sequences,
ϑA,1 : E1 −→ EA,1,
is an isomorphism. In view of the Comparison Theorem the induced
homomorphism in cohomology ϑ#
A : H
(
Ω (M)⊗
∧
Px,∇A
)
−→ H (A)
is an isomorphism.
116 Koszul complexes and Chevalley’s theorems for Lie algebroids
We start by calculating the differential operators d0 in E0 and dA,0 in
EA,0. It is immediate from the definitions that
∇A : Ω (M)
k ⊗
∧l
Px −→
(
Ω (M)⊗
∧
Px
)
k+1
, k, l ≥ 0.
It follows that d0 = 0. On the other hand, recall from [K-M-2, Conclusion
5.2] that EjA,0 = Ωj
(
M ;
∧
ggg∗
)
and that the differential dA,0 becomes the
Chevalley-Eilenberg differential of values at each point.
Now, we show that ϑA,0 : E0 −→ EA,0 simply comes from the inclu-
sion map
j : Ω (M)⊗
∧
Px = Ω (M)⊗
(∧
ggg∗|x
)
Iθx
−→ Ω
(
M ;
∧
ggg∗
)
and its values are dA,0-closed. In fact, j is homogeneous of bidegree zero.
Thus we need only show that
ϑA − j : Ωk (M)⊗
∧
Px −→ Ω (A)k+1 .
But j (z ⊗ Φ) = #∗A ∧ χW,Io (Φ⊗ 1) , and so property (1) yields, for z ∈
Ωk (M),
(ϑA − j) (z ⊗ Φ) = #∗Az ∧ (ϑA (1⊗ Φ)− χW,Io (Φ⊗ 1)) ∈ Ω (A)k+1 .
To prove Step 3 we need only show that (ϑA,0)
#
: H (E0, d0) −→
H (EA,0, dA,0) is an isomorphism. In view of the formulae for d0 and dA,0
it remains to show that the inclusion map j induces an isomorphism
j# : Ω (M)⊗
∧
Px = Ω (M)⊗
(∧
ggg∗|x
)
Iθx
−→
(
Ω
(
M ;
∧
ggg∗
)
, dA,0
)
.
Since the Lie algebras ggg|x are reductive (assumption (A1)), by the
structural theorem for reductive Lie algebras [G-H-V, Vol. III, s.
5.12, Theorem 1] we have
(∧
ggg∗|x
)
Iθx
= H
(
ggg|x
)
. Therefore, the iso-
morphism property of j# follows immediately from assumption (A2):(
Ω
(
M ;
∧
ggg∗
)
, dA,0
)
= Ω (M ; H (ggg)) = Ω
(
M ; H
(
ggg|x
))
. The proof of
the fundamental theorem is now complete.
Problem 20. What can we do in the case when
(
W̃ggg|x
)
Iθx
(
(
Wggg|x
)
Iθx
to calculate H (A) ? [The simplest examples of this case come from con-
nected pfb’s with disconnected structural Lie groups].
Jan Kubarski 117
References
[G-H-V] W. Greub, S. Halperin, R. Vanstone, Connections, Curvature,
and Cohomology, Vol. II 1973, Vol. III, 1976, New York and
London.
[H-S] G.Hochschild, J.-P.Serre, Cohomology of Lie algebras, Ann.
Math. 57, 1953, 591-603.
[I-K-V] V. Itskov, M. Karashev and Y. Vorobjev: Infinitesimal Poisson
Cohomology, Amer. Math. Soc. Transl. (2), Vol. 187, (1998).
[K1] J.Kubarski, The Chern-Weil Homomorphism of regular Lie Al-
gebroids, UNIVERSITE CLAUDE BERNARD – LYON 1, Pub-
lications du Départment de Mathématiques, nouvelle série, 1991,
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|
| id | oai:trim.imath.kiev.ua:article-278 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:04:50Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a6/806b0ffe015d67c424afdecd8e557ea6.pdf |
| spelling | oai:trim.imath.kiev.ua:article-2782018-02-10T20:56:26Z Koszul complexes and Chevalley's theorems for Lie algebroids Комплекс Кошуля і теореми Шевалле для алгеброїдів Лі Kubarski, J. Kubarski, J. We use Koszul complexes and Chevalley-type theorems to calculate the&nbsp;cohomology $\mathbf{H} \left( A\right) $ of a transitive Lie&nbsp;algebroid $A$ under some assumptions on the isotropy Lie algebras. Ми використовуємо комплекси Кошуля та теореми типу Шевалле для обчислення&nbsp;когомологій $ \mathbf {H} \left (A \right) $ транзитивих алгеброїдів Лі $ A $ при деяких припущеннях на&nbsp;ізотропії алгебр Лі. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/278 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 97-117 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 97-117 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 97-117 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/278/282 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Kubarski, J. Kubarski, J. Koszul complexes and Chevalley's theorems for Lie algebroids |
| title | Koszul complexes and Chevalley's theorems for Lie algebroids |
| title_alt | Комплекс Кошуля і теореми Шевалле для алгеброїдів Лі |
| title_full | Koszul complexes and Chevalley's theorems for Lie algebroids |
| title_fullStr | Koszul complexes and Chevalley's theorems for Lie algebroids |
| title_full_unstemmed | Koszul complexes and Chevalley's theorems for Lie algebroids |
| title_short | Koszul complexes and Chevalley's theorems for Lie algebroids |
| title_sort | koszul complexes and chevalley's theorems for lie algebroids |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/278 |
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