Table Of ContentEXTENSIONS OF LIE BRACKETS.
OLIVIERBRAHIC
Abstract. We provide a framework for extensions of Lie algebroids,
includingnon-abelianextensionsandLiealgebroidsoverdifferentbases.
0 Our approach involves Ehresmann connections, which allows straight
1 generalizations of classical constructions. We exhibit a filtration in co-
0 homology and explain the associated spectral sequence. We also give a
2 description of the groupoid integrating an extension in case a complete
connection exists. The integrability is also studied.
n
a
J
8
1
Introduction
]
G
The concept of Lie algebroid proved to be relevant in diverse fields of
D
geometry, with interesting applications in physics as well. However, despite
.
h intensive studies, the notion of extension was never systematically investi-
t gated. It is the aim of this paper to fill this gap. We study extensions of
a
m Lie algebroids from a general perspective, the interest is to get a unified
[ treatment for a large variety of situations.
5 The basic setting will be a surjective Lie algebroid morphism π : AE →
v AB covering a surjective submersion p : E → B where AE and AB are Lie
2 algebroids over E and B, respectively. In this situation, we denote K the
6
kernel of π. It is a Lie algebroid over E and we obtain this way A as an
E
4
extension of A by K:
1 B
0. K ֒→ AE ։ AB.
1
8 In Section 1, we recall basic notion concerning Lie algebroids and their
0 extensions. ThenweintroduceinSection2thenotionofEhresmannconnec-
:
v tion for an extension: it is given by a sub-vector bundle H complementary
Xi to K in AE:
r A = K⊕H.
a E
Such a connection always exists and we can construct the usual horizontal
lifting, it is a C∞(B)-linear application h : Γ(A ) → Γ(H). From these
B
notions, we derive a couple (D,ω) where:
• D : Γ(A ) → Der(K) is a C∞(B)-linear map with values in deriva-
B
tions of K, such that the symbol s of D p -projects onto ♯ (α)
Dα α ∗ B
for any α∈ Γ(A );
B
• ω isa2-formonA withvaluesinsectionsofK: ω ∈ Ω2(A )⊗Γ(K).
B B
Partially supported by the Fundac¸˜ao para a Ciˆencia e a Tecnologia through
the Program POCI 2010/FEDER and by the Projects POCI/MAT/57888/2004 and
POCI/MAT/55958/2004.
1
2 OLIVIERBRAHIC
In order to define an extension, the couple (D,ω) is submitted to the fol-
lowing compatibility conditions:
∂ ω = 0,
H
Curv = adK◦ω.
D
Here, ∂ :Ωk(A )⊗Γ(ΛlK) → Ωk+1(A )⊗Γ(ΛlK) is a covariant derivative
H B B
associated to D by usual formulas and Curv (α,β) := [D ,D ]−D .
D α β [α,β]
The description of extensions we get this way lies in-betwen usual treat-
ments for both Lie algebras and fibrations:
Theorem. Let p : E → B be a submersion, K → E a Lie algebroid whose
characteristic foliation lies in Vert and A → B a Lie algebroid.
B
Extensions of A by K covering p are (up to equivalence) classified by
B
equivalence classes of admissible couples (D,ω) under the relation (D,ω) ∼
(D′,ω′) if there exists ∆ ∈ Ω1(A )⊗Γ(K) such that
B
D′ = D+adK◦∆,
ω′ = ω+∂ ∆+[∆∧∆] ,
H K
where [∆∧∆] ∈ Ω2(A )⊗Γ(K) is defined by:
K B
[∆∧∆] (α,β) = [∆(α),∆(β)] .
K K
Besides, we explain that given a connection, parallel transport along an
A -path a ∈ P(A ) is well defined (see subsection 2.1) and, in the case of
B B
a complete connection, we obtain the usual notion of holonomy along a as
a Lie algebroid morphism
K| Φa // K| .
E E
γ(0) γ(1)
Here, γ :I → B is the base path of a. In general, Φ does not factor through
A -homotopies unless Curv vanishes. Note however the relation Curv =
B D D
adK◦ω suggests that this can by measured by inner automorphisms of K,
this will play an important role. In any case, one can still think of Φ as a
groupoid morphism Φ : P(A ) → Gauge(K) where
B
Gauge(K) := Lie algebroid morphisms Ψ : K| → K|
y,x Ex Ey
(cid:8) (cid:9)
and P(A ) are both groupoids over B.
B
InSection3welookatthecohomology ofanextension. Thereisanatural
filtration, and the choice of a connection allows a rather nice description of
the associated spectral sequence.
The Section 4 is devoted to the description of the Weinstein groupoid
G(A ). We obtain G(A ) as a quotient of P(A ) ⋉ G(K). Here P(A )
E E B B
acts on G(K) by means of holonomy, and P(A )⋉G(K) corresponds to an
B
associated groupoid over E. Theprecise statement is given by Theorem 4.5:
Theorem. Consider a Lie algebroid extension K ֒→ A ։ A endowed
E B
with a Ehresmann connection A = K⊕H, which we assume to be complete.
E
Then the topological source simply connected groupoid G(A ) integrating
E
A is naturally identified with the quotient
E
P(A )⋉ G(K)/∼,
B B
EXTENSIONS OF LIE BRACKETS. 3
where the equivalence relation is given by: (a ,g )∼(a ,g ) if and only if
0 0 1 1
there exists a A -homotopy h = adt+bdǫ : TI2 7→ A between a and a
B B B 0 1
such that:
g ·g−1 = ∂(h ,t(g )).
1 0 B 0
Here, ∂(h ,x ) is the element in G(K) represented by the K-path:
B 0
1
aǫ
ǫ → Z (Φ0,Bs)∗ ω(a,b)s,ǫ ds ∈KγKǫ,
0 (cid:0) (cid:1)
where γǫ := φ−1◦φ (x ).
K aǫ a0 0
B B
In general, the sequence of topological groupoids
˜ι π˜
1 → G(K) −→ G(A )−→ G(A ) → 1.
E B
obtained by integration of a Lie algebroid extension might not be exact.
In order to measure this lack of exactness, we denote M the monodromy
groupoid of the extension. It is defined as the kernel of ˜ι. In the case
there exists a complete Ehresmann connection, we can build a connecting
homomorphismwhoseimage is precisely M, similarly to whatis well known
for fibrations:
Theorem. Let K ֒→ A ։ A be a Lie algebroid extension that admits a
E B
complete Ehresmann connection. Then there exists a homomorphism
∂ : π (A )⋉ E → G(K),
2 2 B B
that makes the following sequence exact:
··· → π (A )⋉E −∂→2 G(K) −→˜ι G(A )։ G(A ).
2 B E B
Here, π (A ) denote the second homotopy groups of A , as defined in
2 B B
the Appendix C. In the case A is integrable, it coincides with the union of
B
all second homotopy groups of the source fibers, seen as a bundle of groups
over B. In this work we do not try to put any topology on π (A ) whose
2 B
role is essentially algebraic.
TheinterestofthemonodromygroupoidMliesinthefactthatitcontrols
theintegrability ofA inthecase ofclean extensions. Byclean, wemeanby
E
definitionthattherestrictions ofπ totheisotropy Liealgebras ofA surject
E
onto those of A . This implies that, pointwise, one has exact sequences of
B
Lie algebras
(ker♯ ) ֒→ (ker♯ ) ։ (ker♯ ) .
K x E x B p(x)
In this case, we show that A is integrable provided A and K are, and if
E B
M is discrete near identities (see 4.17 for the precise definition).
Theorem. Let K ֒→ A ։ A be a clean Lie algebroid extension. Assume
E B
that it admits a complete Ehresmann connection and that both K and A
B
are integrable Lie algebroids.
Then A is integrable if M is discrete near identities in G(K).
E
Acknowledgments. I would like to thank Rui Loja Fernandes for the
many useful discussions and suggestions that helped to improve this work.
4 OLIVIERBRAHIC
Contents
Introduction 1
1. Generalities on Lie algebroids 4
1.1. Derivations. 5
1.2. Extensions of Lie algebroids 6
2. Ehresmann Connections 7
2.1. Parallel transport 8
2.2. Admissible couples 9
2.3. Isotropies 10
2.4. Semi-direct products 11
2.5. Abelian Extensions 11
2.6. Examples 11
3. Cohomology 13
3.1. The spectral sequence. 13
3.2. Interpretation 17
3.3. An example of computation 18
4. Integration 19
4.1. Splitting A -paths and A -homotopies 19
E E
4.2. The Monodromy Groupoid 24
4.3. Integrability 27
Appendix A. The Weinstein groupoid and A-homotopies 28
Appendix B. Fibered Lie groupoids 30
Appendix C. Spheres in a Lie algebroid 31
References 33
1. Generalities on Lie algebroids
We first briefly recall basic definitions concerning Lie algebroids [16] and
refer to [3], [13] for more details.
Definition 1.1. A Lie algebroid A → M is a vector bundle A over a
smooth manifold M, together with vector bundle homomorphism ♯ : A →
TM called the anchor, and a bracket [ , ] defined on the space Γ(A) of
A
sections ofAsuchthatforanysmoothfunctionf ∈C∞(M)andanysection
α,β ∈ Γ(A) of A, the following conditions hold
i) [α,fβ] = f[α,β] +L (f)β;
A A ♯(α)
ii) [α,β] = −[β,α] ;
A A
iii) ♯([α,β] )= [♯(α),♯(β)] ;
A TM
iv) [α,β] ,γ = 0.
I A A
α,β,γ(cid:2) (cid:3)
In other words, the bracket [ , ] induces a structure of real Lie algebra on
A
sections of A such that the Leibniz condition (i) holds and ♯ is a Lie algebra
homomorphism from Γ(A) to X(M) := Γ(TM).
The cohomology associated to a Lie algebroid is defined on the complex
of cochains Ck := Γ(ΛkA∗) which we might also denote Ωk(A) (or Ωk(M)
for short when A = TM). The differential operator is given by the usual
EXTENSIONS OF LIE BRACKETS. 5
formula
(1) hd c,α ...α i := (−1)iL hc,α ...αˆ ...α i
A 0 k ♯αi 0 i k
X
i=0...n
+ (−1)i+jhc,[α ,α ] ,α ...αˆ ...αˆ ...α i.
i j A 0 i j k
X
0≤i<j≤k
Definition 1.2. Given two Lie algebroids A → M and A → N, a Lie
M N
algebroid homomorphism Φ : A → A over φ : M → N is a vector
M N
bundle homomorphism Φ covering φ such that the map induced on forms:
Φ∗ :Ωk(A ) → Ωk(A ),
N M
defined by hΦ∗(u),ai = hu ,Φ(a)i, commutes with the differential:
φ(m)
d ◦Φ∗ = Φ∗◦d .
AM AN
When φ is a diffeomorphism, we shall denote Φ : Γ(A) → Γ(A) the
∗
application induced on sections, Φ (α) := Φ◦α◦φ−1. See for instance [12]
∗
for a detailed discussion on Lie algebroid morphisms.
Recall that the isotropy at a point x ∈ M is defined as the kernel of the
anchor, gA := ker♯ . Itis naturally endowed witha structureof Liealgebra.
x x
In general, the dimension of the isotropy algebra may vary from one point
to another, so all isotropies do not fit in a smooth bundle of Lie algebras in
general.
1.1. Derivations.
Definition 1.3. A derivation of a Lie algebroid A→ M is an application
D :Γ(A) → Γ(A),
togetherwithavectorfields ∈ X(M)(calledthesymbolofthederivation)
D
satisfying for any smooth function f ∈ C∞(M) and any two sections of A
α,β ∈ Γ(A)
D(fα)= fD(α)+s (f)α,
D
D([α,β] )= [D(α),β] +[α,D(β)] ,
A A A
♯(D(α)) = [s ,♯(α)] .
D TM
A derivation is the infinitesimal version of a Lie algebroid automorphism:
any derivation determines a vector field on A whose flow is a Lie algebroid
automorphism et vice versa (see for instance the appendix in [6]). We will
denote Der(A) the space of all derivations, it has a natural structure of Lie
algebra, whose bracket is given by the commutator [D ,D ] = D ◦D −
1 2 1 2
D ◦D .
2 1
Definition 1.4. An inner derivation is a derivation of the form D(α) =
[δ,α] for some smooth section δ ∈ Γ(A), in which case we will write D =
A
adA.
δ
Thereareobviousnotionsoftime-dependentderivationsandinnerderiva-
tions, corresponding to which we get time-dependent vector fields on A.
In particular, given a time-dependent section αt ∈ Γ(A), we shall denote
ψα : A→ A the flow on A associated to adA .
t,s αt
6 OLIVIERBRAHIC
1.2. Extensions of Lie algebroids. The notion of extension we will use
in this work is summarized below.
Definition 1.5. An extension of a Lie algebroid A is a surjective Lie
B
algebroid morphism π : A → A covering a surjective submersion p :E →
E B
B.
With this definition, K := ker π is a smooth sub-vector bundle of A
E
that, as proven below, turns out to be a Lie sub-algebroid of A . Thus, we
E
obtain an exact sequence of Lie algebroids:
K ֒→ A ։ A .
E B
We shall refer to A as an extension of A by K. As generic notations, we
E B
will write ♯ :A → TE, ♯ : A → TB, ♯ : K → TE and [ , ] , [ , ] ,
E E B B K AE AB
[ , ] the respective anchors and brackets. The isotropy algebras will be
K
denoted gE := ker(♯ ) , gB := ker(♯ ) , and gK := ker(♯ ) = gE ∩K, for
x E x b B b x K x x
any x ∈ E,b ∈ B.
Remark 1.6. Note that the injection K ⊂ A does not determine π (not
E
even p) so one should not think of A as a quotient of A by K.
B E
Definition 1.7. Given an extension π : A ։ A , we will refer to a π-
E B
projectable section µ of A any section µ such that, for any form ν ∈
E
Ω1(A ), the contraction hπ ◦ µ,νi is a basic function on E → B. This
B
means that π ◦µ induces a well defined section of A that we will denote
B
π(µ).
Lemma 1.8. Let µ ∈ Γ(A ) be a π-projectable section, then ♯ ◦µ is p -
E E ∗
projectable, moreover:
p (♯ (µ)) = ♯ (π(µ)).
∗ E B
Proof. Recalltheconditionforπ tobeaLiealgebroid morphism: d ◦π∗ =
AE
π∗◦d whereπ∗ : Ωk(A∗ ) → Ωk(A )is naturally inducedby π. For k = 0,
AB B E
this condition writes:
hd ◦π∗(f),µi = hπ∗◦d (f),µi (f ∈C∞(B),µ ∈ Γ(A ))
AE AB E
⇐⇒ hd(π∗f),♯ (µ)i = hd (f),π◦µi
E A
⇐⇒ hd(f ◦p),♯ (µ)i = hdf,♯ ◦π◦µi
E B
⇐⇒ hdf ◦dp,♯ (µ)i = hdf,♯ ◦π◦µi,
E B
which is `a priori an equality in C∞(E). However, if we assume µ to be
π-projectable, the right-hand term is in fact a basic function, which implies
that ♯ (µ) p -projects onto ♯ (π(µ)). (cid:3)
E ∗ B
Lemma 1.9. Let µ,ν ∈ Γ(A ) be π-projectable sections, then [µ,ν] is
E AE
also projectable, moreover:
π([µ,ν] ) = [π(µ),π(ν)] .
AE AB
Proof. Thisisanimmediateconsequenceoftheconditiond ◦π∗ = π∗◦d
AE AB
π∗ :Ωk(A )→ Ωk(A ) for k = 1 and of the preceding lemma. (cid:3)
B E
As a consequence of the two last lemmas, we can state the following.
EXTENSIONS OF LIE BRACKETS. 7
Proposition 1.10. Let π : A → A be a Lie algebroid extension. Then
E B
K := ker π is a Lie algebroid on E whose characteristic foliation lies in
Vert := ker p ⊂ TE.
∗
In particular, if we restrict the vector bundle K to a fiber E := p−1({x})
x
over any x ∈B, we obtain a nice Lie algebroid K| over E .
Ex x
2. Ehresmann Connections
We now introduce a notion of connection [9] for extensions of Lie alge-
broids.
Definition 2.1. An Ehresmann connection on a Lie algebroid extension
K ֒→ A ։ A is a smooth sub-vector bundle H ⊂ A complementary to
E B E
K in A :
E
K⊕H = A .
E
We will call H the horizontal distribution of the connection.
Clearly, such a connection always exists and there is an associated notion
of horizontal lifting h : Γ(A ) 7→ Γ(H): we define h(α) as the unique
B
section of H that π-projects on α.
We might use the terminology of connection in a rather unristricted way,
to refer either to the operator h or to the horizontal sub-bundle H.
Definition 2.2. Given an Ehresmann connection, we define the associated
curvature2-formω ∈ Ω2(A )⊗Γ(K)(otherwisestated,alltensorproducts
B
aretobetaken over thecommutative ringC∞(B))bythefollowingformula:
ω(α,β) := h([α,β] )−[h(α),h(β)] .
AB AE
The fact that ω takes values in Γ(K) is a direct consequence of lemma
1.9. A connection also determines a C∞(B)-linear application D :Γ(A ) →
B
Der(K) defined by:
D (κ) := [h(α),κ] .
α AE
More precisely for any sections α,β ∈ Γ(A ), any function g ∈ C∞(B) and
B
any sections κ,κ ,κ ∈ Γ(K):
1 2
(2) D +D = D ,
α β α+β
(3) D = g.D ,
g.α α
(4) D ([κ ,κ ] ) = [D (κ ),κ ] +[κ ,D (κ )] ,
α 1 2 K α 1 2 K 1 α 2 K
(5) ♯ (D (κ)) = [s ,κ] .
K α Dα TE
Here, the first two equalities express the C∞(B)-linearity of D; equality
(2) is clear and (3) follows from Proposition 1.10. The last two equations
express the fact that D has values in derivations of K: equation (4) is a
consequence of Jacobi identity (iv) and (5) follows from (iii) in Definition
1.1.
As a derivation of K, each D naturally extends to a derivation of the
α
graded algebra Γ(ΛkK) of multi-sections of K:
D (κ ∧···∧κ ) := κ ∧···∧D (κ )∧···∧κ , (κ ∈ Γ(K)),
α 1 k 1 α i k i
X
i=1...k
8 OLIVIERBRAHIC
and by duality to multi-sections of Ωk(K):
hD θ,κi := L hθ,κi−hθ,D κi, θ ∈ Ωk(K), κ ∈ Γ(ΛkK) .
α sDα α
(cid:0) (cid:1)
The covariant differential associated to D will be denoted ∂ : Ωk(A )⊗
H B
Γ(ΛlK) → Ωk+1(A )⊗Γ(ΛlK) to fit with classical notations (note however
B
that it is related to D rather than H since two different horizontal distribu-
tions can induce the same derivations). It is given by the usual formula:
(6) ∂ θ (α ...α ) := (−1)iD θ(α ...αˆ ...α )
H 0 k αk 0 i k
X
i=0...k
+ (−1)i+jθ([α ,α ] α ...αˆ ...αˆ ...α ).
i j AB 0 i j k
X
0≤i<j≤k
This exact formula can be dualized to define a covariant derivative ∂ :
H
Ωk(A )⊗Ωl(K) → Ωk+1(A )⊗Ωl(K) which we will also denote ∂ .
B B H
The curvature Curv of D is defined by Curv (α,β) := D ◦D −D ◦
D D α β β
D −D . By construction, it is related to the curvature 2-form ω by:
α [α,β]AB
(7) Curv (α,β)(κ) = [ω(α,β),κ] .
D K
Moreover, ω is D closed in the sense that ∂ ω = 0, which writes:
H
(8) D ω(β,γ)−ω([α,β] ,γ), (α,β,γ ∈ Γ(A )).
I α AB B
α,β,γ
It is a consequence of Jacobi identity (iv).
In order not to confuse Curv with ω we shall refer to ω as the curvature
D
2-form, and to Curv as the curvature.
D
Remark 2.3. The following fact will play an important role in this work: in
general, D might failto induceaninfinitesimalaction of A onE. However,
B
itisclear in(7)thatthisfailureisstillcontrolled by K. Inotherwords, ifwe
think of K as a defining a homotopy theory on E, one can see an extension
of A by K as an infinitesimal action up to homotopy. Note that this ter-
B
minology corresponds to a different notion than the one of “representations
up to homotopy” appearing for instance in [7].
2.1. Parallel transport. Given a connection, parallel transport along a
small A -path is well defined, at least locally as we briefly explain now.
B
Fix a time-dependent section αt ∈ Γ(A ). We denote Xt the linear
B Dα
vector field on K associated to D and φDα its flow. It is a` priori only
αt t,s
defined in a neighborhood of κ in K and for t close to s.
Now, since the application α 7→ D is C∞(B)-linear, we see that the
α
restriction of Xt to a fiber K over some b ∈ B only depends on αt(b).
Dα |Eb
Moreover, by definition Dt = adAE so it has symbol ♯ (h(αt)) which
α h(αt) E
we know from Lemma 1.8 to p -project onto ♯ (αt). This implies that the
∗ B
restriction of φDα toK onlydependsonαt′ evaluated atthepointφ♯Bα(b)
t,0 |Eb t′,0
for t′ in [0,t].
Thus, if a : I → A is a A -path over γ : I → B, one can extend it
B B
arbitrarilytoatimedependentαtsectionofA anddefineΦa (κ) := φDα(κ)
B t,0 t,0
where κ ∈ K . We know that it is well defined for t small enough
|Eγ(0)
(depending on κ) and independent of the choice of αt.
EXTENSIONS OF LIE BRACKETS. 9
Definition 2.4. An Ehresmann connection is said to be complete if, for
any A -path a, Φa (κ) is well defined for any κ∈ K and all t ∈ I.
B t,0 |Eγ(0)
For a complete connection, we obtain at time t = 1 an application:
K Φa // K .
|Eγ(0) |Eγ(1)
called paralleltransportorholonomy alonga. Clearly Φ isaLiealgebroid
a
morphism. We shall denote φ : E → E the base map covered by Φa
a γ(0) γ(1)
2.2. Admissible couples. Once fixed a connection, one can decompose
any section η of A as a sum of elements of the form κ+fh(α) with κ ∈
E
Γ(K),α ∈ Γ(A ) and f ∈ C∞(E) so we get an isomorphism:
B
Γ(A ) = Γ(K)⊕ C∞(E)⊗Γ(A ).
E B
Thestructureof Lie algebroid on A is thus entirely determined by theones
E
on K and A , and by the couple (D,ω). Indeed, the brackets are given by:
B
(9) [κ ,κ ] = [κ ,κ ] ,
1 2 AE 1 2 K
(10) [h(α),κ] = D κ,
AE α
(11) [h(α),h(β)] = h([α,β] )+ω(α,β),
AE AB
and extended to arbitrary sections using Leibniz rule. The anchor is given
by the following formulas:
♯ (h(α)) = s ,
AE Dα
♯ (κ) = ♯ (κ).
AE K
Reciprocally, once fixed a submersion p : E → B, a Lie algebroid K over
E whose characteristic foliation lies in Vert and a Lie algebroid A over B,
B
one shall consider couples (D,ω) where:
• D : Γ(A ) → Der(K) is a C∞(B)-linear application such that the
B
symbol s of D p -projects onto ♯ (α) for any α ∈ Γ(A );
Dα α ∗ B B
• ω isa2-formonA withvaluesinsectionsofK: ω ∈ Ω2(A )⊗Γ(K).
B B
Definition 2.5. The couple (D,ω) is said to be admissible if ∂ ω = 0 and
H
Curv = adK◦ω.
D
Under the assumption of admissibility, the formulas above define a Lie
algebroid structureonA := K⊕p∗A comingwiththeobviousEhresmann
E B
connection. Note however that given arbitrary Lie algebroids A and K,
B
there might not exist an extension of A by K.
B
Anaturalquestionistoaskhowthechoiceoftheconnectioninfluencesour
construction. Given two connections h and h′, one can form the difference
∆ := h−h′ ∈Ω(A )⊗Γ(K) and one easily gets the following relations:
B
(12) D′ κ = D κ+[∆(α),κ] ,
α α K
(13) ω′(α,β) = ω(α,β)+∂ ∆(α,β)+[∆(α),∆(β)] .
H K
where of course (D,ω) and (D′,ω′) denote the compatible couples respec-
tively associated to h and h′. Thus one can state the following.
10 OLIVIERBRAHIC
Theorem 2.6. Let p : E → B be a submersion, K → E a Lie algebroid
whose characteristic foliation lies in Vert and A → B a Lie algebroid.
B
Extensions of A by K covering p are (up to equivalence) classified by
B
equivalence classes of admissible couples (D,ω) under the relation (D,ω) ∼
(D′,ω′) if there exists ∆ ∈ Ω1(A )⊗Γ(K) such that
B
D′ = D+adK◦∆,
ω′ = ω+∂ ∆+[∆∧∆] ,
H K
where [∆∧∆] ∈ Ω2(A )⊗Γ(K) is defined by:
K B
[∆∧∆] (α,β) = [∆(α),∆(β)] .
K K
Remark 2.7. As for usual considerations concerning extensions, by equiva-
lence of extensions, we mean an algebroid isomorphism A → A′ such that
E E
the following diagram commutes:
K (cid:31)(cid:127) // AE π // // AB
K(cid:15)(cid:15) I(cid:31)d(cid:127)K // A(cid:15)(cid:15)′E π′ // // AB(cid:15)(cid:15) I.dAB
In a practical situation, this theorem might be of no real help for classi-
fication. However, it gives a good intuition on how Lie algebroid extensions
are made.
2.3. Isotropies. For a general extension K ֒→ A ։ A there is no simple
E B
relation between the isotropy Lie algebras gB , gE and gK for any x ∈ E.
p(x) x x
In particular, these do not fit into an extension of Lie algebras in general,
as shows example 2.15. In the next proposition we summarize a few general
properties.
Proposition 2.8. Let K ֒→ A ։ A be a Lie algebroid extension. Then
E B
the following properties hold:
i) π(gE) ⊂ gB;
x y
ii) gK = gE ∩K ;
x x x
iii) π−1(gB) = (♯ )−1(Vert );
x y E x x
for any x ∈E, y = p(x).
Definition 2.9. We will say that an extension is clean provided the re-
striction of π to gE surjects onto gB for any x ∈ E.
x p(x)
As easily seen, in the case of a clean extension the restriction of π to gE
x
induces an extension of Lie algebras:
gK ֒→ gE ։ gB .
x x p(x)
Moreover, clean extensions enjoy the following property: each orbit O
E
of A fibers over an orbit O of A , the typical fiber having as connected
E B B
components orbits of K (this follows from (i) and (iii) in Proposition 2.8).
Once fixed an orbit O of A , one can always choose a Ehresmann con-
E E
nection such thath(gB) ⊂ gE for any y ∈ B,x ∈ p−1(y)∩O . Note however
y x E
that this might not be possible for all e ∈E in general.
Forsuchaconnection, therestrictionofthecurvature2-formω tosections
of the isotropy Lie algebra bundle gB takes values in gK .
|OB |OE
