On the connection theory of transitive Lie groupoids

0.35 MB·English

by  Iakovos Androulidakis

On the
onne
tion theory of extensions of transitive
∗†‡
5
Lie groupoids
0
0
2
Iakovos Androulidakis
n
a C.N.R.S. - Université Pierre et Marie Curie (Paris VI)
J
Institut de Mathématiques de Jussieu - UMR 7586

5
175, Rue du Chevaleret

] F-75013 PARIS
G
Fran
e
D
androulidakismath.jussieu.fr
.
h
t
a February 1, 2008
m
[

2
v
2 Abstra
t
8
2 Due to aresult by Ma
kenzie,extensionsof transitiveLiegroupoidsareequivalentto
7
ertain Lie groupoids whi
h admit an a
tion of a Lie group. This paper is a treatment
0 of the equivariant
onne
tion theory and holonomy of su
h groupoids, and shows that
3
su
h
onne
tions give rise to the transition data ne
essary for the
lassi(cid:28)
ation of their
0
respe
tive Lie algebroids.
/
h
t
a
m 1 Introdu
tion
:
v A groupoid, in the
ategori
al sense, is a
ategory where every arrow is invertible. Lie
i groupoidsarethose oneswhose obje
tand arrowspa
esaremanifolds,and the stru
ture
X s t t s
maps (sour
e , target , multipli
ation and inversion) are smooth, plus and are
r
a submersions. ALiegroup
anbe
onsideredasaLiegroupoid(with asinglepointasthe
obje
t spa
e), but in general Lie groupoids are inherently non
ommutative, hen
e their
extensive use in di(cid:27)erential geometry. In the same fashion, abundle of Lie groupsis also
a Lie groupoid, with its proje
tion playing the role of both the sour
e andΩt−−a→→rgeMt maps.
In this paperwe fo
us on the transitive
ase, namely those Lie groupoids su
h
(t,s):Ω→M ×M
that the an
hor map Ω−−→→Mis a surje
tive submersion.
GivenatransitiveLiegroupoid ,anextensionofthisisapairofLiegroupoid
morphisms
ι ϕ
F >−−−>Ω−−−≫Φ
(1)
Φ−−→→M F →M
where isatransitiveLiegroupoidand isabundleofLiegroups. Mu
h
of the work of Ma
kenzie
on
erns su
h extensions, and for a general Lie groupoid there
∗
†2000 Mathemati
s Subje
t Classi(cid:28)
ation. Primary 53C05. Se
ondary 22A22, 53C10, 55R10.
‡Keywords: groupoids, algebroids, extensions,
onne
tions
Resear
h supportedbytheGreek StateS
holarships Foundation
1
alwaysexistsafoliationwhoserestri
tiontoeveryleafissu
hanextension. Whendealing
with su
h an extension, it is quite di(cid:30)
ult to keep tra
k of the transition fun
tions of
all the groupoids involved. Ma
kenziein [14℄, managed to reformulate su
h an extension
to a single (transitive) Lie groupoid over the total spa
e of a prin
ipal bundle, plus a
group a
tion. These are
alled PBG-groupoids,and an a

ountof this
orresponden
eis
given in Se
tion 1. It is therefore reasonable to expe
t that extensions of transitive Lie
groupoidsare
lassi(cid:28)edbythetransitionfun
tionsoftheir
orrespondingPBG-groupoids,
but it is ne
essary to keep tra
k of the group a
tion as well, and it is not
ertain that
there exist transition fun
tions whi
h enjoy some kind of equivarian
e.
This is where
onne
tions are ne
essary. The transversals of an extension of tran-
sitive Lie groupoids
orrespond to
onne
tions in the respe
tive PBG-groupoid whi
h
are suitably equivariant, and these are the
onne
tions whose in(cid:28)nitesimal theory and
holonomy are given in this paper. The term isometabli
is used for these
onne
tions,
instead of equivariant, in order to highlight the non-standard nature of equivarian
e for
the transition fun
tions that they indu
e.
The
lassi(cid:28)
ationofPBG-groupoids(extensionsoftransitiveLiegroupoids)isaprob-
lem of a di(cid:27)erent orderpresentedin [1℄. In this paper we fo
us on the in(cid:28)nitesimal level,
namely the
lassi(cid:28)
ation of transitive PBG-algebroids. It is shown in [15℄ that every
A TM L
transitive Lie algebroid (i.e. an extension of by a Lie algebra bundle ) lo
ally
TU ⊕L

anbewrittenas i Ui,andin[15℄itisshownthatsu
hLiealgebroidsare
lassi(cid:28)ed
(χ,α)={(χ ,α )} α :U →Aut(h)
ij ij i,j∈I ij ij
by pairs , where arethe transition fun
tions
L χ : TU → U ×h
ij ij ij
of the se
tions of and are
ertain di(cid:27)erential 1-forms, and the
h L
two of them satisfy suitable
ompatibility
onditions. Here is the (cid:28)ber type of and
{U } M
i i∈I
is a simple open
over of . This data arises from the fa
t that lo
ally there
TU → A
exist (cid:29)at
onne
tions, namely Lie algebroid morphisms i Ui. In this paper we
showthatlo
allyPBG-algebroidshave(cid:29)atisometabli

onne
tions,andthis givesriseto
transition data whi
h is suitably equivariant, modulo a simple
onne
tivity assumption.
Namely, we prove that:
P(M,G) G
Theorem 1.1 Let be a prin
ipal bundle whose stru
ture group is simply
{P ≡ U ×G} A
i i i∈I

onne
ted and a lo
al trivialisation of this bundle. If is a PBG-
(χ,α)
algebroid over this bundle then there exist transition data su
h that
χ (Xg)=χ (X)g
ij ij
(i) and
α (ug)=α (u)g
ij ij
(ii)
X ∈TP u∈P g ∈G
ij ij
for all , and .
A
When is integrablethough, the simple
onne
tivity assumptionis no longer ne
es-
sary. It will be shown that:
A P(M,G)
Theorem 1.2 For any integrable PBG-algebroid over a prin
ipal bundle
(χ,α) (i) (ii)
there exist transition data whi
h satisfy and of 1.1.
Note that the in order to prove the above result, the equivarian
e of the se
tions of the
A
PBG-groupoid whi
h integrates is investigated, and this is an important step to the

lassi(cid:28)
ation of PBG-groupoids.
Finally, we give an a

ount of the holonomy of isometabli

onne
tions whi
h leads
to the proof of an Ambrose-Singer theorem for isometabli

onne
tions. The treatment
of holonomy presented here follows the fashion of [15℄, i.e. we use the equivariant ver-
sion of deformable se
tions. This method provides more information for the relation of
isometabli

onne
tions of PBG-Lie algebra bundles with their se
tions.
The stru
ture of this paper is as follows: Se
tions 2 and 3 are an a

ount of the

orresponden
e between extensions and PBG stru
tures, in both the groupoid and the
algebroid level. A number of examples is in
luded, most of whi
h are used in the proofs
2
laterin thepaper. Se
tion4givesthe
onne
tiontheoryofPBG-groupoidsandPBG-Lie
algebra bundles. In Se
tion 5 we give the proofs of 1.1 and 1.2. Se
tion 6 is an a

ount
of the holonomyof isometabli

onne
tionsin the fashionof [15℄, and Se
tion 7givesthe
proof of the Ambrose-Singertheorem for isometabli

onne
tions.
A
knowledgments
I would like to thank Kirill Ma
kenzie for the en
ouragement to write an a

ount of

onne
tion theory in the PBG setting, aswell ashisuseful
omments. Alsomany thanks
to Rui Fernandes and Stavros Papadakis for the dis
ussions we had while I was writing
this paper.
2 PBG-groupoids
C∞
Let us (cid:28)x some
onventions (cid:28)rst. All manifolds
onsidered in the paper are - dif-
ferentiable, Hausdor(cid:27), para
ompa
t and have a
ountable basis for their topology. We
sξ ξ

onsider the arrows in a groupoid "from right to left(cid:17), i.e. the sour
e of an element
Ω tξ
in agroupoid is
onsideredto beon the rightand its target on theleft. The obje
t
Ω M 1 : M → Ω
in
lusion map of a groupoid over a manifold is denoted by , namely
x 7→ 1
x
. In [15℄ the expression `Lie groupoid' stood for a lo
ally trivial di(cid:27)erentiable
groupoid; here we use the expression `transitive Lie groupoid' for
larity.
Υ−−→→P
De(cid:28)nition 2.1 APBG-groupoidisaLiegroupoid whosebaseisthetotalspa
e
P(M,G) G Υ
of a prin
ipal bundle together with a right a
tion of on the manifold su
h
(ξ,η)∈Υ×Υ sξ =tη g ∈G
that for all su
h that and we have:
t(ξ·g)=t(ξ)·g s(ξ·g)=s(ξ)·g
(i) and
1 =1 ·g
u·g u
(ii)
(ξη)·g =(ξ·g)(η·g)
(iii)
(ξ·g)−1 =ξ−1·g
(iv)
Υ P(M,G) Υ−−→→P(M,G)
WedenoteaPBG-groupoid overtheprin
ipalbundle by
Υ G R g ∈G
g
andtheright-translationin
omingfromthe -a
tionby forany . Theright-
P R
translation in will be denoted by g. The properties inethe previous de(cid:28)nition show
G P(M,G) Υ
that the group of the base prin
ipal bundle a
ts on by automorphisms,
R Υ R
g g
gna∈mGely isanautoϕmorphismoftheLiegroupoid overthedi(cid:27)eomΥorphismΥ′ forall
. Aemorphism of Lie groupoidsbetween twoPBG-groupoϕid◦sR a=ndR′ ◦ϕoverthe
same prin
ipal bundle is
alled a morphism of PBG-groupoids, if g g for all
g ∈G
. In the same fashion, a PBG-Lie group bundle (PBG-LGB) is aeLie geroup bundle
F P P(M,G) G
over the total spa
e of a prin
ipal bundle su
h that the group a
ts on
F F → P(M,G)
by Lie group bundle automorphisms. WeIΥdenote a PBG-LGB byΥ−−→→P(M,G) . It
is easy to see that the gauge group bundle of a PBG-groupoid is a
PBG-LGB.
Now let us des
ribe the
orresponden
e of transitive PBG-groupoids with extensions
of Lie groupoids. For any given extension of Lie groupoids (1),
hoose a basepoint and
take its
orresponding extension of prin
ipal bundles
π(id,π)
N >−−−>Q(M,H)−−−≫P(M,G).
π(id,π) H
That is to say that is a surje
tive morphism of prin
ipal bundles and is an
G N Q(P,N,π)
extension of the Lie group by . This gives rise to the prin
ipal bundle ,
3
whi
h Ma
Υke=nzQie×iQn [14℄
alled the transverse bundle of the previous extension. The Lie
groupoid N
orresponding to the transverse bundle admits the following a
tion
G
of :
hv ,v ig =hv h,v hi,
2 1 2 1
v ,v ∈ Q g ∈ G h ∈ H π(h) = g
2 1
for any Υ−−→→aPn(dM,G) , where is any element with . This
a
tion makes a PBGI-gΥro>u−p−o−i>d.ΥO−n−−th≫e Poth×erPhand, wreiting aGtransitive
PBG-groupoid as an exa
t sequen
e , the fa
t that a
ts by
G
Lie groupoid automorphisms allows us to quotient the sequen
e over and obtain the
extension of Lie groupoids
IΥ Υ P ×P
>−−−> −−−≫ .
G G G (2)
The two pro
esses are mutually inverse (for the details of the proof see [14℄). This
establishes the following result:
Υ−−→→P(M,G)
Proposition 2.2 Any transveFrse>−P−B−G>-Ωgro−u−−p≫oidP×P
orresponds exa
tly to
an extension of Lie groupoids G .
Let us give now some examples of PBG-groupoids, whi
h will also be useful in later
se
tions.
P(M,G) H.
Example 2.3 Consider a prin
ipal bundle and a Lie group Suppose given
G H, (g,h) 7→ R (h) g ∈ G h ∈ H.
g
an a
tion by automorphisms of on say for all and
R :H →H H g ∈G.
g
ThatistoPsa×ytHha×t P −−→→P(M,Gis).anautomorphismof forall Formthetrivial
groupoid This is easily seen to be a transitive PBG groupoid.
We will refer to it as the trivial PBG-groupoid
orresponding to the given a
tion.
Example 2.4 For this example, (cid:28)rst let us re
all the de(cid:28)nition of an a
tion groupoid.
G M M>7 G
When a Lie group a
ts on a manifold the a
tion groupoid is the produ
t
M ×G s(x,g)= x, t(x,g) =xg, 1 =
x
manifold with the following groupoid stru
ture:
(x,e ) (xg,h)· (x,g) = (x,gh)
G
(x,g) . (Txhge,gm−1u)ltipli
ation is de(cid:28)ned by and the inverGse of
is . The a
tion groupoid is transitive if and only if the a
tion of on
M P(M,G) G
is transitive. Now
onsider a prin
ipal bundle . There is an a
tion of
P>7 G R (u,h) = (ug,g−1hg)
g
on de(cid:28)nPed>7 bGy −−→→P(M,G) . This a
tion makes the a
tion groupoid a
PBG-groupoid e .
E P
Example 2.5 A PBG ve
tor bundle is a ve
tor bundle over the total spa
e of a
P(M,G) G E
prin
ipal bundle su
h that the stru
ture group a
ts on by ve
tor bundle
E →P(M,G)
automorphisms. We denote a PBG ve
tor bundle by . The isomorphisms
between the (cid:28)bers of an arbitrary ve
tor bundle de(cid:28)ne a transitive Lie groupoid
alled
the frame groupoid. This is the groupoid
orresponding to the frame prin
ipal bundle
Φ(E)
indu
ed by the ve
tor bundle under
onsideration. The frame groupoid of a PBG
E →P(M,G) P(M,G)
ve
tor bundle has a
anoni
al PBG stru
ture over . Namely, for
g ∈ G ξ : E → E
u v
any , an isomorphism between two (cid:28)bers de(cid:28)nes an isomorphism
R (ξ):E →E
g ug vg
by
R (ξ)(V)=ξ(V ·g−1)·g
e g
V ∈ Eug e ǫ : P>7
for all . Moreover, there is a
anoni
al morphism of PBG-groupoids
G → Φ(E) P(M,G) ǫ(u,g)
over the prin
ipal bundle . Namely, is the isomorphism
R :E →E G E ǫ
g u ug
indu
ed by the a
tionof on . It is straightforwardto showthat is
ibndeed a morphism of Lie groupoids. It also preserves the a
tions be
ause:
ǫ((u,h)·g)=ǫ(u·g,g−1hg)=[(Rg−1hg)u·g :Eu·g →Eu·hg]=
b =Rg[(Rh)u :Eu →Eu·h]=Rg(ǫ(u,h)).
e b e
4
LetusnowgiveanexampleofaPBG-groupoidthatarisesfromanextensionofprin
ipal
bundles.
SU(2)(S2,U(1),p)
Example 2.6 We
onsider the prin
ipal bundles and
s t
SO(3)(S2,SO(2),p′)
. For the (cid:28)rst bundle, denote a typi
al element (cid:20)−t s(cid:21) of
SU(2) |s|2 +|t|2 = 1 (s,t) U(1) SU(2)
su
h that by . Regard as a subgroup of by
z ∈U(1) (z,0) p
mapping every to and let be
(s,t)7→(−2Re(s·t),−2Im(s·t),1−2·|t|2).
A 0
SO(2) SO(3) A7→ p′
Forthese
ondbundle,regard asasubgroupof by (cid:20)0 1(cid:21) andlet
A7→A·e (e ,e ,e ) 3
3 1 2 3
be where istheusualbasisof R . Wede(cid:28)neamorphismbetween
q =(s,t)∈SU(2)
these prin
ipal bundles makrin=g(urse,rof,qru)at∈ernio3ns. Consider any q·r·aqs−a1
1 2 3
unit quaternion and every q ·r ·q−1 ∈ R3. as a ve
toqr =qua(ste,rtn)io∈n.STUh(2e)n
iAs a: ve
3t→or qu3aternAio(nr,)i=.e.q·r·q−1 R For every SO(3) ϕ:U(1)w→eSdOe(cid:28)(n2e)
q q
R R as . Thisisanelementin . Let
A U(1) z ∈ U(1) z = eιθ = cosθ+ιsinθ
be the restri
tion of to . If we write a as . for
θ ∈
some R, then it is easy to see that
cos(2θ) sin(2θ)
ϕ(z)= ∈SO(2).
(cid:20)−sin(2θ) cos(2θ)(cid:21)
ϕ
2
Thisshowsboththat isasurje
tivesubmersionandthatitskernelis Z . Wetherefore
get the extension
>−−−>SU(2)(S2,U(1))R−(i−d−S2≫,ϕ)SO(3)(S2,SO(2)).
2
Z
SU(2)(SO(3), ,A)
2
Its transverse bundle is Z and the PBG-groupoid it indu
es is
SU(2)×SU(2)
−−→→SO(3)(S2,SO(2)),
2
Z
SO(2) SU(2)×SU(2)
where the a
tion of on 2 is:
Z
h(s ,t ),(s ,t )i·z =h(s z,t z),(s z,t z)i.
1 1 2 2 1 1 2 2
G H ≤ G
It was shown inG[1×5G, IIŸ7℄ that for any Lie group and
lGos>7edGs,ubgroupG thGe
gaug(eg,ggr′oHu)po7→id(g·Hg′)His isomorphi
to the a
tion groupoid H where a
ts on H
via . The isomorphism is
hg ,g i7→(g ·g−1,g H).
1 2 1 2 2
SU(2)×SU(2) −−→→SO(3)(S2,SO(2))
Therefore 2 is isomorphi
to the a
tion PBG-
Z
SU(2)>7 SU(2)−−→→SO(3)(S2,SO(2)) SU(2) ∼= P3
groupoid 2 . Note that 2 R . The a
tion of
Z Z
SO(2) P3>7 SU(2)
on R is
((w ,w ),(s,t)· )·R =((w ,w ),(s·z,t·z)· ).
1 2 2 θ 1 2 2
Z Z
5
3 PBG-algebroids
This is an a

ount of PBG stru
tures on the algebroid level. We show that every PBG-
Lie algebroid
orrespondsto an extensionofLie algebroids;the a

ountin [14℄ gaveonly
a partial result of this type. Some fundamental results are also proved, whi
h will be
useful in the study of the
onne
tion theory in later se
tions of this paper. Let us begin
by des
ribing the notion of a PBG stru
ture on the algebroid level. We start with the
spe
ial
ase of Lie algebra bundles.
L
De(cid:28)nition 3.1 A PBG-Lie algebra bundle (PBG-LAB) is a Lie algebra bundle over
P P(M,G) G L
the totalspa
e of a prin
ipal bundle , together withan a
tion of on su
h
R : L → L
g
that ea
h right-translation is a Lie algebra bundle automorphism over the
R :P →P
g
right translation .
L → P(M,G)
We denote a PBG-LAB by . A morphism between two PBG-LABs over
the same prin
ipal bundle is a morphism of LABs whi
h preserves the group a
tions.
P(M,G) h
Example 3.2 Let be a prin
ipal bundle and a Lie algebra whi
h admits a
(V,g)7→V ·g G
right a
tion of su
h that
[V,W]·g =[V ·g,W ·g]
V,W ∈h P×h G R (u,V)=(ug,V ·g)
g
forall . NowthetrivialLAB admitsthe -a
tion .
This makes it a PBG-LAB, the trivial PBG-LAB.
A M
De(cid:28)nition 3.3 A Lie algebroid is a ve
tor bundle on base together with a ve
tor
q : A → TM A [ , ]: ΓA×ΓA → ΓA
bundle map ,
alled the an
hor of , and a bra
ket
whi
h is R-bilinear, alternating, satis(cid:28)es the Ja
obi identity, and is su
h that
q([X,Y])=[q(X),q(Y)]
(i) ,
[X,uY]=u[X,Y]+q(X)(u)Y
(ii)
X,Y ∈ΓA u∈C∞(M)
for all and .
P ×h → P(M,G)
Given a trivial PBG-LAB , re
all that the Whitney sum ve
tor
TP ⊕(P ×h) P
bundle has a trivial Lie algebroid stru
ture over . The an
hor map is
the (cid:28)rst proje
tion and the Lie bra
ket on its se
tions is de(cid:28)ned by the formula
[X ⊕V,Y ⊕W]=[X,Y]⊕{X(W)−Y(V)+[V,W]}
X,Y ∈ ΓTP h V,W P G
for all and smooth -valued fun
tions on on . The -a
tion on
P ×h
de(cid:28)nes a PBG stru
ture on this Lie algebroid. Namely, with the notation used in
G TP ⊕(P ×h)
the previous example (3.2) de(cid:28)ne the following a
tion of on :
(X ⊕(u,V),g)7→R (X ⊕(u,V))=T R (X)⊕(ug,V ·g).
g u g
RΓ: Γ(TP ⊕(P ×h))b→ Γ(TP ⊕(P ×h))
Denote by g the
orresponding a
tion on the
TP ⊕(P ×h)
se
tions of b . This is given by the formula
RgΓ(X ⊕V)u =Tug−1Rg(Xug−1)⊕(Vug−1 ·g)
X ∈ ΓTP, V ∈b C∞(P,h) u ∈ P
for all and . This a
tion preserves the Lie bra
ket in
Γ(TP ⊕(P ×h))
and is an a
tion by automorphisms on the trivial Lie algebroid. This

onstru
tion is an example of the notion of a PBG-algebroid.
P(M,G)
De(cid:28)nition 3.4 A PBG-algebroid over the prin
ipal bundle is a Lie algebroid
A P G A (X,g) 7→ R (X)
g
over together with a right a
tion of on denoted by for all
X ∈ A, g ∈ G R : A → A
su
h that ea
h g is a Lie algebroid automorphbism over the
R P
right translation g in . b
6
A P(M,G) A⇒P(M,G) G
We denote aPBG-algebroid over by . The a
tionof on
A G ΓA
indu
es an a
tion of on , namely
X·g =Rg◦X ◦Rg−1.
g ∈ G X ∈ ΓA b
for all and . The right-translation with respe
t to this a
tion is denoted
RΓ: ΓA→ΓA g ∈G
by g for all . With this notation de(cid:28)nition 3.4 implies that
b RΓ([X,Y])=[RΓ(X),RΓ(Y)]
g g g
X,Y ∈ ΓA g ∈ Gb b b
for all and . The following result strengthens 3.2 of [14℄, where the
quotient manifold was assumed to exist.
A P(M,G)
Proposition 3.5 Let be a transitive PBG-Lie algebroid on . Then the quo-
A/G
tient manifold exists and inherits a quotient stru
ture of transitive Lie algebroid
A
from ; further, it is an extension
L A TP
>−−−> −−−≫
G G G
P(M,G) L/G
of the Lie algebroid of the gauge groupoid of by the quotient LAB .
A/G
Proof. The main requirement is to prove that the manifold exists. We apply the
A−→P p
A

riterion of Godement [8℄. Denote the proje
tion by , and write
Γ′ ={(X,Xg) | X ∈A, g ∈G};
Γ′ A×A Γ′ ⊆ (p ×p )−1(Γ)
A A
we must show that is a
losed submanifold of . Now
Γ = {(u,ug) | u ∈ P, g ∈ G}. Γ P ×P
wphe×rep Sin
e is a
losedΓsu′bmanifold of , and
A A
(p ×p i)s−a1(sΓu)rje
tivesubmersion, it su(cid:30)
es to provethat is a
losed submanifold of
A A
. De(cid:28)ne
f: (p ×p )−1(Γ)−→A, (X,Y)7→Xg−Y
A A
p (Y) = p (X)g A f
A A
where . From the lo
al triviality of , it eapsily foΓl′lows that is a
A
surje
Γt′ive submersion. The preimage of the zero se
tion under A−is→A,/aGnd thi♮s shows
that is a
losed submanifold. Denote the quotient proje
tion by .
A A/G
Theve
torbundlestru
tureof quotientsto inastraightforwardfashion. Sin
e
q: A −→ TP G
the an
hor is (cid:21)equivariant, it quotients to a ve
tor bundle morphism
A/G −→ TP/G π
whi
h is again a surje
tive submersion; denote this by , and de(cid:28)ne
r =q◦π q TP/G
where is the an
hor of .
A/G Γ(A/G)
C∞F(eMor)the bra
keet sGtru
ture of , note (cid:28)rsAt that
an be identi(cid:28)ed with the
module of (cid:21)equivariant se
tions of as in the
ase of the Atiyah sequen
e of
ΓA G
a prin
ipal bundle. Sin
e the bra
ket on restri
ts to the (cid:21)equivariant se
tions by
Γ(A/G)
assumption, this bra
ket transfers to . It is now straightforward to
he
k that
A/G M r ♮: A−→A/G
thismakes aLiealgebroidon withan
hor ,and aLiealgebroid
p π L/G
morphismover . That isaLiealgebroidmorphismwithkernel iseasily
he
ked.
It is a straightforward exer
ise to verify that PBG-groupoids di(cid:27)erentiate to PBG-
algebroids. The next result, whi
h appears in [14℄, gives a
onverse.
Υ s
Theorem 3.6 Let be an (cid:21)simply
onne
ted lo
ally trivial Lie groupoid with base the
P(M,G) g ∈G
total spa
e of aprin
ipal bundle . Suppose thatfor all there is given a Lie
R : AΥ−→AΥ
g
algebroid automorphism whi
h de(cid:28)nes the stru
tureof a PBG-algebroid
AΥ Υ AΥ
on . Then there is aenatural stru
ture of PBG-groupoid on whi
h indu
es on
the given PBG-Lie algebroid stru
ture.
7
The examples whi
h follow will be useful later on.
A M
Example 3.7 Consider a transitive Lie algebroid over a manifold su
h that
π (M) 6= 0 M M p: M → M
1
. Let be the
overing spa
e of and the
overing pro-
M(M,π (M),p) p!!A
je
tion. Then we hafve the prin
ipal bundle 1 .fDenote the pullba
k
p∗ f
p!!A - p∗A
q′ p∗(q)
? ?
T(p∗)
TM - p∗TM
p!!A X′f⊕C X′ ∈ΓTM C ∈Γ(p∗A)
Se
tions of are of the form , where , and
T(p∗)(X′)=p∗(q)(C). f
This
onstru
tionis des
ribed in [11℄, but let us re
allhere the Lie algebroidstru
ture of
p!!A C =Σu (X ◦p) u ∈C(M) X ∈ΓA
i i i i
. If we write with and , then
p∗(q)(C)=Σu (qf(X ◦p))
i i
T(p)◦X′ =Σu (q(X ◦p)) p!!A q′
i i
with . We de(cid:28)ne a Lie algebroid stru
ture on with as
the an
hor map and bra
ket given by
[X′⊕Σu (X ◦p),Y′⊕Σu (Y ◦p)]=
i i j j
=[X′,Y′]⊕{Σu u ([X ,Y ]◦p)+ΣX′(u )(Y ◦p)−ΣY′(u )(X ◦p)}.
i j i j j j i i
p!!A M(M,π (M),p)
1
Now the pullba
k Lie algebroid is a PBG-algebroid over .
P>7 G−−→→fP(M,G)
Example 3.8 Consider the a
tion PBG-groupoid
onstru
ted in
P>7 g ⇒ P(M,G) P>7 g
2.4. This di(cid:27)erentiates to the a
tion PBG-algebroid . Here is
P ×g V ∈ g
the produ
t manifold Van†d∈tΓheTPan
hor is the map that asso
iates to every
the fundamental ve
tor (cid:28)eld . The Lie bra
ket is given by the formula
[V,W]=V†(W)−W†(V)+[V,W]v,
(3)
[V,W]v g G
where stands for the point-wise bra
ket in . The -a
tion that makes it a
PBG-algebroid is of
ourse
Rg(u,V)=(ug,Adg−1(V)).
b
L → P(M,G) Φ[L]
Example 3.9 Given a PBG-LAB , let denote the groupoid of Lie
L P(M,G)
algebra isomorphisms between the (cid:28)bers of . This is a PBG-groupoid over
Φ(L) Φ[L]
in the same way as (see 2.5). It was shown in [15℄ that di(cid:27)erentiates to
CDO[L] P CDO
the Lie algebroid over . The notation stands for (cid:16)
ovariant di(cid:27)erential
operator(cid:17). It is a transitive Lie algebroid and its se
tions are those (cid:28)rst or zeroth order
D: ΓL→ΓL
di(cid:27)erential operators su
h that:
D ∈ΓCDO[L] ♯(D)∈ΓTP
(i) For all there is a ve
tor (cid:28)eld su
h that
D(fµ)=fD(µ)+♯(D)(f)µ
µ∈ΓL f ∈C∞(P)
for every and .
8
D
(ii) The operators a
t as derivations of the bra
ket, i.e.
D([µ ,µ ])=[D(µ ),µ ]+[µ ,D(µ )]
1 2 1 2 1 2
µ ,µ ∈ΓL
1 2
for all .
CDO[L] ♯
The an
hor map of is exa
tly the map established in (i). The adjoint bundle
L
of this Lie algebroid is the LAB of endomorphisms of whi
h are derivations of the
bra
ket. We therefore have the exa
t sequen
e
♯
Der(L)>−−−>CDO[L]−−−≫TP.
G Φ[L] (D,g)7→RCDO(D)
The a
tion of on di(cid:27)erentiates to the a
tion g on the se
tions
CDO[L]
of de(cid:28)ned by
[RgCDO(D)](µ)=Rg◦D(Rg−1 ◦µ◦Rg)◦Rg−1
g ∈G,D ∈ΓCDO[L] µ∈ΓL R
g
for all and . (Re
all that denotes the right-translation
L CDO[L] P(M,G)
on the se
tions of .) This a
tion makes a PBG-algebroid over .
CDO[L]
In general, is the analogue of the automorphism group in the Lie algebroid
framework. In the PBG setting, we have the following de(cid:28)nition:
A⇒P(M,G) K →P(M,G)
De(cid:28)nition 3.10 Let be a (transitive) PBG-algebroid and
A K
a PBG-LAB. An equivariant representation of on is an equivariant Lie algebroid
ρ: A→CDO[L]
morphism .
A ⇒ P(M,G)
Proposition 3.11 Let be a transitive PBG-algebroid. Then its adjoint
L CDO[L]
bundle is a PBG-LAB and a PBG-algebroid, both over the prin
ipal bundle
P(M,G)
.
A
Proof. Sin
e is transitive we have the exa
t sequen
e of Lie algebroids
j ♯
L>−−−>A−−−≫TP.
(V,g) 7→ R (V) G ΓL j(RΓ(V)) = RΓ(j(V))
Let g denote the a
tion of on de(cid:28)ned by g g .
A
The result now follows from the fa
t that is a PBG-algebroid. b b
ϕ : A → A′
A Lie algebroid morphism between two PBG-algebroids over the same
P(M,G) ϕ◦R = R ◦ϕ
g g
prin
ipal bundle is a morphism of PBG-algebroids if for all
g ∈ G
. It is easy to verify that morphisms of PBG-groupoids di(cid:27)berentibate to PBG-
algebroids.
Υ Υ′
Proposition 3.12 Let and be PBG-groupoids over the same prin
ipal bundle
P(M,G) Υ s
ϕ :AΥ→suA
hΥ′that is -simply
onne
ted. Then every morphismϕo:fΥPB→GΥ-a′lgebroids
∗
integrates to a unique morphism of PBG-groupoids .
ϕ
∗
Proof. It was shown in [10℄ thϕa:t Υthe→LΥie′algebroid morphism inϕtegrates uniqueGly
to a morphism of Lgie∈gGroupoiRds R′ . It su(cid:30)
es to show that respe
ts the -
a
tions. Forevery let g and g be theright-translationsindu
ed by thea
tions
G Υ Υ′ R R′
of on and respe
tiveely. Theseedi(cid:27)erentiate to the right-translations g and g
ϕ
∗
reevseprye
tgiv∈elGy ownethhaeveLiReg′al◦gϕeb◦roRidg−l1ev=elϕ.∗S.inT
hee moirsphaismmoropfhLisiemgroofuPpBoiGds-aRlgg′ebb◦rϕoi◦dsR,gfb−o1r
di(cid:27)erentiatesto Rg′ ◦ϕb∗◦Rg. Tbheuniquenessof ϕ yields Rg′ ◦ϕ◦Rg−1 =ϕ fborall g ∈bG.
b b e e
9
L→
Example 3.13 ThenaturalmorphismofPBG-groupoidsarisingfromaPBG-LAB
P(M,G) ǫ : P>7 g → CDO[L]
∗
di(cid:27)erentiates to a morphism of PBG-algebroids . Su
h
morphisms areknownas (cid:16)derivativerepresentations(cid:17) in the
ontextof Poisson geometry.
L
We will refer to this one as the natural derivative representation indu
ed by .
A′ A
A′ >A−−L−i>e sAubalgebroid of a PBG-algebroid is a PBG-subalgebroid if the in
lusion
is a morphism of PBG-algebroids.
A ⇒ P(M,G)
Proposition 3.14 If is a transitive PBG-algebroid with adjoint bundle
L ad(A) CDO[L]
then is a PBG-subalgebroid of .
ad(A) CDO[L]
Proof. TheLiealgebroid wasshowntobeaLiesubalgebroidof in[15℄.
G
It su(cid:30)
es to show that it is
losed under the -a
tion de(cid:28)ned in 3.9. Indeed, suppose
X ∈ΓA
. Then:
[RgCDO(adX)](V)=Rg◦adX(Rg−1 ◦V ◦Rg)◦Rg−1 =
=Rg◦[X,Rg−1 ◦V ◦Rg]◦Rg−1 =[Rg(X),V]=adRg(X)(V)
V ∈ΓL RCDO(ad )∈ad(A)
for all . So, g X .
ad(L)
TheLiealgebroid isintegrableasasubalgebroidoftheintegrableLiealgebroid
CDO[L] α Int(A)
. We denote the -
onne
ted Lie groupoid it integrates to by , following
the notation of [15℄. The proof of the following statement is immediate.
A⇒P(M,G) IntA−−→→P(M,G)
Corollary 3.15 If is atransitivePBG-algebroid then
Φ[L]
is a PBG-subalgebroid of .
ad(A) ad : A →
We
an also
onsider as the image of the adjoint representation
CDO[L] ad
. In this sense, is a representation of PBG algebroids, i.e. a morphism of
PBG-algebroids.
A L
De(cid:28)nition 3.16 Let beaPBG-algebroid and aPBG-LAB,bothoverthesameprin-
P(M,G) ρ:A→CDO[L] A L

ipal bundle . A representation of in is a representation
of PBG-algebroids if it is a morphism of PBG-algebroids.
g ∈G,X ∈ΓA µ∈ΓL
In parti
ular, for all and it is required:
[RCDO(ρ(X))](µ)=ρ(RΓ(X))(µ).
g g
b
The developed form of this formula is:
Rg◦ρ(X)(Rg−1 ◦µ◦Rg)=ρ(RgΓ(X))(µ)◦Rg.
b
4 In(cid:28)nitesimal
onne
tions for PBG stru
tures
LiegroupoidsandLiealgebroidsprovideanaturalframeworkforthestudyof
onne
tion
P(M,G,p)
theory. For example, given a prin
ipal bundle ,
onsider its
orresponding
Atiyah sequen
e, namely the exa
t sequen
e of ve
tor bundles
P ×g TP p∗
>−−−> −−−≫TM
G G
G g
(where the a
tion of on TimPplied is simply the adjoint), together with thGe bra
ket of
se
tionsoftheve
torbundle G whi
hisobtainedbyidentifyingthemwith (cid:21)invariant
P p
∗
ve
tor (cid:28)elds on . This bra
ket is preserved by the ve
tor bundle morphism . Thus
the Atiyah sequen
e of a prin
ipal bundle is a transitive Lie algebroid. In this setting,
10