Emad. B. Al-Zangana. Dep. of Math. Dep. of Math. College of Science ... Al-Mustansiriyah University ... Using quotient topology AL-Taai and AL-Janabi in 2001.
On the Symmetry Between Lie Group and Lie Groupoid Action By A. A. Al-Taai Dep. of Math. College of Science Al-Mustansiriyah University
Emad. B. Al-Zangana Dep. of Math. College of Science Al-Mustansiriyah University انمستخهص
عمهنا. ( E, P, B) ( هى تعمٍم نفعم زمر نً نسبٍا إنى حسمه تفاضهٍتG, B) فعم انسمروٌاث انتفاضهٍت ( انتً فعههاG, B) ) ًٌتمحىر حىل أعطاء نىع من انسمروٌاث انتفاضهٍت (زمروٌت ن وانتً منها نحصم عهى تناضر بٍن انفعم األساش( E, P, B) ( حسمت نٍفٍت أساسٍتE, P, B) ٌجعم وانتً بذورها تؤدي إنى تناضر بٍن مىرفسٌهما انفعهٍٍنG ً وزمروٌت ن ًنكم من زمرة ن T , T . األساسٍٍن
ABSTRACT The notion of an action of a differentiable groupoid (G, B) generalize the action of a lie group relative to a differentiable bundle ( E, P, B) . Our work here is to give a special type of differentiable groupoid (lie groupoid) (G, B) in order to make ( E, P, B) a principal fiber bundle ( E, P, B) by which we get a symmetry between principal action of a lie group and that of a lie groupoid G and then a symmetry (conjugation form) between their principal action morphisms T , T respectively.
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INTRODUCTION
The concept of groupoid is one of the means by which the twentieth century reclaims the original domain of application of a group concept. In 1920’s, Brandt and Baer gave the algebraic theory of groupoid. In 1950’s , Ehresmann introduced the groupoid into the differential geometry. Also, the concept of groupoid action is due originally to Ehresmann (1959), generalizing the group action on a set, given in his work on the space of all isomorphisms between the fibers of a fiber bundle. From any action of a group on a set E , the quotient E E / has a groupoid structure with base E / called Ehresmann groupoid [5]. Using Ehresmann groupoid in algebraic case and in category form, AL-Taai [1] proved the existence of the symmetry between principal action’s law of group and groupoid. Using quotient topology AL-Taai and AL-Janabi in 2001 proved the existence of the symmetry between principal action’s
of
topological group and certain topological groupoids (Q-groupoids).When E and has the differentiable structure E / and E E / in general has no differentiable structure[3].This problem solved by theorem of Godement (2.2). We present in this paper the concept of principal lie group action's law (2.4) and its relation with principal fiber bundle is exposing .Also, we present the concept of weak principal differentiable groupoid action’s law (4.1) and the concept of principal differentiable groupoid action’s law(4.2) and show by example they are different. The purpose of this paper is to study and investigate the concept of principal groupoid action’s law with respect to Lie group and Lie groupoid and proved that there exists a symmetry between them (5.2).It should be remarked that the differentiable manifold are supposed be metrizable separable paces.
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1. Preliminaries 1.1: Given a groupoid G with base B , we denote by , the source and target map from G to B , : B G the canonical identification of objects with units, the composition law , the inverse law , the "difference law" (division map) ( g1 , g 2 ) g1 g 21 from G to G (where G is the fiber product of by itself over ). 1.2: C -groupoid A C -groupoid is a groupoid (G, B) together with smooth manifold structure ( C - structure) on G and B such that : (i) : G B is a surmersion (which implies G is a regular submanifold of G G ). (ii)
: B G is a C -map .
(iii) : G G is a C -map . One can see that is a diffeomorphism , , and are surmersion and is regular embedding (which identifies B and (B) ). For any x B , we set Gx 1 ( x) called -fiber at x ,
x
G 1 ( x)
called the -
fiber , both G x , x G closed regular submanifold of G , x B .We set
: ( , ) : ( ) : G B B , which is a C -map called transitor map. When onto (G, B) is called transitive. x G x 1 ( x, x) called the isotropy (vertex) group which is a lie group with unity (x) , x B .For any g 1 ( y, x) , the map Int ( g ):x Gx y G y ; ( h ghg 1 ) is an
isomorphism of Lie groups and the map Rg : G y Gx ; ( h hg ) is a diffeomorphism of -fibers . If G is transitive we have an isomorphy class [ x G x ] xB of Lie groups and diffeomorphy class of -fibers[2][5].
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1.3: A morphism of a C -groupoid is a morphism of groupoids ( f , f 0 ) : G, B) (G, B) such that f and f o are C -maps. When f is
diffeomorphism the pair ( f , f 0 ) is called an isomorphism of C groupoids (which implies that f o is diffeomorphism) [5]. We denote by DG the category of C -groupoids (and their morphisms ) and D the category of C -manifolds (and their C -maps ). 2. Principal action of Lie group 2.1: Let : E E be a law of C -(right)action of a lie group on a C -manifold E ,then: defines a diffeomorphismr : E E ; ( z z r ) for each r [3] and ( E , E ) has a C -groupoid structure [5]. The graph of the equivalence relation E E ① [1] defined by this action has a B groupoid structure but has no (in general) C -structure. Solve this problem by the following Lemma which is well known by theorem of Godement [3]. 2.2 Lemma: Let be a lie group acting differentiably (from the right) on a C -manifold E , then there exist a C -structure on E / for which the
underlying topology is the quotient topology on E / and for which the canonical projection : E B is submersion iff the graph of the equivalence relation defined by this action is closed regular submanifold of the product manifold E E . The C -structure on E / satisfying these requirements is then unique. 2.3 Example: (i) Let
: E E be a law of
C -right action of a lie group on a C -
manifold E . If acts freely and properly on E , then the graph of the equivalence relation defined by this action is closed regular submanifold of ①
E E is the fiber product of the canonical projection B
space
B , fore more details see [2][3][4]. 4
: E B E / by itself over the orbit
the product manifold E E (i.e., there exist on the orbit space E / a unique C -structure for which the canonical projection is submersion).So , ( E E, E ) B
is C -groupoid [3]. (ii) If H is closed subgroup of a Lie group , then H acts differentiably on by law of composition such that the graph of the equivalence relation
defined by this action is closed regular submanifold of H [5] . 2.4 Definition: A Lie group is said to act principally on a C -manifold E if acts freely on E (from the right) and E / B has a C -structure for which : E B is surmersion. In this case E E isomorphic to E in DG by ( z, r ) ( zr, z ) and the orbit B
map z : E ; (r zr) , for each z E is an regular embedding , also the translation function [2][5], T : E E ; (( z, zr) r ) , is a morphism in DG . B It should be remarked that if acts principally on E , then ( E, , E / ) is principal fiber bundle, then acts principally on E such that B E / in D . 2.5 Definition: The C -groupoid ( E , E ) is called action groupoid and a C -morphism T : E E is called (principal )action morphism. B
2.6 Lemma: If
: E E is a law of
C -right action of a lie group on
a C -manifold E , then ( E E / , E / ) is a C -groupoid (called Ehersmann groupoid) [2][5]. 3. Lie groupoid 3.1 Definition: A C -groupoid (G, B) is called locally trivial groupoid (L.T.groupoid) if the transitor is submersion [3]. 3.2 Definition: A C -groupoid (G, B) is called Lie groupoid if the transitor is surmersion [3]. 5
3.3 Example: (1) Ehersmann groupoid is Lie groupoid. (2) Any trivial C -groupoid is Lie groupoid. The following Lemma gives the properties and characterization of the L.T.groupoid (Lie groupoid). 3.4 Lemma [2]: Let (G, B) be a C -groupoid, then the following statements are equivalents: (1) G is L.T.-groupoid . (2) x :
Gx
: Gx B is submersion .
(3) x B , an open neighborhood U of x in B such that the following diagram is commutative in DG : U U x Gx 1 (U U )
Pr1 Pr2 U U
If G is transitive then: (4) G is Lie groupoid . (5) x:Gx Gx G ; ( g , h) gh 1 is surmersion , x B . 3.5 : Let (G, B) be a C -groupoid, then (Gx , x , [ x]) ① is C -bindle and the vertex group x G x acts principally on -fiber at G x (from the right)by law of composition and then x G x (Gx , x , [ x]) is principal fiber bundle. If G is Lie groupoid we have an isomorphy class [ x G x (Gx , x , [ x]) ] xB of principal fiber bundles [2]. 3.6 proposition: Let G be a Lie groupoid, then) Ehersmann groupoid (Gx Gx / x Gx , Gx / x Gx ) is isomorphic to (G, B) in DG , x B .
①
[ x] (Gx ) (orbit of x ) is just submanifold of B [2][6]. 6
Proof: The map x:Gx Gx G and the canonical projection x:Gx Gx Gx Gx / x Gx are both surmersions and constant on the fibers of
each other , hence the dotted arrows in the following diagram:
x G G / G G x G x x x x x
x
Fx
Fx1
exist and unique in D (The
universal property of surmersion ) .
G
The map Fx given above defined by Fx ([( g1 , g 2 )]) g1 g 21 becomes a diffeomorphism. Simple verification show that ( Fx1 , x ) is an isomorphism of C -groupoids.
4. Principal action of Lie groupoid 4.1 Definition: A C -groupoid (G, B) is said to act differentiably on a C bundle ( E, P, B) if (G, B) acts (from the left) on a bundle ( E, P, B) such that the action law : G * E E ① is C -map. The action is called weak principal if the action is free and transitive. Simple verification show that (G * E, E) has a C -groupoid structure (called action groupoid) and T : G * E G ; ( g , z) g is an DG (called principal action morphism), also when the action is transitive (G * E, E) and (G, B) are both transitive C -groupoid. 4.2 Definition: A weak principal action of a C -groupoid on a C - bundle is called principal if the action groupoid is L.T.-groupoid. 4.3 Example: Let (G, B) be a transitive C -groupoid , then the triple
(Gx , x , B) is a C - bundle on which G acts weakly by restriction the law of
①
G E is the fiber product of P and over B .
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composition on G * Gx (fiber product of x and over B ).If G is Lie groupoid then the action is principal . 4.4 Proposition: (1) The action groupoid (G * E, E) is L.T.-groupoid iff the (orbit) map z : Gx E is submersion, z E , P( z) x B .
(2) A weak principal action is principal iff the (orbit) map z : Gx E is submersion, z E , P( z) x B .(in fact diffeomorphism) iff the action groupoid (G * E, E) isomorphic to ( E E, E) in DG (by transitor map). Proof: The prove is an easy consequence of the following commutative diagrams in D : GP ( z )
GP( z ) {z} (G * E) z
z*
G P(z)
GP ( z ) {z}
z*
z* E
E
4.5 Theorem: Let
E {z}
: G * E E be a law of principal action of a L.T.-
groupoid (G, B) on a C - bundle ( E, P, B) , then: (i) ( E, P, B) is a principal fiber bundle , where [ x Gx ] xB . (ii) x G x (Gx , x , B) and x Gx ( E, P, B) are isomorphic principal fiber bundle , x B .
Proof: (i) Let z E , with P( z) x B and = x Gx . I Gx Gx E by ( z , r ) * ( g r ) ,where Define : E 1 z
z
g z* ( z ) and is the induced map by the restriction of on G x whose 1
image is G x . Simple verification show that is a law of C -right free action. (a)Simple verification show that fibers of P equal to the orbits of . (b)To prove that P has a C -local section at z : 8
There is an open neighborhood U of x in B such that 1 (U U ) is isomorphic to U U x Gx in DG (3.4), denote such isomorphism by x . Define S : U Gx by S ( y) ( y, x, ( x)), y U , S is C -local section of x . 1
Now define z : U E by z ( y) * (S ( y)[S ( x) ] , z ), y U . z is C -local section of P containing z on its range we have the following steps: (1) To prove z is C -map: 1
Define f : U G by f ( y) S ( y)[S ( x) ] . f is C -map and the induced map f : U Gx by f whose image is in G x is C -map. So, by using this C -map
z is C -map. 1
(2) p( z ( y)) p( (S ( y)[S ( x) ] , z)) x (S ( y)) y, y U and z ( x) z . Hence ( E, P, B) is a principal fiber bundle. (ii) Let z E , with P( z) x B . From (i) x Gx ( E, P, B) is a principal fiber bundle and , also x Gx (Gx , x , B) is a principal fiber bundle (3.5). The maps
z* : Gx E,
I :x Gx x Gx , I : B B , where I , I are the identity maps,
represent an isomorphism of principal fiber bundles. 4.6 Corollary: If is a law of principal action of L.T.-groupoid (G, B) on a C - bundle ( E, P, B) , then the C -(bundle) map P : E B is surmersion.
5. Symmetry between laws of principal action in DG 5.1 Theorem: Let : E E be a law of principal (right) action of a lie group on a C -manifold E, then: (i) The Ehresmann groupoid ( G E E / , B E / ) has a weak principal (left) action on a C -bundle ( E, , B) .
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(ii) The action groupoid G * E is an isomorphic to the Decartes groupoid E E in DG (i.e. G * E Lie groupoid and Ehresmann groupoid acts principally on ( E, , B) ).
(iii) is isomorphic to the vertex group x Gx of G and G x is diffeomorphic to E for each x B . Proof: (i) Elements of ( G E E / , B E / ) are the orbits [( z , z)] of ( z , z ) under with source
(z) and the fiber product of and : E B is the regular
submanifold G * E of G E of elements ([( z, z)], z1 ) with ( z, z1 ) E E E in DG B (2.4).Define
: G * E E by ([( z, z)], z1 ) ( z,T ( z, z1 )) where
T is the principal
action morphism related to . To prove is C -map we have the following steps: Let
~
~ : E E G E E /
be
the
canonical
projection
map,
then
1
( I ) (G E ) E E E is regular submanifold of E E E and the induced map B
F : E E E G E by restriction of B
~ I E on
E E E whose image is G E is B
surmersion. From the following commutative diagram: I T E E E E B
F
GE
we have that is C -map.
E
One can easily verify the other conditions. So,
is a law of weak principal
action. (ii) The transitor : G * E E E ; is bijective C -map. The following diagram is commutative in D for which the square (A) is a fiber product of
and over B .
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~
EE
G
G*E
(A)
E
B
T
Hence the dotted arrow exists and unique in D making the whole diagram commutative in D (universal property of pull-back).We denoted this morphism by : E E G E which is defined by ( z1 , z2 ) ([( z1 , z2 )], z2 ). One can verify that I G*E and I EE and then diffeomorphism. Hence
is an isomorphism of
will be a
C -groupoids over E .
(iii) Let x B , with x ( z), z E and E x 1 ( x) . Define f : x Gx by f (r ) [( z r, z)] f is C - bijective group homomorphism. Its inverse f 1 : x Gx defined by f 1 ([( z r, z s)]) rs 1 is C -map: Define f * : Ex Ex by f * ( z r, z s) rs 1. f * is surjective C -map. From the following commutative diagram in D :
~ 1 ( x Gx ) Ex Ex f
~ x
Gx
f
1
we have that f 1 is C -map. Therefore f is an isomorphism of Lie groups , and Gx E in D by (4.4,2).
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5.2 Main Theorem: There is a symmetry in DG between the principal action law of a Lie group and that of Lie groupoid on a C -manifold. Proof: Given a principal (right ) action
: E E of a Lie group
on a
C -manifold E , then Ehresmann groupoid ( G E E / , B E / ) is a Lie
groupoid and acts principally (from the left) on the C -bundle ( E, , B) such that the vertex groups of G are isomorphic to and each -fiber is diffeomorphic to E (5.1). conversely; given a principal action
: G * E E of a Lie groupoid
(G, B) on
a C -bundle ( E, , B) , then the vertex group (unique up to isomorphism) acts principally (from the right) on E (4.5)(2.4) and Ehresmann groupoid E E / isomorphic to G DG (3.6).
5.3 Corollary: There is symmetry between laws of principal action morphism of a Lie group and that of Lie groupoid. Proof: Given a principal action morphism T : E E by (5.2) there is a Lie B groupoid (Ehresmann groupoid) (G, B) acts principally on a C -bundle ( E, , B) , hence we have a principal action morphism T : G E G (4.1).
Define F : E E G by F ( z1 , z2 , r ) [( z1 r, z2 r )] . F is clearly a morphism of C -groupoids and the following diagram is commutative in DG .
E EE
G*E E E
B
T*
T
K
G
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I E E F G and where K = E E , E EE E E Pr 3
( z, z ) ( z, z, e) [( z, z)] .
Conversely; given a principal action morphism T : G E G of a Lie groupoid (G, B) on a C -bundle ( E, , B) , then by (5.2) there is a Lie group (vertex group unique up to isomorphism ) acts principally (from the right) on E , hence we have a principal action morphism T : E E definition (2.4) B
making the following diagram commutative in DG : EE E
E E G*E
B
T
T*
K
G
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References [1] A. A. Al –Taai; Symmetry between principal action’s law of group and groupoid, Al- Mustansiriyah Journal of Science, Vol.9, No. 3, 1998. [2] A. A. Al –Taai; Caracterisation universell du groupe fondamental d’un feuilletage, Thesis, University of Paul Sabatier, Toulouse-France, 1988. [3] J. Diéudonne; Treatise on analysis, Vol. 3, Translater I.G. Macdonald, Academic Press, New York, 1972.
[4] K. Mackenzie; Lie groupoids and Lie algebroids in differential geometry, Lond. Math. Soc. Lecture note series 124,Camberidge University Press, 1987.
[5] P.J. Oliver; Applications of Lie groups to differential equations, Springer-Verlag, New York, Inc. 1986.
[6] J. Pradines; How to define the graph of a singular foliation, Cahires de Topol. et Geom. Diff. 26, (339-380), 1986.
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