SYMMETRIC SPACES OF THE NON COMPACT TYPE LIE GROUPS
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Niveau: Supérieur, Doctorat, Bac+8
SYMMETRIC SPACES OF THE NON-COMPACT TYPE : LIE GROUPS by Paul-Emile PARADAN Abstract. — In these notes, we give first a brief account to the theory of Lie groups. Then we consider the case of a smooth manifold with a Lie group of symmetries. When the Lie group acts transitively (e.g. the manifold is homogeneous), we study the (affine) invariant connections on it. We end up with the particuler case of homogeneous spaces which are the symmetric spaces of the non-compact type. Resume (Espaces symetriques de type non-compact : groupes de Lie) Dans ces notes, nous introduisons dans un premier les notions fon- damentales sur les groupes de Lie. Nous abordons ensuite le cas d'une variete differentiable munie d'un groupe de Lie de symetries. Lorsque le groupe de Lie agit transitivement (i.e. la variete est homogene) nous etudions les connexions (affines) invariantes par ce groupe. Finalement, nous traitons le cas particulier des espaves symetriques de type non- compact. Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Lie groups and Lie algebras: an overview.

  • paul-emile paradan

  • groupe de lie de symetries

  • g?g ??

  • topological space

  • ?? xy

  • morphism ?

  • levi-civita connection

  • all bijective

  • lie group


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SYMMETRIC SPACES OF THE NON-COMPACT TYPE : LIE GROUPS
by
Paul-Emile PARADAN
Abstract. —In these notes, we give first a brief account to the theory of Lie groups. Then we consider the case of a smooth manifold with a Lie group of symmetries. When the Lie group acts transitively (e.g. the manifold is homogeneous), we study the (affine) invariant connections on it. We end up with the particuler case of homogeneous spaces which are the symmetric spaces of the non-compact type.
Re´sum´ groupes de type non-compact :(Espa ´triques eces syme de Lie) Dans ces notes, nous introduisons dans un premier les notions fon-damentales sur les groupes de Lie. Nous abordons ensuite le cas d’une varie´t´edi´erentiablemuniedungroupedeLiedesym´etries.Lorsque legroupedeLieagittransitivement(i.e.lavarie´te´esthomoge`ne)nous e´tudionslesconnexions(anes)invariantesparcegroupe.Finalement, noustraitonslecasparticulierdesespavessyme´triquesdetypenon-compact.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Lie groups and Lie algebras: an overview . . . . . . . . . . . . . 2 3. Semi-simple Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4. Invariant connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5. Invariant connections on homogeneous spaces . . . . . . . . 36 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2000Mathematics Subject Classification. —22E15, 43A85, 57S20. Key words and phrases. —Lie group, connection, curvature, symmetric space.
2
PAUL-EMILE PARADAN
1. Introduction
This note is meant to give an introduction to the subjects of Lie groups and of equivariant connections on homogeneous spaces. The final goal is the study of the Levi-Civita connection on a symmetric space of the non-compact type. An introduction to the subject of “symmetric spaces” from the point of view of differential geometry is given in the course by J. Maubon [5].
2. Lie groups and Lie algebras: an overview
In this section, we review the basic notions concerning the Lie groups and the Lie algebras. For a more complete exposition, the reader is invited to consult standard textbooks, for example [1], [3] and [6].
Definition 2.1. —A Lie groupGis a differentiable manifold(1)which is also endowed with a group structure such that the mappings
G×G−→G,(x, y)7xymultiplication G−→G, x7x1inversion
are smooth.
We can define in the same way the notion of atopological group is: it a topological space(2)which is also endowed with a group structure such that the ‘multiplication’ and ‘inversion’ mappings are continuous. The most basic examples of Lie groups are (R,+), (C− {0},×), and the general linear group GL(V) of a finite dimensional (real or complex)
(1)All manifolds are assumed second countable in this text. (2)Here “topological space” means Hausdorff and locally compact.
SYMMETRIC SPACES : LIE GROUPS
3
vector spaceV classical groups like. The SL(n,R) ={gGL(Rn),det(g) = 1}, O(n,R) ={gGL(Rn),tgg= Idn}, U(n) ={gGL(Cn),tgg= Idn}, O(p, q) ={gGL(Rp+q),tgIpqg= Ipq},where Ipq=dI0p0IdqIdn Sp(R2n) ={gGL(R2n),tgJ g=J},whereJ=dI0n0are all Lie groups. It can be proved by hand, or one can use an old Theorem of E. Cartan.
Theorem 2.2. —LetGbe aclosedsubgroup ofGL(V). ThenGis an embedded submanifold ofGL(V), and equipped with this differential structure it is a Lie group.
The identity element of any groupGwill be denoted bye write. We the tangent spaces of the Lie groupsG, H, Kat the identity elemente respectively as:g=TeG,h=TeH,k=TeK. Example: The tangent spaces at the identity element of the Lie groups GL(Rn),SL(n,R),O(n,R) are respectively gl(Rn) ={endomorphisms ofRn}, sl(n,R) ={Xgl(Rn),Tr(X) = 0}, o(n,R) ={Xgl(Rn),tX+X= 0}, o(p, q) ={Xgl(Rn),tXIdpq+ IdpqX= 0},wherep+q=n 2.1. Group action. —A morphismφ:GHof groups is by defi-nition a map that preserves the product :φ(g1g2) =φ(g1)φ(g2). Exercise 2.3. —Show thatφ(e) =eandφ(g1) =φ(g)1 .
Definition 2.4 (left) action of a group. — AGon a setMis a mapping
(2.1)
α:G×M−→M
such thatα(e, m) =m,mM mMandg, hG.
, andα(g, α(h, m)) =α(gh, m)for all
4
PAUL-EMILE PARADAN
Let Bij(M) be the group of all bijective maps fromMontoM. The conditions onαare equivalent to saying that the mapGBij(M), g7→ αgdefined byαg(m) =α(g, m) is a group morphism. IfGis a Lie (resp. group and topological)Mis a manifold (resp. topological space), the action ofGonMis said to be smooth (resp. continuous) if the map (2.1) is smooth (resp. continuous). When the notations are understood we will writegm, or simplygm, forα(g, m).
Arepresentationof a groupG complex)on a real (resp. vector space Vis a group morphismφ:GGL(V group the) :Gacts onVthrough linear endomorphisms.
Notation: Ifφ:MNis a smooth map between differentiable manifolds, we denote byTmφ:TmMTφ(m)Nthe differential ofφat mM.
2.2. Adjoint representation. —LetGbe a Lie group and letgbe the tangent space ofGate consider the conjugation action of. WeGon itself, defined by (h) =h1 h, g,G cgg g The mappingscg:GGare smooth andcg(e) =efor allgG, so one can consider the differential ofcgate Ad(g) =Tecg:ggSincecgh=cgchwe have Ad(gh) = Ad(g)Ad(h is, the mapping). That (2.2) Ad :G−→GL(g)
is a smooth group morphism which is called theadjoint representation ofG. The next step is to consider the differential of the map Ad ate:
(2.3) ad =TeAd :g−→gl(g)This is theadjoint representation ofg (2.3), the vector space. Ingl(g) denotes the vector space of all linear endomorphisms ofg, and is equal to the tangent space of GL(g) at the identity.
Lemma 2.5. —We have the fundamental relations ad(Ad(g)X) = Ad(g)ad(X)Ad(g)1forgG, Xg.
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