RUCTURE Of THEco · Basic Theory · Representations of Lie groups · Classification of Lie groups (and Lie algebras) · Homogeneous spaces , klein geometries I. BUNDLES · Fiberbundles , vector bundles and principal bundles · Associated vector bundles · Homogeneous vector bundles III . CONNECTIONS · Linear connections on vector bundles (in particular , affine connections · Principal connections on principal bundles · Geometric structures determining (classes of) distinct connections (e. g. Riew-refds , conformal structures, projective structures, ... ( · Holonoury · Cortan geometrics #Lie groups (Cap , Lie groups , lecture notes) #Basic Theory For a group G we write : · M : GxG > G for the multiplication map · U : G > G for the inversion , r(g) = gt . eeG for the identity/neutral element in G. Def. 1 . 1 A topological group is a topological space G equipped with a group structure (M , 0 , 2) s . t . pe and r are continuous . Remark Any abstract group can be mode into a topological group by equipping it with the discret topology . De01. 2 A lie group is a smooth manifold G equipped with a group (M. W , e) s. + . M and w are smooth. Zeue. In Def . 1 . 2 it is enough to require that M is smooth , since w is then automatically swooth by applying the Inverse Function/Umplicit Function Thus to the equation M(g . w(g) = e. #f. 1 . 3 ① A homomorphism between topological groups (resp . Lie greups) Gand It is a continuous (resp. smooth) map D : G-H that is also a group homomorphism (4(g. h) = y(g)4(h) -gintG) ② A homomorphism y : G-H as in ① is called an isomorphism between top- groups Cresp . Lie graeps) , if i is a homeomorphism Cresp . diffeomorphism). Note that in this case , " is also a group homomorphisms . #station Two lie groups G andH are called isomorphic , if 7 a lie group isomorphine between them. We write G-H in this lose. Groups of greatest interest in mathematics and physics consist of bijections f : M -+ M of a set M (space) with group multiplication given by camposition : M(fif) = f - E = for fif e Bij (M) = Ebijections of M] w(f) = fe = Im Examples If M has some extra structure , one can consider subgroceps of (Bij(M) , 0) consisting of bijections preserving that extre structure . · M topolog . Space Homeo (M) : = { fe Bij(M) : f is a homean ? · M Smooth wifd (resp . Smooth briented ufel) Diff(M) = St : M + M : f is a differn . 3 Diff(M) = [f : M + M : Orientation preserv differs] - suppose M is a smooth refe equipped with a geometric structure like a Riemannian metric g or a symplectic form wi kam (Mig) = [f : M + M : differu . s . 1 . f * g = g] Symp(Mig) = 37 eDiffIM) : f = w] With the exception of kom (Mig) , these groups are infinite-dimensional and neuce not lie groups in our sense . These groups are all naturally topological groups. Isom (M,g) is a lie group of die . Lumin(M) Now some examples of actual lie groups. Typical examples arise from linear transformations of finite-dim . Vector spaces . Examples ① IR , K with respect to addition + is a lie group and so is any finite-dim . Vector space over IR or & W . r . to +. ⑪ and any complex vector space is even a complex lie group Li . e . a complex wife with holoma purc group structure) . ② IRK50S , Clos are lie groups W . r . to multiplication Also , U(1) = S = Eze D : (z) = 13 is a lie group . ③ The product Ext of lie groups is again a lie group. In particular , the n-dim . torus TV =. W(1) is a lie group. - n times Also , for min natural numbers , IRM Th is a hie group . The latter exhoust all connected commentative lie groups . ④ If G is a lie group , then a lie Subgroup I of G is a subgroup HIG that is also a sunmetal. Since the multipl on It is just the restrictio of the one from G , it smooth , and so It is a lie group . ⑤ Suppose V is a real or complex finite-dim. vector space. EL(V) = Elinear isomorphismus ofVGEEnd(v)= Y = Slinear mops v + 13 open Subset is a lie group with respect to composition. Vin Choice of bocis , we may identify V= IR" Cresp . (h) and GL(V) with GL(n , (K) = 5 AeMen(ik) : A invertible] with (k = IR orc and composition of linaor mops because matix multiplication . Agoin , GLIn , K) is alsom complex lie group . It is called theGeneral Linear group. ⑥ Matrix groups (also called linear lie grps) are Lie subgroups G of GL(mIR) and GL(n . C). · Special linear group G = SL(n , (k) = &AEGL(n , (4) : det(A) = 13 K = IRND (SLInik) is n lauplex wie group) . · Orthogonal groups : Ipiq = (Edeld) Mach) = (i...) n = p+ q defines inner product of Signature (19) on IR TigniM(Sin) = Tongm(9. 0) + T(0.n)Let c : (-3 , a) + G Co-curve representing a : c(0) = gc'(0) = 3 Then tr(c(D . h) represents (9 . 0) - ToGxG) M(c(t), n) = fh(c(t) => Tigin)M(3 , 0)M(CH, h)- - Similarly , Tign/M(0, ) =Mg, c)le for a curve C : (-5 . c) > G = Tudg4 with c(0) = h , c'(d = y. ② e = (g , w(g) => 0 = Trgig)M (5 , Tgrs) i differentiale => Tp9" +TSt => TTgrs = Tgggg El TgWS = -TelgoTg99g Second formula follows similarly from diff . e = M(wig) , g) * Recall that for any (local) differu . f : M + M on a wefe , f * T(TM) > TCTM) , given by f* = [Tf) Sof , is linear and f * [siy] = [fs, Ei 16SuppseGegroup.Thenuvetnient, = (t(G) if Jg* => FgeG (resp. (99)* =s VgeG). Denote by Jtz(G) (resp. Ctr(G) the set of left-Cresp. right) invariant rectorfields on G. By linearity of the pull-back and its compatibility with [ , J , it is a subalgesia of the infinite-dic. Lie algebra (It(G) , [ , J) .