By Werner Hildbert Greub
Read Online or Download Connections, Curvature, and Cohomology Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes PDF
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Extra info for Connections, Curvature, and Cohomology Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes
Example text
If M is projective and finitely generated, then N can be chosen so that M @ N has a finite basis. If M is finitely generated and projective, then so is M*, and for all R-modules N , the homomorphism a given just above is an isomorphism. In particular, the isomorphism M* ORM HomR(M;M ) 7 0. Algebraic and Analytic Preliminaries 8 specifies a unique tensor tME M* Q R M such that (Y(tM) = LM . It is called the unit tensor for M . A graded module is a module M in which submodules Mp have been distinguished such that M = C Mp, P>O The elements of Mp are called homogeneous of degree p .
Lie algebra of a Lie group. T h e Lie algebra of a Lie group G is the vector space, T,(G), together with the Lie algebra structure induced from S L ( G )by the isomorphism of Corollary I to Proposition I, sec. 2. Thus, for h, k E T,(G), [h, kl = [Xh Xkl(4 9 (Note that the isomorphism S,(G) 5 T,(G) determines a second Lie product [ , 3" in T,(G). In view of sec. 2, we have [h, k] = h, k -[h, k]", E Te(G). Thus the map h ++ -h defines an. ) Now consider a homomorphism of Lie groups, s,: G H . Since y(e) = e ( e denotes the unit of both groups), the derivative ds, restricts to a linear map --f (dq)e: Te(G) + This map will be denoted by s,'.
Proposition X: T h e correspondence Ad: a representation of G in E. Proof: Evidently r, 0 A d a defines a rb = r a b ,and so AdaoAdb =Adab. Thus Ad is a group homomorphism. I t remains to show that Ad is smooth. Define a smooth map T : G x G -+ G by setting w, 4 = TdX), y , x E G. Its derivative, dT,is smooth. e)(O, h) = AdY (h). Hence, for each h E E, the maps y Ad is smooth. I-+ Ad y ( h ) are smooth. D.