By Werner Hildbert Greub
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Additional resources for Connections, Curvature, and Cohomology Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes
E, expc(tnen). + By Corollary I1 to Proposition VI, sec. 6, maps a neighbourhood V of 0 diffeomorphically onto a neighbourhood U of e. , n) are continuous homomorphisms [w + H ; thus they are smooth by the argument above. )tn) = dexpG(tlel)) ’” dexPdtnen)) + is smooth. In particular, y is smooth in U. But for any a P’(4 = E G, d4P ’ ( 4 Thus g, is smooth in a neighbourhood of a and hence in G. D. s3. Representations In this article G denotes a fixed Lie group with Lie algebra E. 8. The derivative of a representation.
A O O O arbitrary, we obtain = Xu,s-o,a * I n particular, the Lie algebra structure of L , induced from the Lie group structure of GL( V ) is given by [a,8] = a o / 3 - p o a . 3. The group of invertibles of an associative algebra: Let A be an associative finite-dimensional algebra over [w, with unit element. For a E A , define p(u): A + A to be left multiplication by a. , if and only if det p(a) # 0. T h e invertible elements of A form a group G ( A )under composition; the condition above shows that G ( A )is open in A.
De Rham cohomology. Let M be an n-manifold. Then ( A ( M ) ,6) is a graded differential algebra; its cohomology is denoted by H ( M ) = C= :, HP(M)and is called the de Rham cohomology algebra of M . T h e homomorphism v*: A ( M ) +- A ( N ) determined by a smooth map induces a homomorphism v#: H ( M ) t H ( N ) . If dim H ( M ) < 00, then the pth Bettinumber, b, ,of Mis dim Hp(M). T h e polynomialf(t) number = Ep bptp is called the Poincare'polynomial and the c (-l)Pbp n x, = V=O is called the Euler-PoincarC characteristic of M .