By Werner Hildbert Greub
Read or Download Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes (Pure and Applied Mathematics Series; v. 47-II) PDF
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Additional resources for Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes (Pure and Applied Mathematics Series; v. 47-II)
P(x-’)w,), w iE W , X E G, @E TP(W). , w P ) , h E E. i=l 4. DiSferential spaces: Let (W, d ) be a differential space (cf. sec. 7) and denote its homology by H ( W). Assume that P is a representation of G in W such that P(x) d 0 = d 0 P(x), x E G. I. Lie Groups 42 Then P(x) determines a linear map P(x),: H( W )+ H( W ) and P,: x F+ P(x), is a representation of G in H ( W ) . On the other hand, the representation, P’, of E satisfies P’(h) o d = hEE d o P’(h), (differentiate the relation above).
A subspace I'C W is called stable for P (respectively, stable fw 8) if each of the operators P (x), x E G (respectively 8(h), h E E ) maps V to itself. Now fix h E E. Then P(exp th), and P'(h) are linear transformations of W. In particular, we regard the 1-parameter group P,: t H P(exp th) 39 40 I. Lie Groups as a path in the vector space i),(t) inL, . Lw . Thus differentiation yields a path On the other hand recall from Example 2, sec. 4, that T,,(,, GL(W) x Lw . Moreover, x p * ( h ) ( ~= ) (7, T Applying this formula with 0 T P‘(h)), = T = GL(w), h E A!?.
81 = a8 - 8% [a, 4. Direct products: Let G, H be Lie groups. T h e product manifold G x H can be made into a Lie group by setting (x, y ) * (x’, y’) = (X . x’, y . J J ’ ) , X, X’ E G y , y’ E H . This Lie group is called the direct product of G and H . T h e projections nG: G x H -+ G and nH: G x H -+ H , and the inclusions G, H -+ G x H , opposite e, are all homomorphisms of Lie groups. T h e Lie algebra homomorphisms nb , are given by ~ k ( hk,) =h &(A, k ) and = k. It follows that the Lie product in T,(G x H ) is given by [(A, k), (h’, k’)] 5.