By Werner Hildbert Greub
Greub W., Halperin S., Vanstone R. Connections, curvature, and cohomology. Vol.2.. Lie teams, primary bundles, and attribute sessions (AP, )(ISBN 0123027020)(543s)_MD_
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L, (cf. Example 2. Let H be a second Lie group with Lie algebra F. Then the exponential map for G x H is given by h E E, K E F. 7. Homomorphisms. Proposition VII: Let T: G + H be a homomorphism of Lie groups. Then the induced homomorphism, T', of Lie algebras satisfies cp o exp, = exp, o cp'. Proof: Fix h E T,(G). Then a: t H T(exp,(th)) and /3: t are 1-parameter subgroups of H. Moreover, i(0) = v'(h)= j(O), ++ expH(tT'(h)) 36 I. Lie Groups and hence (Proposition V, sec. 5) 01 = 8. D. Corollary I: Assume +: G --f H is a second homomorphism of Lie groups and that cp' = #'.
T h e 1-parameter subgroup generated by a vector h E C is given by q , ( t ) = exp th. 6. The exponential map. Define a set map #:R x E + G by # ( t , h) = ah(t), t R, h E E. E Lemma 11: $ is a smooth map. I t satisfies #(st, h) = # ( t , sh), S, t E R, h E E. Proof: T h e equation holds because both sides define the l-parameter subgroup generated by sh (cf. sec. 5). T o show that $ is smooth, define a vector field 2 on the manifold E x G b y Z(h, 4 = (0,&(a)). I n view of Theorem 11, sec. :I x ( V x U ) + E x G such that +(t, h, a ) = Z(&, h, a)), d o , h, 4 = (A, t E I, hE v, a E u, and 4.
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). Hence P’(h) determines an operator P’(h), in H ( W ) and (P‘),: h i--t P’(h)# is a representation of E in H( W). 10. The adjoint representation. Each a E G determines the inner automorphism, r, , of G given by T,(x) x E G. = ax&, Hence the derivative, r; , of r, is an automorphism of the Lie algebra E.