# Connections, curvature and cohomology. Volume 2, Lie groups, by Werner Hildbert Greub

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.