Download A Fundamental Theorem on One-Parameter Continuous Groups of by Kennison L. S. PDF

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By Kennison L. S.

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Fix t ∈ R and put β(h) := α(t + h) (h ∈ R). Since β(h) = α(t + h) = α(t) α(h) = ℓα(t) (α(h)), we have α′ (t) = β ′ (0) = (dℓα(t) )e (α′ (0)) = (dℓα(t) )e (A). 1). 5 Conversely, let be given A ∈ Te G. 1) for t in some interval (−ε, ε). Now we will show that α(s + t) = α(s)α(t) if |s|, |t|, and |s + t| are < ε. Let |s| < ε and put β(t) := α(s + t), γ(t) := α(s) α(t). Then β(0) = α(s), γ(0) = γ(s) and, on the one hand, β ′ (t) = α′ (s + t) = (dℓα(s+t) )e A = (dℓβ(t) )e A, on the other hand, γ ′ (t) = (dℓα(s))α(t) α′ (t) =(dℓα(s) )α(t) (dℓα(t) )e A = d(ℓα(s) ◦ ℓα(t) )e A = (dℓα(s)α(t) )e A = (dℓγ(t) )e A.

F ) . )))(x) k 1 ! . km ! + O(|t|n+1 ) as t → 0. 22 have Proposition Let G and g be as above. f ). Proof Let x ∈ G. We will expand f (x exp(tA) exp(tB) exp(−tA) exp(−tB)) in two different ways as a Taylor series in t up to degree 2, where t → 0 in R. Then we obtain the result by equality of second degree terms in both expansions. f ))(x) + O(|t|3 ). f )(x) + O(|t|3 ). 23 Corollary Let G be a Lie group with g := Te G and Lie(G) the Lie algebra of left invariant vector fields on G. Put [A, B] := b(A, B) (A, B ∈ g).

19 Corollary With G, g, b(A, B) as above and A, B → 0 in g we have log(exp(A) exp(B) exp(−A) exp(−B)) = b(A, B) + O((|A| + |B|)3 ). Proof Fix A, B ∈ g and let t → 0 in R. 3) three times, log(exp(tA) exp(tB) exp(−tA) exp(−tB)) = log exp(tA + tB + 21 t2 b(A, B) + O(|t|3 )) exp(−tA − tB + 12 t2 b(A, B) + O(|t|3 )) = t2 b(A, B) + O(|t|3 ). 20 Proposition C ∞ (G) by Let G, g be as above. f )(x) := d f (x exp(tA)) dt t=0 (x ∈ G). f is a left invariant vector field V on G such that Ve = A. 7 Ex. 21 Let G, g be as above.

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