New PDF release: Abelian Group Theory

By D. Arnold, R. Hunter, E. Walker

ISBN-10: 3540084479

ISBN-13: 9783540084471

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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. e’s, so they must be equal. 1) on (−ε, ε) which satisfies α(s+t) = α(s)α(t) = α(t)α(s) for s, t, s + t in the definition interval.

Hence d exp0 = id. 18 Proposition Let G be a Lie group and put g := Te G. 3) as |A|, |B| → 0 in g. In particular, if G ⊂ GL(n, C ) is a linear Lie group (and thus g ⊂ gl(n, C )) then b(A, B) = AB − BA (A, B ∈ g). Proof Clearly, by the uniqueness of Taylor expansion, the bilinear map b is unique if it exists. For the existence proof consider the Taylor expansion bi,j (A, B) + O((|A| + |B|)3 ). log(exp(A) exp(B)) = i+j≤2 Here bi,j (A, B) ∈ g, with each coordinate being a polynomial in the coordinates of A and B, homogeneous of degree i in A and homogeneous of degree j in B.

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. Let A, B, A1 , . . , Am ∈ g, x ∈ G, t ∈ R and f ∈ C∞ (G). f )(x) + O(|t|n+1 ) as t → 0, k! f ))(x) + O(|t|n+1 ) as t → 0, k!

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Abelian Group Theory by D. Arnold, R. Hunter, E. Walker


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