By Roger W. Carter

Emphasizing the vast advances that experience taken position within the thought of finite easy teams, this monograph considers the teams of Lie sort as fastened issues of reductive algebraic teams over an algebraically closed box of major attribute less than the motion of a Frobenius map.

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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.