By E. Grosswald

ISBN-10: 3540091041

ISBN-13: 9783540091042

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Ii) Define ϕ˜ : V0 → U1 , ϕ˜ := expG |V0 . It follows by (i) that ϕ˜ is a local coordinate chart on G around 1 ∈ G. Then denote g = L(G), pick r1 > 0 such that ϕ˜ Bg (0, r1 ) · ϕ˜ Bg (0, r1 ) ⊆ U1 , and define µ : Bg (0, r1 ) × Bg (0, r1 ) → g by µ(x, y) = ϕ˜−1 ϕ(x), ˜ ϕ(y) ˜ = ϕ˜−1 m(ϕ(x), ˜ ϕ(y)) ˜ . For 0 = x ∈ Bg (0, r1 ), t, s ∈ R and max{|t|, |s|} < 2 rx1 , we have µ(tx, sx) = ϕ˜−1 expG (tx) · expG (sx) = ϕ˜−1 expG ((t + s)x) = (t + s)x. 12 to deduce that µ is real analytic on some neighborhood of (0, 0) ∈ g × g.

Consequently, for = n/|n| ∈ {−1, +1} we have lim z→z0 |ht0 /(2|n|) (z)| |ht0 (z)|1/(2n) = lim = |k(z0 )|1/(2|n|) = 0, /2 z→z0 |z − z0 | /2 |z − z0 | which is impossible since ht0 /(2|n|) is a rational function. Consequently both polynomials ft0 and gt0 are constant, and the proof ends. 38 For the Lie group A× we have D(expA× ) = K× 1 = A = L(A× ). Copyright © 2006 Taylor & Francis Group, LLC Lie Groups and Their Lie Algebras 43 Indeed, v ∈ D(expA× ) if and only if there exists a one-parameter subgroup f : R → A× such that f˙(0) = v.

In the left-hand side, the coefficient is (n + 1)zn+1 , while the coefficient in the right-hand side is a linear combination of expressions of the form (ad zn1 ) · · · (ad znk )(umk ) = [zn1 , . . , [znk , umk ] . ], where umk ∈ {xmk , ymk } and n1 + · · · + nk + mk = n + 1. 3) belongs to L(V). 3), that is (n + 1)zn+1 , in turn belongs to L(V). Thus zn+1 ∈ L(V), and the induction is complete. The series that shows up in the following statement will be called the BakerCampbell-Hausdorff series.

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Bessel Polynomials by E. Grosswald


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