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By P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi
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Additional resources for Analyse Harmonique sur les Groupes de Lie, 1st Edition
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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). f . Ex. 24 Let G be the Lie group defined as the set {(a, b) ∈ R2 | a = 0} with multiplication rule (a, b)(c, d) = (ac, ad + b).
Finally, the definition is independent of k since α(t/k)k = α(t/(kl))kl = α(t/l)l if |t/k|, |t/l| < ε. 1) for all t ∈ R. 13). 1) as a system of differential equations on Mn (C ) and, by restriction, also as a system of differential equations on the submanifold G. 14 asssociates in the case of a linear Lie groups G with A ∈ g the C ∞ -homomorphism t → etA . This suggests the definition of the abstract exponential mapping in the case of a general Lie group. 16 Definition Let G be a Lie group. We now put g := Te G.
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.