Seminar Sophus Lie 2 (1992) 59

Abstract

Extending the Campbell-Hausdorff multiplication

Anselm Eggert

In the Lie Theory of groups and semigroups, one sometimes needs an analytic extension of the locally defined Campbell-Hausdorff multiplication onto much larger domains (cf. e.g. [2]). The following Theorem generalizes an old result by Dixmier [1] significantly. Its proof, which uses a method different from that of Dixmier, will appear in [3].

Theorem . Let 𝔤 be a finite dimensional real Lie algebra and 𝔧 an ideal of 𝔤. Assume that for x ∈ 𝔧 the mapping ad x|𝔧 does not have an eigenvalue contained in 2πiℤ \ {0}. Then there is an open ball V around 0 in 𝔤 and an analytic extension ∘ : (𝔧 + V) × (𝔧 + V) → 𝔤 of the Campbell-Hausdorff-multiplication.

References

[1] Dixmier, J., L’application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. France 85 (1957), 113–121.

[2] Eggert, A., ,,Über Liesche Semialgebren“, Mitteilungen aus dem Mathematischen Seminar Gießen 204 (1991), iv+91pp.

[3] Eggert, A., Extending the Campbell-Hausdorff multiplication, Geometriae Dedicata, to appear.

Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstraße 7 W-6100 Darmstadt Germany e-mail: XMATag05@ddathd21.bitnet

Received March 30, 1992