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Abstract. In every splittable Lie group (resp. Lie algebra), every element can be decomposed by the multiplicative (resp. additive) Jordan decomposition. The aim of the paper in hand is to give a generalization of this concept for arbitrary Lie groups and finite-dimensional Lie algebras. The main result describes a decomposition which is close to the Jordan decomposition. 2000 Mathematics Subject Classification: 17B05, 22E15. It has been impossible so far to establish necessary and sucient conditions for the exponential function of a real Lie group to be surjective. In many studies devoted to finding such conditions, the additive and multiplicative Jordan decompositions turned out to be e¤ective tools (see for example [4], [12], [14], [15], [16]). However, by their very definition, which we shall review below, their proper field of application is the class of so-called Mal'cev splittable Lie-algebras and Lie groups; this is clearly illustrated in Lemma 2.5 and Theorem 3.7 below. It is not obvious, how additive and multiplicative Jordan decompositions might be generalized to arbitrary real Lie algebras and Lie groups. Yet our principal result, Theorem 4.6, shows how such a generalization can be accomplished. 1 Results on Cartan subalgebras and Cartan subgroups First we will
Forum Mathematicum – de Gruyter
Published: May 20, 2003
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