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A generalization of the Jordan decomposition

A generalization of the Jordan decomposition 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

A generalization of the Jordan decomposition

Forum Mathematicum , Volume 15 (3) – May 20, 2003

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Publisher
de Gruyter
Copyright
Copyright © 2003 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2003.021
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Forum Mathematicumde Gruyter

Published: May 20, 2003

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