Journal of Lie Theory Vol. 13, No. 1, pp. 291306 (2003) 

The Exponential Map and Differential Equations on Real Lie GroupsMartin Moskowitz and Richard SackstederMartin MoskowitzDepartment of Mathematics CUNY Graduate Center 365 Fifth Ave. New York NY 10016, USA mmoskowi@axp.mat.uniroma2.it and Richard Sacksteder Department of Mathematics CUNY Graduate Center 365 Fifth Ave. New York NY 10016, USA sackbix@prodigy.net Abstract: Let $G$ be a connected Lie group with Lie algebra $\g$, $\exp_G\colon\g\to G$ the exponential map and $E(G)$ its range. $E^n(G)$ will denote the set of all $n$fold products of elements of $E(G)$. $G$ is called {\it exponential} if $E(G) = E^1(G) = G$. Since most real (or complex) connected Lie groups are not exponential, it is of interest to know that the weaker conclusion $E^2(G)=G$ is always true (Theorem 5.3). This result will be applied to prove Theorem 6.1, a generalized version of FloquetLyapunov theory for Lie groups. It will then be seen the property that a Lie group is exponential is equivalent to the existence of a special form of FloquetLyapunov theory for it (Corollary 6.1). We generalize the wellknown fact that connected nilpotent Lie groups are exponential. Our methods also provide alternative proofs of some known results by arguments which seem simpler and more natural than the usual ones. Among these is part of the classical DixmierSaito result, Theorem 5.4. The method employed here stems from the earliest techniques of Lie theory. It exploits connections between the exponential map and differential equations on $G$, starting from the observation that the one parameter subgroup $g(t) = \exp_G(t\gamma)$ corresponding to an element $\gamma\in\g$ satisfies the differential equations $$ g'(t)=dL_{g(t)}\gamma,\hbox{ and }g'(t)=dR_{g(t)}\gamma $$ on $G$ with the initial condition $g(0) = e_G$. More generally here it will be necessary to consider differential equations corresponding to certain time dependent vector fields on $G$, or equivalently, certain time dependent crosssections of the tangent bundle of $G$. Full text of the article:
Electronic fulltext finalized on: 22 Nov 2002. This page was last modified: 3 Jan 2003.
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