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Commuting exponentials in a Lie group

Published online by Cambridge University Press:  28 September 2006

KARL H. HOFMANN
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-63289 Darmstadt, Germany. e-mail: [email protected]
WALTER J. MICHAELIS
Affiliation:
Department of Mathematics, University of New Orleans, Lakefront, New Orleans, LA 70148, U.S.A. e-mail: [email protected]

Abstract

Two commuting real matrices $A$ and $B$ have commuting exponentials $\exp A$ and $\exp B$, a fact observed for instance in linear algebra or differential equations courses. The converse implication is false. A clarification of this phenomenon is proposed that makes use of the theory of the exponential function $\exp\colon{\fam\euffam g}\to G$ of a real Lie group $G$ and its singularities. In Section 1, a catalog of low-dimensional examples illustrates various ways that, for two elements $X, Y\in{\fam\euffam g}$, the commuting of $\exp X$ and $\exp Y$ in $G$ may fail to entail the commuting of $X$ and $Y$ in ${\fam\euffam g}$. In Section 2, consequences of the relation $[\exp X,\exp Y]={\bf 1}$ are inspected, whereby certain regularity assumptions on $X$ and $Y$ are made. A regular element $Y$ of the Lie algebra ${\fam\euffam g}$ determines a Cartan subalgebra ${\fam\euffam h}={\fam\euffam g}^0(Y)$ of ${\fam\euffam g}$ and a certain subgroup ${\cal W}_Y$ of the (finite!) Weyl group of ${\fam\euffam g}$ with respect to the Cartan subalgebra ${\fam\euffam h}$. If, additionally, the exponential function is regular at $X$ and at $Y$, then the ordered pair $(X,Y)$ is said to be in general position. If $(X,Y)$ is in general position, then the relation $[\exp X,\exp Y]={\bf 1}$ in $G$ permits the definition of a certain element $w(X,Y)\in{\cal W}_Y$. Let ${\fam\euffam z}({\fam\euffam g})$ denote the center of ${\fam\euffam g}$. It is shown that, if $\exp X$ and $\exp Y$ commute in $G$ for $(X,Y)$ in general position, then $[X,Y]\in{\fam\euffam z}({\fam\euffam g})\cap[{\fam\euffam h},{\fam\euffam h}]$ iff $w(X,Y)={\bf 1}$. Write $H\defi\exp{\fam\euffam h}$, and let $Z(G)$ denote the center of $G$. If the identity component of $Z(G)\cap[H,H]$ is simply connected, and if $\exp X$ and $\exp Y$ commute for $(X,Y)$ in general position, then $[X,Y]=0$ iff $w(X,Y)={\bf 1}$. If $G$ is simply connected compact, then $[\exp X,\exp Y]={\bf 1}$ and $[X,Y]=0$ are equivalent for all pairs $(X,Y)$ in general position. In ${\mathop{\rm SO}\nolimits}(3)$ this is not the case; here $|{\cal W}_Y|=2$. In Section 3, examples show that the validity of the equation $\exp X\exp Y=\exp\!(X+Y)$ has no implications whatsoever in the direction of the commuting of $\exp X$ and $\exp Y$. Finally, in Section 4, it is shown that, for a simply connected Lie group $G$, the commuting of $X, Y\in{\fam\euffam g}$ and that of $\exp X,\exp Y\in G$ are equivalent properties for all$X$ and $Y$ if and only if the exponential function is injective. This class of Lie groups was characterized in terms of other properties by Dixmier and by Saito, independently, in 1957.

Type
Research Article
Copyright
2006 Cambridge Philosophical Society

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