On the exponential function of real splittable and real semisimple Lie groups
A Lie group \(G\) is called exponential if \(G= \exp{\mathfrak g}\), and weakly exponential if \(G\) is equal to the closure of \(\exp{\mathfrak g}\). As a generalization of his previous paper [J. Algebra 184, 1082-1092 (1996; Zbl 0870.22004)], the author gives a characterization of the exponential real analytic splittable groups and their exponential covering groups. Let \(G\subset GL(V)\) be a splittable analytic group, that is, for all \(g\in G\), the Jordan decomposition \(g=g_ug_s\) satisfies \(g_u,g_s\in G\). Then \(G\) is exponential if and only if for each nilpotent \(x\in{\mathfrak g}\) the centralizer \(Z_G(x)\) is weakly exponential. Furthermore, this characterization is even true for exponential covering groups \(\widetilde G\) of \(G\). As a corollary, we see that, if \(G\) is a real semisimple Lie group and \({\mathfrak g}\subset{\mathfrak g}{\mathfrak l}(V)\), then \(G\) is exponential if and only if for each nilpotent \(x\in{\mathfrak g}\), \(Z_G(x)\) is weakly exponential.
scientific article; zbMATH DE number 1127637