Lie groups with surjective exponential function (Q2707376)

It is an old problem in Lie theory to determine for a given Lie group \(G\) whether its exponential function is surjective. Let us call these groups exponential. The Habilitationsschrift under review describes the present state of the art concerning this problem. NEWLINENEWLINENEWLINEA reasonable first approach to the problem is to ask when the image of the exponential function is dense; these groups are called weakly exponential. This notion was introduced by \textit{K. H. Hofmann} and \textit{A. Mukherjea} [Math. Ann. 234, 263-273 (1978; Zbl 0382.22005)] and one can say that the problem is essentially settled by the characterization that \(G\) is weakly exponential if and only if all its Cartan subgroups are connected [\textit{K.-H. Neeb}, J. Algebra 179, 331-361 (1996; Zbl 0851.22009)]. From that one in particular derives a classification of all simply connected weakly exponential Lie groups.NEWLINENEWLINENEWLINEA general criterion for the exponentiality of a connected Lie group \(G\) does still not exist, but there are criteria which apply to large classes of groups, and a discussion of these criteria is at the center of Wüstner's Habilitationsschrift. The most general criterion can be formulated as follows. Let us call a connected Lie group \(G\) Malcev splittable if the group \(Ad(G)\) contains with each element its unipotent and its semisimple Jordan factor. One of the main results of the thesis is that a connected Malcev-splittable group is exponential if and only if for each \(X \in g\) for which \(ad X\) is nilpotent the centralizer \(Z_G(X)\) is weakly exponential. For solvable groups the latter condition is equivalent to the connectedness of \(Z_G(X)\). NEWLINENEWLINENEWLINETo derive from the preceding result information on general connected Lie groups, Wüstner defines the concept of a Malcev splittable radical \(A\) of \(G\). It is shown that if \(G\) is exponential, then \(A\) is exponential. Moreover, it is always true that if \(G\) is exponential, then for each \(ad\)-nilpotent element \(X\) the centralizer \(Z_G(X)\) is weakly exponential, and conversely, this condition implies that \(A\) is exponential. These results still leave a little room containing some non-Malcev splittable Lie groups for which exponentiality cannot be decided with the presented methods, but ``most'' Lie groups are Malcev splittable.