A short survey on the surjectivity of exponential Lie groups (Q2762075)

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 paper under review surveys the present state of the knowledge on this problem with an emphasis on the role played by centralizers of nilpotent elements of the Lie algebra. A 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. The most general criterion can be formulated as follows. Let us call a connected Lie group \(G\) Malcev splittable if the group \(\text{Ad}(G)\) contains with each element its unipotent and its semisimple Jordan factor. Then a connected Malcev splittable group is exponential if and only if for each \(X \in \mathbf g\) for which \(\text{ad} X\) is nilpotent and for each element \(g_s \in Z_G(X)\) for which \(\text{Ad}(g_s)\) is semisimple there is an \(ad\)-semisimple \(X_s \in \mathbf z_\mathbf g(X)\) with \(\exp X_s =g_s\). The latter condition is equivalent to \(Z_G(X)\) being weakly exponential, i.e., the image of the exponential function is dense. For solvable groups and for complex algebraic groups the latter condition is equivalent to the connectedness of \(Z_G(X)\). Most of the paper under review explains the context and the different variations of this result. The present survey complements a survey by \textit{D. Đoković} and \textit{K. H. Hofmann} [J. Lie Theory 7, 171-199 (1997; Zbl 0888.22003)].NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].