WEAKLY EXPONENTIAL LIE-GROUPS

In this paper we list all simple real Lie algebras g for which there e xist connected Lie groups with dense images of the exponential functio n. We also describe the simple real Lie algebras for which the exponen tial functions of the associated simply connected Lie groups have dens e images. Let us say that a Lie group is weakly exponential if the ima ge of its exponential function is dense. Hofmann and Mukherjea (On the density of the image of the exponential function, Math. Ann. 234 (197 8), 263-273) show how to reduce the problem of determining whether G i s weakly exponential to the semisimple case. We also give some methods which are useful in determining whether a reductive Lie group is weak ly exponential or not. Our method is based on the fact that a maximal rank subgroup of a weakly exponential Lie group inherits the property of being weakly exponential. This finally permits us to characterize t he reductive Lie algebras having a weakly exponential group of inner a utomorphisms as those where the centralizer of the compact part of a m aximally non-compact Cartan subalgebra has a commutator algebra isomor phic to a product of sl(2, R)-factors. For the groups Sl(n, R), Sp(n, W), and SO(p, q)(0), 2 less than or equal to p, q, p, q even, Hofmann and Mukherjea show that they are not weakly exponential. For the other classical groups the results of Dokovic (The interior and the exterio r of the image of the exponential map in classical Lie groups, J. Alge bra 112 (1985), 90-109) provide information as to whether they are wea kly exponential or not. It is a classical results that complex and com pact simple Lie groups are weakly exponential. (C) 1996 Academic Press , Inc.