Weakly Exponential Lie Groups

Abstract

In this paper we list all simple real Lie algebras g for which there exist connected Lie groups with dense images of the exponential function. We also describe the simple real Lie algebras for which the exponential functions of the associated simply connected Lie groups have dense images. Let us say that a Lie group is weakly exponential if the image of its exponential function is dense. Hofmann and Mukherjea (On the density of the image of the exponential function, Math. Ann. 234(1978), 263–273) show how to reduce the problem of determining whether Gis weakly exponential to the semisimple case. We also give some methods which are useful in determining whether a reductive Lie group is weakly 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 the reductive Lie algebras having a weakly exponential group of inner automorphisms as those where the centralizer of the compact part of a maximally non-compact Cartan subalgebra has a commutator algebra isomorphic to a product of s l (2, R )-factors. For the groups Sl( n, R ), Sp( n, R ), and SO( p, q) 0, 2≤ p, q, p, qeven, Hofmann and Mukherjea show that they are not weakly exponential. For the other classical groups the results of Đokovic (The interior and the exterior of the image of the exponential map in classical Lie groups, J. Algebra 112(1988), 90–109) provide information as to whether they are weakly exponential or not. It is a classical results that complex and compact simple Lie groups are weakly exponential.