Splittable Lie groups and Lie algebras (Q1974171)

Let us call a finite-dimensional Lie algebra \(\mathfrak g\) over a field of characteristic zero Malcev splittable if \(\operatorname {ad}{\mathfrak g}\) is splittable in the sense that for each element of \(\operatorname {ad}{\mathfrak g}\) the semisimple and nilpotent Jordan component belong to \(\operatorname {ad}{\mathfrak g}\). Typical examples are Lie algebras of algebraic groups. Correspondingly we call a connected Lie group Malcev splittable if its Lie algebra has this property. One of the main results of the present paper is that a connected Malcev splittable linear Lie group can be realized as a subgroup of some \(\text{GL}(n,\mathbb R)\) which is splittable in the sense that it contains with each element its semisimple and unipotent Jordan multiplicative component. This result is then used to extend the author's characterization of splittable linear Lie groups with surjective exponential function to the significantly larger class of Malcev splittable Lie groups. According to this criterion the exponential function of a connected Malcev splittable Lie group \(G\) is surjective if and only if for each \(\text{ ad}\)-nilpotent element \(x\) in the Lie algebra of \(G\) the image of the exponential function of the centralizer \(Z_G(x)\) of \(x\) in \(G\) is dense.