A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms

Abstract

The realification of the (2n + 1)-dimensional complex Heisenberg Lie algebra is a (4n + 2) -dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra of sp(n) derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the centre admits a compact Lie algebra of derivations. This also gives us the occasion to revisit a series of classic results, with the expressed aim of attracting the interest of a broader audience.