Automorphisms of non-singular nilpotent Lie algebras

Abstract

For a real, non-singular, 2-step nilpotent Lie algebra n, the group Aut(n)/ Aut0(n), where Aut0(n) is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some automorphisms groups of n follows and is related to how close is n to being of Heisenberg type. For example, at least when the dimension of the center is two, dim Aut(n) is maximal if and only if n is of Heisenberg type. The connection with fat distributions is discussed.