Translations in simply transitive affine actions of Heisenberg type Lie groups (Q1864964)

Let \((x,y)\mapsto x\cdot y\) be a complete left-symmetric structure on a nilpotent Lie algebra \({\mathfrak g}\). Let \(T({\mathfrak g})\) be the set of \(x\in {\mathfrak g}\) satisfying \(x\cdot y=0\) for all \(y\in {\mathfrak g}\). The left-symmetric structure induces a simply transitive action of a nilpotent Lie group \(G\) such that \(\exp (T({\mathfrak g}))\) is precisely the subset of \(G\) consisting of those elements acting as pure translations. The authors prove the following theorem. If \({\mathfrak g}\) is a \(2\)-step nilpotent Lie algebra with \(1\)-dimensional commutator algebra, then for any complete left-symmetric structure on \({\mathfrak g}\) we have \(T({\mathfrak g})\neq 0\).