Contact nilpotent Lie algebras

Abstract:

In this work we show that for $n\geq 1$, every finite $(2n+3)$-dimensional contact nilpotent Lie algebra $\mathfrak {g}$ can be obtained as a double extension of a contact nilpotent Lie algebra $\mathfrak {h}$ of codimension 2. As a consequence, for $n\geq 1$, every $(2n+3)$-dimensional contact nilpotent Lie algebra $\mathfrak {g}$ can be obtained from the 3-dimensional Heisenberg Lie algebra $\mathfrak {h}_3$, by applying a finite number of successive series of double extensions. As a byproduct, we obtain an alternative proof of the fact that a $(2n+1)$-nilpotent Lie algebra $\mathfrak {g}$ is a contact Lie algebra if and only if it is a central extension of a nilpotent symplectic Lie algebra.