Contact nilpotent Lie algebras

Abstract

In this work we show that for n ≥ 1 n\geq 1 , every finite ( 2 n + 3 ) (2n+3) -dimensional contact nilpotent Lie algebra g \mathfrak {g} can be obtained as a double extension of a contact nilpotent Lie algebra h \mathfrak {h} of codimension 2. As a consequence, for n ≥ 1 n\geq 1 , every ( 2 n + 3 ) (2n+3) -dimensional contact nilpotent Lie algebra g \mathfrak {g} can be obtained from the 3-dimensional Heisenberg Lie algebra h 3 \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 ( 2 n + 1 ) (2n+1) -nilpotent Lie algebra g \mathfrak {g} is a contact Lie algebra if and only if it is a central extension of a nilpotent symplectic Lie algebra.