Algèbres de Lie kählériennes et double extension Original Research Article

A Kähler Lie algebra is a real Lie algebra carrying a symplectic 2-cocycle ω and an integrable complex structurejsuch that ω(x, j(y)) is a scalar product. We give a process, called Kähler double extension, which realizes a Kähler Lie algebra as the Kähler reduction of another one. We show that every Kähler algebra is obtained by a sequence of such a process from {0} or a flat Kähler algebra; it is obtained from {0} iff it contained a lagrangian sub-algebra. These methods allow us to prove that any completely solvable and unimodular Kähler algebra is commutative.