Kahlerian Lie algebras and double extension

A Kahler Lie algebra is a real Lie algebra carrying a symplectic 2-cocycle omega and an integrable complex structure j such that omega(x, j(y)) is a scalar product. We give a process, called Kahler double extension, which realizes a Kahler Lie algebra as the Kahler reduction of another one. We show that every Kahler algebra is obtained by a sequence of such a process from {0} or a flat Kahler 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 Kahler algebra is commutative. (C) 1996 Academic Press, Inc.