Kähler Lie algebras and double extension
A finite-dimensional real Lie algebra \({\mathfrak g}\) with a nondegenerate 2-cocycle \(\omega\) is called a symplectic Lie algebra. It is called a Kähler-Lie algebra if it also carries an integrable complex structure \(J\) such that \(\omega(X,JY)\) is a scalar product on \({\mathfrak g}\). Kähler-Lie algebras are the infinitesimal analogue of Lie groups with a left invariant Kähler structure. The authors introduce the Kähler double extension process (by means of two exact sequences), which leads to the following structure theorems on Kähler-Lie algebras. Every such algebra \({\mathfrak g}\) can be obtained from a flat Kähler algebra, or from \(\{0\}\), by successive Kähler double extensions. The latter case occurs if and only if \({\mathfrak g}\) admits a Lagrangian subalgebra. Completely solvable Kähler-Lie algebras are obtained from \(\{0\}\) by successive double extensions by means of one-dimensional algebras. They are unimodular (i.e. \(\text{tr(ad }X)\) vanishes identically) if and only if they are abelian.
scientific article; zbMATH DE number 954782