Kählerian oxidation and reduction of Lie algebras with Kähler structures

Abstract

A Kähler structure on a real Lie algebra $\mathfrak{g}$ is a pair $(ω, J)$ of a 2-cocycle and a complex structure on $\mathfrak{g}$ which are compatible in the sense that $J$ is skew-symmetric with respect to $ω$. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either $\{0\}$, $\mathbb{R}^2$, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.