AI 中文总结
本文研究李代数上的Kähler结构，提出Kählerian约化与氧化框架，证明维数大于四的不可分解拟幂零Kählerian幂零李代数可由基代数经有限次中心平面氧化重构。
AI 中文摘要
实李代数$\mathfrak{g}$上的Kähler结构是一对$(\omega, J)$，其中$\omega$是2-上循环，$J$是$\mathfrak{g}$上的复结构，且二者相容，即$J$关于$\omega$是斜对称的。本文研究李代数上的Kähler结构，特别关注幂零情形。我们引入了一个代数框架，用于通过一维和二维理想对李代数进行Kählerian约化和氧化。我们的主要结构结果表明，任意维数大于四的不可分解、拟幂零Kählerian幂零李代数都可以通过有限序列的Kählerian中心平面氧化来归纳重构。该约化序列源于一个基代数，该基代数为$\{0\}$、$\mathbb{R}^2$，或承认强非幂零复结构的代数。此外，若复结构是幂零的，则所有中间约化也具有幂零复结构。
英文摘要
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.
