Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nilpotent Lie algebras which are called the left and right nilpotent algebras of the solvable algebra. The right nilpotent algebra is the smallest ideal of the lower central series of the solvable algebra, while the left nilpotent algebra is the factor algebra of the solvable algebra and its right nilpotent algebra. We show that the solvable algebras are decomposable if its left nilpotent algebra is an Abelian algebra of dimension higher than one and its right algebra is an Abelian algebra of dimension one. We further show that all the solvable algebras are isomorphic if their left nilpotent algebras are Heisenberg algebras of fixed dimension and their right algebras are Abelian algebras of dimension one.

中文翻译：

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可解李代数的幂零分解

半简单李代数已经完全被 Cartan 和 Killing 分类。Levi定理指出，每个有限维李代数同构于其最大可解理想和半单李代数的半直和。这些将可解李代数的分类作为李代数研究的主要挑战之一。完成此任务的一种方法是采用一类幂零李代数并将这些代数的所有扩展构造为可解代数。在本文中，我们提出了另一种方法，即将可解非幂零李代数分解为两个幂零李代数，称为可解代数的左右幂零代数。右幂零代数是可解代数的下中心级数的最小理想，而左幂零代数是可解代数与其右幂零代数的因子代数。我们证明了可解代数是可分解的，如果其左幂零代数是维数高于 1 的阿贝尔代数，而其右代数是维数为 1 的阿贝尔代数。我们进一步证明所有可解代数都是同构的，如果它们的左幂零代数是固定维数的海森堡代数，而它们的右代数是一维阿贝尔代数。