We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we prove that such structures are trivial in several cases. We classify commutative structures on low-dimensional Lie algebras, and study the case of nilpotent Lie algebras.
翻译:我们研究后Lie代数结构,研究的是由Lie组无足轻重行为驱动的Lie代数(mathfrak{g},\mathfrak{n})一对美元(mathfrak{g},\mathfrak{n})一对Lie 代数结构。我们研究的是简单、半简单、再处理、完美、可溶、无能力、ABelian和单模性Lie代数结构。此外,我们考虑在完美的Lie代数结构上进行交替后Lie代数结构。我们利用Lie 代数同系学证明,这类结构在若干情况下是微不足道的。我们将低维立值代数的混合结构分类,并研究无名代数的例子。