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arXiv 1801.05652math.RA
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幂零李代数的后李代数结构

Post-Lie algebra structures for nilpotent Lie algebras

Dietrich Burde, Christof Ender, Wolfgang Alexander Moens

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摘要：

研究了$(\mathfrak{g},\mathfrak{n})$上幂零李代数的后李代数结构.首先证明了如果$\mathfrak{g}$是幂零的，使得$H^0(\mathfrak{g},\mathfrak{n})=0$，那么$\mathfrak{n}$也一定是幂零的，有界类.对于$2$ -步幂零李代数$(\mathfrak{g},\mathfrak{n})$对上的后李代数结构$x\cdot y$，给出了$x\circ y=\frac{1}{2}(x\cdot y+y\cdot x)$在$\mathfrak{g}$或$\mathfrak{n}$上定义CPA-结构的充要条件.作为推论，我们得到了维数为PDMATHAMT的Heisenberg李代数上的每一个LR-结构都是完备的。最后我们对$\mathfrak{g}\cong \mathfrak{n}\cong \mathfrak{n}_3$上的所有后李代数结构进行了分类，其中$\mathfrak{n}_3$是$3$维Heisenberg李代数.

英文摘要：

We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be nilpotent, of bounded class. For post-Lie algebra structures $x\cdot y$ on pairs of $2$-step nilpotent Lie algebras $(\mathfrak{g},\mathfrak{n})$ we give necessary and sufficient conditions such that $x\circ y=\frac{1}{2}(x\cdot y+y\cdot x)$ defines a CPA-structure on $\mathfrak{g}$, or on $\mathfrak{n}$. As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of dimension $n\ge 5$ is complete. Finally we classify all post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for $\mathfrak{g}\cong \mathfrak{n}\cong \mathfrak{n}_3$, where $\mathfrak{n}_3$ is the $3$-dimensional Heisenberg Lie algebra.

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