Rigidity results for Lie algebras admitting a post-Lie algebra structure

D. Burde, K. Dekimpe, Mina Monadjem
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引用次数: 1

Abstract

We study rigidity questions for pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $\mathfrak{g}$ is semisimple and $\mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $\mathfrak{g}$ and $\mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $\mathfrak{g}=\mathfrak{s}_1\dotplus \mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $\mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $\mathfrak{g}\cong \mathfrak{s}_1\oplus \mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$. We prove additional existence results for pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is complete, and for pairs, where $\mathfrak{g}$ is reductive with $1$-dimensional center and $\mathfrak{n}$ is solvable or nilpotent.
承认后李代数结构的李代数的刚性结果
研究了承认后李代数结构的李代数对$(\mathfrak{g},\mathfrak{n})$的刚性问题。我们证明,如果$\mathfrak{g}$是半简单的,$\mathfrak{n}$是任意的,那么我们在$\mathfrak{g}$和$\mathfrak{n}$必须同构的意义上具有刚性。证明使用李代数$\mathfrak{g}=\mathfrak{s}_1\dotplus \mathfrak{s}_2$分解的结果作为两个半简单子代数的直接向量空间和。我们证明$\mathfrak{g}$必须是半简单的,因此与直接李代数和$\mathfrak{g}\cong \mathfrak{s}_1\oplus \mathfrak{s}_2$同构。这解决了李代数对上后李代数结构的一些开放存在性问题$(\mathfrak{g},\mathfrak{n})$。我们证明了对$(\mathfrak{g},\mathfrak{n})$的附加存在性结果,其中$\mathfrak{g}$是完全的,对$\mathfrak{g}$是约化的,以$1$为中心,$\mathfrak{n}$是可解的或幂零的。
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