Rigidity results for Lie algebras admitting a post-Lie algebra structure
| Autor: | Dietrich Burde, Karel Dekimpe, Mina Monadjem |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | International Journal of Algebra and Computation. 32:1495-1511 |
| ISSN: | 1793-6500 0218-1967 |
| DOI: | 10.1142/s0218196722500679 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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