### Abstract

A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $\mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $\mathfrak{sl}_2$.

Original language | English |
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Article number | 1707.02095 |

Number of pages | 28 |

Journal | arXiv |

Volume | 2017 |

Publication status | Published - 7 Jul 2017 |

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## Cite this

Cuypers, H., & Fleischmann, Y. (2017). A geometric characterization of the symplectic Lie algebra.

*arXiv*,*2017*, [1707.02095]. https://arxiv.org/abs/1707.02095