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arXiv · 1711.01964

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras

D. Burde·W. A. Moens·K. Dekimpe
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Abstract

We show that for a given nilpotent Lie algebra $\mathfrak{g}$ with $Z(\mathfrak{g})\subseteq [\mathfrak{g},\mathfrak{g}]$ all commutative post-Lie algebra structures, or CPA-structures, on $\mathfrak{g}$ are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras $F_{g,c}$ and discover a strong relationship to solving systems of linear equations of type $[x,u]+[y,v]=0$ for generator pairs $x,y\in F_{g,c}$. We use results of Remeslennikov and St\"ohr concerning these equations to prove that, for certain $g$ and $c$, the free-nilpotent Lie algebra $F_{g,c}$ has only central CPA-structures.

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BibTeXRIS

D. Burde, W. A. Moens, K. Dekimpe. 2017-11-06. Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras. https://arxiv.org/abs/1711.01964

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