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

TL;DR

This paper investigates commutative post-Lie algebra structures on nilpotent Lie algebras, showing they are complete under certain conditions, and explores their connection to linear equations in free-nilpotent Lie algebras, revealing cases with only central structures.

Contribution

It establishes conditions for the completeness of CPA-structures on nilpotent Lie algebras and links these structures to solving specific linear equations in free-nilpotent Lie algebras, identifying cases with only central structures.

Findings

CPA-structures on certain nilpotent Lie algebras are complete.

Connection between CPA-structures and linear equations in free-nilpotent Lie algebras.

Existence of only central CPA-structures for specific free-nilpotent Lie algebras.

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.