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

Abstract

We show that for a given nilpotent Lie algebra g with Z(g)⊆[g,g] all commutative post-Lie algebra structures, or CPA-structures, on 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 Fg,c and discover a strong relationship to solving systems of linear equations of type [x,u]+[y,v]=0 for generator pairs x,y∈Fg,c. We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra Fg,c has only central CPA-structures.