In this talk we will explain the results we obtained in [1]. In that paper, we study commutative post-Lie algebra structures, or CPA-structures. A CPA structure on a Lie algebra g with Lie bracket [ · , · ] is a bilinear product on g satisfying x · y = y · x, [x, y] · z = x · (y · z) − y · (x ·z), x · [y,z] = [x · y,z] + [y,x · z]. After recalling the geometric motivation to study these CPA structures, we will show that for a given nilpotent Lie algebra g with Z(g)⊆[g, g] all 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 freenilpotent 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 ∈ Fg,c. We use results of Remeslennikov and Stöhr [2] concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra Fg,c has only central CPA-structures.
[1] Burde, Dietrich; Dekimpe, Karel and Moens, Wolfgang Alexander, Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras. J. Algebra 526 (2019), 1229.
[2] V. Remeslennikov, R. Stöhr: The equation [x, u] + [y, v] = 0 in free Lie algebras. Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1165-1187.