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 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 ∈ F g,c . We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra F g,c has only central CPA-structures.

Commutative post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

“…For the induction step we use that L n /Z(L n ) ∼ = L n−1 . Since dim(Z(L n )) = 1 it follows L n · Z(L n ) = 0 from Corollary 3.4 of [14]. Hence (V, ·) induces a CPA-structure on L n−1 , which by induction hypothesis is already of type 1 or of type 2.…”

“…Proof. Since the ideals I j are characteristic and the left multiplications L(e i ) are derivations, it follows that L(e i )(I j ) ⊆ I j for all 1 ≤ i, j ≤ n. Since filiform Lie algebras are nilpotent stem Lie algebras, all left multiplications are nilpotent by Theorem 3.6 in [14]. Hence we have L(e i )(I j ) ⊆ I j+1 for 1 ≤ i, j ≤ n. This implies that g · I 2 ⊆ I 3 , so that g · g ⊆ I 3 follows from e 1 · e 1 ∈ I 3 .…”

“…We have studied central CPA-structures on nilpotent Lie algebras in [14] and shown the following result, see Theorem 4.3. Proof.…”

“…Proof. Since g is a nilpotent stem Lie algebra, all left multiplications are nilpotent by Theorem 3.6 in [14]. Hence they are nilpotent derivations of g. By Theorem 3.2 in [23] there exists for every nilpotent derivation D ∈ Der(g) an u ∈ g and a ψ ∈ Der(g) such that D = ad(u) + ψ and ψ(g) ⊆ g n−3 , ψ([g, g]) = 0.…”

Commutative post-Lie algebra structures on Kac–Moody algebras

“…Commutative post-Lie algebra structures, CPA-structures in short, are much more tractable, and we have obtained several existence and classification results [7,8,9]. Among other things we proved in [8] that any commutative post-Lie algebra structure on a finite-dimensional perfect Lie algebra over field of characteristic zero is trivial.…”