Post-Lie algebra structures for nilpotent Lie algebras

Abstract

We study post-Lie algebra structures on (g, n) for nilpotent Lie algebras. First we show that if g is nilpotent such that H 0 (g, n) = 0, then also n must be nilpotent, of bounded class. For post-Lie algebra structures x · y on pairs of 2-step nilpotent Lie algebras (g, n) we give necessary and sufficient conditions such that x • y = 1 2 (x · y + y · x) defines a CPA-structure on g, or on n. As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of dimension n ≥ 5 is complete. Finally we classify all post-Lie algebra structures on (g, n) for g ∼ = n ∼ = n 3 , where n 3 is the 3-dimensional Heisenberg Lie algebra.

Post-Lie algebra structures on pairs of Lie algebras

“…We summarize the existence results for post-Lie algebra structures from the previous sections and from the papers [5,6,7,8,9,10,12] as follows. A checkmark only means that there is some non-trivial pair (g, n) of Lie algebras with the given algebraic properties admitting a PA-structure.…”

“…PA-structures are a natural generalization of pre-Lie algebra structures on Lie algebras, which arise among other things from affine manifolds and affine actions on Lie groups, crystallographic groups, étale affine representations of Lie algebras, quantum field theory, operad theory, Rota-Baxter operators, and deformation theory of rings and algebras. There is a large literature on pre-Lie and post-Lie algebras, see for example [5,6,7,8,9,12,14,18] and the references therein. For a survey on pre-Lie algebra respectively post-Lie algebra structures see [2,11].…”

Crystallographic actions on Lie groups and post-Lie algebra structures

“…For further results we refer to [20], [22]. For the general case concerning post-Lie algebra structures on pairs of Lie algebras (g, n) we also have several results, see [24], [25], [26], [31], [32], [39]. Let us explain some of them.…”

“…There are only some classifications in low dimensions. We refer to [31] for a classification of post-Lie algebra structures on (g, n), where both g and n are isomorphic to the 3-dimensional Heisenberg Lie algebra. We have much better classification results for commutative post-Lie algebra structures, which will be discussed in the next section.…”