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Affine actions on Lie groups and post-Lie algebra structures. (English) Zbl 1286.17012

Summary: We introduce post-Lie algebra structures on pairs of Lie algebras \((\mathfrak {g,n})\) defined on a fixed vector space \(V\). Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras \(\mathfrak g\) and \(\mathfrak n\). One result is, for example, that if there exists a post-Lie algebra structure on \((\mathfrak {g,n})\), where \(\mathfrak g\) is nilpotent, then \(\mathfrak n\) must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

MSC:

17B30 Solvable, nilpotent (super)algebras
17D25 Lie-admissible algebras
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