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Affine actions on Lie groups and post-Lie algebra structures. (English) Zbl 06063149
Summary: We introduce post-Lie algebra structures on pairs of Lie algebras (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 𝔤 and 𝔫. One result is, for example, that if there exists a post-Lie algebra structure on (𝔤,𝔫), where 𝔤 is nilpotent, then 𝔫 must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.
17Nonassociative rings and algebras
17B30Solvable, nilpotent Lie (super)algebras
17D25Lie-admissible algebras
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