Читать книгу Algebra and Applications 2 - Группа авторов - Страница 51

Post-Lie algebras have been introduced by Vallette (2007), independent of the introduction of the closely related notion of D-algebra in Munthe-Kaas and Wright (2008). A left post-Lie algebra on a field k is a k-vector space A together with a bilinear binary product ⊳ and a Lie bracket [–, –], such that

for any a, b, c ∈ A. In particular, a post-Lie algebra A is a pre-Lie algebra if and only if the Lie bracket vanishes. The space of vector fields on a Lie group is a post-Lie algebra, and the free post-Lie algebra with one generator is the free Lie algebra on the linear span of planar rooted trees. The binary product ⊳ is given by left grafting, which is not pre-Lie anymore because of planarity. This is the starting point to the Lie-Butcher theory (see, for example, Lundervold and Munthe-Kaas (2013); Ebrahimi-Fard et al. (2015); Curry et al. (2019, 2020)).