Pre- and Post-Lie Algebras: The Algebro-Geometric View

Abstract

We relate composition and substitution in pre-and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp. Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre-and post-Lie algebras.

An operadic approach to substitution in Lie–Butcher series

“…There exists an isomorphism [20] (𝑆(𝐿𝑖𝑒(PT )), ) (OF, ⧢). This means that our coproduct Δ Q : Q → Q ⊗ Q can already be seen as a coaction H 𝑁 → Q ⊗ H 𝑁 via identification by this isomorphism.…”

“…The algebraic picture of a bialgebra cointeracting with the Hopf algebra of Munthe-Kaas and Wright is, however, not present. Substitution was also considered in [20], where algebro-geometric methods were used to show that there is a bialgebraic description. That construction was, however, not made explicit.…”

An operadic approach to substitution in Lie-Butcher series

“…Remark 7. There exists an isomorphism [14] (S(Lie(PT )), ) ∼ = (OF , ¡). This means that our coproduct…”

What is a post-Lie algebra and why is it useful in geometric integration

“…To be more precise, U (postLie({ })) consists of all finite linear combinations of this kind, while infinite combinations such as the exponential live in U (postLie({ })) and are obtained by an inverse limit construction [12]. Elements in the space U (postLie({ })) we call Lie-Butcher (LB) series.…”

“…Lie-Butcher series underwent similar developments replacing non-planar trees by planar ones [16,21]. Correspondingly, pre-Lie algebras are to B-series what post-Lie algebras are to Lie-Butcher series [8,12].…”