Lie-Butcher series, Geometry, Algebra and Computation
For researchers in numerical analysis and geometric integration, this work provides a unified algebraic framework for LB-series, but it is primarily a review and reformulation of existing theory.
This paper presents a concise and self-contained overview of the algebraic structures underlying Lie-Butcher series, which generalize B-series from Euclidean spaces to Lie groups and homogeneous manifolds. It reformulates algebraic operations on LB-series as recursive formulae, supporting the development of a Haskell software library for computations.
Lie-Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell.