Abstract
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.
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Notes
- 1.
Einstein summation convention.
- 2.
Technical issues about divergence of the backward error vector field is discussed in [1].
- 3.
In the computer implementations we are relaxing this to allow \({\text {k}}\) more generally to be a commutative ring, such as e.g. polynomials in a set of indeterminates. In this latter case the \({\text {k}}\)-vector space should instead be called a free \({\text {k}}\)-module. We will not pursue this detail in this exposition.
- 4.
This definition is not strictly categorical, since the mappings \({{\text {inj}}}\) and \(\phi \) are not morphisms inside a category, but mappings from a set to an object of another category. A proper categorical definition of a free object, found in any book on category theory, is based on a forgetful functor mapping the given category into the category of sets. The free functor is the left adjoint of the forgetful functor.
- 5.
Trees with different orderings of the branches are considered different, as embedded in the plane.
- 6.
An associative algebra can be defined by commutative diagrams. The co-algebra structure is obtained by reversing all arrows.
- 7.
Splitting with regard to the coproduct \(\varDelta \).
- 8.
Note that the number under the terms are the coefficients to the terms.
- 9.
Since the action of differentiation operators composes contravariantly, the order of right and left is swapped in the mapping from LB-series to differential equations on manifolds.
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Munthe-Kaas, H.Z., Føllesdal, K.K. (2018). Lie–Butcher Series, Geometry, Algebra and Computation. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_3
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