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The Lie Group Structure of the Butcher Group

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Abstract

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge–Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker–Campbell–Hausdorff Lie group modelled on a Fréchet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated with the Butcher group by Connes and Kreimer.

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Notes

  1. The same construction can be performed for tree maps with values in the field of complex numbers. The group \(G_{{\mathrm {TM}}}^{{\mathbb {C}}}\) of complex-valued tree maps obtained in this way will be an important tool in our investigation. In fact, \(G_{{\mathrm {TM}}}^{{\mathbb {C}}}\) is a complex Lie group and the complexification (as a Lie group) of the Butcher group.

  2. The term “ordered” refers to that the subtree remembers from which part of the tree it was cut.

  3. The basic problem is that the bounded linear operators do not admit a good topological structure if the spaces are not normable. In particular, the chain rule will not hold for Fréchet differentiability in general for these spaces (cf. [27, p. 73] or [22]).

  4. If E and F are Fréchet spaces, real analytic maps in the sense just defined coincide with maps which are continuous and can be locally developed into a power series. (see [15, Proposition 4.1])

  5. i.e. every open set in the box topology can be written as a union of boxes. Note that we can not describe this topology via seminorms as it does not turn \({\mathbb {K}}^{{\mathcal {T}}_0}\) into a topological vector space.

  6. While addition is continuous, scalar multiplication fails to be continuous, cf. the discussion of the problem in [18].

  7. The natural choice for this space is a locally convex direct limit topology. Note that as \({\mathcal {T}}_0\) is countable, the box topology coincides with the inductive limit topology by [21, Proposition 4.1.4].

  8. In [10], the authors work over the field \({\mathbb {Q}}\) of rational numbers. However, by applying \(\cdot \otimes _{{\mathbb {Q}}} {\mathbb {R}}\) to the \({\mathbb {Q}}\)-algebras, the same result holds for the field \({\mathbb {R}}\) (cf. [10, p. 41]). The thesis of Mencattini [26] contains an explicit computation for \({\mathbb {R}}\) and \({\mathbb {C}}\).

  9. This follows from [17, Remark 2.12 and Lemma 2.5] for manifolds modelled on Fréchet spaces.

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Acknowledgments

The research on this paper was partially supported by the projects Topology in Norway (Norwegian Research Council Project 213458) and Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics (Norwegian Research Council Project 231632). The second author would also like to thank Reiner Hermann for helpful discussions on Hopf algebras. Furthermore, we thank the anonymous referees for their insightful comments which led to substantial improvements of the paper.

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  1. Department of Mathematical Sciences, NTNU Trondheim, Trondheim, Norway

    Geir Bogfjellmo & Alexander Schmeding

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  1. Geir Bogfjellmo
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  2. Alexander Schmeding
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Correspondence to Geir Bogfjellmo.

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Communicated by Arieh Iserles.

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Bogfjellmo, G., Schmeding, A. The Lie Group Structure of the Butcher Group. Found Comput Math 17, 127–159 (2017). https://doi.org/10.1007/s10208-015-9285-5

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