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Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups

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Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 267))

Abstract

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples:

  • the Butcher group from numerical analysis and

  • character groups which arise from the Connes–Kreimer theory of renormalisation of quantum field theories.

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Notes

  1. 1.

    In these contexts the term “combinatorial Hopf algebra” is often used though there seems to be no broad consensus on the meaning of this term.

  2. 2.

    This Hopf algebra depends on the quantum field theory under consideration, but we will suppress this dependence in our notation.

  3. 3.

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

  4. 4.

    Note that \(\mathfrak {g}\) can be recovered from the Hopf algebra \(\mathscr {U}(\mathfrak {g})\) (see [7, Theorem 3.6.1]) and the Hopf algebras which arise in this way are characterised by the Milnor–Moore theorem (cf. Remark 7).

  5. 5.

    The problem here 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. [22, p. 73] or [23]).

  6. 6.

    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 expanded into a power series. See [28, Proposition 4.1].

  7. 7.

    Note that it is well known that the algebra multiplication as a bilinear map will be continuous if we require it to be continuous with respect to the projective topological tensor product \(B \otimes _\pi B\) (see e.g. [31]). However, one does not gain more information in doing so, whence we avoid discussing this tensor product (or any topology on the tensor product). For this reason one of the authors has been accused of “cheating”.

  8. 8.

    These numbers depend on the structure of a certain ordinary differential equation. We refer to [42] for more details.

  9. 9.

    Contrarily to the situation for finite-dimensional Lie groups, not every closed subgroup of an infinite-dimensional Lie group is again a Lie subgroup. See [30, Remark IV.3.17] for an example.

  10. 10.

    \(G_0\) in [55, Theorem 6.9] is larger than the set of Runge–Kutta methods, however, the statement still holds if \(G_0\) is the group of Runge–Kutta methods. See also [56, Theorem 317A].

  11. 11.

    A metric d is an ultrametric if \(d(\phi , \psi ) \le \max \{ d(\phi , \chi ), d(\chi , \psi )\}\).

  12. 12.

    Here we consider \(C^r([0,1],\mathbf {L}(G))\) as a locally convex vector space with the pointwise operations and the topology of uniform convergence of the function and its derivatives on compact sets.

  13. 13.

    This is possible since the differentiable structure of the Butcher group turns it into a projective limit of finite-dimensional Lie groups. We will return to this phenomenon in Sect. 7.

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Acknowledgements

The research on this paper was partially supported by the project Topology in Norway (NRC project 213458) and Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics (NRC project 231632). We thank J. M. Sanz-Serna for pointing out references to results from numerical analysis which the authors were unaware of. Finally, we thank the anonymous referees for many useful comments which helped to improve the manuscript.

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Authors and Affiliations

  1. Chalmers Technical University & Gothenburg University, Chalmers Tvä rgata 3, Gothenburg, Sweden

    Geir Bogfjellmo

  2. TU Darmstadt, Schloßgartenstr. 7, Darmstadt, Germany

    Rafael Dahmen

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology—NTNU, Trondheim, Norway

    Alexander Schmeding

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

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  1. Department of Mathematical Sciences, Norwegian University of Science and Technology—NTNU, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  2. Institute for the Mathematical Sciences (ICMAT), Madrid, Spain

    María Barbero Liñán

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Bogfjellmo, G., Dahmen, R., Schmeding, A. (2018). Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups. 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_8

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