Abstract
Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coalgebra that is also equipped with an antiautomorphism known as the antipode, satisfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra morphisms from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups.
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Notes
- 1.
Note that this Lie algebra is precisely the one appearing in the famous Milnor-Moore theorem in Hopf algebra theory [41].
- 2.
Here we consider C r([0, 1], 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.
- 3.
This is only a glimpse at Tannaka-Kreĭn duality, which admits a generalisation to compact topological groups (using complex representative functions, see [26, Chapter 7, §30] and cf. [23, p. 46] for additional information in the Lie group case). Also we recommend [28, Chapter 6] for more information on compact Lie groups.
- 4.
To be exact: The class of integrators depending only on the affine structure (cf. [38]).
- 5.
Silva spaces arise as special inductive limits of Banach spaces, see [15] for more information. They are also often called (DFS)-space in the literature, as they can be characterised as the duals of Fréchet-Schwartz spaces.
- 6.
We are not aware of a literature reference of this fact apart from the forthcoming book by Glöckner and Neeb [22]. To roughly sketch the argument from [22]: Using the regular representation of A one embeds \(B \otimes _{\mathbb {K}} A\) in the matrix algebra M n(B) (where n = dim A). Now as A is a commutant in \(\text{End}_{\mathbb {K}} (A)\), the same holds for B ⊗ A in M n(B). The commutant of a set in a CIA is again a CIA, whence the assertion follows as matrix algebras over a CIA are again CIAs (cf. [45, Corollary 1.2]).
- 7.
BCH-Lie groups derive their name from the fact that there is a neighborhood in their Lie algebra on which the Baker–Campbell–Hausdorff series converges and yields an analytic map. See Definition 35.
- 8.
One only has to observe that a function into the Lie subgroup is absolutely continuous if and only if it is absolutely continuous as a function into the larger group. On the author’s request, a suitable version of [20, Theorem G] for L 1-regularity will be made available in a future version of [19].
- 9.
See [11, Section 5.5] for some explicit examples of this procedure, e.g. for the KPZ equation.
- 10.
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. [31]).
- 11.
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 [18, Proposition 4.1].
- 12.
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Acknowledgements
This research was partially supported by the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691070 and by the Knut and Alice Wallenberg Foundation grant agreement KAW 2014.0354. We are indebted to K.-H. Neeb and R. Dahmen for discussions which led to Lemma 10. Further, we would like to thank L. Zambotti and Y. Bruned for explaining their results on character groups in the renormalisation of SPDEs. Finally, we thank K.H. Hofmann for encouraging and useful comments and apologize to him for leaving out [28] at first.
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Bogfjellmo, G., Schmeding, A. (2018). The Geometry of Characters of Hopf Algebras. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_6
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