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In this article groups of controlled characters of a combinatorial Hopf algebra are considered from the perspective of infinite-dimensional Lie theory. A character is controlled in our sense if it satisfies certain growth bounds, e.g.\ exponential growth. We study these characters for combinatorial Hopf algebras. Following Loday and Ronco, a combinatorial Hopf algebra is a graded and connected Hopf algebra which is a polynomial algebra with an explicit choice of basis (usually identified with combinatorial objects such as trees, graphs, etc.). If the growth bounds and the Hopf algebra are compatible we prove that the controlled characters form infinite-dimensional Lie groups. Further, we identify the Lie algebra and establish regularity results (in the sense of Milnor) for these Lie groups. The general construction principle exhibited here enables to treat a broad class of examples from physics, numerical analysis and control theory. Groups of controlled characters appear in renormalisation of quantum field theories, numerical analysis and control theory in the guise of groups of locally convergent power series. The results presented here, generalise the construction of the (tame) Butcher group, aka the controlled character group of the Butcher-Connes-Kreimer Hopf algebra., Comment: 52 pages, uses TikZ, v5: minor rewrite as per referee's suggestions, corrected typos, added section on category of combinatorial Hopf algebras, main results remain unchanged

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor's $\mathcal{FD}$-topology. Then we establish the continuity of certain mappings between spaces of smooth mappings, e.g.\ the continuity of the joint composition map. As a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group (with respect to these topologies). For the reader's convenience the article includes also a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts., Comment: 45 pages, v3: corrected several typos and added some details, results remain unchanged

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge--Kutta methods. Recently, a natural Lie group structure has been constructed for this group. Unfortunately, the associated topology is too coarse for some applications in numerical analysis. In the present paper, we propose to remedy this problem by replacing the Butcher group with the subgroup of all exponentially bounded elements. This "tame Butcher group" turns out to be an infinite-dimensional Lie group with respect to a finer topology. As a first application we then show that the correspondence of elements in the tame Butcher group with their associated B-series induces certain Lie group (anti)morphisms., Comment: 41 pages, uses TikZ v3: major revision, simplified many arguments, strengthened and added results (most main results remain unchanged)

To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories., Comment: 16 pages, LaTex, this is a companion to arXiv:1506.05415

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold., Comment: 45 pages, LaTex, this is a sequel to arXiv:1409.1428

Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects. In this paper, we develop a framework for shape analysis of curves in Lie groups for problems of computer animations. In particular, we will use these methods to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones., Comment: 37 pages, 7 figures, v4: Major revision, corrected typos, simplified and clarified arguments, main results remain unchanged

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker--Campbell--Hausdorff--Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group. If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris., Comment: 47 pages, 1 figure uses TIKZ. v3: corrected several typos, improved one of the main results, the rest of the results remains unchanged

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense., Comment: 33 pages, LaTex, v2: now includes a proof for the regularity of the real analytic diffeomorphism group

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex manifold) is uniquely determined. If M is regular and the complexified modeling space of M is normal, then a regular complexification exists for some neighborhood of K. For each (real or complex) analytic regular manifold M modeled on a metrizable locally convex space and Banach-Lie group H, this allows the group Germ(K,H) of germs of H-valued analytic maps around K in M to be turned into an analytic Lie group which is regular in Milnor's sense. A special case is a regular real analytic Lie group structure on the group of real analytic H-valued maps on a compact real analytic manifold M (which, previously, had only been treated in the convenient setting of analysis). Combining our results concerning groups of germs with an idea by Neeb and Wagemann, one can also obtain a regular Lie group structure on the group of all real analytic H-valued mappings on the real line., Comment: 32 pages

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 Fr\'echet 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 to the Butcher group by Connes and Kreimer., Comment: 34 pages, TiKZ, v3: major revision, rewritten to increase readability, main results remain unchanged

In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor., Comment: 31 pages, LaTex, v2: corrected several typos and added some remarks, the main results remain unchanged

We endow the diffeomorphism group of a paracompact (reduced) orbifold with the structure of an infinite dimensional Lie group modelled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that this Lie group is C^0-regular and thus regular in the sense of Milnor. Furthermore an explicit characterization of the Lie algebra associated to the diffeomorphism group of an orbifold is given., Comment: 184 pp, LaTex and TikZ. V4: updated some remarks and literature, corrected typos and minor errors. The results remain unchanged. For the reader's convenience, the appendix contains some definitions and known facts from A. Pohl's preprint arXiv:1001.0668 and H. Glockner's preprint arXiv:math/0408008 which are used in the text

We develop differential calculus of $C^{r,s}$-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class $C^{r,s}$., Comment: 42 Pages, Latex, v2: Corrected several misprints and typos. The results remain unchanged

We reconsider a classical theorem by Bican and El Bashir, which guarantees the existence of non-trivial relatively pure submodules in a module category over a ring with unit. Our aim is to generalize the theorem to module categories over rings with several objects. As an application we then consider the special case of alpha-pure objects in such module categories., Comment: 11 pages, corrected several typos, some references and explanations have been added, two errors in the proof of the main theorem have been corrected, the results remain unchanged