Searcharxiv⌕ Search

arXiv subjects

Rafael Dahmen

Publications and source records attributed to Rafael Dahmen.

At least 19 recordsLinked to original sources
arXiv2025

On the singularities of the exponential function of a semidirect product

We show that the Fréchet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi--Metzner--Sachs group of symmetries of an asymptotically flat spacetime is not locally exponential.

math.GR↗
arXiv2023

A topological splitting of the space of meromorphic germs in several variables and continuous evaluators

We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators.

math.CV↗
arXiv2022

On the Topology of J-Groups

We introduce the concept of a topological J-group and determine for many important examples of topological groups if they are topological J-groups or not. Besides other results, we show that the underlying topological space of a pathwise connected topological J-group is weakly contractible which is a strong and unexpected obstruction that depends only on the homotopy type of the space.

math.GR↗
arXiv2021

Long colimits of topological groups IV: Spaces with socks

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors develop techniques for showing that products of certain spaces with CSHP, such as the Closed Long Ray and the Long Line, have CSHP again.

math.GN↗
arXiv2021

Long colimits of topological groups III: Homeomorphisms of products and coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

math.GN↗
arXiv2020

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\textit{algebraic colimit property}$ defined in the authors' previous work.

math.GN↗
arXiv2020

Continuity of Chen-Fliess Series for Applications in System Identification and Machine Learning

Model continuity plays an important role in applications like system identification, adaptive control, and machine learning. This paper provides sufficient conditions under which input-output systems represented by locally convergent Chen-Fliess series are jointly continuous with respect to their generating series and as operators mapping a ball in an $L_p$-space to a ball in an $L_q$-space, where $p$ and $q$ are conjugate exponents. The starting point is to introduce a class of topological vector spaces known as Silva spaces to frame the problem and then to employ the concept of a direct limit to describe convergence. The proof of the main continuity result combines elements of proofs for other forms of continuity appearing in the literature to produce the desired conclusion.

math.OC↗
arXiv2019

Long colimits of topological groups I: Continuous maps and homeomorphisms

The union of a directed family of topological groups can be equipped with two noteworthy topologies: the finest topology making each injection continuous, and the finest group topology making each injection continuous. This begs the question of whether the two topologies coincide. If the family is countable, the answer is well known in many cases. We study this question in the context of so-called long families, which are as far as possible from countable ones. As a first step, we present answers to the question for families of group-valued continuous maps and homeomorphism groups, and provide additional examples.

math.GN↗
arXiv2019

The Pro-Lie Group Aspect of Weakly Complete Algebras and Weakly Complete Group Hopf Algebras

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $\mathbb{K}[G]$ is associated functorially so that the functor $G\mapsto \mathbb{K}[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $\mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=\mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathcal{L}(G)$ in $A_{\rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,\mathbb{R})$. The dual of the group algebra $\mathbb{R}[G]$ is the Hopf algebra ${\mathcal R}(G,\mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${\mathcal R}(G,\mathbb{R})\leftrightarrow \mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.

math.GR↗
arXiv2018

On the Component Factor Group G/G_0 of a Pro-Lie Group G

A pro-Lie group $G$ is a topological group such that $G$ is isomorphic to the projective limit of all quotient groups $G/N$ (modulo closed normal subgroups $N$) such that $G/N$ is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group $G_t:= G/G_0$ of $G$ modulo the identity component $G_0$ is compact. In this case it is straightforward that each Lie group quotient $G/N$ of $G$ has finitely many components. However, in spite of a comprehensive literature on pro-Lie groups, the following theorem, proved here, was not available until now: A pro-Lie group $G$ is almost connected if each of its Lie group quotients $G/N$ has finitely many connected components. The difficulty of the proof is the verification of the completeness of $G_t$.

math.GR↗
arXiv2018

Lie groups of controlled characters of combinatorial Hopf algebras

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.

math.RT↗
arXiv2017

Overview of (pro-)Lie group structures on Hopf algebra character groups

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.

math.GR↗
arXiv2016

Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups

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.

math.DG↗
arXiv2015

Character groups of Hopf algebras as infinite-dimensional Lie groups

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.

math.GR↗
arXiv2015

The Lie group of real analytic diffeomorphisms is not real analytic

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.

math.DG↗
arXiv2014

Bounded solutions of finite lifetime to differential equations in Banach spaces

Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $γ\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $γ$ is bounded, then $γ$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is replaced with an infinite-dimensional Banach space.

math.FA↗