The geometry of characters of Hopf algebras

Character groups of Hopf algebras appear in a variety of mathematical contexts such as non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. In these applications, several species of "series expansions" can then be described as characters from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, Chen-Fliess series from control theory and rough paths. In this note we explain and review the constructions for Lie group and topological structures for character groups. The main novel result of the present article is a Lie group structure for characters of graded and not necessarily connected Hopf algebras (under the assumption that the degree zero subalgebra is finite-dimensional). Further, we establish regularity (in the sense of Milnor) for these Lie groups.Comment: 25 pages, notes for the Abelsymposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and Control", v4: corrected typos and mistakes, main results remains valid, updated reference

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.Comment: 33 pages, LaTex, v2: now includes a proof for the regularity of the real analytic diffeomorphism grou

Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where $C$ only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition A.3, which makes no impact on proofs or results. Version submitted for publication. v3 small changes in response to referee comments, title extended. v4 crucial gap filled, results not affected. v5 final version to appear in Annales de l'Institut Fourie

The diffeomorphism group of a non-compact orbifold

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 tex

On the unit component of the Newman-Unti group

Author's accepted version (postprint).This is an Accepted Manuscript of an article published by IOP Publishing in Classical and Quantum Gravity on 18/1/23.Available online: doi.org/10.1088/1361-6382/acb0a9acceptedVersio

Applications of infinite-dimensional geometry and Lie theory

Habilitation thesisHabilitationsschriftInfinite-dimensional manifolds and Lie groups arise from problems related to differential geometry, fluid dynamics, and the symmetry of evolution equations. Among the most prominent examples of infinite-dimensional manifolds are manifolds of (differentiable) mappings and the diffeomorphism groups Diff(K), where K is a smooth and compact manifold. The group Diff(K) is an infinite-dimensional Lie group which arises naturally in fluid dynamics if K is a three-dimensional torus. The motion of a particle in the fluid corresponds, under periodic boundary conditions, to a curve in Diff(K). As a working definition, an infinite-dimensional Lie group will be a group which at the same time is an infinite-dimensional manifold that turns the group operations into smooth mappings. An infinite-dimensional manifold will be a topological space which is locally (in charts) homeomorphic to an open subset of an infinite-dimensional space. Moreover, we require the change of charts to be smooth. Beyond the realm of Banach spaces, the usual concept of smoothness is no longer available and we replace it with the requirement that all directional derivatives exist and induce continuous mappings, the so called Bastiani calculus. Infinite-dimensional Lie groups and their homogeneous spaces will be the objects of our main interest. In conjunction with Lie theory, we exploit tools from (infinite-dimensional) Riemannian geometry. Recall that a Riemannian metric on a manifold is a choice of inner product for every tangent space which ”depends smoothly” on the basepoint. Generalising Riemannian geometry to infinite-dimensional manifolds, one faces in general the problem that there are no (smooth) partitions of unity. Further, the inner products will in general not be compatible with the topology of the tangent spaces as they are not Hilbert spaces. Thus the finite-dimensional definition of a Riemannian metric (what we will call a ’strong Riemannian metric’) has to be relaxed to admit relevant examples beyond the Hilbert manifold setting. This leads to the notion of a ’weak Riemannian metric’, i.e. a smooth choice of inner products on each tangent space which do not necessarily induce the topology of the tangent space. Constructing weak Riemannian metrics on manifolds of mappings from the L2-inner product, the resulting metrics are studied for example in shape analysis, fluid dynamics and optimal transport. The present thesis explores structures from infinite-dimensional Lie theory and Riemannian geometry, their interplay and applications in three main topics: - Connections between infinite-dimensional Lie groups and higher geometry, - Hopf algebra character groups as Lie groups, and - Applications of the interplay between Lie theory and Riemannian geometry.publishedVersio