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.

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm(K , X ) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group n∈N G n where the G n are Banach Lie groups. IntroductionAn infinite dimensional analytic Lie group is a group which is at the same time an analytic manifold modeled on some locally convex topological vector space such that the group operations are analytic. To construct such Lie groups, it is useful to have tools at hand ensuring the analyticity of nonlinear mappings between locally convex spaces. This text provides a sufficient criterion for complex analyticity in the case where the domain is an LB-space, i.e. a locally convex direct limit of an ascending sequence of Banach spaces:Theorem A [Complex analytic mappings defined on LB-spaces] Let E be a C-vector space that is the union of the increasing sequence of subspaces (E n ) n∈N . Assume that a norm · E n is given on each E n such that all bonding maps i n : E n −→ E n+1 : x → x are continuous and have an operator norm at most 1. We give E the locally convex direct limit topology and assume that it is Hausdorff. Let R > 0 and let U := n∈N B E n R (0) be the union of all open balls with radius R around 0. Let f : U −→ F be a function defined on U with values in a locally convex space F, such that each f n := f | B En R (0) is C-analytic and bounded. Then f is C-analytic. R. Dahmen (B)

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.

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

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 R-vector space V (X) on X. We show that V (X) is a quotient of the free abelian topological group on [−1, 1]×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 algebraic colimit property defined in the authors' previous work.