Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball $$B_{𝔄}$$ in a J*-algebra $$𝔄$$ of operators. Let $$𝔉$$ be the family of all collectively compact subsets W contained in $$B_{𝔄}$$ . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family $$𝔉$$ is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when $$𝔄$$ is a Cartan factor.

In this paper, we construct a hyperkähler structure on the complexification ${𝒪}^{ℂ}$ of any Hermitian symmetric affine coadjoint orbit $𝒪$ of a semi-simple $L^{*}$-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kähler structure of $𝒪$. By a relevant identification of the complex orbit ${𝒪}^{ℂ}$ with the cotangent space $T𝒪$ of $𝒪$ induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on $T𝒪$ compatible with...

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of ${\pi }_2$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional...

We give the definition of a kind of building $ℐ$ for a symmetrizable Kac-Moody group over a field $K$ endowed with a discrete valuation and with a residue field containing $ℂ$. Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if $K=ℂ\left(\right(t\left)\right)$, the geodesic segments...

We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^{\times }$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\widehat{G}$ of $G$ by the group ${GL}_A\left(E\right)$ of automorphisms of the $A$-module $E$. This Lie group extension is a “non-commutative” version of the group $Aut\left(𝕍\right)$ of automorphism...

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we...

In this article we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\mathrm{Ext}{(G,N)}_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H_{ss}^2{(G,Z\left(N\right))}_S$. To each $S$ a locally smooth obstruction class $\chi \left(S\right)$ in a suitably defined cohomology group $H_{ss}^3{(G,Z\left(N\right))}_S$ is defined....

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxemburg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used to construct the hull. We use the nonstandard hull construction to give a nonstandard characterization of Enflo's notion of groups that are uniformly free from small subgroups. We...

Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $𝔤$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda$ is integrable in the sense that it is of the form $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of $*$-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.For...

Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...