A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is...

Denote by ${𝒯}_n$, $n\ge 2$, the regular tree whose vertices have valence $n+1$, $\partial {𝒯}_n$ its boundary. Yu. A. Neretin has proposed a group $N_n$ of transformations of $\partial {𝒯}_n$, thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that $N_n$ is generated by two groups: the group $\mathrm{Aut}\left({𝒯}_n\right)$ of tree automorphisms, and a Higman-Thompson group $G_n$. We prove the simplicity of $N_n$ and of a family of its subgroups.

The study of controlled infinite-dimensional systems gives rise to many papers (see for instance [GXL], [GXB], [X]) but it is also motivated by various mathematical problems: partial differential equations ([BP]), sub-Riemannian geometry on infinite-dimensional manifolds ([Gr]), deformations in loop-spaces ([AP], [PS]). The first difference between finite and infinite-dimensional cases is that solutions in general do not exist (even locally) for every given control function. The aim of this paper...

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to...

We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus $T^n$, $n\ge 2$. We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are $\pm \mathrm{Id}$ and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and...

Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, ${u|}_{\partial \Omega }=0$. We show that is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators....

We construct an infinite-dimensional real analytic manifold structure on the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is defined to be 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 that this group is regular in the sense of Milnor. ...

Let $U_c\left(\right)$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{\infty }$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c\left(\right)$. If $u₀,u₁,u\in U_c\left(\right)$ and the geodesic β joining u₀ and u₁ in $U_c\left(\right)$ satisfy $d_{\infty }(u,\beta )<\pi /2$, then the map $f\left(s\right)=d_{\infty }(u,\beta \left(s\right))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_c\left(\right)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_p\left(\right)$ = u: u unitary and u-1 in the p-Schatten class...