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...