Related papers: Hilbert-Schmidt groups as infinite-dimensional Lie…

In this article we give the realization of the Klein's Program for geometrical structures (Riemannian spaces and fiber bundles with connection) with arbitrary variable curvature within the framework of infinite deformed groups. These groups…

Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…

Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are…

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two…

The prescribed Ricci curvature problem involves finding a Riemannian metric g that satisfies the equation ric(g) = T, where T is a fixed symmetric (0, 2)-tensor field on a differential manifold M. In this paper, we introduce the concept of…

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and…

In this paper we use the $Z-$decomposition as a tool to find locally symmetric left invariant Riemannian metrics on some Lie groups. For this purpose, we need to compute the spectrum of the curvature operator. Since the study of this…

The article is devoted to the problem of Hilbert-Schmidt type analytic extensions in Hardy spaces over the infinite-dimensional unitary matrix group endowed with an invariant probability measure. An orthogonal basis of Hilbert-Schmidt…

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous…

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every…

The present document is the draft of a book which presents an introduction to infinite-dimensional differential geometry beyond Banach manifolds. As is well known the usual calculus breaks down in this setting. Hence, we replace it by the…

We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the…

We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

We give an overview of what is known on Lie groups admitting a left-invariant metric of negative Ricci curvature, including many natural questions and conjectures in the solvable case. We also introduce an open and convex cone C(n) of…

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through…

In this paper we extend results by De la Harpe concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions.…

Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems…

We define a numerical quasi-isometry invariant of a finitely generated group, whose values parametrize the difference between the group being uniformly embeddable in a Hilbert space and the reduced C*-algebra of the group being exact.