New structure on unitary group of Hilbert space.
problem No specific problem stated; constructing a new structure.
method Constructing a Banach Poisson-Lie group structure.
result Banach Poisson-Lie group structure on unitary group of Hilbert space.
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48 results for “Hilbert Lie group”
Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert spacemath.DG
Paper studies how certain hypersurfaces collapse in Hilbert space.
problem Collapse of hypersurfaces in Hilbert space.
method Investigates mean curvature flow with specific conditions.
result Invariant hypersurfaces collapse to group orbits under flow.
Study projective representations of infinite-dimensional Hilbert-Lie groups.
problem Characterize and classify representations of Hilbert-Lie groups.
method Use covariance with respect to one-parameter groups of automorphisms and implement perturbation theory.
result Explicit determination of central extensions for projective representations.
No exceptional orbits found in Hilbert spaces actions.
problem Proving the non-existence of exceptional orbits in Hilbert spaces.
method Analyzing polar actions on separable Hilbert spaces by connected Lie groups.
result Proved non-existence of exceptional orbits in Hilbert spaces.
Criterion for polystability in Lie group actions on manifolds.
problem Characterizing orbits intersecting a specific set in Lie group actions.
method Hilbert-Mumford criterion applied to polystability, using Cartan decomposition and gradient maps.
result Characterization of orbits intersecting a specific set in terms of maximal weight functions.
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…
Geometric quantization on symmetric spaces yields Hilbert spaces with flatness implying Lie group structure.
problem Quantization of compact Riemannian symmetric spaces.
method Geometric quantization using Kähler polarizations and half-form correction.
result Projective flatness of quantum Hilbert spaces implies the symmetric space is a compact Lie group.
We solve Hilbert's fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert's fifth problem for global…
Kac-Moody geometrymath.DG
The geometry of symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings is governed by spherical Weyl groups and simple Lie groups. A natural generalization of semisimple Lie groups are affine Kac-Moody groups as they mirror their structure theory and have good explicitely known representati…
Generalizes manifold results for Lie groups, proving equivariant homotopy type.
problem Hilbert-Smith conjecture for topological -manifolds.
method Generalization of previous results, verification of n-classifying spaces.
result Any Palais-proper topological -manifold has the equivariant homotopy type of a countable proper -CW complex.
In this paper, we study the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds, where "almost free" means that the stabilizers of the group action are finite. First w…
Analyzes isometries in metric Lie groups, proving their regularity and composition properties.
problem Analyzing isometries in metric Lie groups, especially Carnot groups.
method Analyticity of isometries and composition properties proved via Hilbert 5th problem and Riemannian case.
result Every isometry between connected and nilpotent metric Lie groups is a composition of a left translation and an isomorphism.
Study uses regularized mean curvature flow on invariant hypersurfaces in Hilbert space.
problem Analyzing invariant hypersurfaces in Hilbert space.
method Regularized mean curvature flow with specific conditions.
result Invariant hypersurfaces collapse to group orbits under flow.
Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.
problem Existence of periodic geodesics on Hilbert half-Lie groups
method Using completeness results and Lyusternik-Fet type theorem
result Periodic geodesics exist whenever the fundamental group is nontrivial
Revises super unitary representations by relaxing Hilbert space definition.
problem Left-regular representations of super Lie groups are not super unitary.
method Weaken super Hilbert space definition and introduce a new metric.
result Left-regular representations of all super Lie groups are super unitary.
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…
Criterion for nilpotent Lie groups to have nilsolitons.
problem Existence of nilsolitons in nilpotent Lie groups.
method Algebraic criterion for nilpotent Lie algebras, proving necessary and sufficient condition for nilsolitons.
result Criterion provides a necessary and sufficient condition for nilpotent Lie groups to admit nilsolitons.
Study on Banach half-Lie groups and their properties.
problem Characterizing and understanding the properties of half-Lie groups in infinite dimensions.
method Investigation of Banach half-Lie groups, their extensions, and right invariant strong Riemannian metrics.
result The full Hopf--Rinow theorem holds for Banach half-Lie groups, a surprising result.
The paper defines and studies isoparametric submanifolds in Riemannian Hilbert manifolds.
problem Defining and studying isoparametric submanifolds in Riemannian Hilbert manifolds.
method Introducing curvature-invariant submanifolds, regularizable submanifolds, and isoparametric submanifolds; proving the constancy of mean curvatures and independence of shape operators and normal Jacobi operators.
result Proving that certain submanifolds are isoparametric under specific conditions.
The paper develops stability criteria for real reductive Lie groups acting on manifolds.
problem Analyzing stability of real reductive Lie group actions on manifolds.
method Introduced a gradient map and maximal weight function to characterize stability conditions.
result Characterized stability, semistability, and polystability using numerical criteria.
New construction of isoparametric submanifolds in Hilbert spaces.
problem Constructing isoparametric submanifolds in Hilbert spaces.
method Analyzing holonomy maps and connections in Hilbert spaces.
result Each component of the inverse image of an equifocal submanifold by the holonomy map is an isoparametric submanifold.
The Lie group of automorphisms of a Courant algebroid and the moduli space of generalized metricsmath.DG
Lie group structure on automorphisms of Courant algebroids, moduli space of metrics.
problem Understanding the moduli space of generalized metrics.
method Endowed Lie group structure on automorphisms of exact Courant algebroids, proved slice theorem, related to usual metrics.
result Moduli space of generalized metrics is stratified by ILH submanifolds.
New representation of curves helps prove complex geometry result.
problem Understanding cohomologically stable curves in projective space.
method Using commuting matrix polynomials to represent curves and show isomorphism to hyperkähler quotient.
result Hilbert scheme isomorphic to a hyperkähler quotient.
We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space by an action of a finite group of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if t…
Survey on deformations of convex structures on manifolds and orbifolds.
problem Understanding the deformation spaces of convex real projective structures.
method Study of representations into Lie groups, focusing on character varieties and geometric structures.
result Survey of basics of character varieties, geometric structures on orbifolds, and Hilbert geometry.
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an de Rham theorem and constructing an integration functor.
result An equivalence between -representations of -algebroids and -representations of Lie -groupoids for foliations.
We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'',…
The Hilbert-Smith Conjecturemath.GT
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'',…
New theorems show non-embeddability of certain Lie groups and sub-Riemannian manifolds.
problem Non-embeddability of Lie groups and sub-Riemannian manifolds in specific metric spaces.
method Proving non-existence of quasi-isometric embeddings and biLipschitz embeddings into certain metric measure spaces.
result Connected nonabelian nilpotent Lie groups and sub-Riemannian manifolds cannot be embedded into specified metric spaces.
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, …
The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated w…
We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the…
GCNNs on homogeneous spaces use vector bundles and Hilbert spaces.
problem Learning data on homogeneous spaces with global symmetry.
method Analysis of -equivariant convolutional layers on homogeneous spaces, using vector bundles and reproducing kernel Hilbert spaces.
result A precise criterion for expressing -equivariant layers as convolutional layers, leading to stronger results for some groups.
The study examines Kähler structures on coadjoint orbits of Lie groups using coherent and squeezed states.
problem Does the coadjoint orbits of Lie groups support a Kähler structure?
method Examined three Lie groups: Weyl-Heisenberg, SU(2), and SU(1,1). Used coherent and squeezed states to explore Kähler structures.
result Coherent states provide Kähler embeddings, while squeezed states only symplectic embeddings.
In this paper, we prove that, if a full irreducible infinite dimensional anti-Kaehler isoparametric submanifold of codimension greater than one has -diagonalizable shape operators, then it is an orbit of the action of a Banach Lie group generated by one-parameter transformation groups induced by holomorphic Killing …
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for -Anosov representations and uses it to prove properties of boundary maps.
result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
Study extends JB-algebra structure group results to infinite dimensions.
problem Extend results for real Jordan algebras to infinite dimensional JB-algebras.
method Prove structure groups, cone preserving groups, and automorphism groups are embedded Banach-Lie groups; describe components via cones, isotopes, and central projections; apply to special JB-algebra of self-adjoint operators.
result Full description of components of structure group and automorphism group, including their Banach-Lie algebras and connected components.
We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such co…
We sharpen the construction of representation space in the paper "Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations". We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather t…
Paper solves Hilbert's fifth problem for specific groupoids.
problem When are transitive groupoids continuously isomorphic to Lie groupoids?
method Investigates proper and transitive groupoids with compact source fibers.
result Solves Hilbert's fifth problem for specified groupoids.
The paper extends Hilbert's fifth problem to transitive groupoids.
problem When is a transitive topological groupoid continuously isomorphic to a Lie groupoid?
method Investigation of transitive topological groupoids, focusing on proper and transitive groupoids with compact source fibers.
result Presentation of results generalizing Hilbert's fifth problem to transitive groupoids, including solutions for proper and transitive groupoids with compact source fibers.
Study third group cohomology and gerbes over Lie groups.
problem Classify gerbes over Lie groups using third cohomology.
method Relate third group cohomology to locally smooth cohomology, construct transgression maps.
result Construct transgression maps linking cohomology groups.
Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}math.RT
For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Mo…
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uni…