Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to Rn. Along the way, we establish various structural properties …
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Researchers find Frobenius manifold structures on orbits spaces of finite groups.
problem Understanding Frobenius manifold structures on orbits spaces of finite groups.
method Applying Dubrovin's method to various orbits spaces of linear representations of finite groups.
result Discoveries of non-trivial Frobenius manifold structures on orbits spaces.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
problem Classifying Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
method Examining orbits as CR submanifolds, proving total geodesy, and analyzing contact structures.
result Completely determine Sasaki-Einstein orbits.
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.
The abstract discusses extensions of Jacobi groups and their orbit space properties.
problem Understanding the properties of extended Jacobi groups and their orbit spaces.
method Proving an analogue of Chevalley Theorem and constructing a Frobenius structure.
result Construction of a Dubrovin Frobenius structure on the orbit space.
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…
This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.
problem Investigating Hamiltonian systems and Kähler structures on complex coadjoint orbits.
method Analyzes (pseudo)-holomorphic Hamiltonian systems, Lefschetz and almost toric fibrations, and introduces pseudo-holomorphic Hamiltonian systems.
result Complex coadjoint orbits exhibit both Hyperkähler and holomorphic Kähler structures, suggesting Kähler duality.
Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.
problem Classify closed G2-structures with T3-symmetry and derive associated hypersymplectic structures.
method Decompose G2-structures into canonical forms, classify structures based on orbit isotropy, and derive hypersymplectic structures.
result Closed G2-structures with T3-symmetry are classified into two types, leading to specific hypersymplectic structures.
This paper explores hyperkähler structures on specific orbits using different methods.
problem Investigate hyperkähler structures on adjoint orbits.
method Infinite dimensional hyperkähler reduction and symplectic geometry.
result Differences and connections between two methods are thoroughly investigated.
Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.
problem Understanding the manifold structures of orbits of normal operators under different norm topologies.
method Unified treatment of unitary and groupoid orbits, using moment maps and conditional expectations.
result Differentiable structures for orbits and necessary spectral conditions for norm closure and submanifold properties.
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…
Let Ĝ be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of Ĝ. The flag manifold Ĝ/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we charac…
Study of adjoint orbits in simplest non-trivial Lie algebra case.
problem Geometric properties of adjoint orbits in sl(2,R). method Analysis of adjoint orbits, showing three possibilities: hyperboloids or cones.
result Just three possibilities for adjoint orbits: hyperboloids or cones.
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G w…
We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposit…
We study holomorphic extensions of Matsuki orbits in complex Grassmannians.
problem Analyzing the analytic continuation of Matsuki orbits in complex Grassmannians.
method Using Rossi's theory of holomorphic extension and the holomorphic fiber bundle structure, we establish that the envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. result The envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. The paper classifies orbit closures of symplectic Lie algebras.
problem Classifying orbit closures of symplectic Lie algebras under the action of Sp(4,R). method Analyzing the natural action of Sp(4,R) on the set of 4-dimensional Lie algebras with symplectic structures. result A complete classification of orbit closures of 4-dimensional symplectic Lie algebras.
We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra g containing some ideal n. It is shown that any coadjoint orbit in g∗ is a bundle with the affine subspace of g∗ as its fibre. This fibre is an isotropic subma…
Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.
problem Tackles constructing Poisson structures on gauge orbits of Maurer-Cartan elements.
method Constructs Poisson structures on gauge orbits of Maurer-Cartan elements of dgla L, associating a compatible Batalin-Vilkovisky algebra to each MC element.
result MCP structures yield a notion of hamiltonian flow of MC elements and define Lie algebroids on gauge orbits.
Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in six dimensions, and we use this correspondence to interpret symplectic fibrations between these orbits, and to analyse moment polytopes associated to the standard Hamiltonian torus action on the coadjoint orbits. The theory is then applied to d…
New stratification reveals intrinsic singularity types of orbit spaces.
problem Understanding the intrinsic structure of orbit spaces under Lie group actions.
method Introduced the isostabilizer decomposition and established a map to Klein strata.
result A new canonical stratification on the manifold clarifies the relationship with classical structures.
Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.
Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.
problem Understanding rolling dynamics of 2D and 3D manifolds with constraints.
method Modeling rolling as a control affine system on a fibered space Q, analyzing reachable sets.
result Identified possible dimensions of non-open rolling orbits: 2, 5, 6, 7.
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
We study the general geometrical structure of the coadjoint orbits of a semidirect product formed by a Lie group and a representation of this group on a vector space. The use of symplectic induction methods gives new insight into the structure of these orbits. In fact, each coadjoint orbit of such a group is obtained b…
In this paper, it is shown that non-isomorphic effective linear circle actions yield non-diffeomorphic differential structures on the corresponding orbit spaces.
Constructs generalized Frobenius manifolds for specific Weyl groups.
problem Creating structures for orbit spaces of Weyl groups.
method Applying a previously established construction method to specific Weyl groups.
result Generalized Frobenius manifold structures constructed for Aℓ,Bℓ,Cℓ and Dℓ. The paper proves rigidity results for Anosov flows and their orbit equivalences.
problem Characterizing orbit equivalences of Anosov flows and their dynamics.
method Using hyperbolic-like dynamics, the paper proves a spectral rigidity theorem and gives efficient criteria for orbit equivalences.
result Characterizes orbit equivalent flows in terms of fundamental group elements represented by periodic orbits.
In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…
Smooth actions on manifolds can be globally defined under certain conditions.
problem Globalizing partial smooth actions of Lie groupoids on smooth manifolds.
method Providing necessary and sufficient conditions for globalizability and analyzing orbit and stabilizer spaces.
result There exists a unique differentiable structure on the quotient space for free and proper actions.
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …
Characterizes CR manifolds in complex flag manifolds.
problem Closed real orbits in complex flag manifolds.
method Characterization through CR manifold structures and real forms.
result Closed orbits are finitely nondegenerate.
The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.
problem Understanding K-contact manifolds with minimal closed Reeb orbits and their homeomorphism properties.
method Using Boothby-Wang fibration and Hamiltonian torus actions, the study constructs and analyzes K-contact manifolds.
result The existence of K-contact manifolds with minimal closed Reeb orbits that are not homeomorphic to spheres and have unique cohomology rings.
In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair o…
Differentiable spaces derived from Lie group actions have vector fields and forms.
problem Understanding the differential structure of orbit spaces of Lie group actions.
method Analyzing the differential structure of orbit spaces of proper Lie group actions on smooth manifolds.
result Orbit spaces of Lie group actions are differentiable spaces with exterior algebra of differential forms.
Let J1 be the real form of complex simple Jordan algebra with the automorphism group G of type F4(−20). Explicitly, we give the orbit decomposition of J1 under the action of G and determine the Lie group structure of stabilizer for each G-orbit on J1.
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open GR--orbits in flag varieties G/P. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, w…
The abstract describes a new manifold structure for NLS type equations.
problem Defining a new manifold structure for NLS type equations.
method Using quotient maps and invariant bilinear forms.
result A Dubrovin-Frobenius manifold structure on the orbit space of Bn. For a Lie groupoid G with Lie algebroid A, we realize the symplectic leaves of the Lie-Poisson structure on A∗ as orbits of the affine coadjoint action of the Lie groupoid JG⋉T∗M on A∗, which coincide with the groupoid orbits of the symplectic groupoid T∗G …
Lectures explore how differential methods improve understanding of algebraic group orbit spaces.
problem Understanding structure of invariants and orbit spaces of algebraic Lie groups.
method Combines algebraic and differential viewpoints to study orbit spaces.
result Differential approach provides deeper insights into invariants and orbit spaces.