The study finds geodesic orbit metrics on Lie groups from flag manifolds.
problem Investigating geodesic orbit metrics on Lie groups.
method Using generalized flag manifolds to form metrics on simple Lie groups.
result All left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Explicit bounds on orbit curvatures for Lie group actions.
problem Bounding orbit curvatures for Lie group actions.
method Explicit construction of orbits with bounded principal curvatures.
result Explicit bounds on orbit curvatures for unit spheres.
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
This paper classifies geodesic orbit metrics on compact Lie group G2.
problem Classifying geodesic orbit metrics on compact Lie groups.
method Using representation theory of Lie subgroups, specifically weakly regular subgroups.
result Left-invariant geodesic orbit metrics on compact Lie group G2 are classified. The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.
problem Classifying foliations formed by generic coadjoint orbits of certain Lie groups.
method Analyzing Lie groups with specific dimensions and nilradicals, proving foliations in the coadjoint representation.
result The family of generic coadjoint orbits forms a measurable foliation in the Lie groups considered.
A Lie group G naturally acts on its Lie algebra ≫, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group G2 in its Lie algebra ≫2. As results, the group G2 has four orbit types in the Lie algebra ≫2 as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …
The abstract describes a foliation of orbits for a specific class of Lie groups.
problem Classifying foliations formed by generic coadjoint orbits of Lie groups.
method Geometric description and topological classification of foliations.
result The family of generic coadjoint orbits forms a measurable foliation.
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.
The paper proves convexity results for a specific type of Lie groups.
problem Convexity results for non-compact real reductive Lie groups.
method Proves convexity results through orbit projection.
result Convexity results for quasi-hermitian Lie groups.
Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.
problem Understanding Casimir functions for free nilpotent Lie groups of steps 3 and 4.
method Construction of Casimir functions for free nilpotent Lie groups of steps 3 and 4.
result For 3-step groups, coadjoint orbits are fully described as affine subspaces or direct products of quadrics.
Study infinitesimally tight Lagrangian orbits in symplectic manifolds.
problem Understanding Lagrangian orbits in symplectic manifolds.
method Analyzing isotropic orbits of Lie group actions with equivariant moment maps.
result Examples of Lagrangian orbits in complex flag manifolds and cotangent bundles.
Study on almost Kaehler geometry of Lie groups orbits.
problem Understanding the geometry of adjoint orbits of Lie groups.
method Explicit formulas for Chern-Ricci form, scalar curvature, and Nijenhuis tensor derived from root data.
result Explicit formulas and conditions for the Chern-Ricci form and Kaehler type quotients.
The paper studies time-optimal problems on specific Lie groups, describing orbits and integrals.
problem Time-optimal control problems on two-step Carnot groups.
method Description of co-adjoint orbits, Casimir functions, and integrals for the Hamiltonian system.
result Characterization of the flow and constancy of solutions for two-dimensional co-adjoint orbits.
Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.
problem Characterizing minimal orbits of semi-simple Lie groups.
method Analyzing projective orbits induced by representations of semi-simple Lie groups and relating them to invariant subspaces of the underlying modules.
result Minimal orbits of semi-simple Lie groups are in bijection with minimal orbits of compact subgroups on invariant subspaces.
Geometric bijection and homotopy equivalence between Lie group orbits.
problem Isomorphic adjoint and coadjoint representations of Lie groups.
method Geometric bijection and homotopy equivalence of orbits.
result Geometrically defined bijection and homotopy equivalence between adjoint and coadjoint orbits.
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.
Abstract not provided enough details, focusing on vector bundles and orbits.
problem Understanding continuous representations of semisimple Lie groups.
method Not specified in the abstract.
result Not specified in the abstract.
Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.
problem Embedding coadjoint orbits and their equivalence to magnetic geodesic flows.
method Review and initiate study of isotropic and Lagrangian embeddings for SO and Sp cases, then apply to magnetic geodesic flows. result Equivalence between magnetic geodesic flows and certain spin chains.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.
problem Characterize geodesic orbit property for pseudo-Riemannian H-type Lie groups.
method Extend results from Riemannian to pseudo-Riemannian H-type Lie groups, focusing on minimal admissible Clifford modules.
result Complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups.
Criterion for periodic orbits convergence proved.
problem Periodic orbits convergence criterion.
method Criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups.
result Criterion for periodic orbits convergence proved.
Smooth approximations for continuous functions on orbit spaces.
problem Approximating continuous functions on orbit spaces.
method Study of subcartesian spaces and proper Lie group actions.
result Continuous functions can be approximated by smooth functions.
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 study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
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.
Approximates symplectic automorphisms of coadjoint orbits using Hamiltonian Carleman methods.
problem Approximating symplectic automorphisms of coadjoint orbits.
method Hamiltonian Carleman approximation for coadjoint orbits of complex Lie groups.
result Established the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
Explains complex adjoint orbits in Lie theory and geometry.
problem Understanding adjoint orbits in Lie theory and geometry.
method Expository introduction to adjoint orbits of complex semisimple groups.
result Provides insights into properties of semisimple and nilpotent orbits.
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.
The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group G. The first chapter is intended to recall some facts about Lie groups. The mos…
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
The study identifies biharmonic submanifolds in compact symmetric spaces and Lie groups.
problem Characterizing biharmonic submanifolds in compact symmetric spaces and Lie groups.
method Using necessary and sufficient conditions derived from symmetric triad with multiplicities.
result Determined all biharmonic submanifolds in irreducible symmetric spaces of compact type and biharmonic hypersurfaces in compact Lie groups.
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.
Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…
We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that the Lie group action must have odd cohomogeneity. We pay special attention to mani…
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…
Classifies homogeneous hypersurfaces in specific 4D geometries.
problem Classifying homogeneous hypersurfaces in 4D Thurston geometries.
method Analyzing subalgebras of Lie algebras and isometry groups.
result Determined all homogeneous hypersurfaces up to ambient isometries.
It is proved that the orbit space of an irreducible representation of a simple connected compact Lie group of type B, C, or D can be a smooth manifold only in two cases.
Study of actions on curved manifolds with boundary results in new geometric invariant.
problem Classifying actions of compact Lie groups on curved manifolds with boundary.
method Introduced a new geometric invariant for compact symmetric spaces.
result List of representations of compact simple Lie groups with non-trivial reductions.
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.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
In this note we prove that whenever a Lie group G acts on a manifold X, then the orbit Gx through any point x of X is a weakly embedded submanifold of X. The investigation of this problem was inspired by an application to Cat astrophe Theory.
Locally convex bialgebroids reconstruct Lie groupoids of orbits.
problem Reconstructing Lie groupoids from their orbits using bialgebroids.
method Constructing a locally convex bialgebroid over the space of orbits.
result Groupoids can be reconstructed from their spectral action Lie groupoids.
New Einstein metric on G2 is found that's not geodesic orbit.
problem Finding left-invariant Einstein metrics on Lie groups that are not geodesic orbit.
method Developed tools for geodesic orbit Riemannian manifolds; used recent results by I. Chrysikos and Y. Sakane.
result Compact Lie group G2 admits a left-invariant Einstein metric that is not geodesic orbit. Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace …