arXiv research

Unified framework for smooth structures on coadjoint orbits.

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 $\mathbf{R}^n$. Along the way, we establish various structural properties …

New manifold structures on Weyl group orbit spaces proven.

Researchers find Frobenius manifold structures on orbits spaces of finite groups.

New Frobenius manifold structures found on Dicyclic group orbits.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

The study examines Kähler structures on coadjoint orbits of Lie groups using coherent and squeezed states.

The abstract discusses extensions of Jacobi groups and their orbit space properties.

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.

Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.

This paper explores hyperkähler structures on specific orbits using different methods.

Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.

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.

Study geodesic orbit Lorentz nilmanifolds, proving structural properties.

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.

The paper classifies orbit closures of symplectic Lie algebras.

We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra ${\mathfrak g}$ containing some ideal ${\mathfrak n}$. It is shown that any coadjoint orbit in ${\mathfrak g}^*$ is a bundle with the affine subspace of ${\mathfrak g}^*$ as its fibre. This fibre is an isotropic subma…

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

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.

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.

Minimal action of mapping class group on character variety.

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.

The paper proves rigidity results for Anosov flows and their orbit equivalences.

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.

Geodesic orbit property studied for Lorentz manifolds.

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…

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfie…

Study of tangent spaces in diffeological spaces under Lie group actions.

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.

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

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…

Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--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.

For a Lie groupoid $\mathcal{G}$ 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 $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ …

Lectures explore how differential methods improve understanding of algebraic group orbit spaces.