New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionle…
We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein-Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein-W…
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
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.
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.
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.
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 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.
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.
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.
Study orbit spaces of equivariant ANEs for proper actions of metrizable groups.
problem Understanding the extension properties of orbit spaces for proper actions.
method Analyzing equivariant absolute neighborhood extensors for proper G-spaces. result Proving conditions under which orbit spaces of metrizable G-orbits are ANEs. Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
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 family of braided Thompson groups introduced using recursive braids.
problem Dehornoy-Brin braided Thompson group generalization.
method Using recursive braids and strand diagrams to define new groups.
result New groups BVn,r(H) are finitely generated if H is finitely generated. Lifts isometries in orbit spaces for compact groups.
problem Isometries in orbit spaces of compact groups.
method Equivariant isometry of original Euclidean space.
result Simple formula for connected component of isometry group.
The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying G-naturally reductive and G-geodesic orbit metrics on M. result Classification of geodesic orbit spaces with simple isotropy groups.
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.
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.
This paper upbuilds the theoretical framework of orbit braids in M×I by making use of the orbit configuration space FG(M,n), which enriches the theory of ordinary braids, where M is a connected topological manifold of dimension at least 2 with an effective action of a finite group G and the action of G…
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.
Reconstruct flows from their orbit spaces using group actions.
problem Reconstructing flows from their orbit spaces.
method Using group actions and pseudo-Anosov flows.
result Reconstruct flows from their orbit spaces.
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.
New measures on orbit spaces for orthogonal groups identified.
problem Characterizing measures on orbit spaces of orthogonal groups.
method Constructing Hilbert measures on orbit spaces of coregular representations of orthogonal groups.
result Hilbert measures have singularities if and only if the number of copies equals the dimension.
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.
Automorphisms of free groups yield invariant posets of lamination orbits.
problem Understanding the structure of free-by-cyclic groups through automorphisms.
method Analyzing the poset of attracting lamination orbits for free group automorphisms.
result The poset of lamination orbits is a commensurability invariant of free-by-cyclic groups.
New framework mated Kleinian groups with complex polynomials, revealing unique group properties.
problem Mating Kleinian groups with complex polynomials dynamics.
method Orbit equivalence framework for holomorphic mating, focusing on Fuchsian groups and higher Bowen-Series maps.
result Only torsion-free Fuchsian groups can be mated, with specific properties of Bowen-Series maps.
We introduce a recursive adaptive group lasso algorithm for real-time penalized least squares prediction that produces a time sequence of optimal sparse predictor coefficient vectors. At each time index the proposed algorithm computes an exact update of the optimal ℓ1,∞-penalized recursive least squares (R…
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.
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 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…
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F0GMightarrowF(M/G,n) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
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.
A 2-manifold's group structure is deduced from orbit configuration spaces.
problem Understanding the fundamental groups of orbit configuration spaces.
method Relating the four-term exact sequence of orbifold pure braid groups to the fundamental groups of the orbit configuration spaces.
result Fundamental groups of orbit configuration spaces form a four-term exact sequence.
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.
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.
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.
The paper computes infinitesimals for group actions on a multispace of curves.
problem Examining group actions on a multispace of curves.
method Formal study using recursion relations, closely mimicking jet space prolongation.
result Produces a recursion relation for infinitesimals in the multispace.
We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group G acting linearly and rationally on a real vector space V. G can be viewed as the real points of a complex reductive group GC which acts on $V…
We study orbital functions associated to Kleinian groups through the heat kernel approach developed in \cite{artmoiheatcounting1}.
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.
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. We prove the ADO invariants are a q-holonomic family and establish recursion relations.
problem Understanding the q-holonomic properties of ADO link invariants. method Proving the ADO invariants are a q-holonomic family and establishing recursion relations. result The ADO invariants for r≥2 are a q-holonomic family, satisfying independent recursion relations. Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
The present paper is a continuation of Le Anh Vu's ones [13], [14], [15]. Specifically, the paper is concerned with the subclass of connected and simply connected MD5-groups such that their MD5-algebras G have the derived ideal G1:=[G,G]≡ R3. We sha…
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.
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.