Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
We solve extending local Lie groupoids to global ones, linking integrability and associativity.
problem Extending local Lie groupoids to global Lie groupoids.
method We characterize integrable local Lie groupoids and establish a relationship between their integrability and associativity failure.
result A precise relationship between integrability of Lie algebroids and associativity failure in local integration.
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…
This thesis explores symplectic foliations and local Lie groupoids, with applications in current theory.
problem Understanding calibratable symplectic foliations and local Lie groupoids.
method Applying de Rham's and Sullivan's theories to symplectic foliations and generalizing Mal'cev and Olver's theorems.
result Generalizations of theorems by Mal'cev and Olver for local Lie groupoids and algebroids.
Book on infinite-dimensional Lie groups, covering basics and various classes.
problem Understanding Lie groups in infinite-dimensional spaces.
method Develops smooth manifolds and Lie groups in locally convex spaces, discussing various classes.
result Detailed exploration of infinite-dimensional Lie groups and their properties.
Lie group integrators improve global error estimates.
problem Global error estimates for Lie group integrators.
method Relate local error to global error, derive from bounds.
result Lie-Butcher theory proves global error estimates for Lie group integrators.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncom…
In this paper, we use the powerful tool Milnor bases to classify all the 3−dimensional connected and locally symmetric Riemannian Lie Groups by solving system of polynomial equations of structure constants of each Lie algebra . Moreover, we showed that E0(2), is the only Lie group with locally symmetric left invar…
For a local Lie group M we define odd order cohomology classes. The first class is an obstruction to globalizability of the local Lie group. The third class coincides with Godbillon-Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes.
New groups contactomorphic to stratified ones found.
problem Understanding contactomorphic relationships between polarised and stratified Lie groups.
method Constructing modifications of stratified groups and proving contactomorphic relationships.
result Polarised groups are contactomorphic to stratified groups under specific conditions.
Geometric proof for lattice rigidity in Lie groups.
problem Local rigidity of lattices in semisimple Lie groups.
method Geometric proof of well-known results.
result Classical local rigidity of lattices proven geometrically.
Infinite Lie groups meet continuity conditions.
problem Continuity conditions for Lie groups.
method Analyzing locally μ-convex Lie groups. result Infinite dimensional Lie groups have the strong Trotter property.
Local norms of Fourier multipliers bounded on discrete subgroups of Lie groups.
problem Bounding Lp norms of Fourier multipliers on discrete subgroups of Lie groups. method Developed tools to find explicit bounds on c(A), reducing the problem to representations of semisimple and radical parts of Lie algebras. result Explicit bounds on c(A) for unimodular connected solvable Lie groups, showing c(G)=1. Conditions for exponentiating Lie algebras on complete locally convex spaces are established.
problem Conditions for exponentiating Lie algebras of linear operators on complete locally convex spaces.
method Focus on equicontinuous case, establishing necessary conditions for exponentiation to compact Lie groups.
result Necessary conditions for exponentiation to compact Lie groups are established.
Study LCP structures on solvmanifolds, complete list up to 5 dimensions.
problem Characterize LCP structures on solvable Lie groups.
method Classify LCP structures on Lie groups, focus on solvable unimodular case.
result Complete list of solvable unimodular Lie algebras up to dimension 5 with LCP structures.
We define and study the invariant linear and nonlinear horizontal double complexes of a local Lie group.
The paper classifies Sasaki and Vaisman manifolds of unimodular Lie groups.
problem Understanding the structure of Sasaki and Vaisman manifolds on unimodular Lie groups.
method Basic structure theorem and complete classification for simply connected manifolds.
result A complete classification of simply connected Sasaki and Vaisman unimodular Lie groups.
Study different types of G_2-structures on Lie groups.
problem Characterize and find examples of G_2-structures on Lie groups.
method Analyze invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
result Complete characterization of invariant exact locally conformal closed G_2-structures on simply connected Lie groups.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
Local and global rigidity results for Lie group actions on pseudo-Riemannian manifolds.
problem Rigidity of isometric actions of simple Lie groups on pseudo-Riemannian manifolds.
method Analyzing normal bundles and using Lie group properties.
result Local and global rigidity theorems for specific Lie groups and manifolds.
We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, wh…
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
The study classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
problem Classifying and constructing symplectic structures on Lie algebras and solvmanifolds.
method Analyzes locally conformally symplectic Lie algebras and solvmanifolds.
result Classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
The study examines fibrations of Lie groupoids and their classification.
problem Understanding and classifying fibrations of Lie groupoids.
method Introduced locally topological product Lie groupoid fibrations and used groupoid cohomology to obstruct and classify these fibrations.
result The existence of locally topological product Lie groupoid fibrations is obstructed by a specific groupoid cohomology.
Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Study LCS structures of the second kind on Lie algebras.
problem Characterize and construct LCS Lie algebras.
method Construct new examples using representations and characterize existing ones.
result Characterize all LCS Lie algebras obtained with the construction.
In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections ar…
This paper classifies Lie groups embeddable into 4D space.
problem Determining the minimum dimensional Euclidean space for Lie group embeddings.
method Used algebraic equations (Gauss and derived Gauss) to classify Lie groups.
result Classified Lie groups embeddable into 4D space.
The study proves a localization theorem and calculates volumes for superspaces.
problem Calculating volumes of homogeneous superspaces for super-Lie groups.
method Proved the Schwarz-Zaboronsky localization theorem and applied it to volumes.
result Volume calculation for homogeneous superspaces of super-Lie groups.
Study on spectral sequence for abelian Lie group actions, with bounds and applications.
problem Understanding the spectral sequence for abelian Lie group actions.
method Provided upper bounds and examples to show these bounds are sharp.
result Sharp bounds on the degeneration page of spectral sequence.
We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…
Study shows certain Lie groups lead to Oeljeklaus-Toma manifolds.
problem Understanding locally conformally Kähler metrics and their relation to Oeljeklaus-Toma manifolds.
method Analyzing solvable Lie groups and their metrics, and relating them to Oeljeklaus-Toma constructions.
result Found a connection between Lie groups, locally conformally Kähler metrics, and Oeljeklaus-Toma manifolds.
Current Lie groupoids generalize mappings to Lie groupoids.
problem Generalizing Lie group properties to mappings into Lie groupoids.
method Study of superposition operators and Lie groupoid properties.
result Current Lie groupoids inherit properties from underlying maps.
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
problem Integrating Lie-Leibniz triples into Lie group structures.
method Defining Lie group-rack triples and integrating finite-dimensional Lie-Leibniz triples.
result Any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple.
We derive the canonical forms of super Riemannian metrics and the local isometry groups of such metrics. For certain super metrics we also compute the simply connected covering groups of the local isometry groups and interpret these as local spin groups of the super metric. Using a generalization of a Theorem of Rogers…
Relations between parameter rigidity of locally free Lie group actions on closed manifolds and the 1st leafwise cohomology of the orbit foliations are discussed. Some computational results of the leafwise cohomology are included.
We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is uni…
B-Spline CNNs on Lie Groups expand G-CNNs to arbitrary groups.
problem Leveraging geometric structure for improved feature learning.
method Lifting feature maps to B-spline expansions on Lie algebra.
result G-CNNs on Lie groups outperform classical 2D CNNs.
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to id…
New principle for supersymmetric localization on Lie groups.
problem Computing supertrace of non-supersymmetric observables.
method Invariant supersymmetric deformations and fermionic zero modes.
result Path integral localizes to periodic orbits.
New Lie group structure on vertical bisections of Lie groupoids.
problem Constructing Lie group structure on vertical bisections of Lie groupoids.
method Construct Lie group structure on the group of vertical bisections of a regular Lie groupoid, identify Lie algebra, discuss regularity properties.
result Established Lie theoretic properties of vertical bisections of Lie groupoids over non-compact bases.
Reduced sh-Lie structures have been studied for the case when a Lie group acts on the fibers of a vector bundle while preserving the base space of the bundle. In this paper we investigate how one obtains a reduced sh-Lie structure using the ideas of symmetry reduction where the action of the Lie group is transversal to…
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxembourg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used…