Researchers classify homomorphisms for a specific algebra using singular vectors and symmetric polynomials.
problem Classifying homomorphisms for conformal Galilei algebras.
method Identifying homomorphisms with singular vectors and coefficients of symmetric polynomial expansions.
result Explicit description and classification of homomorphisms for conformal Galilei algebras.
Weyl-type theorems extended to Galilei and Carroll geometries.
problem Extending Weyl's theorem to non-relativistic and ultra-relativistic spacetimes.
method Defining and analyzing conformal and projective structures in Galilei and Carroll geometries.
result Torsion-free connections in Galilei and Carroll geometries are uniquely determined by their projective structures.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
We start by analysing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a…
Classifies connections on Galilei manifolds, generalizing known results.
problem Classifying general affine connections on Galilei manifolds.
method Classification through tensor fields, extending known Galilei connections.
result Additional freedom in connections not metric-compatible, linked to clock form and space metric.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
Study generalizes non-interaction theorems for relativistic systems.
problem Understanding interactions in relativistic and non-relativistic systems.
method Generalizes non-interaction theorems for Lorentz violating systems and Galilei invariant systems.
result Extends analysis to very special relativity and anisotropic systems.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Explains non-lorentzian theories and their dynamics.
problem Understanding non-lorentzian kinematics and dynamics.
method Review of kinematical spacetimes, construction of particle dynamics actions, discussion of gravity theories and field theories.
result Introduction and analysis of non-lorentzian gravity and field theories.
In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplec…
We define an almost--cosymplectic--contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost--coPoisson--Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated …
In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special funct…
This paper extends Lie algebra contractions to infinite-dimensional spaces for better understanding of group limits.
problem Understanding group limits through Lie algebra contractions, especially in infinite-dimensional settings.
method Using infinite-dimensional Lie algebras and their integration theory, the paper constructs Lie group expansions.
result Explicit descriptions of Lie groups in elementary terms, including applications to Newtonian gravity.
We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a gr…
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.
The paper studies prolongations of Lie algebras associated with pseudo H-type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo H-type Lie algebras. method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Study indefinite bi-invariant metrics and their conformal properties.
problem Characterize indefinite bi-invariant metrics and their conformal relationships.
method Analyzes Lie algebras under different cases and structures.
result Conformally Einstein properties for various Lie algebras are determined.
Classifies special Lie algebras with semisimple types.
problem Classifying Lie algebras of semisimple type.
method Introduced conformal pseudo-subriemannian fundamental graded Lie algebras and provided their classification.
result Classification of conformal pseudo-subriemannian fundamental graded Lie algebras of semisimple type and their prolongations.
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.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.
problem Finite-dimensional group of conformal transformations in higher dimensions.
method Derived deformation theory of ambitwistor space of complex null-geodesics.
result Infinite-dimensional dg-Lie algebra incorporating symmetries and conformal structure deformations.
We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…
New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.
problem Classifying conformally superintegrable systems in arbitrary dimensions.
method Algebraic geometric approach extended to conformally superintegrable systems.
result An algebraic equation governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems.
Locally conformally product Lie algebras are characterized and constructed.
problem Characterizing and constructing compact locally conformally product Lie algebras.
method Characterization via closed 1-forms and non-unimodular Lie algebras acting on abelian ones.
result Explicit examples of compact LCP manifolds that are not solvmanifolds.
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Machine learning explores symmetries in field theory and algebra.
problem Understanding symmetries in field theory and algebra.
method Using neural networks to analyze conformal field theory and Lie algebra representation theory.
result Recent advances in machine learning have uncovered new symmetries.
Establishes a connection between Kähler metrics and vector bundle sections.
problem Finding Kähler metrics in a conformal class.
method One-to-one correspondence between Kähler metrics and parallel sections of a vector bundle with conformally invariant connection.
result Obstructions for a Riemannian metric to be conformal to a Kähler metric.
The Bergman space and conformally flat 2-disk operads are linked to vertex operator algebras.
problem Understanding invariants of 2D Riemannian manifolds using algebraic structures.
method Introducing a suboperad and showing algebraic structures, using conformally flat factorization homology.
result The Bergman space is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.
Review of conformal geometry in irrational rotation algebra.
problem Understanding conformal geometry of irrational rotation algebra.
method Review of recent progress by Connes and others.
result Recent progress in understanding conformal geometry.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.
problem Characterizing CKY forms on almost abelian Lie groups and proving parallelism.
method Analyzing almost abelian metric Lie algebras, proving parallelism for CKY forms, and classifying cases up to dimension 5.
result CKY forms are parallel on almost abelian Lie algebras, with exceptions for p=1 and p=n−1. Using the AdS/CFT correspondence, we identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
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.
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…
The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.
problem Understanding non-Kähler threefolds with algebraic dimension two.
method Examining compact complex non-Kähler threefolds with locally conformally Kähler metrics and proving they are quasi-bundles over projective surfaces under certain assumptions.
result They are blown-up quasi-bundles over a projective surface.
Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.
problem Locally conformal symplectic structures on Lie algebras of type I.
method Analyzing Lie algebras of type I, proving trivial Morse-Novikov cohomology, and constructing solvmanifolds.
result LCS structures on Lie algebras of type I are of the first kind and can be used to construct compact solvmanifolds.
Study classifies LC Kahler structures on 4D solv Lie alg, with applications.
problem Classifying LC Kahler structures on 4D solvable Lie algebras.
method Investigation through linear equivalence and geometric interpretation.
result Produces many examples, including lcK structures on Oeljeklaus-Toma manifolds.
New concept of metric Lie algebras helps classify Lie groups.
problem Classifying Lie groups based on conformal Killing symmetric tensors.
method Introducing metric Lie algebras of Killing type and proving conditions for these algebras.
result Conditions for Lie algebras to be of Killing type with respect to any positive definite metric.
Study G2-structures with non-closed four-forms under local conformal conditions.
problem Characterizing G2-structures with non-closed four-forms under local conformal conditions.
method Investigated G2-structures with non-closed four-forms and non-zero Lee form, focusing on Lie algebra properties.
result Found solutions for G2-structures in preposition1 and theorem1.
Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic suba…
We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein equations with extra fields, as generalized Maurer-Cartan equations.
A common approach to metric-affine, local Poincaré, special-relativistic and Galilei spacetime geometry is developed. Starting from an affine composite bundle, we introduce local reference frames and their evolution along worldlines and we study both, absolute and relative simultaneity postulates, giving rise to altern…
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian…
Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-di…
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as…