Characterizes third-order ODEs with a specific symmetry Lie algebra.
problem Identifying third-order ODEs with a five-dimensional point symmetry Lie algebra.
method Applied Cartan equivalence method to provide invariant characterization and construct equivalent canonical forms.
result Compact invariant characterization and a procedure to reduce to linear form.
The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on Rk and characterize Lie remarkable equations admitted by the …
In this paper, a complete Lie symmetry analysis of the damped wave equation with time-dependent coefficients is investigated. Then the invariant solutions and the exact solutions generated from the symmetries are presented. Moreover, a Lie algebraic classifications and the optimal system are discussed. Finally, using C…
The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation u′"=f(x,u,u′,u") which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point t…
The Killing operator on a Riemannian manifold is a linear differential operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as aris…
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.
Generalizes kinematical Lie algebras for isotropic spacetimes.
problem Classifying relativity algebras in isotropic spacetimes.
method Elementary proof and Lie group analysis.
result Symplectic involutive Lie algebras always exist.
Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.
problem Classifying Lie symmetry algebras for 2D quasilinear equations.
method Classification based on abelian Lie symmetry algebras of dimension and rank.
result Equations with specific symmetry algebras are linearizable.
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the ∇u=f(u) Poisson's equation, which has a subalgebra isomorphic to the 3−dimensional special Euclidean group SE(3) or group of rigid motions of R3. Looking the adjoint representation of ${\rm SE}(3)…
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.
Study of Lie algebras linked to graphs, revealing symmetry groups.
problem Understanding symmetries in Lie algebras associated with graphs.
method Analyzing 2-step nilpotent Lie algebras linked to uniform complete graphs.
result The Lie automorphism group includes the dihedral group of order 2n. We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by…
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.
This research introduces Lie brackets on spaces of biderivations in Lie algebras.
problem Understanding higher-order infinitesimal symmetries in Lie algebras.
method Study of right biderivations and Lie brackets on their spaces.
result New Lie algebra framework for biderivations with applications in deformation theory.
The paper analyzes Lie symmetries in a specific geometric context.
problem Analyzing Lie symmetries in a canonical connection with a special Lie algebra structure.
method Formulated Lie symmetries for a general linear connection, then applied to the canonical connection with a codimension one abelian nilradical.
result Conditions determining Lie symmetries in the specified geometric context are completely integrated.
Classifies Lie algebras and related spacetimes for a specific type of symmetry.
problem Classifying Lie algebras and related spacetimes for a specific type of symmetry.
method Classification of Lie algebras, homogeneous spacetimes, and coadjoint orbits.
result Classification of three types of Lifshitz spacetimes.
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions with respect to this Lie alge…
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.
This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. New internal symmetry found for Lie pair algebra.
problem Understanding Lie pair structures and their associated algebras.
method Introduced a Lie algebra action by Der(L) on the L_{≤3} algebra.
result Found internal symmetry of the L_{≤3} algebra.
Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.
problem Classifying non-integrable distributions with specific Lie superalgebras.
method Classification based on locality assumptions and W-grading.
result 15 series and 7 exceptional Lie superalgebras identified over C, and analogs over K of characteristic p>0. In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these ac…
In this paper we consider symplectic and contact Lie algebras. We define contactization and symplectization procedures and describe its main properties. We also give classification of such algebras in dimensions 3 and 4. The classification in dimension~4 is closely connected with normal forms of nondegenerate elliptic …
Develops methods for conditional symmetries of higher-order PDEs, removing unnecessary assumptions.
problem Formulating conditional symmetries for higher-order PDEs with unnecessary assumptions.
method Geometrical formulation and Lie systems approach to derive Lie algebras of conditional symmetries.
result New insights and methods for solving higher-order PDEs.
Foams have Lie algebra symmetries that simplify web state spaces.
problem Understanding symmetries in foam structures.
method Defined an action of a Lie subalgebra on foams compatible with glN-foam evaluation. result Endows glN-web state spaces with sl2-action. Study of homogeneous spacetimes with isotropic symmetry and their symmetries.
problem Classifying and analyzing the geometry and symmetries of homogeneous spacetimes.
method Detailed calculation of symmetries, soldering form, vielbein, and invariant connections for spatially isotropic homogeneous spacetimes.
result Boosts act with generic non-compact orbits and determine infinite-dimensional symmetries reminiscent of BMS Lie algebras.
Extends symmetry superalgebras to include all hidden symmetries of manifolds.
problem Tackles hidden symmetries of manifolds generated by Killing spinors.
method Defines generalized symmetry superalgebras, constructs Lie algebra structure, and defines symmetry operators.
result Constructs special Killing-Yano and conformal Killing-Yano forms from bilinears of Killing spinors.
In this paper we study a novel class of parabolic geometries which we call parabolic geometries of Monge type. These parabolic geometries are defined by special gradings of simple Lie algebras, namely, gradings with the property that their -1 component contains a nonzero co-dimension 1 abelian subspace whose bracket wi…
We show how the fundamental cocycles on current Lie algebras and the Lie algebra of symmetries for the sigma model are obtained via the current algebra functors. We present current group extensions integrating some of these current Lie algebra extensions.
Find first (0,2) mirror symmetry examples on Hopf surfaces.
problem Find (0,2) mirror symmetry on compact non-Kähler manifolds.
method Use Borisov's approach with vertex algebras and chiral de Rham complex. Study Killing spinors on quadratic Lie algebras and embeddings of superconformal vertex algebras.
result Construct first (0,2) mirror pairs of Hopf surfaces.
Lie symmetry group method is applied to study the Born-Infeld equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
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.
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
Resolves gap problem for quaternion-Hermitian structures.
problem Determine maximal and submaximal symmetry dimensions for quaternion-Hermitian structures.
method Classifies structures with specific symmetry dimensions and studies geometric properties of submaximally symmetric spaces.
result Identifies locally conformally quaternion-Kähler and quaternion-Kähler with torsion structures.
In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form ut=[E(x,u)ux]x+H(x,u) are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie alge…
Classifies flat projective structures with specific symmetries.
problem Classifying local projective structures with non-trivial Lie symmetries.
method Analyzes flat projective structures with positive-dimensional Lie algebra of projective vector fields.
result Obtained a classification of flat projective structures.
Study 2D viscoelastic equations using Lie group theory.
problem Symmetry classification and reduction of 2D viscoelastic equations.
method Investigation through Lie group theory, including algebra of symmetries and optimal subalgebras.
result Classification of reductions of similarities related to Lie subalgebras.
Researchers discover symmetries in Ricci flows and use them to find invariant solutions.
problem Finding symmetries in Ricci flows on manifolds.
method Developed a method to find Lie point symmetries of Ricci flows and particular metrics.
result Invariant solutions of Ricci flow for specific metric families were obtained.
New method finds Lie group representations without explicit groups, enabling new neural network architectures.
problem Building neural networks equivariant to arbitrary Lie groups.
method Algorithm to find Lie group representations from Lie algebra structure constants. Self-contained method for constructing Lie group-equivariant neural networks.
result First object-tracking model equivariant to the Poincaré group.
The paper analyzes symmetries of Vaidya-Bonner geodesics.
problem Investigating invariance properties of Vaidya-Bonner geodesics.
method Classification of Lie point symmetries and Noether symmetries, determination of optimal system of subalgebras.
result Determination of optimal system of subalgebras for Vaidya-Bonner geodesics.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.
We study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.
Study on metrics on specific nilmanifolds, finding new examples and properties.
problem Characteristically solvable nilmanifolds and their metrics.
method Explicit determination of left-invariant metrics and their properties.
result First known examples of Lie groups without positive index of symmetry.
Reduces multisymplectic Lie systems through symmetry analysis.
problem Solving multisymplectic Lie systems using symmetry reduction.
method Using momentum maps for reduction and reconstruction of multisymplectic Lie systems.
result Solves the original problem by analyzing simpler multisymplectic Lie systems.
New knot invariants discovered using mirror symmetry.
problem Categorify knot homology groups and explain their meaning.
method Homological mirror symmetry for new families of manifolds.
result Explicitly computable invariants for any Lie algebra.
The study explores maximal symmetry in Ricci solitons on Lie groups.
problem Maximal symmetry in left-invariant Riemannian metrics and Ricci solitons.
method Analysis of left-invariant metrics and Ricci solitons on Lie groups, using tools from previous work on Einstein metrics.
result Expanding homogeneous Ricci solitons have maximal isometry algebras but not always maximal isometry groups.
This research bridges Killing vectors and Lie algebras through induced vector fields.
problem Understanding the relationship between Killing vector fields and Lie algebras.
method Defining and exploring induced vector fields to connect Killing vector fields with isometry Lie groups.
result Established a new connection between Killing vector fields and Lie algebras.