arXiv research

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

Machine learning explores symmetries in field theory and algebra.

Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unr…

Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.

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.

Researchers compute c-projective symmetry algebras for Kähler surfaces.

Find first (0,2) mirror symmetry examples on Hopf surfaces.

Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.

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…

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…

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.

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…

Study of Lie algebras linked to graphs, revealing symmetry groups.

We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.

We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show …

We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symm…

We describe some recent development on the theory of formal Frobenius manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk (DGBV) algebras and formulate a version of mirror symmetry conjecture: the extended deformation problems of the complex structure and the Poisson structure are described by…

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmet…

Abstract reviews symmetry and reduction in dynamical systems.

We discover a new example of a generic rank 2-distribution on a 5-manifold with a 6-dimensional transitive symmetry algebra, which is not present in Cartan's classical five variables paper. It corresponds to the Monge equation z' = y + (y'')^(1/3) with invariant quartic having root type [4], and a 6-dimensional non-sol…

Foams have Lie algebra symmetries that simplify web state spaces.

Extends symmetry superalgebras to include all hidden symmetries of manifolds.

Study of homogeneous spacetimes with isotropic symmetry and their symmetries.

We bound the symmetry algebra of a vector distribution, possibly equipped with an additional structure, by the corresponding Tanaka algebra. The main tool is the theory of weighted jets.

Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.

Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.

Characterizes third-order ODEs with a specific symmetry Lie algebra.

Develops methods for conditional symmetries of higher-order PDEs, removing unnecessary assumptions.

Defines null G-structures on Lorentzian manifolds and explores their symmetries.

Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.

Classifies Lie algebras and related spacetimes for a specific type of symmetry.

This research introduces Lie brackets on spaces of biderivations in Lie algebras.

The paper analyzes Lie symmetries in a specific geometric context.

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 $\mathbb{R}^k$ and characterize Lie remarkable equations admitted by the …

Researchers find higher symmetries in symplectic Dirac operator.

New method proves identifiability of deep generative models with algebraic contrast principles.

A new algebraic method extracts symmetry anomalies from 5D SCFTs.

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable struc…

Paper studies symmetries in singular foliations using Lie $\infty$-morphisms.

We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries - in parti…

The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.

Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.

Classifies homogeneous Levi non-degenerate hypersurfaces in complex 3-space.

This research extends Lie algebra actions to singular foliations.

We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain $A_{\infty}$ algebra structures and some canonically defined deformations of s…

The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the quaternionic projective space $\mathbb{H}P^n$, which is flat and has the symmetry algebra …

Study on metrics on specific nilmanifolds, finding new examples and properties.

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.