Symmetry algebra for multi-contact structures given by 2n vector fields on $$\mathbb R^{2n+1}$$

2008 ◽  
Vol 341 (3) ◽  
pp. 529-542 ◽  
Author(s):  
Chong-Kyu Han ◽  
Jong-Won Oh ◽  
Gerd Schmalz
Keyword(s):  
Vector Fields ◽  
Symmetry Algebra ◽  
Contact Structures

scholarly journals Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

2017 ◽  
Vol 28 (11) ◽  
pp. 1750080
Author(s):  
Hassan Azad ◽  
Indranil Biswas ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Second Order ◽  
Semisimple Lie Algebra ◽  
Semisimple Algebras ◽  
Generic Orbit ◽  
Cartan Subalgebras ◽  
Local Vector

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.

scholarly journals On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

2019 ◽  
pp. 5-20
Author(s):  
Artem Atanov ◽  
Alexander Loboda
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourth Order ◽  
Algebraic Surface ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Basis Vector ◽  
Invariant Polynomial ◽  
Real Hypersurfaces ◽  
Symmetry Algebras

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

scholarly journals Foliations by minimal surfaces and contact structures on certain closed3-manifolds

2003 ◽  
Vol 2003 (21) ◽  
pp. 1323-1330
Author(s):  
Richard H. Escobales
Keyword(s):  
Vector Field ◽  
Fundamental Group ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Vector Fields ◽  
Contact Structure ◽  
Contact Structures ◽  
Volume Form ◽  
Basic Vector ◽  
Transverse Volume

Let(M,g)be a closed, connected, orientedC∞Riemannian 3-manifold with tangentially oriented flowF. Suppose thatFadmits a basic transverse volume formμand mean curvature one-formκwhich is horizontally closed. Let{X,Y}be any pair of basic vector fields, soμ(X,Y)=1. Suppose further that the globally defined vector𝒱[X,Y]tangent to the flow satisfies[Z.𝒱[X,Y]]=fZ𝒱[X,Y]for any basic vector fieldZand for some functionfZdepending onZ. Then,𝒱[X,Y]is either always zero andH, the distribution orthogonal to the flow inT(M), is integrable with minimal leaves, or𝒱[X,Y]never vanishes andHis a contact structure. If additionally,Mhas a finite-fundamental group, then𝒱[X,Y]never vanishes onM, by the above together with a theorem of Sullivan (1979). In this caseHis always a contact structure. We conclude with some simple examples.

THE ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH INTEGRABLE COUPLINGS

2008 ◽  
Vol 23 (09) ◽  
pp. 1309-1325 ◽  
Author(s):  
LIN LUO ◽  
WEN-XIU MA ◽  
EN-GUI FAN
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Commutation Relations ◽  
Zero Curvature ◽  
Integrable Couplings ◽  
Integrable Coupling

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospectral and nonisospectral AKNS flows are worked out, and thus a τ-symmetry algebra for the AKNS integrable couplings is engendered from this theory.

scholarly journals Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

2011 ◽  
Vol 147 (5) ◽  
pp. 1613-1634 ◽  
Author(s):  
Eveline Legendre
Keyword(s):  
Finite Number ◽  
Scalar Curvature ◽  
Vector Fields ◽  
Existence Result ◽  
Constant Scalar Curvature ◽  
Contact Structures ◽  
Toric Geometry ◽  
Contact Manifolds ◽  
Reeb Vector ◽  
Toric Contact Manifolds

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

scholarly journals On deformations and extensions of Diff(S2)

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Martín Enríquez Rojo ◽  
Tomáš Procházka ◽  
Ivo Sachs
Keyword(s):  
Vector Fields ◽  
Free Field ◽  
Symmetry Algebra ◽  
Null Infinity ◽  
Central Extensions ◽  
Asymptotically Flat ◽  
Free Field Realization ◽  
The One ◽  
Area Preserving ◽  
Parameter Deformation

Abstract We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that hs[λ], the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $$ \mathfrak{vect} $$ vect (S2) into $$ \mathfrak{vect} $$ vect (ℂ*). For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.

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