Lie algebras of vector fields and differential operators on vector bundles

1993 ◽  
Vol 63 (1) ◽  
pp. 53-56
Author(s):  
A. N. Rudakov
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Vector Bundles ◽  
Differential Operators

scholarly journals PROJECTIVE AND CONFORMAL SCHWARZIAN DERIVATIVES AND COHOMOLOGY OF LIE ALGEBRAS VECTOR FIELDS RELATED TO DIFFERENTIAL OPERATORS

2006 ◽  
Vol 03 (04) ◽  
pp. 667-696 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Manifold ◽  
Conformal Class ◽  
Tensor Fields ◽  
Relative Cohomology ◽  
Schwarzian Derivatives ◽  
Cohomology Of Lie Algebras

Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect(M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl(n + 1, ℝ)-relative cohomology group.

On 𝔬𝔰𝔭(1|2)-relative cohomology of the Lie superalgebra of contact vector fields on ℝ1|1

2017 ◽  
Vol 14 (02) ◽  
pp. 1750022
Author(s):  
Ben Fraj Nizar ◽  
Meher Abdaoui ◽  
Raouafi Hamza
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Contact Structure ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Differential ◽  
Standard Contact

We consider the [Formula: see text]-dimensional real superspace [Formula: see text] endowed with its standard contact structure defined by the 1-form [Formula: see text]. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra [Formula: see text]. We classify [Formula: see text]-invariant linear differential operators from [Formula: see text] to [Formula: see text] vanishing on [Formula: see text], where [Formula: see text] is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the first differential [Formula: see text]-relative cohomology of [Formula: see text] with coefficients in [Formula: see text]. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on [Formula: see text] acting on bilinear differential operators, Int. J. Geom. Methods Mod. Phys. 2(1) (2005) 23–40].

Cohomology of the vector fields lie algebras on ℝℙ1 acting on bilinear differential operators

2019 ◽  
Vol 56 (3) ◽  
pp. 280-296
Author(s):  
Abdaoui Meher
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Differential Operators ◽  
Smooth Vector ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Differential

Abstract Let Vect (ℝℙ1) be the Lie algebra of smooth vector fields on ℝℙ1. In this paper, we classify -invariant linear differential operators from Vect (ℝℙ1) to vanishing on , where is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential -relative cohomology of Vect (ℝℙ1) with coefficients in .

Cohomology of the vector fields Lie algebras on ℝ acting on trilinear differential operators, vanishing on 𝔞𝔣𝔣(1)

2017 ◽  
Vol 14 (10) ◽  
pp. 1750150
Author(s):  
Imed Basdouri ◽  
Elamine Nasri ◽  
Hassen Mechi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Main Topic ◽  
Smooth Vector ◽  
Relative Cohomology ◽  
Tensor Densities

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields [Formula: see text], with coefficients in the space of trilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra [Formula: see text].

scholarly journals COHOMOLOGY OF 𝔬𝔰𝔭(1|2) ACTING ON THE SPACE OF BILINEAR DIFFERENTIAL OPERATORS ON THE SUPERSPACE ℝ1|1

2012 ◽  
Vol 09 (04) ◽  
pp. 1250033 ◽  
Author(s):  
MABROUK BEN AMMAR ◽  
AMINA JABEUR ◽  
IMEN SAFI
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Differential Operators ◽  
Lie Superalgebra

We compute the first cohomology of the ortosymplectic Lie superalgebra 𝔬𝔰𝔭(1|2) on the (1, 1)-dimensional real superspace with coefficients in the superspace 𝔇λ, μ of bilinear differential operators acting on weighted densities (λ ∈ ℝ2 and μ ∈ ℝ). This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on ℝℙ1 acting on bilinear differential operators, Int. J. Geom. Meth. Mod. Phys.2(1) (2005) 23–40].

scholarly journals Lie algebras of conservation laws of variational partial differential equations

2018 ◽  
Vol 18 (2) ◽  
pp. 207-228
Author(s):  
Emanuele Fiorani ◽  
Sandra Germani ◽  
Andrea Spiro
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Differential Operators ◽  
Equivalence Classes ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Bijective Correspondence ◽  
Independent Variables ◽  
Geometric Conditions ◽  
Finite Dimensional

Abstract We establish a version of Noether’s first Theorem according to which the (equivalence classes of) conserved quantities of given Euler–Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action from which the Euler–Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such a bijective correspondence Φ͠ between equivalence classes comes from an explicit (non-bijective) linear map Φ from vector fields into conserved differential operators, and not from a map into divergences of conserved operators as it occurs in other proofs of Noether’s Theorem. Where possible, claims are given a coordinate-free formulation and all proofs rely just on basic differential geometric properties of finite-dimensional manifolds.

COHOMOLOGY OF THE VECTOR FIELDS LIE ALGEBRAS ON ℝℙ1 ACTING ON BILINEAR DIFFERENTIAL OPERATORS

2005 ◽  
Vol 02 (01) ◽  
pp. 23-40 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Line ◽  
Smooth Vector ◽  
Relative Cohomology ◽  
First Cohomology Group ◽  
Tensor Densities

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ) with coefficients in [Formula: see text].

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