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 .

Differential operators on the supercircle S1|2 and symbol map

2016 ◽  
Vol 14 (01) ◽  
pp. 1750002
Author(s):  
Raouafi Hamza ◽  
Zeineb Selmi ◽  
Jamel Boujelben
Keyword(s):  
Vector Fields ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Linear Differential Operators ◽  
Classical Setting ◽  
Modules Of Differential Operators ◽  
Linear Differential ◽  
Standard Contact ◽  
Funct Anal

We consider the supercircle [Formula: see text] equipped with the standard contact structure. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the M[Formula: see text]bius superalgebra [Formula: see text]. We study the space of linear differential operators on weighted densities as a module over [Formula: see text]. We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the [Formula: see text]-modules [Formula: see text] of linear differential operators of order [Formula: see text] acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13–18.]

THE LINEAR $\mathfrak{aff}(n|1)$-INVARIANT DIFFERENTIAL OPERATORS ON WEIGHTED DENSITIES ON THE SUPERSPACE ℝ1|n AND $\mathfrak{aff}(n|1)$-RELATIVE COHOMOLOGY

2013 ◽  
Vol 10 (04) ◽  
pp. 1320004 ◽  
Author(s):  
IMED BASDOURI ◽  
ISMAIL LARAIEDH ◽  
OTHMEN NCIB
Keyword(s):  
Vector Fields ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Differential Cohomology ◽  
Invariant Differential Operators ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Invariant Differential ◽  
Linear Differential ◽  
Cohomology Spaces

Over the (1, n)-dimensional real superspace, we classify [Formula: see text]-invariant linear differential operators acting on the superspaces of weighted densities, where [Formula: see text] is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of [Formula: see text] with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra [Formula: see text]. We explicitly give 1-cocycles spanning these cohomology spaces.

The affine cohomology spaces and its applications

2016 ◽  
Vol 13 (02) ◽  
pp. 1650016 ◽  
Author(s):  
Nizar Ben Fraj ◽  
Ismail Laraiedh
Keyword(s):  
Large Class ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Invariant Differential Operators ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Formula ◽  
Invariant Differential ◽  
Linear Differential ◽  
Cohomology Spaces

We compute the [Formula: see text] cohomology space of the affine Lie superalgebra [Formula: see text] on the (1,1)-dimensional real superspace with coefficient in a large class of [Formula: see text]-modules [Formula: see text]. We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. This work is the generalization of a result by Basdouri et al. [The linear [Formula: see text]-invariant differential operators on weighted densities on the superspace [Formula: see text] and [Formula: see text]-relative cohomology, Int. J. Geom. Meth. Mod. Phys. 10 (2013), Article ID: 1320004, 9 pp.]

scholarly journals Cohomology of 𝔞𝔣𝔣(2|1) acting on the space of linear differential operators on the superspace ℝ1|2

2016 ◽  
Vol 13 (10) ◽  
pp. 1650124 ◽  
Author(s):  
Imed Basdouri ◽  
Ammar Derbali ◽  
Mohamed Elkhames Chraygui
Keyword(s):  
Differential Operators ◽  
Lie Superalgebra ◽  
Linear Differential Operators ◽  
Relative Cohomology ◽  
Linear Differential

We compute the first cohomology of the affine Lie superalgebra [Formula: see text] on the (1,2)-dimensional real superspace with coefficients in the superspace [Formula: see text] of linear differential operators acting on weighted densities. We also compute the same, but [Formula: see text]-relative, cohomology. We explicitly give [Formula: see text]-cocycles spanning these cohomology.

Differential operators on the weighted densities on the supercircle S1|1

2015 ◽  
Vol 52 (4) ◽  
pp. 477-503
Author(s):  
Nader Belghith ◽  
Mabrouk Ben Ammar ◽  
Nizar Ben Fraj
Keyword(s):  
Vector Fields ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Linear Differential Operators ◽  
Classical Setting ◽  
Linear Differential

Over the (1, 1)-dimensional real supercircle, we consider the K(1)-modules Dλ,μk of linear differential operators of order k acting on the superspaces of weighted densities, where K(1) is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules. This work is the simplest superization of a result by Gargoubi and Ovsienko.

First cohomology of 𝔞𝔣𝔣(1) and 𝔞𝔣𝔣(1|1) acting on linear differential operators

2016 ◽  
Vol 13 (01) ◽  
pp. 1550130 ◽  
Author(s):  
Imed Basdouri ◽  
Maha Boujelben ◽  
Ammar Derbali
Keyword(s):  
Differential Operators ◽  
Lie Superalgebra ◽  
Module Structure ◽  
Differential Cohomology ◽  
Linear Differential Operators ◽  
Linear Differential

We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We study also the super analogue of this problem getting the same results.

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.

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