𝔞𝔣𝔣(1|1)-Relative cohomology on ℝ1|1

2017 ◽  
Vol 14 (12) ◽  
pp. 1750174
Author(s):  
Hafedh Khalfoun
Keyword(s):  
Vector Fields ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Poisson Superalgebra ◽  
Relative Cohomology ◽  
Cohomology Space ◽  
Cohomology Spaces

Over the [Formula: see text]-dimensional real superspace [Formula: see text], we classify [Formula: see text]-invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the module of [Formula: see text]-densities [Formula: see text] on [Formula: see text], where [Formula: see text] is the Lie superalgebra of contact vector fields on [Formula: see text] and [Formula: see text] is the affine Lie superalgebra. This result allows us to compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the Poisson superalgebra [Formula: see text]. We explicitly give 2-cocycles spanning these cohomology spaces.

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.]

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].

scholarly journals The second cohomology spaces with coefficents in the superspace of weighted densities

2020 ◽  
Vol 72 (10) ◽  
pp. 1323-1334
Author(s):  
O. Basdouri ◽  
A. Braghtha ◽  
S. Hammami
Keyword(s):  
Vector Fields ◽  
Lie Superalgebra ◽  
Cohomology Space ◽  
Cohomology Spaces

UDC 515.1 Over the -dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra of vector fields on the supercircle with coefficients in the space of weighted densities. We explicitly give 2-cocycle spanning these cohomology spaces.  

𝔥-Relative cohomology on S1|1 and deformations

2018 ◽  
Vol 15 (12) ◽  
pp. 1850202
Author(s):  
Thamer Faidi
Keyword(s):  
Central Extension ◽  
Vector Fields ◽  
Central Charge ◽  
Lie Superalgebra ◽  
Relative Cohomology ◽  
Cohomology Space ◽  
Smooth Coefficients ◽  
Orthosymplectic Lie Superalgebra

Over the [Formula: see text]-dimensional supercircle [Formula: see text], we investigate the first [Formula: see text]-relative cohomology space associated with the embedding of the Lie superalgebra [Formula: see text] of contact vector fields in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text] is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra [Formula: see text] instead of [Formula: see text]. We classify [Formula: see text]-trivial deformations of the standard embedding of the Lie superalgebra [Formula: see text] into the Lie superalgebra [Formula: see text]. This approach leads to the deformations of the central charge induced on [Formula: see text] by the canonical central extension of [Formula: see text].

The relative cohomology of the Lie superalgebra of contact vector fields on S1|2

2018 ◽  
Vol 15 (12) ◽  
pp. 1850203
Author(s):  
N. Ben Fraj ◽  
H. Khalfoun ◽  
T. Faidi
Keyword(s):  
Vector Fields ◽  
Lie Superalgebra ◽  
Module Structure ◽  
Relative Cohomology ◽  
Cohomology Space ◽  
Smooth Coefficients ◽  
Orthosymplectic Lie Superalgebra

Over the [Formula: see text]-dimensional supercircle [Formula: see text], we investigate the first [Formula: see text]-relative cohomology space associated with the embedding of the Lie superalgebra [Formula: see text] of contact vector fields in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text] is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra [Formula: see text] instead of [Formula: see text]. We classify generic formal [Formula: see text]-trivial deformations of the [Formula: see text]-module structure on the superspace of the supercommutative algebra [Formula: see text] of pseudodifferential symbols on [Formula: see text].

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.

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.

Sign in / Sign up

Export Citation Format

Share Document