Cohomology and deformation of 𝔞𝔣𝔣(1|1) acting on differential operators

2018 ◽  
Vol 15 (05) ◽  
pp. 1850072
Author(s):  
Khaled Basdouri ◽  
Salem Omri
Keyword(s):  
Differential Operators ◽  
Lie Superalgebra ◽  
Module Structure ◽  
Infinitesimal Deformation ◽  
Differential Cohomology ◽  
Linear Differential Operators ◽  
Formal Deformation ◽  
Pseudo 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 second differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We classify formal deformations of the [Formula: see text]-module structure on the superspaces of symbols of differential operators. We prove that any formal deformation of a given infinitesimal deformation of this structure is equivalent to its infinitesimal part. This work is the simplest superization of a result by Basdouri [Deformation of [Formula: see text]-modules of pseudo-differential operators and symbols, J. Pseudo-differ. Oper. Appl. 7(2) (2016) 157–179] and application of work by Basdouri et al. [First cohomology of [Formula: see text] and [Formula: see text] acting on linear differential operators, Int. J. Geom. Methods Mod. Phys. 13(1) (2016)].

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.

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.

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].

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.]

Cohomology of 𝔞𝔣𝔣(2) acting on some modules and deformations

2017 ◽  
Vol 14 (12) ◽  
pp. 1750180
Author(s):  
Nizar Ben Fraj ◽  
Ismail Laraiedh ◽  
M. Abdaoui
Keyword(s):  
Large Class ◽  
Differential Operators ◽  
Lie Superalgebra ◽  
Natural Action ◽  
Linear Differential Operators ◽  
Direct Sum ◽  
Cohomology Space ◽  
Linear Differential

Over the [Formula: see text]-dimensional real superspace, we compute the cohomology space of the affine Lie superalgebra [Formula: see text] with coefficient in a large class of [Formula: see text]-modules [Formula: see text]. We apply our results to the module [Formula: see text] of weight densities and the module [Formula: see text] of linear differential operators acting on a superspace of weighted densities. We study nontrivial deformations of the natural action of the Lie superalgebra [Formula: see text] on the direct sum of the superspaces of weighted densities.

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