Cohomology of 𝔞𝔣𝔣(m|1) acting on the space of superpseudodifferential operators on the supercircle S1|m

2018 ◽  
Vol 11 (04) ◽  
pp. 1850057
Author(s):  
Hafedh Khalfoun ◽  
Nizar Ben Fraj ◽  
Meher Abdaoui
Keyword(s):  
Lie Superalgebra ◽  
Differential Cohomology ◽  
Formal Deformation ◽  
Cohomology Space ◽  
Smooth Coefficients ◽  
Explicit Expressions

We investigate the first differential cohomology space associated with the embedding of the affine Lie superalgebra [Formula: see text] on the [Formula: see text]-dimensional supercircle [Formula: see text] in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text]. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles. We study the deformations of the structure of the [Formula: see text]-module [Formula: see text]. We prove that any formal deformation is equivalent to its infinitesimal part.

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)].

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].

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.

Cohomology of orthosymplectic Lie superalgebra acting on λ-densities on ℝ1|n

2016 ◽  
Vol 14 (01) ◽  
pp. 1750016
Author(s):  
Nabila El Gomdi ◽  
Rim Messaoud
Keyword(s):  
Lie Superalgebra ◽  
Acta Math ◽  
Differential Cohomology ◽  
Orthosymplectic Superalgebra ◽  
Orthosymplectic Lie Superalgebra

We compute the first differential cohomology of the orthosymplectic Lie superalgebra [Formula: see text] with coefficients in the superspace of weighted densities [Formula: see text] on the (1, 2)-dimensional real superspace. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result by Basdouri and Essayari [On cohomology of the orthosymplectic superalgebra, Acta Math. Hungar. 130(1–2) (2011) 155–166].

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.

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.  

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.

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.

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