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].

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.

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].

On the 𝔞𝔣𝔣(m|1)-relative cohomology of the orthosymplectic superalgebra 𝔬𝔰𝔭(m|2) and 𝔞𝔣𝔣(m|1)-trivial deformation

2017 ◽  
Vol 14 (02) ◽  
pp. 1750027
Author(s):  
Hafedh Khalfoun ◽  
Thamer Faidi
Keyword(s):  
Lie Superalgebra ◽  
Lie Derivative ◽  
Relative Cohomology ◽  
Orthosymplectic Superalgebra ◽  
Orthosymplectic Lie Superalgebra ◽  
Cohomology Spaces

Over the [Formula: see text]-dimensional supercircle [Formula: see text], we consider the action of the orthosymplectic Lie superalgebra [Formula: see text], by the Lie derivative on the superpseudodifferential operators [Formula: see text]. We compute the [Formula: see text]-relative cohomology spaces [Formula: see text], where [Formula: see text] is the affine Lie superalgebra on [Formula: see text]. We explicitly give cocycles spanning these cohomology spaces. We study the [Formula: see text]-trivial deformations of the structure of the [Formula: see text]-modules [Formula: see text].

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(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.

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