Agrebaoui Boujemaa

University of Sfax | US · Department of Mathematics

The diamond cone is a combinatorial description for a basis of a natural indecom-posable n-module, where n is the nilpotent factor of a complex semisimple Lie algebra g. After N. J. Wildberger who introduced this notion, this description was achieved for g = sl(n) , the rank 2 semisimple Lie algebras and g = sp(2n). In this work, we generalize thes...

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor n of a complex semi-simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achieved for sl}(n), the rank 2 semi-simple Lie algebras and sp(2n). In this work, we generalize these constructions to the L...

The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss the representations structure of simple multiplicative Hom-Lie algebras. Moreover, we study weight modules and r...

The relative cohomology Hdiff1(K(1|3), osp(2, 3);Dγ,µ(S1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators Dγ,µ(S1|3) acting on tensor densities on S1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian...

In the present paper, we define the diamond cone for the Lie superalgebra spo(2m,1), considering the (covariant) tensor representation of spo(2m,1). The diamond cone is no more indecomposable. Nevertheless, we give a basis for each indecomposable component, using quasistandard Young tableaux for spo(2m,1). We realize a bijection between the set of...

We investigate the first cohomology space associated with the embedding of the Lie Orthosymplectic superalgebra \({\mathfrak {osp}}(3|2)\) on the (1,3)-dimensional superspace \({\mathbb {R}}^{1|3}\) in the Lie superalgebra \({\mathcal {S}}\Psi {\mathcal {DO}}(3)\) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and...

In this paper, we recall combinatorial basis for shape and reduced shape algebras of the Lie algebras gl(n)gl(n), sp(2n)sp(2n) and so(2n+1)so(2n+1). They are given by semistandard and quasistandard tableaux. Then we generalize these constructions to the case of the Lie superalgebra spo(2n,2m+1)spo(2n,2m+1). The main tool is an extension of the Schü...

We give an explicit construction of the 11-cocycles of the group of contactomorphisms on the supercircle S1|mS1|m for m=1,2, with coefficients in the space of differential operators acting on S1|mS1|m-tensor densities. We show that they satisfy properties similar to those of the super-Schwarzian derivative.

In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableau...

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank 2 semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we gen...

We classify the nontrivial deformations of the standard embedding of the Lie superalgebra K(2) of contact vector fields on the (1,2)-dimensional supercircle into the Lie superalgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(2) by the canonical central extensi...

The main result of this article is the explicit calculation of the first cohomology space H 1((3), Ψ(S 1|3)) of the Lie superalgebra (3) of contact vector fields on the supercircle S 1|3 with coefficients in the module of superpseudodifferential operators Ψ(S 1|3). For the supercicles of dimensional 1 | 0, 1 | 1, and 1 | 2, the first cohomology spa...

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\mathfrak g$. Such a description is...

We investigate the first cohomology space associated with the embedding of the Lie superalgebra $\cK(2)$ of contact vector fields on the (1,2)-dimensional supercircle $S^{1\mid 2}$ in the Lie superalgebra $\cS\Psi \cD \cO(S^{1\mid 2})$ of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we show that this spa...

We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie superalgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(1) by the canonical central extension o...

We study non-trivial deformations of the natural action of the Lie algebra $\mathrm{Vect}({\mathbb R}^n)$ on the space of differential forms on ${\mathbb R}^n$. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in th...

The space of differential operators acting on skewsymmetric tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and show how they are related to the cohomology with coefficients in ther space of smooth functions of the manifold...

The space of symbols of differential operators on a smooth manifold (i.e., the space of symmetric contravariant tensor fields) is naturally a module over the Lie algebra of vector fields. We study, in the case of $\bf R^n$ with $n\geq2$, multi-parameter formal deformations of this module. The space of linear differential operators on $\bf R^n$ prov...

In this paper we give a proof of the following statement: “Every irreducible integrable representation of level l > 0 of affine Kac-Moody algebra occurs in the tensor product of l highest weight modules of level 1.” The techniques of the proof use a result due to Kac and Wakimoto (1986) which is a particular case of the Parthasaraty-Ranga Rao-Varad...

Thèse (de doctorat)--Université Louis-Pasteur (Strasbourg I), 1995.