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Bechir Dali

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Papers by Bechir Dali

ON KIRILLOV'S LEMMA AND THE BENSON-RATCLIFF INVARIANT

On Kirillov’s lemma and the Benson–Ratcliff invariant, 2021

In this paper we study the conjecture of Benson and Ratcliff, which deals with the class of nilpo... more In this paper we study the conjecture of Benson and Ratcliff, which deals with the class of nilpotent Lie algebras of one-dimensional center. We show that this conjecture is true for any nilpotent Lie algebra g with dim g ≤ 5 but it fails for the dimensions greater or equal to 6. To this end, we produce counterexamples to the Benson-Ratcliff conjecture in all dimensions n ≥ 6. Finally, we show that this conjecture is true for the class of three-step nilpotent Lie algebras and for some other classes of nilpotent Lie algebras.

On Kirillov’s lemma and the Benson–Ratcliff invariant

Journal of Mathematical Physics, May 1, 2021

In this paper, we study the conjecture of Benson and Ratcliff, which deals with the class of nilp... more In this paper, we study the conjecture of Benson and Ratcliff, which deals with the class of nilpotent Lie algebras of a one-dimensional center. We show that this conjecture is true for any nilpotent Lie algebra g with dimg≤5, but it fails for the dimensions greater or equal to 6. To this end, we produce counter-examples to the Benson–Ratcliff conjecture in all dimensions n ≥ 6. Finally, we show that this conjecture is true for the class of three-step nilpotent Lie algebras and for some other classes of nilpotent Lie algebras.

Infinitely small orbits in two-step nilpotent Lie algebras

Journal of Algebra, 2016

Abstract In this paper, we consider the open question: is the cortex of the dual of a nilpotent L... more Abstract In this paper, we consider the open question: is the cortex of the dual of a nilpotent Lie algebra an algebraic set? We give a partial answer by considering the class of two-step nilpotent Lie algebra g . For this class of Lie algebras we give an explicit algorithm for finding its corresponding cortex. By the way, we prove that either the cortex coincides with the zero set of invariant homogeneous polynomials and in this case is z ⊥ where z denotes the center of g , or it is a proper projective algebraic subset of z ⊥ . Finally we materialize our algorithm on examples.

Local convergence of Newton’s method on the Heisenberg group

Journal of Computational and Applied Mathematics, 2016

In the present paper, we study Newton's method on the Heisenberg group for solving the equation f... more In the present paper, we study Newton's method on the Heisenberg group for solving the equation f (x) = 0, where f is a mapping from Heisenberg group to its Lie algebra. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton's method and the estimation of the uniqueness ball of the zero point of f. Some applications to special cases including Kantorovich's condition and γ-condition are provided. The determination of an approximate zero point of an analytic mapping is also presented. Concrete examples are given to illustrate applications of our results.

The spaces Hn(osp(1∣2),M) for some modules M

Journal of Mathematical Physics, Sep 1, 2010

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Déformations polarisées d'algèbres sur les orbites coadjointes des groupes exponentiels

Annales de la Faculté des Sciences de Toulouse, 2000

On considère un groupe de Lie exponentiel G, d'algèbre de Lie g, et une orbite coadjointe 0 d'un ... more On considère un groupe de Lie exponentiel G, d'algèbre de Lie g, et une orbite coadjointe 0 d'un point ~ de g* dual de g, munie d'une carte de Darboux globale qi). On désigne par A l'ensemble des restrictions à 0 des fonctions polynomiales complexes sur g* qu'on identifie à un sous espace d'une complétion ,Â* de et on montre que la complétion de l'ensemble des fonctions quantifiables contient les générateurs pi et qi. Enfin on montre que deux cartes globales (p~,) et q~ de l'orbite C~, associées à une même polarisation ~ définissent la même complétion ,~ de A et deux produits star * et *' équivalents sur ,Â. ABSTRACT.-Let G be an exponential Lie group of Lie algebra g and 0 a coadjoint orbit of ~ in g*,the dual of g,with global Darboux coordinates (pt) We define A as the set of the restrictions on O of complex polynomial functions on g*, which we identify with a subspace of a completion Ã* of (C(pi, q1j , we prove that the complet ion of the subset of quantizable functions on 0 contains the generators pi and Finally we prove that two global coordinates (pi, qz and (p~, qZ), associated to the same polarisation ~ define the same complétion .~4 of A and two équivalents star products * and *'. (*) Reçu le 2 juin 1999, accepté le 21 mars 2000

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Journal of Geometry and Physics, Mar 1, 2010

We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie deriva... more We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols S n δ = n j=0 F δ−j. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H 2 diff (vect(1), D ν,µ) where D ν,µ is the space of differential operators from F ν to F µ. Necessary second-order integrability conditions of any infinitesimal deformations of S n δ are given. We describe completely the formal deformations for some spaces S n δ and we give concrete examples of non trivial deformations.

Rational elimination algorithm and applications

Operators and Matrices, 2021

Given a matrix A ∈ R m×n , we develop an algorithm called the rational elimination algorithm whic... more Given a matrix A ∈ R m×n , we develop an algorithm called the rational elimination algorithm which ressembles the algorithm elimination except that the pivot (the leading coefficient) is a sequence of real independent numbers over Q. This algorithm is used to calculate the rank of A over Q and to seek rational solutions to any linear system Ax = b with b ∈ R m. We also present a criterion for testing the density of additive subgroups of R n which needs the rank of a certain matrix over Q. Finally, we apply such algorithm for testing the regularity of an orbit under the linear continuous action of some subgroup of GL(V) where V is finite dimensional real vector space.

$Research paper thumbnail of Linear family of Lie brackets on the space of matrices $ Mat (n\ times m,\ K) $ and Ado's Theorem$

Linear family of Lie brackets on the space of matrices $ Mat (n\ times m,\ K) $ and Ado's Theorem

Arxiv preprint arXiv:0807.1851, 2008

Abstract: In this paper we classify a linear family of Lie brackets on the space of rectangular m... more

To cite this version

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. One decade of biomedical problems using ICA: a full comparative study

Construction of canonical coordinates for exponential Lie groups

Transactions of the American Mathematical Society, 2009

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through f... more Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Hölder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ ⊂ Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g *. The cross-section mapping σ : Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras E 0 (Ω) and E 1 (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 ,. .. , q d } and {p 1 , p 2 ,. .. , p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by dp k ∧ dq k. The functions {q 1 , q 2 ,. .. , q d } belong to E 0 (Ω), and the functions {p 1 , p 2 ,. .. , p d } belong to E 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 ,. .. , p d , q 1 ,. .. , q d) (restricted to O). Finally, the linear evaluation functions → (X) are shown to be quantizable as well.

Deformation of -modules of symbols

Journal of Geometry and Physics, 2010

We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie deriva... more We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols S n δ = n j=0 F δ−j. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H 2 diff (vect(1), D ν,µ) where D ν,µ is the space of differential operators from F ν to F µ. Necessary second-order integrability conditions of any infinitesimal deformations of S n δ are given. We describe completely the formal deformations for some spaces S n δ and we give concrete examples of non trivial deformations.

The Plancherel Formula for an Inhomogeneous Vector Group

Journal of Fourier Analysis and Applications, 2019

We give a concrete realization of the Plancherel measure for a semi-direct product N H where N an... more We give a concrete realization of the Plancherel measure for a semi-direct product N H where N and H are vector groups for which the linear action of H on N is almost everywhere regular. A procedure using matrix reductions produces explicit (orbital) parameters by which a continuous field of unitary irreducible representations is realized and the almost all of the dual space of N H naturally has the structure of a smooth manifold. Using the simplest possible field of positive semi-invariant operators, the Plancherel measure is obtained via an explicit volume form on a smooth cross-section for almost all H-orbits. The associated trace characters are also shown to be tempered distributions.

The spaces H[sup n](osp(1∣2),M) for some modules M

Journal of Mathematical Physics, 2010

We entirely compute the cohomology for a natural and large class of osp(1|2) modules M . We study... more

To cite this version

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Note on the cortex of some exponential Lie groups

In this paper, we built a family of 4d-dimensional two-step nilpotent Lie algebras (g d) d≥2 so t... more

Deformation of Vect(R)-Modules of Symbols

We consider the action of Vect(R) by Lie derivative on the space Dn λ,µ of symbols of differentia... more We consider the action of Vect(R) by Lie derivative on the space Dn λ,µ of symbols of differential operators. We study deformations of this action. We give integrability conditions of infinitesimal deformations and we exhibit explicit expression of some 2-cocycles generating the second cohomology space H2 diff (Vect(R), Dn λ,µ). Finally, we prove that, in many cases, these integrability conditions are necessary and sufficient and we give examples of non trivial deformations.

THE SPACES H n (osp(1|2),M) FOR SOME WEIGHT MODULES M

Abstract. We entirely compute the cohomology for a natural and large class of osp(1|2) modules M.... more

$Research paper thumbnail of The spaces $\mathrm{H}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$$

The spaces $\mathrm{H}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$

arXiv: Quantum Algebra, 2009

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules... more We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{\lambda,\mu}$ of differential operators on the super circle, acting on densities.

R T ] 1 9 A ug 2 00 9 Deformation of vect ( 1 )-Modules of Symbols

We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie deriva... more We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols Sn δ = ⊕n j=0 Fδ−j. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H diff (vect(1),Dν,μ) where Dν,μ is the space of differential operators from Fν to Fμ. Necessary second-order integrability conditions of any infinitesimal deformations of Sn δ are given. We describe completely the formal deformations for some spaces Sn δ and we give concrete examples of non trivial deformations. Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie. E.mail:basdourimed@yahoo.fr Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie. E.mail: mabrouk.benammar@fss.rnu.tn Département de Mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisie. E.mail: bechir.dali@fss.rnu.tn Département de Mathématiques, Faculté des...