Géométrie différentielle et algèbres non associatives Ecole de recherche CIMPA-Maroc Marrakech, Avril 2015 Programme Première semaine

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1 Cadi-Ayyad University FST Gueliz Mathematics department Request letter for a conference support With my colleague Said Benayadi, we are organizing a CIMPA research school titled "Differential geometry and non associative algebras". This school will hold at the University Cadi-Ayyad in Marrakesh Morocco from 13 to 24 Avril The aim of this school is to give Ph. D. students many introductory courses to some topics where non-associative algebras are used to solve geometric problems. This school is mainly supported by the International Center for Pure and Applied Mathematics. However, we received about forty demands from students With different nationalities: Algeria, Benin, Niger, Cameroun, Palestine and Morocco. With the financial support we have succeeded to have until now we cannot give to the all of them the opportunity to attend the school. A support from the IMU of an amount of 3000 Euros would permit us to do so. Thanks in advance. Mohamed Boucetta Géométrie différentielle et algèbres non associatives Ecole de recherche CIMPA-Maroc Marrakech, Avril 2015 Programme Première semaine Lundi 13 Avril 08h30-09h00: Accueil 09h00-09h45 : Séance d'ouverture 09h45-10h30 : Pause café 10h30-11h30 : Ignacio Bajo (Université de Vigo, Espagne), Lie groups and Lie algebras(i) 11h45-12h45 : Camille Laurent-Gengoux (Université de Lorraine-Metz, France), Introduction à la géométrie riemannienne et à la géométrie symplectique(i) Pause déjeuner 15h30-16h30 : Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(i) 16h30-17h00 : Pause café 17h00-18h00 : Martin Bordemann (Université de Mulhouse, France) Classes d'atiyah en Algèbre et en géométrie(i)

2 18h15-19h05 : Helena Albuquerque (Titre à préciser) Mardi 14 Avril 09h00-10h00 : Martin Bordemann (Université de Mulhouse, France) Classes d'atiyah en Algèbre et en géométrie(ii) 10h00-10h30 : Pause café 10h30-11h30 : Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(ii) 11h45-12h35 : Vayachlef Futony (Titre à préciser) Pause déjeuner 15h30-16h30 : Camille Laurent-Gengoux (Université de Lorraine-Metz, France), Introduction à la géométrie riemannienne et à la géométrie symplectique(ii) 16h30-17h00 : Pause café 17h00-18h00 : Ignacio Bajo (Université de Vigo, Espagne), Lie groups and Lie algebras(ii) 18h15-18h45 : Mohamed Karmouni (Titre à préciser) Mercredi 15 Avril 09h00-10h00 : Ignacio Bajo (Université de Vigo, Espagne), Lie groups and Lie algebras(iii) 10h00-10h30 : Pause café 10h30-11h30 : Camille Laurent-Gengoux (Université de Lorraine-Metz, France), Introduction à la géométrie riemannienne et à la géométrie symplectique(iii) 11h45-12h35 : Abdenacer Makhlouf (Titre à préciser) Jeudi 1 6 Avril 09h00-10h00 : Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(iii) 10h00-10h30 : Pause café 10h30-11h30 : Martin Bordemann (Université de Mulhouse, France) Classes d'atiyah en Algèbre et en géométrie(iii)

3 11h45-12h35 : Ignacio Bajo (Titre à préciser) Pause déjeuner 15h30-16h30 : Ignacio Bajo (Université de Vigo, Espagne), Lie groups and Lie algebras(iv) 16h30-17h00 : Pause café 17h00-18h00 : Camille Laurent-Gengoux (Université de Lorraine-Metz, France), Introduction à la géométrie riemannienne et à la géométrie symplectique(iv) 18h15-18h45 : Abdelouahab Chikh Salah (Titre à préciser) Vendredi 1 7 Avril 09h00-10h00 : Camille Laurent-Gengoux (Université de Lorraine-Metz, France), Introduction à la géométrie riemannienne et à la géométrie symplectique(v) 10h00-10h30 : Pause café 10h30-11h30 : Ignacio Bajo (Université de Vigo, Espagne), Lie groups and Lie algebras(v) 11h45-12h35 : Saïd Benayadi (Titre à préciser) Pause déjeuner 15h30-16h30 : Martin Bordemann (Université de Mulhouse, France) Classes d'atiyah en Algèbre et en géométrie(iv) 16h30-17h00 : Pause café 17h00-18h00 : Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(iv) 18h15-18h45 : Saïd Boulmane(à préciser) Deuxième semaine Lundi 20 Avril 09h00-10h00: Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(v) 10h00-10h30 : Pause café

4 10h30-11h30 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (I) 11h45-12h35 : Cristina Draper (Titre à préciser) Pause déjeuner 15h30-16h30 : Damien Calaque (Université de Montpellier II, France), Connexions, classes d'atiyah et théorème de PBW pour les algébroïdes de Lie(I) 16h30-17h00 : Pause café 17h00-18h00 : Mohamed Boucetta (Université Cadi Ayyad, Marrakech, Maroc), Courbure des métriques invariantes à gauche sur les groupes de Lie(I) 18h15-18h45 : Abdelhadi Mouttassim (Titre à préciser) Mardi 21 Avril 09h00-10h00 : Mohamed Boucetta (Université Cadi Ayyad, Marrakech, Maroc), Courbure des métriques invariantes à gauche sur les groupes de Lie(II) 10h00-10h30 : Pause café 10h30-11h30 : Damien Calaque (Université de Montpellier II, France), Connexions, classes d'atiyah et théorème de PBW pour les algébroïdes de Lie(II) 11h45-12h35 : Hicham Lebzioui (Titre à préciser) Pause déjeuner 15h30-16h30 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (II) 16h30-17h00 : Pause café 17h00-18h00 : Alberto Elduque (Université de Zaragoza, Espagne), Espaces homogènes et algèbres non associatives(vi) 18h15-18h45 : Samir Bekkara (Titre à préciser) Mercredi 22 Avril 09h00-10h00 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (III)

5 10h00-10h30 : Pause café 10h30-11h30 : Damien Calaque (Université de Montpellier II, France), Connexions, classes d'atiyah et théorème de PBW pour les algébroïdes de Lie(III) 11h45-12h35 : Zouhair Saassai /Mohamed Boucetta (Titre à préciser) Jeudi 23 Avril 09h00-10h00 : Mohamed Boucetta (Université Cadi Ayyad, Marrakech, Maroc), Courbure des métriques invariantes à gauche sur les groupes de Lie(III) 10h00-10h30 : Pause café 10h30-11h30 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (IV) 11h45-12h45 : Cmille Laurent-Gengoux (Titre à préciser) Pause déjeuner 15h30-16h30 : Damien Calaque (Université de Montpellier II, France), Connexions, classes d'atiyah et théorème de PBW pour les algébroïdes de Lie(IV) 16h30-17h00 : Pause café 17h00-18h00 : Mohamed Boucetta (Université Cadi Ayyad, Marrakech, Maroc), Courbure des métriques invariantes à gauche sur les groupes de Lie(IV) 18h15-18h45 : Benroumamane (Titre à préciser) Vendredi 24 Avril 09h00-10h00 : Mohamed Boucetta (Université Cadi Ayyad, Marrakech, Maroc), Courbure des métriques invariantes à gauche sur les groupes de Lie(V) 10h00-10h30 : Pause café 10h30-11h30 : Damien Calaque (Université de Montpellier II, France), Connexions, classes d'atiyah et théorème de PBW pour les algébroïdes de Lie(V) 11h45-12h35 : Seddik Oukkas (Titre à préciser) Pause déjeuner

6 15h30-16h30 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (V) 16h30-17h00 : Pause café 17h00-18h00 : Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic), Geometry of homogeneous manifolds (VI) 18h15-18h45 : Une communication (à préciser) Samedi 18/04/2014 : Visite de la ville Dimanche 19/04/2014 : Excursion 1. Contact details for the organizer of the conference: Last Name: Boucetta First Name: Mohamed Department: Mathematics Adress: Cadi-Ayyad University, Facult_e des Sciences et Techniques Gueliz, BP 549 City; Marrakesh, Country: Morocco Telephone no: Telafax no: m.boucetta@uca.ma 2. Contact details for the co-organizer of the conference: Last Name: Benayadi First Name: Said Department: Mathematics Adress: Lorraine University, LMAM, Ile du Saulcy, METZ City; Metz, Country: France Telephone no: Telafax no: said.benayadi@univ-lorraine.fr 3. Application details: Conference name: Research cimpa school " Differential geometry and non associative algebras". Conference start date: Conference end date: Amount requested: 3000 Euros 4. Scientific Programme The main purpose of this school is to give to PhD students and to young researchers some introductory courses on subjects using at the same time differential geometry and non-associative algebras. The program of this school will contain conferences of researchers in these domains with the aim of showing to the participants some current subjects of researches in the themes of this school. In particular, the contents of the courses and the conferences shall show the narrow links between problems of the differential geometry and the non-associative algebras. 4.1 List of invited speakers: 4.1 List of invited speakers:

7 Name of institution No. Title of lectures contacted accepted of Hours D. V. Alekseevsky 6 Geometry of homogeneous yes yes University of Brno, manifolds Czech Republic M. Bordemann 4h30 Atiyah Classes in algebra yes yes University of Mulhouse and geometry France M. Boucetta 5 Curvature of left invariant yes yes Univ. Cadi-Ayyad Morocco Riemannian metrics on Lie groups D. Calaque 4h30 Atiyah classes in algebra yes yes Univ. Montpellier II and geometry (France) A. Elduque 6 Homogeneous spaces yes yes University Of Zaragoza and non associative algebras (Spain) C. Laurent-Gengoux 5 Introduction to Riemannian yes yes Univ. of Lorraine and symplectic geometries France Ingacio Bajo 5 Lie groups and Lie algebras yes yes Univ. of Vigo (Spain) 4.3. Outline of the scienti_c program: Dmitri Alekseevsky (Masaryk University, Brno, Czec Republic) Title: Geometry of homogeneous manifolds Plan of the mini-course: I. Lie groups, Lie algebras and Lie correspondence between Lie groups and Lie algebras This paragraph aims at _xing the notations for the continuation of this minicourse and at considering some results which will be not, possibly, seen in the mini-course of Angela Pasquale. I.1. Lie algebra of a Lie group. Examples : classical groups, linear groups, parabolic subgroups. I.2. Linear representation and linear groups. Reconstruction of a linear Lie group from its Lie algebra I.3. Ado theorem. Locally isomorphic Lie groups. Lie correspondence between Lie algebras and Lie groups II. G-manifolds and homogeneous manifolds II.1. Invariants of a G-manifold as the main object of geometry according to Erlangen Program. Examples : Euclidean,spherical, conformal, projective, Grassman geometries II.2. Homogeneous manifolds as coset spaces. Isotropy representation and invariant tensor _elds II.3. In_nitezimal description of homogeneous Riemannian, symplectic, complex and Kahler geometries III. Semisimple Lie algebras and parabolic subalgebras III.1. Root decomposition of a complex semisimple Lie algebra g and its reconstruction from the root system R III.2. Simple root system _ and Dynkin diagrams. Reconstruction of R from _ III.3. Compact semisimple Lie algebras III.4. Parabolic subalgebras and painted Dynkin diagrams IV. Geometry of compact ag manifolds IV.1. Invariant symplectic structures on ag manifolds. Flag manifolds as homogeneous symplectic manifolds of a compact semisimple Lie group IV.2.Invariant complex structures on ag manifolds

8 IV.3 Invariant Kahler and Kahler-Einstein structures on ag manifolds Martin Bordemann & Damien Calaque Title: Atiyah classes in Algebra and Geometry Part 1 : Invariant connections for homogeneous spaces, multidi_erential operators and associated Atiyah classes, by Martin Bordemann. Abstract. We recall _rst the theory of G-equivariant _bered manifolds over an arbitrary homogeneous space G=H (which are all associated bundles). We then pass to (invariant) connections in principal bundles over G=H (H.-C.Wang, 1958) and its obstructions in terms of Atiyah classes (Nguyen- Van Hai, 1965) with some examples and the evident translation to arbitrary (graded) Lie algebra pairs g _ h. Finally we give a description of multidi_erential operators on G=H _a la Alekseev-Lakhovska, 2005, in terms of quotients of envelopping algebras, and deduce symbol calculus and star-products on G=H. Part 2 : Connections, Atiyah classes and PBW theorems for Lie algebroids, by Damien Calaque. Abstract.Motivated by the _rst part of the mini-lecture and by some examples coming from foliation theory we introduce connections and Atiyah classes for Lie algebroid after having recalled some essentials of the theory of Lie groupoids and Lie algebroids. We shall give an interpretation of the Atiyah classes in terms of obstructions to existence of generalized Poincar_e- Birkho_-Witt isomorphisms. Mohamed Boucetta Title: Curvature of left invariant Riemannian metrics on Lie groups Abstract. This course aims to introduce students to the properties of curvature of left invariant Riemannian metrics on Lie groups as they have been developed in the classic paper of Milnor: J. Milnor, Curvatures of left invariant metrics on Lie Groups, Advances in Mathematics 21, (1976), The course is divided in three parts, each one is devoted to the study of left invariant Riemannian metrics on Lie groups having negative, positive of zero curvature. We distinguish three types of curvatures, namely, sectional curvature, Ricci curvature or scalar curvature. At the end of the course, we give some recent results to show to students that this topic generates a massiveresearch activity among the mathematical community. Alberto Elduque (University of Zaragoza, Spain) Title: Homogeneous spaces and nonassociative algebras Abstract: Many geometric properties of a reductive homogeneous space are encoded in a binary and a ternary products de_ned on the tangent space at any point. In particular, a classical Theorem by Nomizu relates the invariant a_ne connections on these spaces with nonassociative algebras de_ned on this tangent space. The aim of this course is the presentation of the basic concepts of reductive homogeneous spaces, the associated Lie-Yamaguti algebras, and a review of Nomizu's Theorem and related nonassociative algebras. Content: _ Reductive homogeneous spaces and Lie-Yamaguti algebras. _ Invariant a_ne connections and nonassociative algebras. _ Examples. Camille Laurent-Gengoux Title: Introduction to Riemannian Geometry and Symplectic Geometry Abstract: We shall teach the basics of riemanian and symplectic geometry. To start with, we shall recall several points about manifolds, introduced as being objects locally isomorphic to Rn, then we shall introduce symplectic manifolds as being objects locally isomorphic to R2n equipped with a nondegennerated bivector _eld. We prefer the approach by bivector _elds, rather

9 than by 2-forms, that makes possible to move quickly towards action-angle variables and Liouville theorem. We shall speak in particular of Poisson structure, a straightforward generalization, and shall explain in which context they are more natural. We shall then show the di_erence with riemannian manifolds, whose classi_cation is way more problematic. We shall then compare natural connections on both types of manifolds. Ignacio Bajo Title: Lie groups and Lie algebras Abstract: A Lie group is a di_erential manifold G endowed with a group structure so that the applications of multiplication and inversion are smooth. The tangent space at the unit element of G admits then the structure of a Lie algebra. The objective of these lectures is to give an introduction to Lie groups and Lie algebras as well as their interactions. We shall mostly consider the case where G is a linear Lie group, that is a closed subgroup of the group GL(n;R) of n _ n invertible matrices with real coe_cients. In this situation,the Lie algebra of G is a vector subspace of the space of n _ n matrices with real coe_cients. It has the property that it contains the commutator AB BA of every couple of its elements A and B. This Lie algebra is linked to the Lie group G by the exponential of matrices. We shall also present the notions of nilpotent, solvable and semi-simple Lie algebras (respecively, Lie groups), as well as the theorems of Engel, Lie and Cartan providing some criteria for a given Lie algebra to possess these properties. Program: _ The group GL(n;R); linear Lie groups. _ A short revision of di_erentiable manifolds. Abstract Lie groups. _ Abstract Lie algebras. Examples. _ The exponential map, one-parameter subgroups. _ The Lie algebra of a Lie group. _ The adjoint representation. _ The Killing form. _ Nilpotent and solvable Lie algebras. _ Semisimple Lie algebras. 5. Budget: List of supporting organizations and the support they are providing: International Centre of Pure and Applied Mathematic: Euros Lorraine University: 1500 Euros Cadi-Ayyad University: 1500 Euros Hassan II Academy of sciences and technology: 3500 Euros Number of participant: 61 Accommodation: Euros Travel: 6500 Euros Total conference budget: Euros Amount requested from CDC: 3000 Euros Statement on the intended use of this amount: To cover the cost of accommodation for some participants coming from developing countries. Indeed there is 18 participants coming form Algeria, Palestine, Benin, Niger, Cameroun. I organized a cimpa school in 2008 and received a support from the IMU.

10 Curriculum Vitae Personal Data Name: Saïd BENAYADI Date and place of birth: October 15, 1965 in Marrakech (Morocco) Nationality: French and Moroccan Married, one children (Younès) Adress: 41, Rue Saint André, F-5700 Metz, France Phone: +33(0) Academic Position: Maître de Conférences( Hors classe, Habilité à diriger des recherches), University of Lorraine-Metz. Professional Data Affiliation: Laboratoire IECL, CNRS UMR 7502, Université de Lorraine-Metz, Ile du Saulcy, F Metz cedex 01, France. Phone: +33(0) Fax: +33(0) Higher Education : Graduate studies-université de Bourgogne(Dijon, France). Phd: «Etude d'une famille d'algèbres de Lie généralisant les algèbres de Lie semi-simples». 1998: Thèse d'état, Spécialité: Mathématiques. Sujet: «Structures des (Super-)Algèbres de Lie quadratiques», University Cadi Ayyad (Marrakech, Marocco) 2003: Habilitation: «Superalgèbres de Lie quadratiques: structures et classifications inductives». Research area Lie (super-)algebras and Lie(super-)groups, Jordan (super-)algebras and (super-)symmetric spaces, Lie (super-)bialgbras, Hopf(super-)algebras, Poisson(super-)algebras, Geometries of homogeneous spaces.

11 Publications 1. S. Benayadi and A. Elduque, Classification Of quadratic Lie algebras of low dimension, J. Math. Phys. 55, (2014). 2. S. Benayadi and M. Boucetta, Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures, Differential Geometry and Applications 36 (2014), pages S. Benayadi and S. Hidri, Quadratic Leibniz Algebras, Journal of Lie Theory, volume 24, Number 3, 2014, pages H. Albuquerque, E. Barreiro and S. Benayadi, Homogeneous Symmetric Antiassociative Quasialgebras, Communications in Algebra, volume 42, Issue 7, 2014, pages S. Benayadi and A. Makhlouf, Hom-Lie algebras with symmetric invariant bilinear forms, Journal of Geometry and Physics, volume 76, issue2, 2014, pages I. Ayadi, H. Benamor and S. Benayadi, Lie superalgebras with some homogeneous structure s, Journal of Algebra and its Applications, Vol. 11, N.5(2012) (30 pages). 7.Ayadi and S. Benayadi, Associative superalgebras with homogeneous symmetric structures, Communications in Algebra 40(2012), N.4, A. Baklouti and S. Benayadi, Symmetric Symplectic Commutative Associative Algebras and Related Lie Algebras, Algebras Colloquium 18(Spec 1)(2011), pages I. Bajo and S. Benayadi, Abelian para-kähler structures on Lie algebras, Differential Geometry and its Applications 29(2011), pages S. Benayadi and M. Chopp, Lie-admissible structures on Witt type algebras, Journal of Geometry and Physics, volume 61, issue2, 2011, pages H. Albuquerque, E. Barreiro and S. Benayadi, Quadratic Malcev superalgebras with reductive even part, Communications in Algebra, volume 38, Issue 2, 2010, pages I. Ayadi and S. Benayadi, Symmetric Novikov superalgebras, J. Math. Phys (2010), ( )--( ). 13. H. Albuquerque, E. Barreiro and S. Benayadi, Odd-Quadratic Lie superalgebras, Journal of Geometry and Physics 60(2010) H. Albuquerque, E. Barreiro and S. Benayadi, Quadratic Lie superalgebras with reductive even part, Journal of pure and applied algebra, Volume 321, Issue 2, (2009), H. Albuquerque, E. Barreiro and S. Benayadi, Quadratic Lie superalgebras with reductive even part, Journal of pure and applied algebra, Volume 321, Issue 2, (2009), E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bisuperalgebras, Journal of Algebra, Volume 321, Issue 2, (2009), I. Bajo, S. Benayadi and A. Medina, Symplectic structures on quadratic Lie algebras, Journal

12 of Algebra, Volume 316, Issue 1, (2007), I. Bajo and S. Benayadi, Quadratic Lie algebras of quadratic dimension equal to $2$, Journal of pure and applied algebra, 209(2007), H. Albuquerque and S. Benayadi, Quadratic Malcev superalgebras, Journal of pure and applied algebra, 187(2004), S. Benayadi, Socle and some invariants of quadratic Lie superalgebras, Journal of Algebra,Volume 261, 2003, pages H. Benamor and S. Benayadi, Fitting decomposition of Casimir operators of quadratic Lie superalgebras, Journal of Lie Theory, Volume 10, 2000, pages S. Benayadi, The root space decomposition of quadratic Lie superalgebras, Beitre zur Algebra und Geometrie, Volume 41, 2000, pages S. Benayadi, Quadratic Lie superalgebras with the completely reducible action of even part on the odd part, Journal of Algebra,Volume 223, 2000, pages S. Benayadi, Inductive classification of quadratic Lie superalgebras, Research and Exposition in Mathematics, Volume 25, 2002, H. Benamor and S. Benayadi, Relation between invertibility of Casimir operators and semisimplicity of quadratic Lie superalgebras, Journal of Lie Theory, Volume 9, 1999, pages H. Benamor and S. Benayadi, Double extension of quadratic Lie superalgebras,communications in Algebra, 27 (1), 1999, pages M. Ameur, S. Benayadi, C. Bennis et A. Roux, Bicommutativity for a class of graded connected Hopf algebras, Journal of pure and applied algebra, Volume 135, 1999, Pages S. Benayadi, Characterization of quadratic Lie algebras with nilpotent Casimir operator, Algebras, groups and geometries, Volume 16, 1999, pages I. Bajo and S. Benayadi, Lie algebras admitting a unique quadratic structure, Communications in Algebra, 25 (9), 1997, pages S. Benayadi, A new characterization of semisimple Lie algebras, Proceedings of the American Mathematical Society, Volume 125, Number 3, March 1997, Pages S. Benayadi, Sur une famille d' algèbres de Lie idéalement finies, Ann. Math. BlaisePascal, Vol. 3, numéro 2, 1996, pages M. Ameur et S. Benayadi, Sur les invariants d'un module sur une algèbre de Hopf, Communications in Algebra, 24(10), 1996, pages M. Ameur, S. Benayadi, C. Bennis et A. Roux, Sur les algèbres de Hopf graduées connexes, C.R. Acad. Sci. Paris, t. 322, 1996, pages

13 34. S. Benayadi, Structure of perfect Lie algebras without center and outer derivations, Ann. Fac. Sci. Toulouse, Vol. V, number 2, 1996, pages S. Benayadi, Structures de certaines algèbres de Lie quadratiques, Communications in Algebra, 23 (10), 1995, pages S. Benayadi, Une propriété nécessaire et suffisante pour qu'une algèbre de Lie sympathique quadratique soit semi-simple, C.R. Acad. Sci. Paris, t. 319, 1994, pages E. Angelopoulos and S. Benayadi, Construction d'algèbres de Lie sympathiques non semisimples munies de produits scalaires invariants, C.R. Acad. Sci. Paris, t. 317, 1993, pages D. Arnal, H. Benamor, S. Benayadi, G. Pinczon, Une algèbre de Lie non semi-simple vérifiant :H^1(g) = H^2(g) = H^0(g,g) = H^1(g,g) = H^2(g,g) = { 0 }, C.R. Acad. Sci. Paris, t. 315, 1992, pages S. Benayadi, Certaines propriétés d'une classe d'algèbres de Lie qui généralisent les algèbres de Lie semi-simples, Ann. Fac. Sci. Toulouse, Vol. XII, numéro 1, 1991, pages Preprints 1. S. Benayadi and M. Boucetta, On para-kähler and hyper-para-kähler Lie algebras; arxiv: (to appear in Journal of Algebra). 2. A. Baklouti and S. Benayadi, Pseudo-euclidean Jordan algebras; arxiv: (to appear in Communications in Algebra). 3. I. Bajo, S. Benayadi and M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras; arxiv: Recent talks in Conferences and colloquia 2014 : Journées de Géométrie et d'algèbres non-associatives, Université My Isamail, Meknès, Morocco : Workshop On Hom-Algebras, Monastir, Tunisia, october : Séminaire Algèbre du département de Mathématiques de l'université de Coimbra (Portugal). 2013: Séminaire Georges Reeb, université de Haute Alsace-Mulhouse 2012: Euro-aghreb Conference on Algebra, Geometry and Lie Theory, Sousse, Tunisia, December, Colloque à la mémoire de Georges Pinczon, '''Algèbres et super-algèbres de Lie et quantification par déformations '', Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, Octobre Symmetries in Differential Geometry and Mathematical Physics, Université of Luxembourg, 3-6 september 2012.

14 2012: Séminaire de Physique Mathématique du Laboratoire de Mathématiques Painlevé de l'université de Lille. 2011: Séminaire Georges Reeb, université de Haute Alsace-Mulhouse 2011: ''Algebra Geometry Mathematical Physics'', University of Haute Alsace-Mulouse, october : '' Non-associative algebras and related topics'' University of Coimbra (Portugal), July 2010: Euro-Maghreb conference on Geometry and Lie theory, Monastir (Tunisia) November 1-5, : Colloque: Sixième rencontres Alsaciennes de Mathématiques et Physique (26-28 novembre 2009 à l'iphc, Strasbourg. 2009:Séminaire de département de Mathématiques de l'université De Vigo (Espagne). 2009: Rencontre Euro-Maghreb d'algèbre, Théorie des nombres et calcul Formel à l'université Cadi Ayyad de Marrakech (Maroc). Ongoing Phd projects Said Boulmane, Algèbres Alternatives et Algèbres de Malcev-Poisson (with Malika Aït Benhaddou, University Molay Isamail, Meknès (Morocco). Samiha Hidri: Etude de certaines systèmes triples (With A. Baklouti, University of Sfax, Tunisia). Complete dissertations 1. A. Baklouti: Structures et descriptions inductives des algèbres de Jordan Pseudo-euclidiennes (Metz, 2007). 2. M.E. Barreiro Carvalho: Quadratic Lie superalgebras and Lie bi-superalgebras (with Helena Albuquerque, Universidade de Coimbra, Portugal) (Coimbra, 2007). 3. Imen Ayadi: Super-algèbres non associatives avec des structures homogènes (Metz, 2011). 4. Mikaël Chopp: Lie-admissible structures on Witt type algebras and automorphic algebras (with M. Schlichenmaier, University of Luxemboug)(Metz 2011). Recent Invitation at Research Institutes (At least one week) 2013:University of Coimbra.(Portugal). 2010: University of Marrakech (Morocco); University of Vigo (Spain). 2009: University of Vigo(Spain). Recent Organisation of Conference and Colloquia Septembre 2011: Scientific days: '' Non associative Superalgebras and applications'' University

15 Paul Verlaine-Metz. 2009: Colloquium on super-geometry (With Angela Pasquale), Unijversity of Metz. 2009: Member of the organizing committee of the colloquium in Algebra..., University of Marrakech (Morroco) 2007: Organizer (with I. Bajo from university of Vigo) of the special session Lie algebras and their applications in Geometry ein the «First French-Spanish Congress of Mathematics», Zaragoza(Spain), July : Days of Algebra and Geometry, University of Metz.

16 CURRICULUM VITAE Family Name: BOUCETTA Personal Name: Mohamed Sex: M Date of Birth: 16 June 1961 Present Nationality: Morocco Nationality at Birth: Morocco Place of Birth: Marrakech/Morocco Marital Status: Married Number of Children: 2 Full Name and Address of Permanent Institute: Faculté des Siences et Techniques Marrakech BP 549, Marrakech Morocco. Fax: Phone: Home Address: B7, Résidence Marjana I, Koudiat Laabid Marrakech Morocco. Phone: Mailing Address: Permanent: m.boucetta@uca..ma EDUCATION Name of Institute Date attended Qualification Obtained University Montpellier II Ph. D University Montpellier II June 1984 Master 1

17 Invited main lecturer at international conference Date Name Place October Conference Euro-Maghrebine Geometry Nice May Conference Egypte France Cairo May 19 to 30, 2008 CIMPA School Marrakech May 28-June 02, 2007 Troisième Colloque Maghrébin Marrakech Géométrie, Toplogie et système Dynamiques May 8 to 11, 2006 Deuxième Colloque Maghrébin Bizerte Tunisie Géométrie, Toplogie et système Dynamiques July 2005 Summer School and conference ICTP February 20 to March 3, 2005 School CIMPA El Oued Algeria, May 19 to 30, 2004 CIMPA School Marrakech June 4-9, 2001 Workshop on Seiberg-Witten invariants Marrakech Scientific employment and academic responsability Name of Institute Period of Service Duties Université Montpellier II September 1987 to September 1992 Post doctoral fellowship Present Position Position: Permanent Professor at Faculté des Sciences et Techniques Marrakech Starting date: September 1992 I have been a member of the committee of the institute from December 1993 to December Also I have been a Head of the Department of Mathematics from January 1999 to December Now I am Director of Geometry, Topology and Applications Equip. Field of Interest 30. Mathematics 34. Differential Geometry 2

18 Publications. [1] M. Boucetta, Courbure sectionnelle et intégrales premières symétriques du flot géodésique... C. R. Acad. Sci. Paris, t. 326, Série I, pp , (1998). [2] M. Boucetta, Sur la variation des métriques à géodésiques périodiques, Ann. Sci. Math. Québec 22, no.1, pp , (1998). [3] M. Boucetta, Métriques de Lorentz-Zoll sur la surface de de Sitter, Proyecciones Vol. 17, No 1, pp (1998). [4] M. Boucetta, Spectre des Laplaciens de Lichnerowicz sur les sphères et les projectifs réels, Publicacions Matemàtiques, Vol 43, pp (1999). [5] M. Boucetta, Spectre des Laplaciens de Lichnerowicz sur les projectifs complexes, C. R. Acad. Sci. Paris, t. 333, pp , (2001). [6] M. Boucetta, Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C. R. Acad. Sci. Paris, t. 333, pp , (2001). [7] M. Boucetta, Riemann-Poisson manifolds and Kähler-Riemann foliations, C. R. Acad. Sci. Paris, Ser. I 336, pp (2003). [8] A. Abouqateb, M. Boucetta, The modular class of regular Poisson manifold and the Reeb class of its symplectic foliations, C. R. Acad. Sci. Paris, Ser. I 337, pp (2003). [9] M. Boucetta, Poisson manifolds with compatible pseudo-metric and pseudo-riemannian Lie algebras, Differential Geometry and its Applications, Vol. 20, Issue 3, pp (2004). [10] M. Boucetta, On the Riemann-Lie algebras and Riemann-Poisson Lie groups, Journal of Lie theory, Volume 15, pp (2005). [11] M. Boucetta, Solutions of the Classical Yang-Baxter Equation and Noncommutative Deformations, Letters in Mathematical Physics (2008) 83: [12] M. Boucetta, Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on S n, Osaka Journal of Math. 46 (2009) [13] A. Bahayou and M. Boucetta, Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups, C. R. Acad. Sci. Paris, Série I 347, (2009) [14] M. Boucetta, The Lichnerowicz Laplacian on compact symmetric spaces, C. R. Acad. Sci. Paris, Série I 347, (2009) [15] A. Bahayou & M. Boucetta, Metacurvature of Riemannian Poisson- Lie groups, Journal of Lie Theory, Volume 19 (2009)

19 [16] M. Boucetta, Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on P n (C), Journal of Geometry and Physics, Volume 60, Issue 10 (2010) [17] M. Boucetta and A. Medina, Polynomial Poisson structures on some affine solvmanifolds, Journal of Symplectic Geometry, Volume 9, Number 3, 1-15, [18] M. Boucetta and A. Medina, Solutions of the Yang-Baxter equations on orthogonal groups : the case of oscillator groups, Journal of Geometry and Physics 61 (2011) pp [19] M. Boucetta and Z. Sassaai, A class of Poisson structures compatible with the canonical Poisson structure on the cotangent bundle, C. R. Acad. Sci. Paris, Ser. I 349 (2011) [20] Boucetta, Mohamed, Riemannian geometry of Lie algebroids. J. Egyptian Math. Soc. 19 (2011), no. 1-2, [21] M. Ait Haddou, M. Boucetta, H. Lebzioui, Left-invariant Lorentzian flat metrics on Lie groups, Journal of Lie Theory 22 (2012), No. 1, , [22] Saassai, Zouhair; Boucetta, Mohamed; On the local structure of noncommutative deformations. J. Geom. Phys. 82 (2014), [23] Benayadi, Sad; Boucetta, Mohamed, Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures. Differential Geom. Appl. 36 (2014), 66?89. [24] S. Benayadi and M. Boucetta, On Para-Kähler and Hyper-Kähler Lie algebras, arxiv: (To appear in Journal of Algebra). [25] M. Boucetta and Jean-Marie Morvan, (Eds.) Differential Geometry and Topology, Discrete and Computational Geometry, IOS Press, XIII. Books [1] Mohamed Boucetta & Jean-Marie Morvan, Réduction des endomorphismes, Edition Cepadues (2007). [2] Mohamed Boucetta, Dualité, formes quadratiques et formes hermitiennes, Edition Cepadues (2009). [3] Mohamed Boucetta, Analyse I et II pour la licence scientifique, Top Edition (2010). [4] Mohamed Boucetta, Eléments d Analyse Réelle, Edition Ellipse, [5] Mohamed Boucetta, Intégrales généralisées, intégrales dépendant d un paramètre, Edition Cepadues (2013). 4

20 List foreign institutes visited Host Institute Duration of visit Purpose of visit MSRI (Berkley) 15 days Participation to a Seminar Elie Cartan Institute 15 days Scientific stay of Mathematics ( Nancy) Institut Girard Desargues (Lyon) 15 days Scientific stay Instituto Técnico ( Lisbonne) 7 days Participation to a congress University Valenciennes 7 days Scientific setting University Montreal 15 days participation to a congress University El Oued, Algeria 15 days Participation to a school ICTP 23 days Participation to a school University Charle Rabelais (Tours) 6 months Invited Professor University Montpellier II 1 month Invited Professor. University Paul Sabatier 1 month Invited Professor. ICTP 1 month Mathematics Research followship. University Lorraine Many times Invited Professor. XIV. Ph.D students supervised: 1. Amine Bahayou. Sassai Zouhair and Hicham Lebzioui. 5