Dans le premier chapitre de la thèse, nous résumons d’abord les définitions des algèbres Hom-Nambu n-aires (resp. Hom-Nambu- Lie) et algèbres Hom-Nambu n-aires multiplicatives (resp. Hom-Nambu-Lie multiplicatives). Ensuite, on donne,quelques exemples d'algèbres Hom-Nambu de dimension finie. Dans la troisième section du chapitre on rappellela classication des algèbres Hom-Nambu-Lie ternaires de dimension 3 correspondant auxhomomorphismes diagonaux donnée par Ataguema, Makhlouf et Silvestrov dans [12]. Laquatrième section est consacrée aux différentes manières de construire des algèbres n-airesde type Hom-Nambu. On rappelle la construction par twist initiée par Yau. Ensuite on la généralise en une construction d'algèbre n-aire de Hom-Nambu à partir d'une algèbre n-aire de Hom-Nambu et d'un morphisme faible. On s'intéresse aussi à des constructions d'arité plus grande ou plus petite et par produit tensoriel. On montre par ailleurs comment obtenir de nouvelles algèbres n-aires de Hom-Nambu en utilisant les éléments du centroide. La cinquième section est consacrée aux notions de dérivations et de représentationspour les algèbres n-aires. On étudie les αk-dérivations, les dérivations centrales et dansle cas général, la théorie des représentations des algèbres Hom-Nambu n-aires. Nousdiscutons en particulier les cas des représentations adjointes et coadjointes. Les résultatsobtenus dans cette section généralisent ceux donnés pour le cas binaire dans [16, 57]. / The aim of this thesis is to study representation theory and cohomology of n-ary Hom-Nambu-Lie algebras, as well as quadratic structures on these algebras. It is organized as follows.• Chapter 1. n-ary Hom-Nambu algebras : in the first section we recall the definitions of n-ary Hom-Nambu algebras and n-ary Hom-Nambu-Lie algebras, introduced by Ataguema, Makhlouf and Silvestrov and provide some key constructions. These algebras correspond to a generalized version by twisting of n-ary Nambu algebras and Nambu-Lie algebras which are called Filippov algebras. We deal in this chapter with a subclass of n-ary Hom-Nambu algebras called multiplicative n-ary Hom-Nambu algebras. In Section 1.2, we recall the list of 3-dimensional ternary Hom-Nambu-Lie algebras of special type corresponding to diagonal homomorphisms. In Section 1.4 we show different construction procedures. We recall the construction procedures by twisting principles and provide some new constructions using for example the centroid. The first twisting principle, introduced for binary case, was extend to n-ary case. The second twisting principle was introduced for binary algebras. We will extend it to n-ary case in the sequel. Also we recall a construction by tensor product of symmetric totally n-ary Hom-associative algebra by an n-ary Hom-Nambu algebra. In Section 1.5, we extend representation theory of Hom-Lie algebras to the n-ary case and discuss the derivations, αk-derivations and central derivations. The last section of chapter 1 is dedicated to ternary q-Virasoro-Witt algebras. We recall constructions of infinite dimensional ternary Hom-Nambu algebras.• Chapter 2. Cohomology of n-ary multiplicative Hom-Nambu algebras : InSection 2.1. We define a central extension. In the second Section we show that for an n-ary Hom-Nambu-Lie algebra N, the space ∧n−1 N carries a structure of Hom-Leibniz algebra and we dene a cohomology which is suitable for the study of one parameter formal deformations of n-ary Hom-Nambu-Lie algebras. In Section 2.4, we extend to n-ary multiplicative Hom-Nambu-Lie algebras the Takhtajan's construction of a cohomology of ternary Nambu-Lie algebras starting from Chevalley-Eilenberg cohomology of binary Lie algebras. The cohomology of multiplicative Hom-Lie algebras. The cohomology complex for Leibniz algebras was defined by Loday and Pirashvili.• Chapter 3. Quadratic n-ary Hom-Nambu algebras : In the first section we introduce a class of Hom-Nambu-Lie algebras which possess an inner product. In Section 3.3, we provide some constructions of Hom-quadratic Hom-Nambu-Lie algebras starting from an ordinary Nambu-Lie algebra and from tensor product of Hom-quadratic commutative Hom-associative algebra and Hom-quadratic Hom-Nambu-Lie algebra. In Section 3.5, we provide a construction of n-ary Hom-Nambu algebra L which is a generalization of the trivial T∗-extension. In Section 3.6, we give a construction of ternary algebra arising from quadratic Lie algebra. In Section 3.7, we construct quadratic n-ary Hom-Nambu algebras involving elements of the centroid of n-ary Nambu algebras.
| Identifer | oai:union.ndltd.org:theses.fr/2012MULH7311 |
| Date | 15 December 2012 |
| Creators | Mabrouk, Sami |
| Contributors | Mulhouse, Makhlouf, Abdenacer |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | French |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
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