Saïd Benayadi

Double extension of quadratic lie superalgebras

In this paper we study Lie superalgebras g with a non-degenerate super-symmetric, consistentg-invariant bilinear form B. Such a (g,B) is called quadratic Lie superalgebra. Our first result generalizes the notion of double extension to quadratic Lie superalgebras. This notion was introduced by Medina and Revoy [8] to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie superalgebra to be a double extension. Any nonsimple quadratic Lie superalgebra such that dim is a double extension; also we give an inductive classification of this class of quadratic Lie superalgebras.

Socle and some invariants of quadratic Lie superalgebras

Abstract We construct some new invariants of the quadratic Lie superalgebras. These invariants are closely related to the socle and the decomposability of quadratic Lie superalgebras. Next, we establish some relations between these invariants. We use these relations in order to characterize the simple Lie algebras and the basic classical Lie superalgebras among the quadratic Lie superalgebras with completely reducible action of the even part on the odd part and to discuss the problem of characterization of quadratic Lie superalgebras having a unique (up to a constant) quadratic structure. We give a characterization of the socle of a quadratic Lie superalgebra. Several examples are included to show that the situations in the super case change drastically. Lower and upper bounds of dimension of the vector space of even supersymmetric invariant bilinear forms on a quadratic Lie superalgebra are obtained. Finally, we give converses of Koszuls theorems.

Lie algebras admitting a unique quadratic structure

We prove that any Lie algebra g over a field K of characteristic zero admitting a unique up to a constant quadratic structure is necessarily a simple Lie algebra. If the field K is algebraically closed, such condition is also sufficient. Further, a real Lie algebra g admits a unique quadratic structure if and only if its complexification gC is a simple Lie algebra over C

A new characterization of semisimple Lie algebras

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartans second criterion.

Pseudo-Euclidean Jordan Algebras

A pseudo-euclidean Jordan algebra is a Jordan algebra 𝔍 with an associative non-degenerate symmetric bilinear form B. We study the structure of the pseudo-euclidean Jordan algebras over a field 𝕂 of characteristic not two, and we obtain an inductive description of these algebras in terms of double extensions and generalized double extensions. Next, we study the symplectic pseudo-euclidean Jordan 𝕂-algebras, and we give some informations on a particular class of these algebras, namely the class of symplectic Jordan–Manin Algebras.

LIE SUPERALGEBRAS WITH SOME HOMOGENEOUS STRUCTURES

We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [2]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [6]. We study the structures of even (resp. odd)-quadratic odd (resp. even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [6]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superargebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [6].

Associative Superalgebras with Homogeneous Symmetric Structures

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e., even or odd), and associative bilinear form on A. In this article, we show that any associative superalgebra with non-null product cannot admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure, and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.

Symmetric Symplectic Commutative Associative Algebras and Related Lie Algebras

A commutative associative algebra is called symmetric symplectic if it is endowed with both an associative non-degenerate symmetric bilinear form B and an invertible B-antisymmetric derivation D. We give a description of the commutative associative symmetric symplectic 𝕂-algebras by using the notion of T*-extension. Next, we introduce the notion of double extension of symmetric symplectic commutative associative algebras in order to give an inductive description of these algebras. Moreover, much information on the structure of symmetric commutative associative algebras is given in this paper.

Quadratic Malcev Superalgebras with Reductive Even Part

It is our goal to give an inductive description of quadratic Malcev superalgebras with reductive even part. We use the notion of double extension of Malcev superalgebras presented by Albuquerque and Benayadi in [4] and transfer to Malcev superalgebras the concept of generalized double extension given in [6] for Lie superalgebras.

Classification of quadratic Lie algebras of low dimension

In this paper, we give the classification of the irreducible nonsolvable Lie algebras of dimensions ≤13 with nondegenerate, symmetric, and invariant bilinear forms.