Models of some simple modular Lie superalgebras

Communications in Mathematical Physics, 1977

This article deals with the structure and representations of Lie superalgebras (Z 2-graded Lie algebras). The central result is a classification of simple Lie superalgebras over IR and C.

Revista De La Union Matematica Argentina, 2014

We give an expansion of two notions of double extension and T∗-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case, and the obtained Lie superalgebras are indecomposable.

Commentarii Mathematici Helvetici, 2004

The forms of the exceptional simple classical Lie superalgebras are determined over arbitrary fields of characteristic = 2, 3.

Journal of Algebra and Its Applications, 2015

The maximal graded subalgebras for four families of Lie superalgebras of Cartan type over a field of prime characteristic are studied. All maximal graded subalgebras are described completely by a constructive method and their isomorphism classes, dimension formulas are found except for maximal irreducible graded subalgebras. The classification of maximal irreducible graded subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie superalgebras 𝔤𝔩(m, n), 𝔰𝔩(m, n) and 𝔬𝔰𝔭(m, n).

Doklady Mathematics, 2010

If Ꮽ is a Jordan algebra, then Str(Ꮽ) = L Ꮽ ⊕ Der(Ꮽ) is a Lie algebra, called the structure algebra for Ꮽ; here, L Ꮽ denotes the operator of left multipli cation by an element a from Ꮽ, and Der(Ꮽ) is the der ivation algebra of Ꮽ. The Tits-Koecher-Kantor con struction endows the vector space К( ) = ⊕ ⊕ Str(Ꮽ) ⊕ Ꮽ with the Lie algebra structure; here, is an isomorphic copy of Ꮽ. The properties of К(Ꮽ) are closely related to those of Ꮽ.

Revista de la Unión Matemática Argentina

In this paper, we give an expansion of two notions of double extension and $T^*$-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case and the obtained Lie superalgebras are indecomposable.

2010

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.

Background: First examples of simple nonassociative superalgebras were constructed by Shestakov in (1991 and 1992). Since then many authors showed interest towards the study of superalgebras and superalgebras of vector type. Materials and Methods: Multiplication in M is uniquely defined by a fixed finite set of derivations and by elements of A. The types of derivations used in this article to obtain the results are the near derivation , the derivation and the derivation Results: The flexible Lie - admissible superalgebra over a 2, 3 – torsion free field on one odd generator e is isomorphic to the twisted superalgebra with the free generator In a 2, 3 – torsion free flexible Lie - admissible superalgebras of vector type F, the even part A is differentiably simple, associative and commutative algebra and the odd part M is a finitely generated associative and commutative A – bimodule. Conclusions: A connection between the integral domains, the finitely generated projective modules over them, the derivations of an integral domain and the flexible Lie – admissible superalgebras of vector type has been established. Main conclusions: If A is an integral domain and be a finitely generated projective A-module of rank 1, then is a flexible Lie - admissible superalgebra with even part A and odd part M provided that the mapping is a nonzero derivation of A into the A - module , is a set of derivations of A where .

In this paper, we give a generalization of results in [PU07] and [DPU] by applying the tools of graded Lie algebras to quadratic Lie superalgebras. In this way, we obtain a numerical invariant of quadratic Lie superalgebras and a classification of singular quadratic Lie superalgebras, i.e. those with a nonzero invariant. Finally, we study a class of quadratic Lie superalgebras obtained by the method of generalized double extensions.

Proceedings of the Edinburgh Mathematical Society, 2004

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a $\mu$-algebra.AMS 2000 Mathematics subject cla...