This paper presents the classification over fields of real and complex numbers, minimal Z2×Z2-graded Lie algebras superalgebras spanned by four generators with no empty graded sector. The inequivalent (super)algebras are obtained solving constraints imposed respective Jacobi identities. A motivation for this mathematical result is to systematically investigate properties dynamical systems invar...

We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of “local system of vertex operators” for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space M has a natural vertex (super)algebra structure with M as a module. Then we prove that for a vertex (operat...

We classify all simple supermodules over the queer Lie superalgebra q2 up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.

Many work was done for filiform Lie algebras defined by M. Vergne [8]. An interesting fact is that this algebras are obtained by deformations of the filiform Lie algebra Ln,m. This was used for classifications in [4]. Like filiform Lie algebras, filiform Lie superalgebras are obtained by nilpotent deformations of the Lie superalgebra Ln,m. In this paper, we recall this fact and we study even co...

The Lie superalgebra psl(2|2) is recognized as a quite special algebra. In mathematics, it has the vanishing Killing form and allows for three-dimensional central extension. physics, shows up symmetry of one-dimensional Hubbard model asymptotic S-matrix anti-de Sitter/conformal field theory (AdS/CFT) correspondence. this paper, we present Drinfeld realization Yangian algebra associated with cen...

In this paper complexes of generalized Verma modules over the infinitedimensional exceptional Lie superalgebras E(3, 8) and E(5, 10) are constructed and studied. 0. Introduction. In our papers [KR1]–[KR3] we constructed all degenerate irreducible modules over the exceptional Lie superalgebra E(3, 6). In the present paper we apply the same method to the exceptional Lie superalgebras E(3, 8) and ...

Generalized quantum statistics such as para-Bose and para-Fermi statistics are related to the basic classical Lie superalgebras B(0|n) and Bn. We give a quite general definition of “a generalized quantum statistics associated to a Lie superalgebra G”. This definition is closely related to a certain Z-grading of G. The generalized quantum statistics is determined by a set of root vectors (the cr...