Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a superalgebra. prove that finite-dimensional superalgebra is obtained as the result one these procedures. For algebras, in addition known “classical” restrictedness, we introduce (2,4)-structure on non-alternating series: orthogonal and Hamiltonian vector fields. superal...

Abstract. An explicit construction of all finite-dimensional irreducible representations of classical Lie algebras is a solved problem and a Gelfand-Zetlin type basis is known. However the latter lacks the orthogonality property or does not consist of weight vectors for so(n) and sp(2n). In case of Lie superalgebras all finite-dimensional irreducible representations are constructed explicitly o...

Let V be a finite-dimensional superspace over C and g a simple (or a “close” to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in gl(V ). Under the classical invariant theory for g we mean the description of g-invariant elements of the algebra A p,q k,l = S . (V k ⊕Π(V ) ⊕ V ∗p ⊕Π(V )). We give such description for gl(V ), sl(V ) and osp(V ) and their “odd” analogs: q(V ), sq(V );...

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

We classify anti-involutions of Lie superalgebra ŜD preserving the principal gradation, where ŜD is the central extension of the Lie superalgebra of differential operators on the super circle S. We clarify the relations between the corresponding subalgebras fixed by these anti-involutions and subalgebras of ĝl∞|∞ of types OSP and P . We obtain a criterion for an irreducible highest weight modul...

Let g be a simple, finite-dimensional Lie superalgebra over C. These have been classified by V. Kac. Unless g is a Lie algebra or a Lie superalgebra of type osp(1, 2n), the category of finite-dimensional representations of g is not semisimple; q.v. [6]. This leads to a classification problem. For example, in [3], the representation theory of sl(m,n) is worked out by showing it is wild when m,n ...

Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan supe...

There are well-known constructions of the exceptional simple Lie algebras of type E8 and F4 which go back to Witt [Wit41], as Z2-graded algebras g = g0̄⊕g1̄ with even part the orthogonal Lie algebras so16 and so9 respectively, and odd part given by their spin representations (see [Ada96]). Brown [Bro82] found a new simple finite dimensional Lie algebra over fields of characteristic 3 which presen...