Defining Relations for Lie Superalgebras with Cartan Matrix

We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple -graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. For symmetrizable Cartan matrices similar (but not identical) presentations are obtained by Yamane who also considered q-quantized versions of these relations (q-alg 9603015). Yamane did not try to find out which of the relations he offers constitute a minimal set generating all other relations; contrariwise we give minimal sets of defining relations and in various cases our relations look much simpler than Yamane's (though still very complicated in some cases).     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

Composition factors of Kac modules for the general linear Lie superalgebras

The composition factors of Kac modules for the general linear Lie superalgebras are explicitly determined. In particular, a conjecture of Hughes, King and van der Jeugt in [J. Math. Phys. 33 (1992), 470–491] is proved.     

The classification of filiform Leibniz superalgebras of nilindex <Emphasis Type="Italic">n</Emphasis> + <Emphasis Type="Italic">m</Emphasis>

In this work the classification of filiform Leibniz superalgebras of nilindex n ＋ m, where n and m （m≠0） are dimensions of even and odd parts, respectively, is obtained.