Models of Some Simple Modular Lie Superalgebras

Models of the exceptional simple modular Lie superalgebras in characteristic p ≥ 3, that have appeared in the classification due to Bouarroudj, Grozman and Leites [BGLb] of the Lie superalgebras with indecomposable symmetrizable Cartan matrices, are provided. The models relate these exceptional Lie superalgebras to some low dimensional nonassociative algebraic systems. Introduction The finite dimensional modular Lie superalgebras with indecomposable symmetrizable Cartan matrices over algebraically closed fields are classified in [BGLb] under some extra technical hypotheses. Their results assert that, for characteristic ≥ 3, apart from the Lie superalgebras obtained as the analogues of the Lie superalgebras in the classification in characteristic 0 [Kac77], by reducing the Cartan matrices modulo p, there are the following exceptions that have to be added to the list of known simple Lie superalgebras: (i) Two exceptions in characteristic 5: br(2; 5) and el(5; 5). (The superalgebra el(5; 5) first appeared in [Eld07b].) (ii) A family of exceptions given by the Lie superalgebras that appear in the Supermagic Square in characteristic 3 considered in [CE07a, CE07b]. With the exception of g(3, 6) = g(S1,2, S4,2) these Lie superalgebras first appeared in [Eld06b] and [Eld07b]. (iii) Another two exceptions in characteristic 3, similar to the ones in characteristic 5: br(2; 3) and el(5; 3). The Lie superalgebra el(5; 5) was shown in [Eld07b] to be related to Kac’s 10dimensional exceptional Jordan superalgebra, by means of the Tits construction of Lie algebras in terms of alternative and Jordan algebras [Tit66]. The purpose of this paper is to provide models of the other three exceptions: br(2; 3) and el(5; 3) in characteristic 3, and br(2; 5) in characteristic 5. Actually, the superalgebra br(2; 3) already appeared in [Eld06b, Theorem 3.2(i)] related to a symplectic triple system of dimension 8. Here it will be shown to be related to a nice five dimensional orthosymplectic triple system. The Lie superalgebra el(5; 3) will be shown to be a maximal subalgebra of the Lie superalgebra g(8, 3) = g(S8, S1,2) in the Supermagic Square. Furthermore, it will be shown to be related to an orthogonal triple system defined on the direct sum of two copies of the octonions and, finally, it will be proved to be the Lie superalgebra of derivations of a specific orthosymplectic triple system, and this Date: May 9, 2008. 2000 Mathematics Subject Classification. Primary 17B50; Secondary 17B60, 17B25.