Some New Simple Modular Lie Superalgebras

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 presents the same pattern, but with g0̄ = so7. For the simple Lie superalgebras in Kac’s classification [Kac77b], only the orthosymplectic Lie superalgebra osp(1, 4) presents the same pattern, since g0̄ = sp4 here, and g1̄ is its natural four dimensional module. But sp4 is isomorphic to so5 and, as such, g1̄ is its spin module. Quite recently, the author [Eld05] found another instance of this phenomenon. There exists a simple Lie superalgebra over fields of characteristic 3 with even part isomorphic to so12 and odd part its spin module. This paper is devoted to settle the question of which other simple either Z2-graded Lie algebras or Lie superalgebras present this same pattern: the even part being an orthogonal Lie algebra and the odd part its spin module. It turns out that, besides the previously mentioned examples, and of so9, which is the direct sum of so8 and its natural module, but where, because of triality, this natural module can be substituted by the spin module, there appear exactly two other possibilities for Lie superalgebras, one in characteristic 3 with even part isomorphic to so13, and the other in characteristic 5, with even part isomorphic to so11. These simple Lie superalgebras seem to appear here for the first time. The characteristic 5 case will be shown to be strongly related to the ten dimensional simple exceptional Kac Jordan superalgebra, by means of a construction due to Tits. As has been proved by McCrimmon [McCpr], and indirectly hinted in [EO00], the Grassmann envelope of this Jordan