Simple Lie superalgebras and nonintegrable distributions in characteristic~$p$

S. Bouarroudj ; D. A. Leites

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 15-29

Résumé

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for $p\le 5$) and several other little-known simple Lie algebras over algebraically closed fields for $p=3$ as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by $G(2)$, $O(7)$, $Sp(4)$, and $Sp(10)$. The description was performed in terms of Cartan–Tanaka–Shchepochkina prolongs using Shchepochkina's algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over $\mathbb C$; for $p=3$, we obtain a series of new simple Lie superalgebras and an exceptional one.