Structures of G(2) type and nonintegrable distributions in characteristic p

Abstract

Lately we observe: (1) an upsurge of interest (in particular, triggered by a paper by Atiyah and Witten) to manifolds with G(2)-type structure; (2) classifications are obtained of simple (finite dimensional and graded vectorial) Lie superalgebras over fields of complex and real numbers and of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p>3; (3) importance of nonintegrable distributions in (1) -- (2). We add to interrelation of (1)--(3) an explicit description of several exceptional simple Lie algebras for p=2, 3 (Melikyan algebras; Brown, Ermolaev, Frank, and Skryabin algebras) 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 is performed in terms of Cartan-Tanaka-Shchepochkina prolongs and is similar to descriptions of simple Lie superalgebras of vector fields with polynomial coefficients. Our results illustrate usefulness of Shchepochkina's algorithm and SuperLie package; two families of simple Lie algebras found in the process might be new.