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MiS Preprint
77/2006

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

Pavel Grozman and Dimitry A. Leites

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 $\Zee$-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 $p=5$ (Melikyan algebras), for $p=3$ (Brown, Ermolaev, Frank, and Skryabin algebras) as subalgebras of Lie algebras of vector fields preserving certain 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 --- a main tool in constructing simple Lie superalgebras of vector fields with polynomial coefficients --- and is similar to descriptions of these superalgebras. We give presentations of some algebras. Our results illustrate usefulness of Shchepochkina's algorithm and SuperLie package: one family of simple Lie algebras found in the process might be new.

Received:
Aug 22, 2006
Published:
Aug 22, 2006
MSC Codes:
17B50, 37J60
Keywords:
simple Lie algebras, nonholonomic structures

Related publications

inJournal
2005 Repository Open Access
Pavel Grozman and D. A. Leites

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

In: Letters in mathematical physics, 74 (2005) 3, pp. 229-262