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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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