MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

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

Pavel Grozman and Dimitry A. Leites


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.

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

Related publications

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