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

Pavel Grozman and Dimitry A. Leites

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Submission date: 22. Aug. 2006
Pages: 27
published in: Letters in mathematical physics, 74 (2005) 3, p. 229-262 
DOI number (of the published article): 10.1007/s11005-005-0026-6
Bibtex
with the following different title: Structures of G(2) type and nonintegrable distributions in characteristic p
MSC-Numbers: 17B50, 37J60
Keywords and phrases: simple Lie algebras, nonholonomic structures
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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=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.