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 $p$ greater than 3; (3) importance of non-integrable distributions in observations (1) and (2). We add to interrelation of (1)--(3) an explicit description of several exceptional simple Lie algebras for $p=2,3$ (Brown, Ermolaev, Frank, and Skryabin algebras, and analogs of Melikyan algebras) as subalgebras of Lie algebras of vector fields preserving non-integrable distributions analogous to (or identical with) those preserved by $\mathfrak{g}(2)$, $\mathfrak{o}(7)$, $\mathfrak{sp}(4)$, $\mathfrak{sp}(10)$ and the Brown algebra $\mathfrak{br}(3)$. 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; at least two families of simple Lie algebras found in the process are new.

34 pages; references added; exposition edited

References in corpus (2)

• Invariant operators on supermanifolds and Standard Models
• How to realize Lie algebras by vector fields

Cited by in corpus (7)

• Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations
• Classification of simple Lie superalgebras in characteristic $2$
• Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras
• Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras
• On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2
• Restricted simple Lie (super)algebras in characteristic $3$
• Deformations of Symmetric Simple Modular Lie (Super)Algebras