How to realize Lie algebras by vector fields

An algorithm for embedding finite dimensional Lie algebras into Lie algebras of vector fields (and Lie superalgebras into Lie superalgebras of vector fields) is offered in a way applicable over ground fields of any characteristic. The algorithm is illustrated by reproducing Cartan's interpretations of the Lie algebra of G(2) as the Lie algebra that preserves certain non-integrable distributions. Similar algorithm and interpretation are applicable to other exceptional simple Lie algebras, as well as to all non-exceptional simple ones and many non-simple ones, and to many Lie superalgebras.Comment: 17 pages, LaTe

Explicit bracket in an exceptional simple Lie superalgebra

Abstract. This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, cvect(0|3)∗, a subalgebra of vect(4|3). We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of cvect(0|3)∗. Moreover we realize cvect(0|3) ∗ by ”glued ” pairs of generating functions on a (3|3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly