How to realize Lie algebras by vector fields

Abstract

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.

arXiv comment17 pages, LaTeX

References in corpus (3)

• Structures Preserved by Consistently Graded Lie Superalgebras
• Structures Preserved by Exceptional Lie Algebras
• Explicit Realization of Induced and Coinduced modules over Lie Superalgebras by Differential Operators

Cited by in corpus (16)

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• Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix
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• Classification of simple Lie superalgebras in characteristic $2$
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• Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras
• Second-Order Conformally Equivariant Quantization in Dimension 1|2
• Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
• Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras
• Polynomial representations of classical Lie algebras and flag varieties
• Restricted simple Lie (super)algebras in characteristic $3$
• Deformations of Symmetric Simple Modular Lie (Super)Algebras
• Gardner's deformation of the Krasil'shchik-Kersten system
• On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications
• Two problems in the theory of differential equations