Differential systems associated with tableaux over Lie algebras

TL;DR

This paper explores the construction and properties of exterior differential systems derived from tableaux over Lie algebras, extending the formalism with Spencer cohomology and analyzing involutiveness.

Contribution

It revisits and extends the definition of tableaux over Lie algebras using Spencer cohomology, and investigates involutiveness of the associated differential systems.

Findings

Extended the formalism of tableaux over Lie algebras.

Analyzed involutiveness of the differential systems.

Discussed examples illustrating the theory.

Abstract

We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.