The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra

TL;DR

This paper explores the structure and integrability of Lie-invariant geometric objects derived from ideals in the Grassmann algebra, extending Cartan's theory to nonlinear dynamical systems and providing explicit Lax representations.

Contribution

It generalizes Cartan's theory for Lie-invariant objects to Lax integrable systems using prolongation structures and Cartan-Ehresmann connections.

Findings

Detailed analysis of integrable one-forms in Grassmann algebra

Construction of effective Maurer-Cartan forms for applications

Explicit Lax representation for Burgers system derived

Abstract

We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation.