Exterior differential calculus in generalized Lie algebras(algebroids) category with applications to interior and exterior algebraic(differential) systems

TL;DR

This paper introduces a new category of generalized Lie algebras and algebroids, develops an exterior differential calculus, and extends classical results like Maurer-Cartan and Frobenius theorems to this setting, with applications to algebraic differential systems.

Contribution

It defines generalized Lie algebras and algebroids, develops exterior calculus on them, and generalizes classical theorems to this new framework, opening new research directions.

Findings

Established a Maurer-Cartan type theorem for generalized Lie algebras.

Characterized involutivity of algebraic differential systems in this new context.

Presented new research directions in algebraic differential symplectic spaces.

Abstract

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the category of generalized Lie algebroids are presented. An exterior differential calculus on generalized Lie algebras is pre- sented and a theorem of Maurer-Cartan type is obtained. Supposing that any submodule(vector subbundle) of a generalized Lie algebra(algebroid) is an interior algebraic(differential) system (IAS(IDS)) for that generalized Lie algebra/algebroid, then the involutivity of the IAS(IDS) in a result of Frobenius type is characterized. Introducing the notion of exterior algebraic(differential) system of a generalized Lie algebra(algebroid), the involutivity of an IAS(IDS) is characterized in a result of Cartan type. Finally, new directions by research in algebraic(differential) symplectic spaces theory are presented.