Moufang symmetry X. Generalized Lie and Maurer-Cartan equations of continuous Moufang transformations

TL;DR

This paper develops generalized Maurer-Cartan equations for continuous Moufang transformations, extending classical Lie theory to Moufang loops and establishing foundational differential equations for their birepresentations.

Contribution

It introduces a minimal generalization of Maurer-Cartan equations for Moufang loops, applicable to their continuous birepresentations, expanding the Lie theory framework.

Findings

Derived differential equations for Moufang loop birepresentations

Established commutation relations as a generalization of Maurer-Cartan equations

Provided a basis for analyzing continuous Moufang transformations

Abstract

The differential equations for a continuous birepresentation of a local analytic Moufang loop are established. The commutation relations for the infinitesimal operators of the representation are found. These commutation relations can be seen as a (minimal) generalization of the Maurer-Cartan equations and do not depend on the particular birepresentation.