Generating Higher-Order Lie Algebras by Expanding Maurer Cartan Forms

TL;DR

This paper introduces a generalized method for expanding higher-order Lie algebras using Maurer-Cartan forms, with potential applications in higher-spin gauge theories and differential algebra generalizations.

Contribution

It develops a new procedure to generate expanded higher-order Lie algebras and relates it to the S-expansion formalism, extending the mathematical framework.

Findings

Derived higher-order Maurer-Cartan equations for specific algebra decompositions

Established a dual formulation linking expansion methods and semigroup choices

Suggested potential applications in higher-spin gauge theories

Abstract

By means of a generalization of the Maurer-Cartan expansion method we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher order Maurer-Cartan equations for the case $\mathcal{G}=V_{0}\oplus V_{1}$ are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher order Maurer Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories.