Dual Formulation of the Lie Algebra S-expansion Procedure

TL;DR

This paper introduces a dual formulation of the S-expansion method for Lie algebras, utilizing Maurer-Cartan forms, which enhances understanding and application in gauge theories and gravity models.

Contribution

It presents a novel dual approach to the S-expansion procedure, enabling generalizations to gauge free differential algebras and providing insights into gravity Lagrangians.

Findings

Dual formulation based on Maurer-Cartan forms

Application to gauge free differential algebras

Clarification of relations between gravity Lagrangians

Abstract

The expansion of a Lie algebra entails finding a new, bigger algebra G, through a series of well-defined steps, from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method which is based on the dual picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which in turn is relevant for physical applications in, e.g., Supergravity. It also sheds new light on the puzzling relation between two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely the Einstein-Hilbert Lagrangian and the one for the so-called "exotic gravity".