On the number of possible resonant algebras

TL;DR

This paper investigates the classification of all possible resonant algebras derived from the Maxwell algebra and S-expansion, which are relevant for gravity and supergravity theories, by linking them to specific commutative monoids.

Contribution

It systematically determines all resonant algebras based on generator content and their relation to commutative monoids, expanding the algebraic framework for gravity models.

Findings

Identified all possible resonant algebras with given generators.

Established a connection between resonant algebras and commutative monoids.

Provided a classification framework for algebraic structures in gravity theories.

Abstract

We explore the question concerning the number of distinct resonant algebras depending on the generator content, which consists of the Lorentz generator, translation, and new additional Lorentz-like and translation-like generators. Such algebra enlargements originate directly from the so-called Maxwell algebra and implementation of the S-expansion framework. Resonant algebras, being a sub-class of the S-expanded algebras, similarly should find use in the construction of gravity and supergravity models and in some other applications. The undertaken task of establishing all the possible resonant algebras is closely related to the subject of finding commutative monoids (semigroups with the identity element) of the particular order, were we additionally enforce the parity condition.