A Java library to perform S-expansions of Lie algebras

TL;DR

The paper presents a Java library that automates S-expansion of Lie algebras by coupling a target algebra with finite abelian semigroups. It formalizes the construction of expanded, resonant, and reduced algebras, and provides tools to analyze invariants such as the Killing-Cartan metric and the semigroup metric to assess property preservation like semisimplicity and compactness. The work documents extensive implementations (11 classes) and 45 example programs, enabling users to perform semigroup computations, expansions, and reductions without deep Java expertise. It also discusses the enumeration of finite semigroups, resonances, and zero-elements, and demonstrates how specific semigroups preserve semisimplicity, with implications for generalized contractions and gravity theories built from expanded algebras. The library thus offers a practical, open-source platform for systematic exploration of S-expansions and their applications in mathematical physics.

Abstract

The contraction method is a procedure that allows to establish non-trivial relations between Lie algebras and has had succesful applications in both mathematics and theoretical physics. This work deals with generalizations of the contraction procedure with a main focus in the so called S-expansion method as it includes most of the other generalized contractions. Basically, the S-exansion combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in order to define new S-expanded algebras. After giving a description of the main ingredients used in this paper, we present a Java library that automatizes the S-expansion procedure. With this computational tool we are able to represent Lie algebras and semigroups, so we can perform S-expansions of Lie algebras using arbitrary semigroups. We explain how the library methods has been constructed and how they work; then we give a set of example programs aimed to solve different problems. They are presented so that any user can easily modify them to perform his own calculations, without being necessarily an expert in Java. Finally, some comments about further developements and possible new applications are made.

Figures (2)

• Figure 1: Action of the expansion methods on the scheme of the Theory of the Classification of Lie Algebras.
• Figure 2: Expansions of $\mathfrak{sl}(2,\mathbb{R})$ with abelian semigroups of order $3$