A Java Library to Perform <i>S</i>-Expansions of Lie Algebras

• Published in: Axioms
• Main Authors: Carlos Inostroza, Igor Kondrashuk, Nelson Merino, Felip Nadal
• Format: Article
• Language: English
• Published: MDPI AG 2025-09-01
• Subjects: <i>S</i>-expansion; Lie algebra; semigroup; Java programming
• Online Access: https://www.mdpi.com/2075-1680/14/10/735

The contraction method is a procedure that allows to establish non-trivial relations between Lie algebras and has had successful applications in both mathematics and theoretical physics. This work deals with generalizations of the contraction procedure, with a main focus on the so-called <i>S</i>-expansion method, as it includes most of the other generalized contractions. Basically, the <i>S</i>-expansion combines a Lie algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> with a finite abelian semigroup <i>S</i> in order to define new <i>S</i>-expanded algebras. After giving a description of the main ingredients used in this paper, we present a Java library that automates the <i>S</i>-expansion procedure. With this computational tool, we are able to represent Lie algebras and semigroups, so we can perform <i>S</i>-expansions of Lie algebras using arbitrary semigroups. We explain how the library methods have 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 their own calculations, without necessarily being an expert in Java. Finally, some comments about further developments and possible new applications are made.