Infinite S-Expansion with Ideal Subtraction and Some Applications

TL;DR

This paper introduces an infinite S-expansion method with ideal subtraction for Lie (super)algebras, generalizing finite S-expansions and providing new insights into algebra contractions and invariant tensor construction.

Contribution

It proposes a novel infinite S-expansion approach involving ideal subtraction, extending the finite S-expansion framework and offering a new perspective on algebra contractions.

Findings

Infinite S-expansion generalizes finite procedures

Subtraction of infinite ideal corresponds to algebra reduction

Method enables construction of invariant tensors for expanded algebras

Abstract

According to the literature, the S-expansion procedure involving a finite semigroup is valid no matter what the structure of the original Lie (super)algebra is; However, when something about the structure of the starting (super)algebra is known and when certain particular conditions are met, the S-expansion method (with its features of resonance and reduction) is able not only to lead to several kinds of expanded (super)algebras, but also to reproduce the effects of the standard as well as the generalized In\"on\"u-Wigner contraction. In the present paper, we propose a new prescription for S-expansion, involving an infinite abelian semigroup $S^{(\infty)}$ and the subtraction of an infinite ideal subalgebra. We show that the subtraction of the infinite ideal subalgebra corresponds to a reduction. Our approach is a generalization of the finite S-expansion procedure presented in the literature, and it offers an alternative view of the generalized In\"on\"u-Wigner contraction. We then show how to write the invariant tensors of the target (super)algebras in terms of those of the starting ones in the infinite S-expansion context presented in this work. We also give some interesting examples of application on algebras and superalgebras.