Expanding Lie (super)algebras through abelian semigroups

TL;DR

This paper extends the expansion method for Lie algebras using abelian semigroups, providing systematic conditions, new algebra examples, and explicit invariant tensors relevant for supergravity theories.

Contribution

It introduces a systematic framework for expanding Lie algebras with abelian semigroups, recovering known cases and discovering new algebra structures with explicit invariant tensors.

Findings

Reproduces known expanded algebras with specific semigroups

Identifies new expanded algebra structures from different semigroups

Provides explicit invariant tensors for S-expanded algebras

Abstract

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S \times g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit, non-trace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary space-time dimensions.