Expanding Lie (super)algebras through abelian semigroups

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.