Expanding Lie (super)algebras through Abelian semigroups

We propose an outgrowth of the expansion method introduced by de Azcárraga et al. [Nucl. Phys. B 662, 185 (2003)]. 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×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-Fré-like superalgebra, are considered. Finally, we find explicit, nontrace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of supergravity theories in arbitrary space-time dimensions.

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