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Expanding Lie and Gauge Free Differential Algebras through abelian Semigroups

The relevance of Lie algebras in physics can hardly be overemphasized. The Poincaré algebra, for instance, lies at the core of Special Relativity, and serves also as a local symmetry in General Relativity. Supergravity, on the other hand, lacks so far a clear-cut gauge theory interpretation, and this is often regarded as a major drawback. It is the purpose of this talk to introduce a method for the so called expansion of Lie algebras, namely the abelian semigroup expansion, S-expansion for short. For every Lie algebra g and finite abelian semigroup S, it is shown that the direct product G = S x g satisfies the requirements for a Lie algebra as well. When certain conditions are met, ``resonant subalgebras'' and ``reduced algebras'' can be extracted from the S-expanded algebra G. Crucially for its potential applications in physics (particularly in gauge formulations for gravity and supergravity), the method produces an invariant tensor for the S-expanded algebra G from one for the original algebra g. We also review the dual version of the method in terms of the Lie algebra's Maurer-Cartan forms. This dual version permits the generalization of the method to the case of a gauge free differential algebra, which is relevant again for applications in supergravity.

Expanding Lie and Gauge Free Differential Algebras through abelian Semigroups

The relevance of Lie algebras in physics can hardly be overemphasized. The Poincaré algebra, for instance, lies at the core of Special Relativity, and serves also as a local symmetry in General Relativity. Supergravity, on the other hand, lacks so far a clear-cut gauge theory interpretation, and this is often regarded as a major drawback. It is the purpose of this talk to introduce a method for the so called expansion of Lie algebras, namely the abelian semigroup expansion, S-expansion for short. For every Lie algebra g and finite abelian semigroup S, it is shown that the direct product G = S x g satisfies the requirements for a Lie algebra as well. When certain conditions are met, ``resonant subalgebras'' and ``reduced algebras'' can be extracted from the S-expanded algebra G. Crucially for its potential applications in physics (particularly in gauge formulations for gravity and supergravity), the method produces an invariant tensor for the S-expanded algebra G from one for the original algebra g. We also review the dual version of the method in terms of the Lie algebra's Maurer-Cartan forms. This dual version permits the generalization of the method to the case of a gauge free differential algebra, which is relevant again for applications in supergravity.

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