The metric of the Minkowski superspace as an invariant of the Poincaré supergroup (Q1896731)

The aim of this work is to build a formal mathematical framework for describing supergravitation -- a theory in which the supergeometry is determined by supersymmetry. Supersymmetry is defined as an automorphism of the supergeometric structure and, in part, as an infinitesimal supertransformation preserving the metric of the superspace. In turn the metric itself is defined as the invariant of the transformation supergroup. Starting with a \(\mathbb{Z}_2\)-graded space a \(\mathbb{Z}_2\)-graded Lie algebra is defined and given the name Lie superalgebra with Grassmannian structure. This algebra is used to build a sheaf of algebras on a differentiable manifold and the resulting structure is a supermanifold or a superspace -- the two terms being synonymous. The superspace is made Riemannian by defining a nondegenerate metric form on it. Minkowski superspace is defined as a superspace endowed with a supermetric invariant with respect to transformations belonging to the Poincaré supergroup. Certain calculations are done to illustrate certain aspects of the formalism.