On the hidden symmetries of $D=11$ supergravity

Abstract: We report on recent developments regarding the supersymmetric Free Differential Algebra describing the vacuum structure of $D=11$ supergravity. We focus on the emergence of a hidden superalgebra underlying the theory, explaining the group-theoretical role played by the nilpotent fermionic generator naturally appearing for consistency of the construction. We also discuss the relation between this hidden superalgebra and other superalgebras of particular relevance in the context of supergravity and superstring, involving a fermionic generator with 32 components.