Resonant superalgebras for supergravity

Considering supergravity theory is a natural step in the development of gravity models. This paper follows the ``algebraic`` path and constructs possible extensions of the Poincar\'e and Anti-de-Sitter algebras, which inherit their basic commutation structure. Previously achieved results of this type are fragmentary and show only a limited fraction of possible algebraic realizations. Our paper presents the newly obtained symmetry algebras, evaluated within an efficient pattern-based computational method of generating the so-called 'resonating' algebraic structures. These supersymmetric extensions of algebras, going beyond the Poincar\'e and Anti-de Sitter ones, contain additional bosonic generators $Z_{ab}$ (Lorentz-like), and $U_a$ (translational-like) added to the standard Lorentz generator $J_{ab}$ and translation generator $P_{a}$. Our analysis includes all cases up to two fermionic supercharges, $Q_{\alpha}$ and $Y_{\alpha}$. The delivered plethora of superalgebras includes few past results and offers a vastness of new examples. The list of the cases is complete and contains all superalgebras up to two of Lorentz-like, translation-like, and supercharge-like generators $(JP+Q)+(ZU+Y)=JPZU+QY$. In the latter class, among $667$ founded superalgebras, the $264$ are suitable for direct supergravity construction. For each of them, one can construct a unique supergravity model defined by the Lagrangian. As an example, we consider one of the algebra configurations and provide its Lagrangian realization.