2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186912In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group ${A}(4,R)$ and of its subgroup ${GL}(4,R)$ in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the $\overline{SL}(4,R)$ covering subgroup. --- We set up a (first--order) Lagrangian formalism and build up the corresponding Noether machinery. For an arbitrary gauge Lagrangian, the three gauge field equations come out in a suggestive Yang-Mills like form. The conservation--type differential identities for energy--momentum and hypermomentum and the corresponding complexes and superpotentials are derived. Limiting cases such as the Einstein--Cartan theory are discussed. In particular we show, how the ${A}(4,R)$ may ``break down'' to the Poincaré (inhomogeneous Lorentz) group. In this context, we present explicit models for a symmetry breakdown in the cases of the Weyl (or homothetic) group, the ${SL}(4,R)$, or the ${GL}(4,R)$.197 pagesGeneral Relativity and Quantum CosmologyHigh Energy Physics - Theorytext