Resonant JPZ algebras and 3D Chern--Simons gravity

In this work we construct three-dimensional Chern–Simons gravity theories for the six resonant $JPZ$ algebras generated by the Lorentz generators $J_a$, translations $P_a$, and an additional Lorentz-like generator $Z_a$. The resonance condition together with the Jacobi identities leaves precisely six configurations arising from the semigroup expansions. Our main result is that three of them, $B_4$, ${\tilde{C}}_4$, and $C_4$, are isomorphic to $iso(2,1)\oplus so(2,1)$, although they yield different gravitational realizations when we keep the original $JPZ$ field identification fixed. For each case $JPZ$, we determine the invariant tensors, gauge curvatures, Chern–Simons action, field equations, and the corresponding maximal-rank coupling sectors. The field equations can be written in the compact form ${\mathrm{\Omega }}_{\sigma }\mathbf{F}=0$, which makes explicit how the additional field $h^a$ contributes to the curvature, torsion, and the remaining gauge-curvature components. We use the direct-sum isomorphisms to classify the local metric sectors and to relate them to known global and asymptotic solutions, while keeping the gravitational interpretation tied to the original $(\omega ,e,h)$ fields.