Super-Poincaré algebra

The super-Poincaré algebra extends the Poincare algebra by odd spinor generators. In four-dimensional $\mathcal{N}=1$ supersymmetry, $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ and the equal-chirality anticommutators vanish.

Under standard assumptions on a nontrivial analytic relativistic S-matrix, the Coleman–Mandula theorem says that every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra.

The Haag–Łopuszański–Sohnius theorem allows a Z2-graded Lie superalgebra and shows that supersymmetry, including possible central charges and R-symmetries, is the nontrivial fermionic extension compatible with an interacting relativistic S-matrix.

The superspin Casimir classifies massive supermultiplets analogously to the Pauli-Lubanski Casimir for Poincare representations. One construction contracts the tensor $C_{\mu \nu }=B_{\mu }{P}_{\nu }-B_{\nu }{P}_{\mu }$, where $B_{\mu }$ is the supersymmetric correction of the Pauli-Lubanski pseudovector.