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Various kinds of supersymmetric QFTs are studied in the physics literature. A typical physics talk describes a Lie "superalgebra" by a huge list of operators (with many supercharges, internal rotations among supercharges and other Poincare group generators) and commutator relations.

I am wishing that we can start with Spec$(\mathbb{R}^{p|q})$ (with some equipped bilinear form perhaps) and study its automorphisms (and the Lie algebra of the automorphism group) to recover the Lie "superalgebra" of physicists.

Is there an exposition of these ideas in the literature?

Arctic Char
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  • As a vector space, the physicist's Lie superalgebra is simply $V\oplus \Pi S$, where $V$ is an ordinary vector space, and $\Pi S$ is a parity reversed spinor module. Except for $\wedge^2\Pi S \to V, [s,t] = \Gamma(s,t)$, all brackets are zero (here, $\Gamma$ is dual to Clifford multiplication) – Johannes Moerland Jun 02 '25 at 11:08
  • To add more "charges", you can twist the spinor module by further vector spaces. Of course, you can also take the semidirect product with $Spin(n)$. – Johannes Moerland Jun 02 '25 at 11:11

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