Super-Poincaré algebra

Definition

Let $\mathfrak t=V_{\bar0}\oplus S_{\bar1}$ be a supertranslation algebra whose bracket $\Gamma:\operatorname{Sym}^2S\to V$ is $\mathfrak{so}(V,\eta)$ -equivariant. The associated super-Poincaré algebra is

\mathfrak{spoin}(V,S,\Gamma) =\mathfrak{so}(V,\eta)\ltimes\mathfrak t.

Its even part is the Poincaré algebra

\mathfrak{spoin}_{\bar0} =\mathfrak{so}(V,\eta)\ltimes V,

and its odd part is $S$ .

For $A,B\in\mathfrak{so}(V,\eta)$ , $v,w\in V$ , and $s,t\in S$ , the nonzero brackets are

[A,B]_{\mathfrak{so}},\qquad [A,v]=Av,\qquad [A,s]=A\cdot s,\qquad [s,t]=\Gamma(s,t).

Equivariance of $\Gamma$ is precisely the mixed Jacobi identity involving a Lorentz generator and two supercharges.

Terminology and enrichments

In physics, “the supersymmetry algebra” often means a dimension- and signature-specific extension of this algebra. Common enrichments include multiple supercharge copies ( $\mathcal N>1$ ), $R$ -symmetry derivations, central charges, and tensorial brane charges. These choices change the algebra and must be stated explicitly.

The Lie superalgebra is infinitesimal data. Its integration to a global Lie supergroup additionally requires a compatible choice of global reduced Poincaré or spin-cover group, naturally expressed as a super Harish–Chandra pair.

References

J. Wess and J. Bagger, Supersymmetry and Supergravity, second edition, Princeton University Press, 1992. Relevant: Chapter 3.
D. S. Freed, Five Lectures on Supersymmetry, American Mathematical Society, 1999. Publisher record. Relevant: Lectures 1–2.