nLab
super Poincare Lie algebra

Idea

A super Lie algebra extension of a Poincare Lie algebra.

The corresponding super Lie group is the super Euclidean group (except for the signature of the metric).

Cohomology

The super Poincareacute; Lie algebra has, on top of the Lie algebra cocycles that it inherits from $\mathrm{𝔰𝔬}(n)$, a discrete number of special cocycles bilinear in the spinors, on the super translation algebra, that exist only in very special dimensions.

For the statement of the following theorem, we use the relation between division algebra and supersymmetry, as described there.

Theorem

• In dimensional $d=\mathrm{3,4,6},10$, $\mathrm{𝔰𝔦𝔰𝔬}(d-\mathrm{1,1})$ has a nontrivial 3-cocycle given by

  $$(\psi ,\varphi ,A)\mapsto g(\psi \cdot \varphi ,A)$$(\psi, \phi, A) \mapsto g(\psi \cdot \phi, A)

  for spinors $\psi ,\varphi \in 𝒮$ and vectors $A\in 𝒯$, and 0 otherwise.

• In dimensional $d=\mathrm{4,5,7},11$, $\mathrm{𝔰𝔦𝔰𝔬}(d-\mathrm{1,1})$ has a nontrivial 4-cocycle given by

  $$(\Psi ,\Phi ,𝒜,ℬ)\mapsto ⟨\Psi ,(𝒜ℬ-ℬ𝒜)\Phi ⟩$$(\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle

  for spinors $\Psi ,\Phi \in 𝒮$ and vectors $𝒜,ℬ\in 𝒱$, with the commutator taken in the Clifford algebra.

The 4-cocycle in $d=11$ is the one that induces the supergravity Lie 3-algebra.

References

A comprehensive account and classification of super Poincaré Lie algebras is in

• D. V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen, Polyvector Super-Poincaré Algebras (arXiv)

The exceptional fermionic cocycles of the super Poincaré Lie algebra are discussed in

• John Baez, John Huerta, Division algebras and supersymmetry II (pdf)