For a classification of spin representations with this property see at spin representations the sections real irreducible spin representations in Lorentz signature and super Poincaré brackets. For explicit construction in components see at Majorana spinor the section The spinor pairing to vectors.
The super Poincaré Lie algebra of -dimensional Minkowski spacetime with respect to the spin representation with symmetric and -equivariant pairing is the super Lie algebra extension of the Poincaré Lie algebra by (the vector space underlying taken in odd degree)
It is precisely the symmetry and -equivariant assumption on that makes this a well defined super Lie algebra: the symmetry corresponds to the graded skew-symmetry of the Lie bracket on elements in , which are regarded as odd, and the -equivariance yields the nontrivial Jacobi identity for and :
The Chevalley-Eilenberg algebra is generated on
elements and of degree
and elements of degree
with the differential defined by
The super Poincaré Lie algebra has, on top of the Lie algebra cocycles that it inherits from , a discrete number of exceptiona cocycles bilinear in the spinors, on the super translation algebra, that exist only in very special dimensions.
The following theorem has been stated at various placed in the physics literature (known there as the brane scan for -symmetry in Green-Schwarz action functionals for super--branes on super-Minkowski spacetime). A full proof is in Brandt 12-13. The following uses the notation in terms of division algebras (Baez-Huerta 10).
In dimensional , has a nontrivial 3-cocycle given by
for spinors and vectors , and 0 otherwise.
In dimensional , has a nontrivial 4-cocycle given by
for spinors and vectors , with the commutator taken in the Clifford algebra.
The 4-cocycle in is the one that induces the supergravity Lie 3-algebra.
All these cocycles are controled by the relevant Fierz identities.
The super-Poincaré Lie algebra has a class of super Lie algebra extensions called extended supersymmetry algebras or polyvector extensions , because they involve additional generators that transforn as skew-symmetric tensors. A complete classification is in (ACDP).
For instance the “M-theory Lie algebra” is a polyvector extension of the super Poincaré Lie algebra by polyvectors of rank and (the M2-brane and the M5-brane in the brane scan), see below Polyvector extensions as automorphism Lie algebras.
For instance the automorphisms of the supergravity Lie 3-algebra gives the “M-theory Lie algebra”-extension of super-Poincaré in 11-dimensions (FSS 13). This is also discussed at supergravity Lie 3-algebra – Polyvector extensions.
The seminal classification result of simple supersymmetry algebras is due to
Lecture notes include
Super spacetimes and super Poincaré-group (pdf)
The Polyvector extensions of (the “M-theory super Lie algebra”) were first considered in
Riccardo D'Auria, Pietro Fré Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140
reviewed in section 8.8. of
and specifically for super-D-branes this discussion is in
A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in
A rigorous classification of these cocycles was later given in
Friedemann Brandt, Supersymmetry algebra cohomology
I: Definition and general structure J. Math. Phys.51:122302, 2010, (arXiv:0911.2118)
II: Primitive elements in 2 and 3 dimensions, J. Math. Phys. 51 (2010) 112303 (arXiv:1004.2978)
III: Primitive elements in four and five dimensions, J. Math. Phys. 52:052301, 2011 (arXiv:1005.2102)
IV: Primitive elements in all dimensions from to , J. Math. Phys. 54, 052302 (2013) (arXiv:1303.6211)
A classification of some special cases of signature/supersymmetry of this is also in the following (using a computer algebra system):
Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 (arXiv:1106.0335)
An introduction to the exceptional fermionic cocycles on the super Poincaré Lie algebra, and their description using normed division algebras, are discussed here:
This subsumes some of the results in (Azcárraga-Townend)
A direct constructions of ordinary (Lie algebraic) extensions of the super Poincare Lie algebra by means of division algebras is in
For more on this see at division algebra and supersymmetry.