They are examples of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras.
Thus a super-Poincaré algebra is a Z2-graded vector space with a graded Lie bracket such that the even part is a Lie algebra containing the Poincaré algebra, and the odd part is built from spinors on which there is an anticommutation relation with values in the even part.
The Poincaré algebra describes the isometries of Minkowski spacetime.
From the representation theory of the Lorentz group, it is known that the Lorentz group admits two inequivalent complex spinor representations, dubbed
; such decompositions of tensor products of representations into direct sums is given by the Littlewood–Richardson rule.
Normally, one treats such a decomposition as relating to specific particles: so, for example, the pion, which is a chiral vector particle, is composed of a quark-anti-quark pair.
This leads to a natural question: if Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry be extended to the fundamental representation?
That is, the Coleman–Mandula theorem is a no-go theorem that states that the Poincaré algebra cannot be extended with additional symmetries that might describe the internal symmetries of the observed physical particle spectrum.
However, the Coleman–Mandula theorem assumed that the algebra extension would be by means of a commutator; this assumption, and thus the theorem, can be avoided by considering the anti-commutator, that is, by employing anti-commuting Grassmann numbers.
The simplest supersymmetric extension of the Poincaré algebra contains two Weyl spinors with the following anti-commutation relation: and all other anti-commutation relations between the Qs and Ps vanish.
The Pauli matrices can be considered to be a direct manifestation of the Littlewood–Richardson rule mentioned before: they indicate how the tensor product
The tensor product then gives an algebraic relation to the Minkowski metric
It is a closed algebra, since all Jacobi identities are satisfied and can have since explicit matrix representations.
Just as the Poincaré algebra generates the Poincaré group of isometries of Minkowski space, the super-Poincaré algebra, an example of a Lie super-algebra, generates what is known as a supergroup.
That is, we can view superspace as the direct sum of Minkowski space with 'spin dimensions' labelled by coordinates
generates translations in the direction labelled by the coordinate
In (3 + 1) Minkowski spacetime, the Haag–Łopuszański–Sohnius theorem states that the SUSY algebra with N spinor generators is as follows.
The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra and a reductive Lie algebra B (such that its self-adjoint part is the tangent space of a real compact Lie group).
The Lie bracket for the odd part is given by a symmetric equivariant pairing {.,.}
to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from
charges and A being a complex intertwiner with the real part mapping to
We see that there is more than one N = 2 supersymmetry; likewise, the SUSYs for N > 2 are also not unique (in fact, it only gets worse).
It is theoretically allowed, but the multiplet structure becomes automatically the same with that of an N=4 supersymmetric theory.
[citation needed] This is the maximal number of supersymmetries in a theory without gravity.
In 0 + 1, 2 + 1, 3 + 1, 4 + 1, 6 + 1, 7 + 1, 8 + 1, and 10 + 1 dimensions, a SUSY algebra is classified by a positive integer N. In 1 + 1, 5 + 1 and 9 + 1 dimensions, a SUSY algebra is classified by two nonnegative integers (M, N), at least one of which is nonzero.
The structure of supersymmetry algebra is mainly determined by the number of the fermionic generators, that is the number N times the real dimension of the spinor in d dimensions.
, which admits a unique theory called eleven-dimensional supergravity which is the low-energy limit of M-theory.
There is also N = (2, 0) SUSY algebra, which is called the type IIB supersymmetry.
Type IIA / IIB / I superstring theory has the SUSY algebra of the corresponding name.
The supersymmetry algebra for the heterotic superstrings is that of type I.