Intheoretical physics, asupersymmetry algebra (orSUSY algebra) is a mathematical formalism for describing the relation betweenbosons andfermions. The supersymmetry algebra contains not only thePoincaré algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum ofN realspinor representations of thePoincaré group. Such symmetries are allowed by theHaag–Łopuszański–Sohnius theorem. WhenN>1 the algebra is said to haveextended supersymmetry. The supersymmetry algebra is asemidirect sum of acentral extension of thesuper-Poincaré algebra by a compactLie algebraB of internal symmetries.
Bosonic fieldscommute whilefermionic fields anticommute. In order to have a transformation that relates the two kinds of fields, the introduction of aZ2-grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called aLie superalgebra.
Just as one can have representations of aLie algebra, one can also haverepresentations of a Lie superalgebra, calledsupermultiplets. For each Lie algebra, there exists an associatedLie group which isconnected andsimply connected, unique up toisomorphism, and the representations of the algebra can be extended to creategroup representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of aLie supergroup.
The general supersymmetry algebra for spacetime dimensiond, and with the fermionic piece consisting of a sum ofN irreducible real spinor representations, has a structure of the form
where
The terms "bosonic" and "fermionic" refer to even and odd subspaces of the superalgebra.
The terms "scalar", "spinor", "vector", refer to the behavior of subalgebras under the action of the Lorentz algebraL.
The numberN is the number of irreducible real spin representations. When the signature of spacetime is divisible by 4 this is ambiguous as in this case there are two different irreducible real spinor representations, and the numberN is sometimes replaced by a pair of integers (N1,N2).
The supersymmetry algebra is sometimes regarded as a real super algebra, and sometimes as a complex algebra with a hermitian conjugation. These two views are essentially equivalent, as the real algebra can be constructed from the complex algebra by taking the skew-Hermitian elements, and the complex algebra can be constructed from the real one by taking tensor product with thecomplex numbers.
The bosonic part of the superalgebra is isomorphic to the product of the Poincaré algebraP.L with the algebraZ×B of internal symmetries.
WhenN>1 the algebra is said to haveextended supersymmetry.
WhenZ is trivial, the subalgebraP.Q.L is thesuper-Poincaré algebra.