Supersymmetry Algebra

In theoretical physics, a supersymmetry algebra (or SUSY algebra) is one of the symmetry algebras incorporating supersymmetry, a relation between bosons and fermions, allowed by the Haag–Lopuszanski–Sohnius theorem as supersymmetries of the S-matrix. The supersymmetry algebra contains not only the Poincare algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of N real spinor representations of the Poincare group. When N>1 the algebra is said to have extended supersymmetry. The supersymmetry algebra is a semidirect product of a central extension of the super Poincare algebra by a compact Lie algebra B of internal symmetries.

Bosonic fields commute while fermionic fields anticommute. In order to relate the two kinds of fields in a single algebra, the introduction of a Z₂-grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called a Lie superalgebra. The spin-statistics theorem shows that bosons have integer spin, while fermions have half-integer spin. Consequently, the odd elements in a supersymmetry algebra need to have half-integer spin, in contrast to the tensorial symmetries which are more traditional symmetries in physics.

Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra, called supermultiplets. For each Lie algebra, there exists an associated Lie group which is connected and simply connected, unique up to isomorphism, and the representations of the algebra can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.