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In theoretical physics, a supersymmetry algebra (or SUSY algebra) is a symmetry algebra incorporating supersymmetry, a relation between bosons and fermions. In a supersymmetric world, every boson would have a partner fermion of equal rest mass.

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.

On the other hand, 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. For each Lie algebra, there exists an associated Lie group which is connected and simply connected. Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, 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.

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Supersymmetry algebra