Lie superalgebras

In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notion of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading used here is distinct from a second Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading having cohomological origins. A graded Lie algebra (say, graded by Z {\displaystyle \mathbb {Z} } or N {\displaystyle \mathbb {N} } ) that is anticommutative and has a graded Jacobi identity also has a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } grading; this is the "rolling up" of the algebra into odd and even parts. This rolling-up is not normally referred to as "super". Thus, supergraded Lie superalgebras carry a pair of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑gradations: one of which is supersymmetric, and the other is classical. Pierre Deligne calls the supersymmetric one the super gradation, and the classical one the cohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded.