Lie superalgebra

A Lie superalgebra is a Z2-graded vector space with a graded-antisymmetric bracket satisfying the graded Jacobi identity. The bracket of two odd elements is symmetric and is conventionally written as an anticommutator.

A Lie superalgebra representation acts on a parity-graded vector space by linear operators preserving the graded brackets. Odd generators reverse parity, and their brackets are represented by anticommutators.

The Jacobi identity of a Lie superalgebra includes parity signs. In particular, for odd $c$ and even $\varphi$ it gives $[c,[c,\varphi ]]=\frac{1}{2}[[c,c],\varphi ]$.