Supercommutative algebra (original) (raw)
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have where |x| denotes the grade of the element and is 0 or 1 (in Z2) according to whether the grade is even or odd, respectively. Equivalently, it is a superalgebra where the supercommutator The even subalgebra of a supercommutative algebra is always a commutative algebra. That is, even elements always commute. Odd elements, on the other hand, always anticommute. That is,