# Supercommutative algebra

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In mathematics, a **supercommutative (associative) algebra** is a superalgebra (i.e. a **Z**_{2}-graded algebra) such that for any two homogeneous elements *x*, *y* we have^{[1]}

where |*x*| denotes the grade of the element and is 0 or 1 (in **Z**_{2}) according to whether the grade is even or odd, respectively.

Equivalently, it is a superalgebra where the supercommutator

always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as **skew-commutative associative algebras** to emphasize the anti-commutation, or, to emphasize the grading, **graded-commutative** or, if the supercommutativity is understood, simply **commutative**.

Any commutative algebra is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The **supercenter** of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra.

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,

for odd *x* and *y*. In particular, the square of any odd element *x* vanishes whenever 2 is invertible:

Thus a commutative superalgebra (with 2 invertible and nonzero degree one component) always contains nilpotent elements.

A **Z**-graded anticommutative algebra with the property that *x*^{2} = 0 for every element *x* of odd grade (irrespective of whether 2 is invertible) is called an alternating algebra.^{[2]}

## See also

[edit]## References

[edit]**^**Varadarajan, V. S. (2004).*Supersymmetry for Mathematicians: An Introduction*. American Mathematical Society. p. 76. ISBN 9780821883518.**^**Nicolas Bourbaki (1998).*Algebra I*. Springer Science+Business Media. p. 482.