Editing Supercommutative algebra (section)

{{Short description|Type of associative algebra that "almost commutes"}}{{refimprove|date=October 2014}}
In [[mathematics]], a '''supercommutative (associative) algebra''' is a [[superalgebra]] (i.e. a '''Z'''<sub>2</sub>-[[graded algebra]]) such that for any two [[homogeneous element]]s ''x'', ''y'' we have<ref name=Varadarajan>{{cite book|last1=Varadarajan|first1=V. S.|title=Supersymmetry for Mathematicians: An Introduction|publisher=American Mathematical Society|year=2004|isbn=9780821883518|page=76}}</ref>

:<math>yx = (-1)^{|x| |y|}xy ,</math>

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

Equivalently, it is a superalgebra where the [[supercommutator]]

:<math>[x,y] = xy - (-1)^{|x| |y|}yx</math>

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 algebra]]s (also known as [[exterior algebra]]s) 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,
:<math>xy + yx = 0\,</math>
for odd ''x'' and ''y''. In particular, the square of any odd element ''x'' vanishes whenever 2 is invertible:
:<math>x^2 = 0 .</math>
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 {{nowrap|1=''x''{{sup|2}} = 0}} for every element ''x'' of odd grade (irrespective of whether 2 is invertible) is called an [[alternating algebra]].<ref name="bourbaki">{{cite book|author=Nicolas Bourbaki|year=1998|title=Algebra I|publisher=[[Springer Science+Business Media]]|page=482}}</ref>