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In [[mathematics]], a '''supercommutative 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


:<math>yx = (-1)^{|x| |y|}xy.\,</math>
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Equivalently, it is a superalgebra where the [[supercommutator]]
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:<math>[x,y] = xy - (-1)^{|x| |y|}yx\,</math>
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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'''.
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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.
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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,
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:<math>xy + yx = 0\,</math>
 
for odd ''x'' and ''y''. In particular, the square of any odd element ''x'' vanishes whenever 2 is invertible:
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:<math>x^2 = 0.\,</math>
 
Thus a commutative superalgebra (with 2 invertible and nonzero degree one component) always contains [[nilpotent]] elements.
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==See also==
  <li>[http://jz.eeeqi.com/news/html/?203595.html http://jz.eeeqi.com/news/html/?203595.html]</li>
*[[Commutative algebra]]
 
*[[Lie superalgebra]]
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{{DEFAULTSORT:Supercommutative Algebra}}
</ul>
[[Category:Algebras]]
[[Category:Super linear algebra]]

Latest revision as of 04:19, 2 January 2015

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