Editing Anticommutative property

{{short description|Property of math operations which yield an inverse result when arguments' order reversed}}

In [[mathematics]], '''anticommutativity''' is a specific property of some non-[[commutative]] mathematical [[Operation (mathematics)|operations]]. Swapping the position of [[Binary operation|two arguments]] of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion ''[[inverse element|inverse]]'' refers to a [[group (mathematics)|group structure]] on the operation's [[codomain]], possibly with another operation. [[Subtraction]] is an anticommutative operation because commuting the operands of {{nowrap|1=''a'' − ''b''}} gives {{nowrap|1=''b'' − ''a'' = −(''a'' − ''b'');}} for example, {{nowrap|1=2 − 10 = −(10 − 2) = −8.}} Another prominent example of an anticommutative operation is the [[Lie algebra|Lie bracket]].

In [[mathematical physics]], where [[symmetry (physics)|symmetry]] is of central importance, or even just in [[multilinear algebra]] these operations are mostly (multilinear with respect to some [[vector space|vector structures]] and then) called '''antisymmetric operations''', and when they are not already of [[arity]] greater than two, extended in an [[associative]] setting to cover more than two [[Argument of a function|arguments]].

== Definition ==
If <math>A, B</math> are two [[abelian group]]s, a [[bilinear map]] <math>f\colon A^2 \to B</math> is '''anticommutative''' if for all <math>x, y \in A</math> we have

:<math>f(x, y) = - f(y, x).</math>

More generally, a [[multilinear map]] <math>g : A^n \to B</math> is anticommutative if for all <math>x_1, \dots x_n \in A</math> we have
:<math>g(x_1,x_2, \dots x_n) = \text{sgn}(\sigma) g(x_{\sigma(1)},x_{\sigma(2)},\dots x_{\sigma(n)})</math>

where <math>\text{sgn}(\sigma)</math> is the [[Parity of a permutation|sign]] of the [[permutation]] <math>\sigma</math>.

== Properties ==
If the abelian group <math>B</math> has no 2-[[Torsion (algebra)|torsion]], implying that if <math>x = -x</math> then <math>x = 0</math>, then any anticommutative bilinear map <math>f\colon A^2 \to B</math> satisfies

:<math>f(x, x) = 0.</math>

More generally, by [[Transposition (mathematics)|transposing]] two elements, any anticommutative multilinear map <math>g\colon A^n \to B</math> satisfies

:<math>g(x_1, x_2, \dots x_n) = 0</math>

if any of the <math>x_i</math> are equal; such a map is said to be '''[[Alternating multilinear map|alternating]]'''. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if <math>f\colon A^2 \to B</math> is alternating then by bilinearity we have

:<math>f(x+y, x+y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, y) + f(y, x) = 0</math>

and the proof in the multilinear case is the same but in only two of the inputs.

== Examples ==

Examples of anticommutative binary operations include:

* [[Cross product]]
* Lie bracket of a [[Lie algebra]]
* Lie bracket of a [[Lie ring]]
* [[Subtraction]]

==See also== 
* [[Commutativity]]
* [[Commutator]]
* [[Exterior algebra]]
* [[Graded-commutative ring]]
* [[Operation (mathematics)]]
* [[Symmetry in mathematics]]
* [[Particle statistics]] (for anticommutativity in physics).

== References ==
*{{Citation
 | last = Bourbaki
 | first = Nicolas
 | author-link = Nicolas Bourbaki
 | title = Algebra. Chapters 1–3
 | place = [[Berlin]]-[[Heidelberg]]-[[New York City]]
 | publisher = [[Springer-Verlag]]
 | chapter = Chapter III. [[Tensor algebra]]s, [[exterior algebra]]s, [[symmetric algebra]]s
 | series = Elements of Mathematics
 | year = 1989
 | edition = 2nd printing
 | isbn = 3-540-64243-9
 | mr = 0979982
 | zbl = 0904.00001
}}.

== External links ==
{{Wiktionary}}
*{{springer
| title= Anti-commutative algebra
| id= A/a012580
| last= Gainov
| first= A.T.
| author-link= 
}}. Which references the [http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5238&option_lang=eng Original Russian work]
*{{MathWorld 
|title=Anticommutative 
|urlname=Anticommutative}} 

[[Category:Properties of binary operations]]