# Anticommutativity

In mathematics, anticommutativity is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result, which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation, such as addition.

A prominent example of an anticommutative operation is the Lie bracket.

## Definition

An ${\displaystyle n}$-ary operation is antisymmetric if swapping the order of any two arguments negates the result. For example, a binary operation "∗" is anti-commutative (with respect to addition) if for all x and y,

xy = −(yx).

More formally, a map ${\displaystyle *\;:A^{n}\to A}$ from the set of all n-tuples of elements in a set A (where n is a non-negative integer) to a group ${\displaystyle {\mathfrak {G}}=(A,+,0)}$ is anticommutative with respect to the group operation "+" if and only if

${\displaystyle *(x_{1},x_{2},\dots x_{n})=\operatorname {sgn} _{\sigma }(*(x_{\sigma (1)},x_{\sigma (2)},\dots x_{\sigma (n)}))\qquad \forall \;(x_{1},x_{2},\dots ,x_{n})\in A^{n},}$

where ${\displaystyle (\sigma (1),\dots \sigma (n))}$ is the result of permuting ${\displaystyle (1,2,\dots n)}$ with the permutation ${\displaystyle \sigma ,}$ and ${\displaystyle \operatorname {sgn} _{\sigma }}$ is the identity map for even permutations ${\displaystyle \sigma }$ and maps each element of A to its inverse for odd permutations ${\displaystyle \sigma }$. In an associative setting it is convenient to denote this with a binary operation "∗":

${\displaystyle x_{1}*x_{2}*\dots *x_{n}=\operatorname {sgn} _{\sigma }(x_{\sigma (1)}*x_{\sigma (2)}*\dots *x_{\sigma (n)}).}$

This equality expresses the following concept:

Particularly important is the case n = 2. A binary operation ${\displaystyle *:A\times A\to {\mathfrak {G}}}$ is anticommutative if and only if

${\displaystyle x_{1}*x_{2}=-(x_{2}*x_{1})\qquad \forall (x_{1},x_{2})\in A\times A}$

This means that x1x2 is the additive inverse of the element x2x1 in ${\displaystyle {\mathfrak {G}}}$.

In the most frequent cases in physics, where ${\displaystyle A}$ carries already a field structure, the fact

${\displaystyle x*x=-(x*x)\qquad \forall x\in A}$

implies that applying an anticommutative operation to any collection of operands yields zero, if any two operands are equal (provided the characteristic of the field is not ${\displaystyle 2}$). That is

${\displaystyle x_{1}*\dots *y*\dots *y*\dots *x_{n-2}=0.}$

## Properties

If the group ${\displaystyle \scriptstyle {\mathfrak {G}}}$ is such that

${\displaystyle {\mathfrak {-a}}={\mathfrak {a}}\iff {\mathfrak {a}}={\mathfrak {0}}\qquad \forall {\mathfrak {a}}\in {\mathfrak {G}}}$

i.e. the only element equal to its inverse is the neutral element, then for all the ordered tuples such that ${\displaystyle x_{j}=x_{i}}$ for at least two different index ${\displaystyle i,j}$

${\displaystyle x_{1}*x_{2}*\dots *x_{n}={\mathfrak {0}}}$

In the case ${\displaystyle n=2}$ this means

${\displaystyle x_{1}*x_{1}=x_{2}*x_{2}={\mathfrak {0}}}$

## Examples

Examples of anticommutative binary operations include: