Anticommutative property

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.

Subtraction is an anticommutative operation because -(a - b) = b - a. For example, 2 - 10 = -(10 - 2) = -8.

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

Definition

If A,B are two abelian groups, a bilinear map \( {\displaystyle f:A^{2}\to B} \) is anticommutative if for all \( x,y\in A \) we have

\( {\displaystyle f(x,y)=-f(y,x).}

More generally, a multilinear map g : A n → B {\displaystyle g:A^{n}\to B} {\displaystyle g:A^{n}\to B} is anticommutative if for all x 1 , … x n ∈ A {\displaystyle x_{1},\dots x_{n}\in A} {\displaystyle x_{1},\dots x_{n}\in A} we have

\( {\displaystyle g(x_{1},x_{2},\dots x_{n})={\text{sgn}}(\sigma )g(x_{\sigma (1)},x_{\sigma (2)},\dots x_{\sigma (n)})} \)

where \( \text{sgn}(\sigma) \) is the sign of the permutation σ {\displaystyle \sigma } \sigma .
Properties

If the abelian group B has no 2-torsion, implying that if \( {\displaystyle x=-x} \) then x=0, then any anticommutative bilinear map \( {\displaystyle f:A^{2}\to B} \) satisfies

\( {\displaystyle f(x,x)=0.} \)

More generally, by transposing two elements, any anticommutative multilinear map \( {\displaystyle g:A^{n}\to B} \) satisfies

\( {\displaystyle g(x_{1},x_{2},\dots x_{n})=0} \)

if any of the \( x_{i} \) are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if \( {\displaystyle f:A^{2}\to B} \) is alternating then by bilinearity we have

\( {\displaystyle f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x)=0} \)

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