Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. 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. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket.

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.

Definition
If $$A, B$$ are two abelian groups, a bilinear map $$f\colon A^2 \to B$$ is anticommutative if for all $$x, y \in A$$ we have


 * $$f(x, y) = - f(y, x).$$

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

where $$\text{sgn}(\sigma)$$ is the sign of the permutation $$\sigma$$.

Properties
If the abelian group $$B$$ has no 2-torsion, implying that if $$x = -x$$ then $$x = 0$$, then any anticommutative bilinear map $$f\colon A^2 \to B$$ satisfies


 * $$f(x, x) = 0.$$

More generally, by transposing two elements, any anticommutative multilinear map $$g\colon A^n \to B$$ satisfies


 * $$g(x_1, x_2, \dots x_n) = 0$$

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 $$f\colon A^2 \to B$$ is alternating then by bilinearity we have


 * $$f(x+y, x+y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, y) + f(y, x) = 0$$

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