Anticommutative property

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. If e_i^2 = 1, \quad e_i e_j + e_j e_i = 0, i \ne j\ , then (\sum_{i=1}^n x_i e_i )^2 = \sum_{i=1}^n x_i^2 . == Examples ==

Examples of anticommutative binary operations include: • Cross product • Lie bracket of a Lie algebra • Lie bracket of a Lie ring • Subtraction ==See also==