# anticommutative

A binary operation$\star$” is said to be anticommutative if it satisfies the identity

 $\displaystyle y\!\star\!x\;=\;-(x\!\star\!y),$ (1)

where the minus denotes the element in the algebra in question.  This implies that  $x\!\star\!x\;=\;-(x\!\star\!x)$,  i.e. $x\!\star\!x$ must be the neutral element of the addition of the algebra:

 $\displaystyle x\!\star\!x\;=\;\textbf{0}.$ (2)

Using the distributivity of “$\star$” over “$+$” we see that the indentity (2) also implies (1):

 $\textbf{0}\;=\;(x\!+\!y)\!\star\!(x\!+\!y)\;=\;x\!\star\!x+x\!\star\!y+y\!% \star\!x+y\!\star\!y\;=\;x\!\star\!y+y\!\star\!x$

A well known example of anticommutative operations is the vector product in the algebra  $(\mathbb{R}^{3},\,+,\,\times)$,  satisfying

 $\vec{b}\!\times\!\vec{a}\;=\;-(\vec{a}\!\times\!\vec{b}),\qquad\vec{a}\!\times% \!\vec{a}\;=\;\vec{0}.$

Also we know that the subtraction of numbers obeys identities

 $b\!-\!a\;=\;-(a\!-\!b),\qquad a\!-\!a\;=\;0.$

An important anticommutative operation is the Lie bracket.

Title anticommutative Anticommutative 2014-02-04 7:50:58 2014-02-04 7:50:58 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 17A01 anticommutative operation anticommutativity Supercommutative AlternativeAlgebra Subcommutative