# 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[edit]

If are two abelian groups, a bilinear map is **anticommutative** if for all we have

More generally, a multilinear map is anticommutative if for all we have

where is the sign of the permutation .

## Properties[edit]

If the abelian group has no 2-torsion, implying that if then , then any anticommutative bilinear map satisfies

More generally, by transposing two elements, any anticommutative multilinear map satisfies

if any of the 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 is alternating then by bilinearity we have

and the proof in the multilinear case is the same but in only two of the inputs.

## Examples[edit]

Examples of anticommutative binary operations include:

- Cross product
- Lie bracket of a Lie algebra
- Lie bracket of a Lie ring
- Subtraction

## See also[edit]

- Commutativity
- Commutator
- Exterior algebra
- Graded-commutative ring
- Operation (mathematics)
- Symmetry in mathematics
- Particle statistics (for anticommutativity in physics).

## References[edit]

- Bourbaki, Nicolas (1989), "Chapter III. Tensor algebras, exterior algebras, symmetric algebras",
*Algebra. Chapters 1–3*, Elements of Mathematics (2nd printing ed.), Berlin-Heidelberg-New York City: Springer-Verlag, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.

## External links[edit]

Look up in Wiktionary, the free dictionary.anticommutative property |

- Gainov, A.T. (2001) [1994], "Anti-commutative algebra",
*Encyclopedia of Mathematics*, EMS Press. Which references the Original Russian work - Weisstein, Eric W. "Anticommutative".
*MathWorld*.