# Anticommutativity

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.

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

## Contents

## DefinitionEdit

An -ary operation is antisymmetric if swapping the order of any two arguments negates the result. For example, a binary operation "∗" is anti-commutative (with respect to addition) if for all *x* and *y*,

*x*∗*y*= −(*y*∗*x*).

More formally, a map from the set of all *n*-tuples of elements in a set *A* (where *n* is a non-negative integer) to a group is anticommutative with respect to the group operation "+" if and only if

where is the result of permuting with the permutation and is the identity map for even permutations and maps each element of *A* to its inverse for odd permutations . In an associative setting it is convenient to denote this with a binary operation "∗":

This equality expresses the following concept:

- the value of the operation on some fixed ordered n-tuple is unchanged when applying any even permutation to the arguments, and
- the value of the operation is the additive inverse of this value, whenever an odd permutation is applied to the arguments. The need for the existence of this additive inverse element is the main reason for requiring the codomain of the operation "∗" to be at least a group.

Particularly important is the case *n* = 2. A binary operation is anticommutative if and only if

This means that *x*_{1} ∗ *x*_{2} is the additive inverse of the element *x*_{2} ∗ *x*_{1} in .

In the most frequent cases in physics, where carries already a field structure, the fact

implies that applying an anticommutative operation to any collection of operands yields zero, if any two operands are equal (provided the characteristic of the field is not ). That is

## PropertiesEdit

If the group is such that

i.e. *the only element equal to its inverse is the neutral element*, then for all the ordered tuples such that for at least two different index

In the case * * this means

## ExamplesEdit

Examples of anticommutative binary operations include:

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

See also: graded-commutative ring

## See alsoEdit

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

## ReferencesEdit

- 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, pp. xxiii+709, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.

## External linksEdit

Look up in Wiktionary, the free dictionary.anticommutativity |

- Gainov, A.T. (2001) [1994], "Anti-commutative algebra", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Anticommutative".
*MathWorld*.