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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,

xy = −(yx).

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:

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

 

This means that x1x2 is the additive inverse of the element x2x1 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:

See also: graded-commutative ring

See alsoEdit

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.

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