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.
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).
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, whenver 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.
This means that x1 ∗ x2 is the additive inverse of the element x2 ∗ x1 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
If the group is such that
In the case this means
Examples of anticommutative binary operations include:
See also: graded-commutative ring
- Exterior algebra
- Operation (mathematics)
- Symmetry in mathematics
- Particle statistics (for anticommutativity in physics).
- 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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