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.

DefinitionEdit

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  .

PropertiesEdit

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.

ExamplesEdit

Examples of anticommutative binary operations include:

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, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.

External linksEdit