Property of math operations which yield an inverse result when arguments' order reversed
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. 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. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket.
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.
Definition
If
are two abelian groups, a bilinear map
is anticommutative if for all
we have
![{\displaystyle f(x,y)=-f(y,x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5d8cb40f11dc4a8d94b6d1d04245cde4de03bf)
More generally, a multilinear map
is anticommutative if for all
we have
![{\displaystyle g(x_{1},x_{2},\dots x_{n})={\text{sgn}}(\sigma )g(x_{\sigma (1)},x_{\sigma (2)},\dots x_{\sigma (n)})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/694b15ba65d74aea92f5466c63e7fb3d108163d5)
where
is the sign of the permutation
.
Properties
If the abelian group
has no 2-torsion, implying that if
then
, then any anticommutative bilinear map
satisfies
![{\displaystyle f(x,x)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/500f81b50726950d68ff21c0cde6547df4be6635)
More generally, by transposing two elements, any anticommutative multilinear map
satisfies
![{\displaystyle g(x_{1},x_{2},\dots x_{n})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56b79fe393022b3cbccc24e4c0f43301135d94d0)
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
![{\displaystyle f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96603c535a5fd634fe41b6c5a6574fa6eb74f70d)
and the proof in the multilinear case is the same but in only two of the inputs.
Examples
Examples of anticommutative binary operations include:
See also
References
- 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