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

In mathematical physics, where symmetry is of central importance, or even just in multilinear algebra these operations are mostly (multilinear with respect to some vector structures and then) called antisymmetric operations, and when they are not already of arity greater than two, extended in an associative setting to cover more than two arguments.

Definition

are two abelian groups, a bilinear map is anticommutative we have

More generally, a multilinear map we have

is the sign of the permutation.

Properties

has no 2-torsion satisfies

More generally, by transposing satisfies

are equal; such a map is said to be alternating is alternating then by bilinearity we have

and the proof in the multilinear case is the same but in only two of the inputs.

Examples

Examples of anticommutative binary operations include:

• Cross product

• Lie bracket of a Lie algebra

• Lie bracket of a Lie ring

• Subtraction

• Products of i, j, and k in quaternions

See also

• Commutativity

• Commutator

• Exterior algebra

• Graded-commutative ring

• Operation (mathematics)

• Symmetry in mathematics

• Particle statistics (for anticommutativity in physics).

References

• .

External links

• . Which references the Original Russian work