Anticommutativity

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. 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 n {displaystyle n} -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, More formally, a map ∗ : A n → A {displaystyle *;:A^{n} o A} from the set of all n-tuples of elements in a set A (where n is a non-negative integer) to a group G = ( A , + , 0 ) {displaystyle {mathfrak {G}}=(A,+,0)} is anticommutative with respect to the group operation '+' if and only if where ( σ ( 1 ) , … σ ( n ) ) {displaystyle (sigma (1),dots sigma (n))} is the result of permuting ( 1 , 2 , … n ) {displaystyle (1,2,dots n)} with the permutation σ , {displaystyle sigma ,} and sgn σ {displaystyle operatorname {sgn} _{sigma }} is the identity map for even permutations σ {displaystyle sigma } and maps each element of A to its inverse for odd permutations σ {displaystyle sigma } . 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 ∗ : A × A → G {displaystyle *:A imes A o {mathfrak {G}}} is anticommutative if and only if This means that x1 ∗ x2 is the additive inverse of the element x2 ∗ x1 in G {displaystyle {mathfrak {G}}} . In the most frequent cases in physics, where A {displaystyle A} carries already a field structure, the fact