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.

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

Definition

An $n$ -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,

x ∗ y = −(y ∗ x).

More formally, a map $*\;:A^{n}\to A$ from the set of all n-tuples of elements in a set A (where n is a non-negative integer) to a group ${\mathfrak {G}}=(A,+,0)$ is anticommutative with respect to the group operation "+" if and only if

*(x_{1},x_{2},\dots x_{n})=\operatorname {sgn} _{\sigma }(*(x_{\sigma (1)},x_{\sigma (2)},\dots x_{\sigma (n)}))\qquad \forall \;(x_{1},x_{2},\dots ,x_{n})\in A^{n},

where $(\sigma (1),\dots \sigma (n))$ is the result of permuting $(1,2,\dots n)$ with the permutation $\sigma ,$ and $\operatorname {sgn} _{\sigma }$ is the identity map for even permutations $\sigma$ and maps each element of A to its inverse for odd permutations $\sigma$ . In an associative setting it is convenient to denote this with a binary operation "∗":

x_{1}*x_{2}*\dots *x_{n}=\operatorname {sgn} _{\sigma }(x_{\sigma (1)}*x_{\sigma (2)}*\dots *x_{\sigma (n)}).

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, whenever 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 ${\mathfrak {G}}$ of the operation "∗" to be at least a group.

Particularly important is the case n = 2. A binary operation $*:A\times A\to {\mathfrak {G}}$ is anticommutative if and only if

x_{1}*x_{2}=-(x_{2}*x_{1})\qquad \forall (x_{1},x_{2})\in A\times A

This means that x₁ ∗ x₂ is the additive inverse of the element x₂ ∗ x₁ in ${\mathfrak {G}}$ .

In the most frequent cases in physics, where $A$ carries already a field structure, the fact

x*x=-(x*x)\qquad \forall x\in A

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 $2$ ). That is

x_{1}*\dots *y*\dots *y*\dots *x_{n-2}=0.

Properties

If the group $\scriptstyle {\mathfrak {G}}$ is such that

{\mathfrak {-a}}={\mathfrak {a}}\iff {\mathfrak {a}}={\mathfrak {0}}\qquad \forall {\mathfrak {a}}\in {\mathfrak {G}}

i.e. the only element equal to its inverse is the neutral element, then for all the ordered tuples such that $x_{j}=x_{i}$ for at least two different index $i,j$

x_{1}*x_{2}*\dots *x_{n}={\mathfrak {0}}

In the case $n=2$ this means

x_{1}*x_{1}=x_{2}*x_{2}={\mathfrak {0}}

Examples

Examples of anticommutative binary operations include:

Subtraction
Cross product
Lie bracket of a Lie algebra
Lie bracket of a Lie ring

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, pp. xxiii+709, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001 .

External links

Gainov, A.T. (2001) [1994], "Anti-commutative algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Anticommutative". MathWorld.

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