Anticommutativity

In mathematics, anticommutativity is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric operations.

Definition

An $n$ -ary operation is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation ∗ is anti-commutative if for all x and y,

x ∗ y = −(y ∗ x).

More formally, a map $\scriptstyle *:A^{n}\to {\mathfrak {G}}$ from the set of all n-tuples of elements in a set A (where n is a general integer) to a group $\scriptstyle {\mathfrak {G}}$ is anticommutative 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 {\boldsymbol {x}}=(x_{1},x_{2},\dots ,x_{n})\in A^{n}

where $\scriptstyle \sigma :(n)\to (n)$ is an arbitrary permutation of the set (n) of the first n positive integers and $\mathrm {sgn} (\sigma )$ is its sign. This equality expresses the following concept:

the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation of the elements of a fixed one.
the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples constructed by odd permutation to the elements of the fixed one. The need for the existence of this inverse element is the main reason for requiring the codomain $\scriptstyle {\mathfrak {G}}$ of the operation to be at least a group.

Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: "−1" does not have a precise meaning since a multiplication is not necessarily defined on $\scriptstyle {\mathfrak {G}}$ .

Particularly important is the case n = 2. A binary operation $\scriptstyle *:A\times A\to {\mathfrak {G}}$ 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 inverse of the element x₂ ∗ x₁ in $\scriptstyle {\mathfrak {G}}$ .

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

External links

Look up anticommutativity in Wiktionary, the free dictionary.

Gainov, A.T. (2001), "Anti-commutative algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Anticommutative", MathWorld.

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