Anticommutativity

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In mathematics, anticommutativity refers to the property of an operation being anticommutative, i.e. being non commutative in a precise way. 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 anticommutative if for all x and y, x*y = −y*x.

More formally, a map $scriptstyle\; *:A^n\; longrightarrow\; mathfrak\{G\}$ from the set of all n-tuples of elements in a set $A$ (where $n$ is a general integer) to a group $scriptstylemathfrak\{G\}$ (whose operation is written in additive notation for the sake of simplicity), is anticommutative if and only if

$x\_1*x\_2*dots*x\_n\; =\; sgn(sigma)\; x\_\{sigma(1)\}*x\_\{sigma(2)\}*dots*\; x\_\{sigma(n)\}\; qquad\; forallboldsymbol\{x\}\; =\; (x\_1,x\_2,dots,x\_n)\; in\; A^n$

where $scriptstylesigma:(n)longrightarrow(n)$ is an arbitrary permutation of the set $(n)$ of first $n$ non-zero integers and $mathrm\{sgn\}(sigma)$ is its sign. This equality express 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 $scriptstylemathfrak\{G\}$ 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: "$-1$" has not a precise meaning since a multiplication is not necessarily defined on $scriptstylemathfrak\{G\}$.

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

$x\_1\; *\; x\_2\; =\; -x\_2\; *\; x\_1\; qquadforall(x\_1,x\_2)in\; Atimes\; A$

This means that $scriptstyle\; x\_1\; *\; x\_2$ is the inverse of the element $scriptstyle\; x\_2\; *\; x\_1$ in $scriptstylemathfrak\{G\}$.

Properties

If the group $scriptstylemathfrak\{G\}$ 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

Anticommutative operators include:

• Subtraction
• Cross product
• Lie algebra
• Lie ring

See also

• Commutativity
• Commutator
• exterior algebra
• Operation (mathematics)
• Symmetry in mathematics
• Particle statistics (for anticommutativity in physics).

References

• , ISBN 3-540-64243-9, chapter III, "Tensor algebras, exterior algebras, symmetric algebras".

  External links

  • Weisstein, Eric W." Anticommutative". From MathWorld--A Wolfram Web Resource.
  • A.T. Gainov, " Anti-commutative algebra", Springer-Verlag Online Encyclopaedia of Mathematics.

  
  
  

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