Anticommutative property

 ( Properties Of Binary Operations ) 

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of Binary operation of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse element refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the Lie algebra.

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 space 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.

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Definition If $A,\; B$ are two , a bilinear map $f\backslash colon\; A^2\; \backslash to\; B$ is anticommutative if for all $x,\; y\; \backslash in\; A$ we have

$f(x,\; y)\; =\; -\; f(y,\; x).$

More generally, a multilinear map $g\; :\; A^n\; \backslash to\; B$ is anticommutative if for all $x\_1,\; \backslash dots\; x\_n\; \backslash in\; A$ we have

$g(x\_1,x\_2,\; \backslash dots\; x\_n)\; =\; \backslash text\{sgn\}(\backslash sigma)\; g(x\_\{\backslash sigma(1)\},x\_\{\backslash sigma(2)\},\backslash dots\; x\_\{\backslash sigma(n)\})$

where $\backslash text\{sgn\}(\backslash sigma)$ is the sign of the permutation $\backslash sigma$.

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Properties If the abelian group $B$ has no 2-torsion, implying that if $x\; =\; -x$ then $x\; =\; 0$, then any anticommutative bilinear map $f\backslash colon\; A^2\; \backslash to\; B$ satisfies

$f(x,\; x)\; =\; 0.$

More generally, by transposing two elements, any anticommutative multilinear map $g\backslash colon\; A^n\; \backslash to\; B$ satisfies

$g(x\_1,\; x\_2,\; \backslash dots\; x\_n)\; =\; 0$

if any of the $x\_i$ are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if $f\backslash colon\; A^2\; \backslash to\; B$ is alternating then by bilinearity we have

$f(x+y,\; x+y)\; =\; f(x,\; x)\; +\; f(x,\; y)\; +\; f(y,\; x)\; +\; f(y,\; y)\; =\; f(x,\; y)\; +\; f(y,\; x)\; =\; 0$

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

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Examples Examples of anticommutative binary operations include:

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

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See also

• Commutativity
• Commutator
• Exterior algebra
• Graded-commutative ring
• Operation (mathematics)
• Symmetry in mathematics
• Particle statistics (for anticommutativity in physics).

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External links

• . Which references the Original Russian work

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