In
mathematics, anticommutativity is a specific property of some non-
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
mathematical
operations. 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
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
, 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 bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.
In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, 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
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
.
Definition
If
are two
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, a
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
is anticommutative if for all
we have
:
More generally, a
multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W ar ...
is anticommutative if for all
we have
:
where
is the
sign of the
permutation .
Properties
If the abelian group
has no 2-
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
, implying that if
then
, then any anticommutative bilinear map
satisfies
:
More generally, by
transposing two elements, any anticommutative multilinear map
satisfies
:
if any of the
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
is alternating then by bilinearity we have
:
and the proof in the multilinear case is the same but in only two of the inputs.
Examples
Examples of anticommutative binary operations include:
*
Cross product
* Lie bracket of a
Lie algebra
* Lie bracket of a
Lie ring
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
*
Subtraction
See also
*
Commutativity
*
Commutator
*
Exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
*
Graded-commutative ring In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy
:xy = (-1)^ yx,
where , ''x'' , and , ''y'' , d ...
*
Operation (mathematics)
In mathematics, an operation is a function which takes zero or more input values (also called "'' operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.
The most commonly studied opera ...
*
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
Given a structured obje ...
*
Particle statistics
Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble (an idealization comprising the state space of possible states of a system, each labeled w ...
(for anticommutativity in physics).
References
*.
External links
*. Which references th
Original Russian work*{{MathWorld
, title=Anticommutative
, urlname=Anticommutative
Properties of binary operations