Commutator

For an electrical switch that periodically reverses the current, see commutator (electric). In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable A commutator is an electrical Switch that periodically reverses the current direction in an Electric motor or Electrical generator

For Telemetry multiplexer-demultiplexer, see decommutator. Telemetry (synonymous with Telematics) is a Technology that allows the remote measurement and reporting of Information of interest to the system designer In Telemetry, decommutator decom or decommutation is the process of removing multiple channels of encoded data from a frameworked and synchronized serial Bit

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, commutativity is the ability to change the order of something without changing the end result There are different definitions used in group theory and ring theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

1 Group theory
- 1.1 Identities
2 Ring theory
- 2.1 Identities
3 Graded Rings and Algebras
4 Derivations
5 See also
6 References

Group theory

The commutator of two elements g and h of a group G is the element

[g, h] = g⁻¹h⁻¹gh

It is equal to the group's identity if and only if g and h commute (i. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element e. if and only if gh = hg). The subgroup of $G$ generated by all commutators is called the derived group or the commutator subgroup of G. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups. In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally

N. B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as

[g, h] = ghg⁻¹h⁻¹

Identities

In the sequel the expression a^x denotes the conjugated (by x) element x⁻¹a x.

$[y,x] = [x,y]^{-1}\,$ .
$[[x, y^{-1}], z]^y\cdot[[y, z^{-1}], x]^z\cdot[[z, x^{-1}], y]^x = 1$ .
$[x y, z] = [x, z]^y\cdot [y, z]$ .
$[x, y z] = [x, z]\cdot [x, y]^z$ .

The second identity is also known under the name Hall-Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). The fourth identity follows from the first and third.

N. B. The above definition of the conjugate of a by x is used by group theorists. Many other mathematicians define the conjugate of a by x as xax⁻¹. This is usually written $x a$ .

Ring theory

The commutator of two elements a and b of a ring or an associative algebra is defined by

[a, b] = ab − ba

It is zero if and only if a and b commute. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In linear algebra, if two endomorphisms of a space are represented by commuting matrices with respect to one basis, then they are so represented with respect to every basis. Linear algebra is the branch of Mathematics concerned with By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. Lie bracket can refer to Lie algebra Lie bracket of vector fields In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. This article assumes some familiarity with Analytic geometry and the concept of a limit. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical The uncertainty principle is ultimately a theorem about these commutators. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements

Identities

The commutator has the following properties:

Lie-algebra relations:

$[A,A] = 0 \,\!$
$[A,B] = - [B,A] \,\!$
$[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 \,\!$

The second relation is called anticommutativity, while the third is the Jacobi identity. In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation

Additional relations:

$[A,BC] = [A,B]C + B[A,C] \,\!$
$[AB,C] = A[B,C] + [A,C]B \,\!$
$[A,BC] = [AB,C] + [CA,B] \,\!$
$[ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC \,\!$
$[[[A,B], C], D] + [[[B,C], D], A] + [[[C, D], A], B] + [[[D, A], B], C] = [[A, C], [B, D]] \,\!$

If $A$ is a fixed element of a ring $\scriptstyle\mathfrak{R}$ , the first additional relation can also be interpreted as a Leibniz rule for the map $\scriptstyle D_A: R \rightarrow R$ given by $\scriptstyle B \mapsto [A,B]$ . In other words: the map $D A$ defines a derivation on the ring $\scriptstyle\mathfrak{R}$ . In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator

The following identity involving commutators, a special case of the Baker-Campbell-Hausdorff formula, is also useful:

$e^{A}Be^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...$

Graded Rings and Algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as $\ [\omega,\eta]_{gr} := \omega\eta - (-1)^{\deg \omega \deg \eta} \eta\omega$

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful involving the adjoint representation:

$\operatorname{ad} (x)(y) = [x, y] .$

Then $ad(x)$ is a derivation and $ad$ is linear, i. In Mathematics, the Baker-Campbell-Hausdorff formula is the solution to Z = \log(e^X e^Y\ for non- commuting X In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator e. , $ad(x + y) = ad(x) + ad(y)$ and ${\rm ad} (\lambda x)=\lambda\,\operatorname{ad} (x)$ , and a Lie algebra homomorphism, i. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie e, $ad([x, y]) = [ad(x),ad(y)]$ , but it is not always an algebra homomorphism, i. e the identity $\operatorname{ad}(xy) = \operatorname{ad}(x)\operatorname{ad}(y)$ does not hold in general.

Examples:

${\rm ad} (x){\rm ad} (x)(y) = [x,[x,y]\,]$
${\rm ad} (x){\rm ad} (a+b)(y) = [x,[a+b,y]\,]$

References

Griffiths, David J. In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator In Mathematics, the Pincherle derivative of a Linear operator \scriptstyle{ T\mathbb K \longrightarrow \mathbb K } on the Vector space of In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is (2004). Introduction to Quantum Mechanics, 2nd ed. , Prentice Hall. ISBN 0-13-805326-X.
Liboff, Richard L. (2002). Richard L Liboff is a US physicist who has authored five books and nearly 150 other publications in variety of fields including Plasma physics, Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

Dictionary

-noun

an electrical switch, in a generator or motor, that periodically reverses the direction of an electric current
(mathematics) (of a group) an element of the form ghg⁻¹h⁻¹ where g and h are elements of the group; it is equal to the group's identity if and only if g and h commute
(mathematics) (of a ring) an element of the form ab-ba, where a and b are elements of the ring, it is identical to the ring's zero element if and only if a and b commute

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents

Group theory

Identities

Ring theory

Identities

Graded Rings and Algebras

Derivations

See also

References

Dictionary

commutator

-noun