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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 in an electric motor or electrical generator. ...
In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java...
In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
Group theory is that branch of mathematics concerned with the study of 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 familiar from the integers. ...
Group theory
The commutator of two elements g and h of a group G is the element In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
- [g, h] = g−1h−1gh
It is equal to the group's identity if and only if g and h commute (i.e. if and only if gh = hg). The subgroup generated by all commutators is called the derived group or the commutator subgroup of G. 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 groups. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. The commutator subgroup can also be defined as the set of elements g of the group...
In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...
N.B. Some authors choose to define the commutator as
- [g, h] = ghg−1h−1
Identities In the sequel the expression ax denotes the conjugated (by x) element x−1a x. - [y,x] = [x,y] −1
- [[x,y−1],z] y [[y,z−1],x] z [[z,x−1],y]x = 1
- [xy,z] = [x,z]y [y,z]
- [x,yz] = [x,z] [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).
Ring theory The commutator of two elements a and b of a rings or associative algebra is defined by In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...
In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ...
- [a, b] = ab − ba
It is zero if and only if a and b commute. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. 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. The uncertainty principle is ultimately a theorem about these commutators. A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
Fig. ...
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 operations. ...
In quantum physics, the Heisenberg uncertainty principle states that one cannot assign with full precision values for certain pairs of observable variables, including the position and momentum, of a single particle at the same time even in theory. ...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
Likewise, the anticommutator is defined as ab + ba, often written { a, b }. See also Poisson algebra. A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ...
Identities The commutator has the following properties:
Lie-algebra relations:
- [A,B] = − [B,A]
- [A,A] = 0
- [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0
Additional relations: - [A,BC] = [A,B]C + B[A,C]
- [AB,C] = A[B,C] + [A,C]B
- [ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC
If A is a fixed element of a ring R, the first additional relation can also be interpreted as a Leibniz rule for the map given by In other words: the map DA defines a derivation on the ring R. At least two results in calculus are called Leibnizs rule or the Leibniz rule, in honor of Gottfried Leibniz. ...
In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...
See also In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...
In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...
References - Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X.
- Liboff, Richard L. (2002). Introductory Quantum Mechanics, Addison-Wesley. ISBN 0805387145.
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