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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
and ring theory.


Group theory

The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
and solvable groups and the largest
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as :.


Identities (group theory)

Commutator identities are an important tool in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
. The expression denotes the conjugate of by , defined as . # x^y = x , y # , x= ,y. # , zy= , ycdot , zy and z, y= , yz \cdot , y # \left , y^\right= , x and \left ^, y\right= , x. # \left left[x,_y^\right_z\right.html" ;"title=",_y^\right.html" ;"title="left[x, y^\right">left[x, y^\right z\right">,_y^\right.html" ;"title="left[x, y^\right">left[x, y^\right z\righty \cdot \left[\left[y, z^\right], x\right]^z \cdot \left[\left[z, x^\right], y\right]^x = 1 and \left[\left[x, y\right], z^x\right] \cdot \leftz ,x], y^z\right] \cdot \lefty, z], x^y\right] = 1. Identity (5) is also known as the ''Hall–Witt identity'', after Philip Hall and Ernst Witt. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). N.B., the above definition of the conjugate of by is used by some group theorists. Many other group theorists define the conjugate of by as . This is often written ^x a. Similar identities hold for these conventions. Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well: :(xy)^2 = x^2 y^2 , x , x y]. If the
derived subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
is central, then :(xy)^n = x^n y^n , x\binom.


Ring theory

Rings often do not support division. Thus, the commutator of two elements ''a'' and ''b'' of a ring (or any associative algebra) is defined differently by : , b= ab - ba. The commutator is zero if and only if ''a'' and ''b'' commute. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. The anticommutator of two elements and of a ring or associative algebra is defined by : \ = ab + ba. Sometimes ,b+ is used to denote anticommutator, while ,b- is then used for commutator. The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras and in the derivation of the Dirac equation in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. The commutator of two operators acting on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is a central concept in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation. In phase space, equivalent commutators of function star-products are called
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a len ...
s and are completely isomorphic to the Hilbert space commutator structures mentioned.


Identities (ring theory)

The commutator has the following properties:


Lie-algebra identities

# + B, C= , C+ , C/math> #
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= 0 # , B= -
, A The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> # ,_[B,_C_+_[B,_[C,_A.html"_;"title=",_C.html"_;"title=",_[B,_C">,_[B,_C_+_[B,_[C,_A">,_C.html"_;"title=",_[B,_C">,_[B,_C_+_[B,_[C,_A_+_[C,_[A,_B.html" ;"title=",_C">,_[B,_C_+_[B,_[C,_A.html" ;"title=",_C.html" ;"title=", [B, C">, [B, C + [B, [C, A">,_C.html" ;"title=", [B, C">, [B, C + [B, [C, A + [C, [A, B">,_C">,_[B,_C_+_[B,_[C,_A.html" ;"title=",_C.html" ;"title=", [B, C">, [B, C + [B, [C, A">,_C.html" ;"title=", [B, C">, [B, C + [B, [C, A + [C, [A, B = 0 Relation (3) is called anticommutativity, while (4) is the Jacobi identity.


Additional identities

# [A, BC] = , B + B , C/math> #
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= , BD + B , C + BC
, D The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> # , BCDE= , BDE + B , CE + BC
, D The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
+ BCD
, E The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> # B, C= A , C+ , C # BC, D= AB
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+ A
, D The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
+
, D The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
C # BCD, E= ABC
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+ AB
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+ A
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D +
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CD # , B + C= , B+ , C/math> # + B, C + D= , C+
, D The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
+ , C+
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/math> # B, CD= A , C + , CD + CA
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+ C
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=A , C + AC ,D+ ,CB + C
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# A, C
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= [A,_B_C.html"_;"title=",_B.html"_;"title="[A,_B">[A,_B_C">,_B.html"_;"title="[A,_B">[A,_B_C_D.html" ;"title=",_B">[A,_B_C.html" ;"title=",_B.html" ;"title="[A, B">[A, B C">,_B.html" ;"title="[A, B">[A, B C D">,_B">[A,_B_C.html" ;"title=",_B.html" ;"title="[A, B">[A, B C">,_B.html" ;"title="[A, B">[A, B C D+ [B, C], D], A] + [C, D], A], B] + [D, A], B], C] If is a fixed element of a ring ''R'', identity (1) can be interpreted as a product rule, Leibniz rule for the map \operatorname_A: R \rightarrow R given by \operatorname_A(B) = , B/math>. In other words, the map ad''A'' defines a derivation on the ring ''R''. Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation. Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express Z- bilinearity. Some of the above identities can be extended to the anticommutator using the above ± subscript notation. For example: # B, C\pm = A , C- + , C\pm B # B, CD\pm = A , C- D + AC
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- + , C- DB + C
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\pm B # A,B ,D= [B,C+,A.html"_;"title=",C.html"_;"title="[B,C">[B,C+,A">,C.html"_;"title="[B,C">[B,C+,A+,D.html" ;"title=",C">[B,C+,A.html" ;"title=",C.html" ;"title="[B,C">[B,C+,A">,C.html" ;"title="[B,C">[B,C+,A+,D">,C">[B,C+,A.html" ;"title=",C.html" ;"title="[B,C">[B,C+,A">,C.html" ;"title="[B,C">[B,C+,A+,D[B,D]_+,A]_+,C]+[A,D]_+,B]_+,C]- ,C+,B]_+,D] #\left[A, , C\pm\right] + \left[B, [C, A]_\pm\right] + \left[C, [A, B]_\pm\right] = 0 # ,BC\pm = ,B- C + B ,C\pm # ,BC= ,B\pm C \mp B ,C\pm


Exponential identities

Consider a ring or algebra in which the exponential e^A = \exp(A) = 1 + A + \tfracA^2 + \cdots can be meaningfully defined, such as a Banach algebra or a ring of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
. In such a ring, Hadamard's lemma applied to nested commutators gives: e^A Be^ \ =\ B + , B+ \frac ,_[A,_B_+_\frac[A,__,_[A,_B.html"_;"title=",_B.html"_;"title=",_[A,_B">,_[A,_B_+_\frac[A,__,_[A,_B">,_B.html"_;"title=",_[A,_B">,_[A,_B_+_\frac[A,__,_[A,_B+_\cdots __\_=\__e^(B). _(For_the_last_expression,_see_''Adjoint_derivation''_below.)_This_formula_underlies_the_Baker–Campbell–Hausdorff_formula#An_important_lemma.html" ;"title=",_B">,_[A,_B.html" ;"title=",_B.html" ;"title=", [A, B">, [A, B + \frac[A, , [A, B">,_B.html" ;"title=", [A, B">, [A, B + \frac[A, , [A, B+ \cdots \ =\ e^(B). (For the last expression, see ''Adjoint derivation'' below.) This formula underlies the Baker–Campbell–Hausdorff formula#An important lemma">Baker–Campbell–Hausdorff expansion of log(exp(''A'') exp(''B'')). A similar expansion expresses the group commutator of expressions e^A (analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets), e^A e^B e^ e^ = \exp\!\left( , B+ \frac[AB, [A, B + \frac \left(\frac [A, [B, [B, A] + [AB, [AB, [A, B]\right) + \cdots\right).


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 := \omega\eta - (-1)^ \eta\omega.


Adjoint derivation

Especially if one deals with multiple commutators in a ring ''R'', another notation turns out to be useful. For an element x\in R, we define the adjoint mapping \mathrm_x:R\to R by: :\operatorname_x(y) = , y= xy-yx. This mapping is a derivation on the ring ''R'': :\mathrm_x\!(yz) \ =\ \mathrm_x\!(y) \,z \,+\, y\,\mathrm_x\!(z). By the Jacobi identity, it is also a derivation over the commutation operation: :\mathrm_x ,z\ =\ mathrm_x\!(y),z\,+\, ,\mathrm_x\!(z). Composing such mappings, we get for example \operatorname_x\operatorname_y(z) = ,_[y,_z,.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z,">,_z.html" ;"title=", [y, z">, [y, z, and \operatorname_x^2\!(z) \ =\ \operatorname_x\!(\operatorname_x\!(z)) \ =\ [x, [x, z]\,]. We may consider \mathrm itself as a mapping, \mathrm: R \to \mathrm(R) , where \mathrm(R) is the ring of mappings from ''R'' to itself with composition as the multiplication operation. Then \mathrm is a Lie algebra homomorphism, preserving the commutator: :\operatorname_ = \left \operatorname_x, \operatorname_y \right By contrast, it is not always a ring homomorphism: usually \operatorname_ \,\neq\, \operatorname_x\operatorname_y .


General Leibniz rule

The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation: :x^n y = \sum_^n \binom \operatorname_x^k\!(y)\, x^. Replacing ''x'' by the differentiation operator \partial, and ''y'' by the multiplication operator m_f : g \mapsto fg, we get \operatorname(\partial)(m_f) = m_, and applying both sides to a function ''g'', the identity becomes the usual Leibniz rule for the ''n''-th derivative \partial^\!(fg).


See also

* Anticommutativity * Associator * Baker–Campbell–Hausdorff formula * Canonical commutation relation * Centralizer a.k.a. commutant * Derivation (abstract algebra) *
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a len ...
* Pincherle derivative * Poisson bracket * Ternary commutator * Three subgroups lemma


Notes


References

* * * * * * *


Further reading

*


External links

* {{Authority control Abstract algebra Group theory Binary operations Mathematical identities