In 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 and ring theory.
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Calculating Commutators

GT7. The Commutator Subgroup

Commutator Subgroup of a Group  Group Theory  lesson 49

Commutator , commutator subgroup  Theoretical part  abstract algebra  math with Akash Tripathi

Commutator of group and their theroms . And its property
Transcription
Group theory
The commutator of two elements, g and h, of a group G, is the element
 [g, h] = g^{−1}h^{−1}gh.
This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg).
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 and solvable groups and the largest abelian quotient group.
The definition of the commutator above is used throughout this article, but many group theorists define the commutator as
 [g, h] = ghg^{−1}h^{−1}.^{[1]}^{[2]}
Using the first definition, this can be expressed as [g^{−1}, h^{−1}].
Identities (group theory)
Commutator identities are an important tool in group theory.^{[3]} The expression a^{x} denotes the conjugate of a by x, defined as x^{−1}ax.
 and
 and
 and
Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. It is a grouptheoretic analogue of the Jacobi identity for the ringtheoretic commutator (see next section).
N.B., the above definition of the conjugate of a by x is used by some group theorists.^{[4]} Many other group theorists define the conjugate of a by x as xax^{−1}.^{[5]} This is often written . Similar identities hold for these conventions.
Many identities that are true modulo certain subgroups are also used. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:
If the derived subgroup is central, then
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
The commutator is zero if and only if a and b commute. In linear algebra, 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 a and b of a ring or associative algebra is defined by
Sometimes is used to denote anticommutator, while is then used for commutator.^{[6]} 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.
The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, 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.^{[7]} In phase space, equivalent commutators of function starproducts are called Moyal brackets and are completely isomorphic to the Hilbert space commutator structures mentioned.
Identities (ring theory)
The commutator has the following properties:
Liealgebra identities
Relation (3) is called anticommutativity, while (4) is the Jacobi identity.
Additional identities
If A is a fixed element of a ring R, identity (1) can be interpreted as a Leibniz rule for the map given by . 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 Zbilinearity.
From identity (9), one finds that the commutator of integer powers of ring elements is:
Some of the above identities can be extended to the anticommutator using the above ± subscript notation.^{[8]} For example:
Exponential identities
Consider a ring or algebra in which the exponential can be meaningfully defined, such as a Banach algebra or a ring of formal power series.
In such a ring, Hadamard's lemma applied to nested commutators gives: (For the last expression, see Adjoint derivation below.) This formula underlies the Baker–Campbell–Hausdorff expansion of log(exp(A) exp(B)).
A similar expansion expresses the group commutator of expressions (analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets),
Graded rings and algebras
When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as
Adjoint derivation
Especially if one deals with multiple commutators in a ring R, another notation turns out to be useful. For an element , we define the adjoint mapping by:
This mapping is a derivation on the ring R:
By the Jacobi identity, it is also a derivation over the commutation operation:
Composing such mappings, we get for example and We may consider itself as a mapping, , where is the ring of mappings from R to itself with composition as the multiplication operation. Then is a Lie algebra homomorphism, preserving the commutator:
By contrast, it is not always a ring homomorphism: usually .
General Leibniz rule
The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:
Replacing by the differentiation operator , and by the multiplication operator , we get , and applying both sides to a function g, the identity becomes the usual Leibniz rule for the nth derivative .
See also
 Anticommutativity
 Associator
 Baker–Campbell–Hausdorff formula
 Canonical commutation relation
 Centralizer a.k.a. commutant
 Derivation (abstract algebra)
 Moyal bracket
 Pincherle derivative
 Poisson bracket
 Ternary commutator
 Three subgroups lemma
Notes
 ^ Fraleigh (1976, p. 108)
 ^ Herstein (1975, p. 65)
 ^ McKay (2000, p. 4)
 ^ Herstein (1975, p. 83)
 ^ Fraleigh (1976, p. 128)
 ^ McMahon (2008)
 ^ Liboff (2003, pp. 140–142)
 ^ Lavrov (2014)
References
 Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: AddisonWesley, ISBN 0201019841
 Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 013805326X
 Herstein, I. N. (1975), Topics In Algebra (2nd ed.), Wiley, ISBN 0471010901
 Lavrov, P.M. (2014), "Jacobi type identities in algebras and superalgebras", Theoretical and Mathematical Physics, 179 (2): 550–558, arXiv:1304.5050, Bibcode:2014TMP...179..550L, doi:10.1007/s1123201401612, S2CID 119175276
 Liboff, Richard L. (2003), Introductory Quantum Mechanics (4th ed.), AddisonWesley, ISBN 0805387145
 McKay, Susan (2000), Finite pgroups, Queen Mary Maths Notes, vol. 18, University of London, ISBN 9780902480179, MR 1802994
 McMahon, D. (2008), Quantum Field Theory, McGraw Hill, ISBN 9780071543828
Further reading
 McKenzie, R.; Snow, J. (2005), "Congruence modular varieties: commutator theory", in Kudryavtsev, V. B.; Rosenberg, I. G. (eds.), Structural Theory of Automata, Semigroups, and Universal Algebra, NATO Science Series II, vol. 207, Springer, pp. 273–329, doi:10.1007/1402038178_11, ISBN 9781402038174
External links
 "Commutator", Encyclopedia of Mathematics, EMS Press, 2001 [1994]