The commutator of two elements, g and h, of a
groupG, is the element
g, h] = g−1h−1gh.
This element is equal to the group's identity if and only if g and h commute (from the definition gh = hg [g, h, being g, h] equal to the identity 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 Ggenerated 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
abelianquotient group.
The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as
Commutator identities are an important tool in
group theory.[3] The expression ax denotes the
conjugate of a by x, defined as x−1ax.
and
and
and
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 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 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:
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
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 adA 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.
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:
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:
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 n-th derivative .