Editing Commutator (section)

== Ring theory ==
<!-- This section is linked from [[Lie algebra]] -->
[[ring (algebra)|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
: <math>[a, b] = ab - ba.</math>

The commutator is zero if and only if ''a'' and ''b'' commute. In [[linear algebra]], if two [[endomorphism]]s 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 algebra|Lie bracket]], every associative algebra can be turned into a [[Lie algebra]].

The '''anticommutator''' of two elements {{mvar|a}} and {{mvar|b}} of a ring or associative algebra is defined by
: <math>\{a, b\} = ab + ba.</math>

Sometimes <math>[a,b]_+</math> is used to denote anticommutator, while <math>[a,b]_-</math> is then used for commutator.<ref>{{harvtxt|McMahon|2008}}</ref> The anticommutator is used less often, but can be used to define [[Clifford algebra]]s and [[Jordan algebra]]s 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 [[observable]]s described by these operators can be measured simultaneously. The [[uncertainty principle]] is ultimately a theorem about such commutators, by virtue of the [[Uncertainty relation|Robertson–Schrödinger relation]].<ref>{{harvtxt|Liboff|2003|pp=140–142}}</ref> In [[phase space]], equivalent commutators of function [[Moyal product|star-products]] are called [[Moyal bracket]]s and are completely isomorphic to the Hilbert space commutator structures mentioned.

=== Identities (ring theory) ===
The commutator has the following properties:

==== Lie-algebra identities ====
# <math>[A + B, C] = [A, C] + [B, C]</math>
# <math>[A, A] = 0</math>
# <math>[A, B] = -[B, A]</math>
# <math>[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0</math>

Relation (3) is called [[anticommutativity]], while (4) is the [[Jacobi identity]].

==== Additional identities ====
# <math>[A, BC] = [A, B]C + B[A, C]</math>
# <math>[A, BCD] = [A, B]CD + B[A, C]D + BC[A, D]</math>
# <math>[A, BCDE] = [A, B]CDE + B[A, C]DE + BC[A, D]E + BCD[A, E]</math>
# <math>[AB, C] = A[B, C] + [A, C]B</math>
# <math>[ABC, D] = AB[C, D] + A[B, D]C + [A, D]BC</math>
# <math>[ABCD, E] = ABC[D, E] + AB[C, E]D + A[B, E]CD + [A, E]BCD</math>
# <math>[A, B + C] = [A, B] + [A, C]</math>
# <math>[A + B, C + D] = [A, C] + [A, D] + [B, C] + [B, D]</math>
# <math>[AB, CD] = A[B, C]D + [A, C]BD + CA[B, D] + C[A, D]B =A[B, C]D + AC[B,D] + [A,C]DB + C[A, D]B</math>
# <math>[[A, C], [B, D]] = [[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C]</math>

If {{mvar|A}} is a fixed element of a ring ''R'', identity (1) can be interpreted as a [[product rule|Leibniz rule]] for the map <math>\operatorname{ad}_A: R \rightarrow R</math> given by <math>\operatorname{ad}_A(B) = [A, B]</math>.  In other words, the map ad<sub>''A''</sub> defines a [[derivation (abstract algebra)|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'''-[[Bilinear map|bilinearity]].

From identity (9), one finds that the commutator of integer powers of ring elements is:
: <math>[A^N, B^M] = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} A^{n}B^{m} [A,B] B^{N-n-1}A^{M-m-1} = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} B^{n}A^{m} [A,B] A^{N-n-1}B^{M-m-1}</math>

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.<ref>{{harvtxt|Lavrov|2014}}</ref>
For example:
# <math>[AB, C]_\pm = A[B, C]_- + [A, C]_\pm B</math>
# <math>[AB, CD]_\pm = A[B, C]_- D + AC[B, D]_- + [A, C]_- DB + C[A, D]_\pm B</math>
# <math>[[A,B],[C,D]]=[[[B,C]_+,A]_+,D]-[[[B,D]_+,A]_+,C]+[[[A,D]_+,B]_+,C]-[[[A,C]_+,B]_+,D]</math>
# <math>\left[A, [B, C]_\pm\right] + \left[B, [C, A]_\pm\right] + \left[C, [A, B]_\pm\right] = 0</math>
# <math>[A,BC]_\pm = [A,B]_- C + B[A,C]_\pm = [A,B]_\pm C \mp B[A,C]_-</math>
# <math>[A,BC] = [A,B]_\pm C \mp B[A,C]_\pm</math>

==== Exponential identities ====
Consider a ring or algebra in which the [[exponential function|exponential]] <math>e^A = \exp(A) = 1 + A + \tfrac{1}{2!}A^2 + \cdots</math> 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: <math display="inline">e^A Be^{-A}
  \ =\  B + [A, B] + \frac{1}{2!}[A, [A, B]] + \frac{1}{3!}[A, [A, [A, B]]] + \cdots
  \ =\  e^{\operatorname{ad}_A}(B).
</math> (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 <math>e^A</math> (analogous to elements of a [[Lie group]]) in terms of a series of nested commutators (Lie brackets),
<math display="block">e^A e^B e^{-A} e^{-B} =
\exp\!\left(  [A, B] + \frac{1}{2!}[A{+}B, [A, B]] + \frac{1}{3!} \left(\frac{1}{2} [A, [B, [B, A]]] + [A{+}B, [A{+}B, [A, B]]]\right) + \cdots\right). </math>