Editing Commutator (section)

== Adjoint derivation ==
Especially if one deals with multiple commutators in a ring ''R'', another notation turns out to be useful. For an element <math>x\in R</math>, we define the [[adjoint representation of a Lie algebra|adjoint]] mapping <math>\mathrm{ad}_x:R\to R</math> by:
: <math>\operatorname{ad}_x(y) = [x, y] = xy-yx.</math>

This mapping is a [[Derivation (differential algebra)|derivation]] on the ring ''R'': 
: <math>\mathrm{ad}_x\!(yz) \ =\ \mathrm{ad}_x\!(y) \,z \,+\, y\,\mathrm{ad}_x\!(z).</math>
By the [[Jacobi identity]], it is also a derivation over the commutation operation: 
: <math>\mathrm{ad}_x[y,z] \ =\ [\mathrm{ad}_x\!(y),z] \,+\, [y,\mathrm{ad}_x\!(z)] .</math>

Composing such mappings, we get for example <math>\operatorname{ad}_x\operatorname{ad}_y(z) = [x, [y, z]\,] </math> and <math display="block">\operatorname{ad}_x^2\!(z) \ =\ 
\operatorname{ad}_x\!(\operatorname{ad}_x\!(z)) \ =\ 
[x, [x, z]\,].</math> We may consider <math>\mathrm{ad}</math> itself as a mapping, <math>\mathrm{ad}: R \to \mathrm{End}(R) </math>, where <math>\mathrm{End}(R)</math> is the ring of mappings from ''R'' to itself with composition as the multiplication operation. Then <math>\mathrm{ad}</math> is a [[Lie algebra]] homomorphism, preserving the commutator:
: <math>\operatorname{ad}_{[x, y]} = \left[ \operatorname{ad}_x, \operatorname{ad}_y \right].</math>

By contrast, it is '''not''' always a ring homomorphism: usually <math>\operatorname{ad}_{xy} \,\neq\, \operatorname{ad}_x\operatorname{ad}_y </math>.

=== General Leibniz rule ===
The [[general Leibniz rule]], expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:
: <math>x^n y = \sum_{k = 0}^n \binom{n}{k} \operatorname{ad}_x^k\!(y)\, x^{n - k}.</math>

Replacing <math>x</math> by the differentiation operator <math>\partial</math>, and <math>y</math> by the multiplication operator <math>m_f : g \mapsto fg</math>, we get <math>\operatorname{ad}(\partial)(m_f) = m_{\partial(f)}</math>, and applying both sides to a function ''g'', the identity becomes the usual Leibniz rule for the ''n''th derivative <math>\partial^{n}\!(fg)</math>.