Editing Adjoint representation (section)

== Adjoint representation of a Lie algebra ==
Let <math>\mathfrak{g}</math> be a Lie algebra over some field. Given an element {{mvar|x}} of a Lie algebra <math>\mathfrak{g}</math>, one defines the adjoint action of {{mvar|x}} on <math>\mathfrak{g}</math> as the map 
:<math>\operatorname{ad}_x : \mathfrak{g} \to \mathfrak{g} \qquad\text{with}\qquad \operatorname{ad}_x (y) = [x, y]</math>
for all {{mvar|y}} in <math>\mathfrak{g}</math>. It is called the '''adjoint endomorphism''' or '''adjoint action'''. (<math>\operatorname{ad}_x</math> is also often denoted as <math>\operatorname{ad}(x)</math>.) Since a bracket is bilinear, this determines the [[linear map|linear mapping]]
:<math>\operatorname{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) = (\operatorname{End}(\mathfrak{g}), [\;,\;])</math>
given by {{math|''x'' ↦ ad<sub>''x''</sub>}}. Within End<math>(\mathfrak{g})</math>, the bracket is, by definition, given by the commutator of the two operators:
:<math>[T, S] = T \circ S - S \circ T</math>
where <math>\circ</math> denotes composition of linear maps. Using the above definition of the bracket, the [[Jacobi identity]]
:<math>[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0</math> 
takes the form 
:<math>\left([\operatorname{ad}_x, \operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x, y]}\right)(z)</math>
where {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are arbitrary elements of <math>\mathfrak{g}</math>.

This last identity says that ad is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets. Hence, ad is a [[representation of a Lie algebra]] and is called the '''adjoint representation''' of the algebra <math>\mathfrak{g}</math>.

If <math>\mathfrak{g}</math> is finite-dimensional and a basis for it is chosen, then <math>\mathfrak{gl}(\mathfrak{g})</math> is the Lie algebra of square matrices and the composition corresponds to [[matrix multiplication]].

In a more module-theoretic language, the construction says that <math>\mathfrak{g}</math> is a module over itself.

The kernel of ad is the [[center (algebra)|center]] of <math>\mathfrak{g}</math> (that's just rephrasing the definition). On the other hand, for each element {{mvar|z}} in <math>\mathfrak{g}</math>, the linear mapping <math>\delta = \operatorname{ad}_z</math> obeys the [[general Leibniz rule|Leibniz' law]]:
:<math>\delta ([x, y]) = [\delta(x),y] + [x, \delta(y)]</math>
for all {{mvar|x}} and {{mvar|y}} in the algebra (the restatement of the Jacobi identity). That is to say, ad<sub>''z''</sub> is a [[Lie algebra extension#Derivations|derivation]] and the image of <math>\mathfrak{g}</math> under ad is a subalgebra of Der<math>(\mathfrak{g})</math>, the space of all derivations of <math>\mathfrak{g}</math>.

When <math>\mathfrak{g} = \operatorname{Lie}(G)</math> is the Lie algebra of a Lie group ''G'', [[#Derivative of Ad|ad is the differential of Ad]] at the identity element of ''G''.

There is the following formula similar to the [[General Leibniz rule|Leibniz formula]]: for scalars <math>\alpha, \beta</math> and Lie algebra elements <math>x, y, z</math>,
:<math>(\operatorname{ad}_x - \alpha - \beta)^n [y, z] = \sum_{i = 0}^n \binom{n}{i} \left[(\operatorname{ad}_x - \alpha)^i y, (\operatorname{ad}_x - \beta)^{n - i} z\right].</math>