Editing Lie algebra (section)

=== Derivations ===
For an [[algebra over a field|algebra]] ''A'' over a field ''F'', a [[derivation (abstract algebra)|''derivation'']] of ''A'' over ''F'' is a linear map <math>D\colon A\to A</math> that satisfies the [[product rule|Leibniz rule]]
:<math>D(xy) = D(x)y + xD(y)</math>
for all <math>x,y\in A</math>. (The definition makes sense for a possibly [[non-associative algebra]].) Given two derivations <math>D_1</math> and <math>D_2</math>, their commutator <math>[D_1,D_2]:=D_1D_2-D_2D_1</math> is again a derivation. This operation makes the space <math>\text{Der}_k(A)</math> of all derivations of ''A'' over ''F'' into a Lie algebra.<ref>{{harvnb|Humphreys|1978|p=4.}}</ref>

Informally speaking, the space of derivations of ''A'' is the Lie algebra of the [[automorphism group]] of ''A''. (This is literally true when the automorphism group is a Lie group, for example when ''F'' is the real numbers and ''A'' has finite dimension as a vector space.) For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of ''A''. Indeed, writing out the condition that
:<math>(1+\epsilon D)(xy) \equiv (1+\epsilon D)(x)\cdot (1+\epsilon D)(y) \pmod{\epsilon^2}</math>
(where 1 denotes the identity map on ''A'') gives exactly the definition of ''D'' being a derivation.

'''Example: the Lie algebra of vector fields.''' Let ''A'' be the ring <math>C^{\infty}(X)</math> of [[smooth function]]s on a smooth manifold ''X''. Then a derivation of ''A'' over <math>\mathbb{R}</math> is equivalent to a [[vector field]] on ''X''. (A vector field ''v'' gives a derivation of the space of smooth functions by differentiating functions in the direction of ''v''.) This makes the space <math>\text{Vect}(X)</math> of vector fields into a Lie algebra (see [[Lie bracket of vector fields]]).<ref>{{harvnb|Varadarajan|1984|p=49.}}</ref> Informally speaking, <math>\text{Vect}(X)</math> is the Lie algebra of the [[diffeomorphism group]] of ''X''. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An [[group action|action]] of a Lie group ''G'' on a manifold ''X'' determines a homomorphism of Lie algebras <math>\mathfrak{g}\to \text{Vect}(X)</math>. (An example is illustrated below.)

A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra <math>\mathfrak{g}</math> over a field ''F'' determines its Lie algebra of derivations, <math>\text{Der}_F(\mathfrak{g})</math>. That is, a derivation of <math>\mathfrak{g}</math> is a linear map <math>D\colon \mathfrak{g}\to \mathfrak{g}</math> such that
:<math>D([x,y])=[D(x),y]+[x,D(y)]</math>.
The ''inner derivation'' associated to any <math>x\in\mathfrak g</math> is the adjoint mapping <math>\mathrm{ad}_x</math> defined by <math>\mathrm{ad}_x(y):=[x,y]</math>. (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras, <math>\operatorname{ad}\colon\mathfrak{g}\to \text{Der}_F(\mathfrak{g})</math>. The image <math>\text{Inn}_F(\mathfrak{g})</math> is an ideal in <math>\text{Der}_F(\mathfrak{g})</math>, and the Lie algebra of ''outer derivations'' is defined as the quotient Lie algebra, <math>\text{Out}_F(\mathfrak{g})=\text{Der}_F(\mathfrak{g})/\text{Inn}_F(\mathfrak{g})</math>. (This is exactly analogous to the [[outer automorphism group]] of a group.) For a [[semisimple Lie algebra]] (defined below) over a field of characteristic zero, every derivation is inner.<ref>{{harvnb|Serre|2006|loc=Part I, section VI.3.}}</ref> This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.<ref>{{harvnb|Fulton|Harris|1991|loc=Proposition D.40.}}</ref>

In contrast, an abelian Lie algebra has many outer derivations. Namely, for a vector space <math>V</math> with Lie bracket zero, the Lie algebra <math>\text{Out}_F(V)</math> can be identified with <math>\mathfrak{gl}(V)</math>.