Editing Lie algebra (section)

==Definitions==
=== Subalgebras, ideals and homomorphisms ===
The Lie bracket is not required to be [[associative]], meaning that <math>[[x,y],z]</math> need not be equal to <math>[x,[y,z]]</math>. Nonetheless, much of the terminology for associative [[ring (mathematics)|rings]] and algebras (and also for groups) has analogs for Lie algebras. A '''Lie subalgebra'''  is a linear subspace <math>\mathfrak{h} \subseteq \mathfrak{g}</math> which is closed under the Lie bracket.  An '''ideal''' <math>\mathfrak i\subseteq\mathfrak{g}</math> is a linear subspace that satisfies the stronger condition:<ref>By the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.</ref>
:<math>[\mathfrak{g},\mathfrak i]\subseteq \mathfrak i.</math>

In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, and [[normal subgroup]]s correspond to ideals.

A Lie algebra '''homomorphism''' is a linear map compatible with the respective Lie brackets:
:<math> \phi\colon \mathfrak{g}\to\mathfrak{h}, \quad \phi([x,y])=[\phi(x),\phi(y)]\ \text{for all}\ x,y \in \mathfrak g. </math>
An '''isomorphism''' of Lie algebras is a [[bijective]] homomorphism.

As with normal subgroups in groups, ideals in Lie algebras are precisely the [[kernel (algebra)|kernels]] of homomorphisms. Given a Lie algebra <math>\mathfrak{g}</math> and an ideal <math>\mathfrak i</math> in it, the ''quotient Lie algebra'' <math>\mathfrak{g}/\mathfrak{i}</math> is defined, with a surjective homomorphism <math>\mathfrak{g}\to\mathfrak{g}/\mathfrak{i}</math> of Lie algebras. The [[first isomorphism theorem]] holds for Lie algebras: for any homomorphism <math>\phi\colon\mathfrak{g}\to\mathfrak{h}</math> of Lie algebras, the image of <math>\phi</math> is a Lie subalgebra of <math>\mathfrak{h}</math> that is isomorphic to <math>\mathfrak{g}/\text{ker}(\phi)</math>.

For the Lie algebra of a Lie group, the Lie bracket is a kind of infinitesimal commutator. As a result, for any Lie algebra, two elements <math>x,y\in\mathfrak g</math> are said to ''commute'' if their bracket vanishes: <math>[x,y]=0</math>.

The [[centralizer]] subalgebra of a subset <math>S\subset \mathfrak{g}</math> is the set of elements commuting with ''<math>S</math>'': that is, <math>\mathfrak{z}_{\mathfrak g}(S) = \{x\in\mathfrak g : [x, s] = 0 \ \text{ for all } s\in S\}</math>. The centralizer of <math>\mathfrak{g}</math> itself is the ''center''  <math>\mathfrak{z}(\mathfrak{g})</math>. Similarly, for a subspace ''S'', the [[normalizer]] subalgebra of ''<math>S</math>'' is <math>\mathfrak{n}_{\mathfrak g}(S) = \{x\in\mathfrak g : [x,s]\in S \ \text{ for all}\ s\in S\}</math>.<ref>{{harvnb|Jacobson|1979|p=28.}}</ref> If <math>S</math> is a Lie subalgebra, <math>\mathfrak{n}_{\mathfrak g}(S)</math> is the largest subalgebra such that <math>S</math> is an ideal of <math>\mathfrak{n}_{\mathfrak g}(S)</math>.

==== Example ====
The subspace <math>\mathfrak{t}_n</math> of diagonal matrices in <math>\mathfrak{gl}(n,F)</math> is an abelian Lie subalgebra. (It is a [[Cartan subalgebra]] of <math>\mathfrak{gl}(n)</math>, analogous to a [[maximal torus]] in the theory of [[compact Lie group]]s.) Here <math>\mathfrak{t}_n</math> is not an ideal in <math>\mathfrak{gl}(n)</math> for <math>n\geq 2</math>. For example, when <math>n=2</math>, this follows from the calculation:
<blockquote><math>\begin{align}
\left[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix},
\begin{bmatrix}
x & 0 \\
0 & y
\end{bmatrix}
\right] &= \begin{bmatrix}
ax & by\\
cx & dy \\
\end{bmatrix} - \begin{bmatrix}
ax & bx\\
cy & dy \\
\end{bmatrix} \\
&= \begin{bmatrix}
0 & b(y-x) \\
c(x-y) & 0
\end{bmatrix}
\end{align}</math></blockquote>
(which is not always in <math>\mathfrak{t}_2</math>).

Every one-dimensional linear subspace of a Lie algebra <math>\mathfrak{g}</math> is an abelian Lie subalgebra, but it need not be an ideal.

=== Product and semidirect product ===
For two Lie algebras <math>\mathfrak{g}</math> and <math>\mathfrak{g'}</math>, the ''[[direct product|product]]'' Lie algebra is the vector space <math>\mathfrak{g}\times \mathfrak{g'}</math> consisting of all ordered pairs <math>(x,x'), \,x\in\mathfrak{g}, \ x'\in\mathfrak{g'}</math>, with Lie bracket<ref>{{harvnb|Bourbaki|1989|loc=section I.1.1.}}</ref>
:<math> [(x,x'),(y,y')]=([x,y],[x',y']).</math>
This is the product in the [[product (category theory)|category]] of Lie algebras. Note that the copies of <math>\mathfrak g</math> and <math>\mathfrak g'</math> in <math>\mathfrak{g}\times \mathfrak{g'}</math> commute with each other: <math>[(x,0), (0,x')] = 0.</math>

Let <math>\mathfrak{g}</math> be a Lie algebra and <math>\mathfrak{i}</math> an ideal of <math>\mathfrak{g}</math>. If the canonical map <math>\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}</math> splits (i.e., admits a section <math>\mathfrak{g}/\mathfrak{i}\to \mathfrak{g}</math>, as a homomorphism of Lie algebras), then <math>\mathfrak{g}</math> is said to be a [[semidirect product]] of <math>\mathfrak{i}</math> and <math>\mathfrak{g}/\mathfrak{i}</math>, <math>\mathfrak{g}=\mathfrak{g}/\mathfrak{i}\ltimes\mathfrak{i}</math>. See also [[Lie algebra extension#By semidirect sum|semidirect sum of Lie algebras]].

=== 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>.