Editing Lie algebra (section)

== Structure theory and classification ==
Lie algebras can be classified to some extent. This is a powerful approach to the classification of Lie groups.

=== Abelian, nilpotent, and solvable ===
Analogously to [[abelian group|abelian]], [[nilpotent group|nilpotent]], and [[solvable group]]s, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra <math>\mathfrak{g}</math> is ''abelian{{anchor|abelian}}'' if the Lie bracket vanishes; that is, [''x'',''y''] = 0 for all ''x'' and ''y'' in <math>\mathfrak{g}</math>. In particular, the Lie algebra of an abelian Lie group (such as the group <math>\mathbb{R}^n</math> under addition or the [[torus|torus group]] <math>\mathbb{T}^n</math>) is abelian. Every finite-dimensional abelian Lie algebra over a field <math>F</math> is isomorphic to <math>F^n</math> for some <math>n\geq 0</math>, meaning an ''n''-dimensional vector space with Lie bracket zero.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the ''commutator subalgebra'' (or ''derived subalgebra'') of a Lie algebra <math>\mathfrak{g}</math> is <math>[\mathfrak{g},\mathfrak{g}]</math>, meaning the linear subspace spanned by all brackets <math>[x,y]</math> with <math>x,y\in\mathfrak{g}</math>. The commutator subalgebra is an ideal in <math>\mathfrak{g}</math>, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the [[commutator subgroup]] of a group.

A Lie algebra <math>\mathfrak{g}</math> is ''[[nilpotent Lie algebra|nilpotent]]'' if the [[lower central series]]
:<math> \mathfrak{g} \supseteq [\mathfrak{g},\mathfrak{g}] \supseteq [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] \supseteq [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] \supseteq \cdots</math>
becomes zero after finitely many steps. Equivalently, <math>\mathfrak{g}</math> is nilpotent if there is a finite sequence of ideals in <math>\mathfrak{g}</math>,
:<math>0=\mathfrak{a}_0 \subseteq \mathfrak{a}_1 \subseteq \cdots \subseteq \mathfrak{a}_r = \mathfrak{g},</math>
such that <math>\mathfrak{a}_j/\mathfrak{a}_{j-1}</math> is central in <math>\mathfrak{g}/\mathfrak{a}_{j-1}</math> for each ''j''. By [[Engel's theorem]], a Lie algebra over any field is nilpotent if and only if for every ''u'' in <math>\mathfrak{g}</math> the adjoint endomorphism
:<math>\operatorname{ad}(u):\mathfrak{g} \to \mathfrak{g}, \quad \operatorname{ad}(u)v=[u,v]</math>
is [[nilpotent endomorphism|nilpotent]].<ref>{{harvnb|Jacobson|1979|loc=section II.3.}}</ref>

More generally, a Lie algebra <math>\mathfrak{g}</math> is said to be ''[[solvable Lie algebra|solvable]]'' if the [[derived series]]:
:<math> \mathfrak{g} \supseteq [\mathfrak{g},\mathfrak{g}] \supseteq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] \supseteq [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]]  \supseteq \cdots</math>
becomes zero after finitely many steps. Equivalently, <math>\mathfrak{g}</math> is solvable if there is a finite sequence of Lie subalgebras,
:<math>0=\mathfrak{m}_0 \subseteq \mathfrak{m}_1 \subseteq \cdots \subseteq \mathfrak{m}_r = \mathfrak{g},</math>
such that <math>\mathfrak{m}_{j-1}</math> is an ideal in <math>\mathfrak{m}_{j}</math> with <math>\mathfrak{m}_{j}/\mathfrak{m}_{j-1}</math> abelian for each ''j''.<ref>{{harvnb|Jacobson|1979|loc=section I.7.}}</ref>

Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called its [[radical of a Lie algebra|radical]].<ref>{{harvnb|Jacobson|1979|p=24.}}</ref> Under the [[Lie correspondence]], nilpotent (respectively, solvable) Lie groups correspond to nilpotent (respectively, solvable) Lie algebras over <math>\mathbb{R}</math>.

For example, for a positive integer ''n'' and a field ''F'' of characteristic zero, the radical of <math>\mathfrak{gl}(n,F)</math> is its center, the 1-dimensional subspace spanned by the identity matrix. An example of a solvable Lie algebra is the space <math>\mathfrak{b}_{n}</math> of upper-triangular matrices in <math>\mathfrak{gl}(n)</math>; this is not nilpotent when <math>n\geq 2</math>. An example of a nilpotent Lie algebra is the space <math>\mathfrak{u}_{n}</math> of strictly upper-triangular matrices in <math>\mathfrak{gl}(n)</math>; 
this is not abelian when <math>n\geq 3</math>.

=== Simple and semisimple ===
{{main|Semisimple Lie algebra}}
A Lie algebra <math>\mathfrak{g}</math> is called ''[[Simple Lie algebra|simple]]'' if it is not abelian and the only ideals in <math>\mathfrak{g}</math> are 0 and <math>\mathfrak{g}</math>. (In particular, a one-dimensional—necessarily abelian—Lie algebra <math>\mathfrak{g}</math> is by definition not simple, even though its only ideals are 0 and <math>\mathfrak{g}</math>.) A finite-dimensional Lie algebra <math>\mathfrak{g}</math> is called ''[[semisimple Lie algebra|semisimple]]'' if the only solvable ideal in <math>\mathfrak{g}</math> is 0. In characteristic zero, a Lie algebra <math>\mathfrak{g}</math> is semisimple if and only if it is isomorphic to a product of simple Lie algebras, <math>\mathfrak{g} \cong \mathfrak{g}_1 \times \cdots \times \mathfrak{g}_r</math>.<ref>{{harvnb|Jacobson|1979|loc=Ch. III, § 5.}}</ref>

For example, the Lie algebra <math>\mathfrak{sl}(n,F)</math> is simple for every <math>n\geq 2</math> and every field ''F'' of characteristic zero (or just of characteristic not dividing ''n''). The Lie algebra <math>\mathfrak{su}(n)</math> over <math>\mathbb{R}</math> is simple for every <math>n\geq 2</math>. The Lie algebra <math>\mathfrak{so}(n)</math> over <math>\mathbb{R}</math> is simple if <math>n=3</math> or <math>n\geq 5</math>.<ref>{{harvnb|Erdmann|Wildon|2006|loc=Theorem 12.1.}}</ref> (There are "exceptional isomorphisms" <math>\mathfrak{so}(3)\cong\mathfrak{su}(2)</math> and <math>\mathfrak{so}(4)\cong\mathfrak{su}(2) \times \mathfrak{su}(2)</math>.)

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field ''F'' has characteristic zero, every finite-dimensional representation of a semisimple Lie algebra is [[semisimple representation|semisimple]] (that is, a direct sum of irreducible representations).<ref name="reducibility" />

A finite-dimensional Lie algebra over a field of characteristic zero is called [[reductive Lie algebra|reductive]] if its adjoint representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a semisimple Lie algebra.<ref>{{harvnb|Varadarajan|1984|loc=Theorem 3.16.3.}}</ref>

For example, <math>\mathfrak{gl}(n,F)</math> is reductive for ''F'' of characteristic zero: for <math>n\geq 2</math>, it is isomorphic to the product
:<math>\mathfrak{gl}(n,F) \cong F\times \mathfrak{sl}(n,F),</math>
where ''F'' denotes the center of <math>\mathfrak{gl}(n,F)</math>, the 1-dimensional subspace spanned by the identity matrix. Since the special linear Lie algebra <math>\mathfrak{sl}(n,F)</math> is simple, <math>\mathfrak{gl}(n,F)</math> contains few ideals: only 0, the center ''F'', <math>\mathfrak{sl}(n,F)</math>, and all of <math>\mathfrak{gl}(n,F)</math>.

=== Cartan's criterion ===
[[Cartan's criterion]] (by [[Élie Cartan]]) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of the [[Killing form]], the symmetric bilinear form on <math>\mathfrak{g}</math> defined by
:<math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),</math>
where tr denotes the trace of a linear operator. Namely: a Lie algebra <math>\mathfrak{g}</math> is semisimple if and only if the Killing form is [[nondegenerate form|nondegenerate]]. A Lie algebra <math>\mathfrak{g}</math> is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.</math><ref>{{harvnb|Varadarajan|1984|loc=section 3.9.}}</ref>

=== Classification ===
The [[Levi decomposition]] asserts that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its solvable radical and a semisimple Lie algebra.<ref>{{harvnb|Jacobson|1979|loc=Ch. III, § 9.}}</ref> Moreover, a semisimple Lie algebra in characteristic zero is a product of simple Lie algebras, as mentioned above. This focuses attention on the problem of classifying the simple Lie algebras.

The simple Lie algebras of finite dimension over an [[algebraically closed field]] ''F'' of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, using [[root system]]s. Namely, every simple Lie algebra is of type A<sub>''n''</sub>,  B<sub>''n''</sub>, C<sub>''n''</sub>,  D<sub>''n''</sub>, E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub>, F<sub>4</sub>, or G<sub>2</sub>.<ref>{{harvnb|Jacobson|1979|loc=section IV.6.}}</ref> Here the simple Lie algebra of type A<sub>''n''</sub> is <math>\mathfrak{sl}(n+1,F)</math>, B<sub>''n''</sub> is <math>\mathfrak{so}(2n+1,F)</math>, C<sub>''n''</sub> is <math>\mathfrak{sp}(2n,F)</math>, and D<sub>''n''</sub> is <math>\mathfrak{so}(2n,F)</math>. The other five are known as the [[exceptional Lie algebra]]s.

The classification of finite-dimensional simple Lie algebras over <math>\mathbb{R}</math> is more complicated, but it was also solved by Cartan (see [[simple Lie group]] for an equivalent classification). One can analyze a Lie algebra <math>\mathfrak{g}</math> over <math>\mathbb{R}</math> by considering its complexification <math>\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}</math>.

In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic <math>p>3</math> were classified by [[Richard Earl Block]], Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See [[restricted Lie algebra#Classification of simple Lie algebras]].) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.