Editing Lie algebra (section)

=== 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>.