Editing Lie algebra (section)

=== 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>.