Editing Semisimple Lie algebra (section)

== Classification ==
{{See also|Root system}}
[[File:Connected Dynkin Diagrams.svg|thumb|The simple Lie algebras are classified by the connected [[Dynkin diagram]]s.]]
Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a [[direct sum]] of [[simple Lie algebra]]s (by definition), and the finite-dimensional simple Lie algebras fall in four families – A<sub>n</sub>, B<sub>n</sub>, C<sub>n</sub>, and D<sub>n</sub> – with five exceptions
[[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E8 (mathematics)|E<sub>8</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]]. Simple Lie algebras are classified by the connected [[Dynkin diagram]]s, shown on the right, while semisimple Lie algebras correspond to not necessarily connected Dynkin diagrams, where each component of the diagram corresponds to a summand of the decomposition of the semisimple Lie algebra into simple Lie algebras.

The classification proceeds by considering a [[Cartan subalgebra]] (see below) and its [[adjoint representation of a Lie algebra|adjoint action]] on the Lie algebra. The [[root system]] of the action then both determines the original Lie algebra and must have a very constrained form, which can be classified by the Dynkin diagrams. See the section below describing Cartan subalgebras and root systems for more details.

The classification is widely considered one of the most elegant results in mathematics – a brief list of axioms yields, via a relatively short proof, a complete but non-trivial classification with surprising structure. This should be compared to the [[classification of finite simple groups]], which is significantly more complicated.

The enumeration of the four families is non-redundant and consists only of simple algebras if <math>n \geq 1</math> for A<sub>n</sub>, <math>n \geq 2</math> for B<sub>n</sub>, <math>n \geq 3</math> for C<sub>n</sub>, and <math>n \geq 4</math> for D<sub>n</sub>. If one starts numbering lower, the enumeration is redundant, and one has [[exceptional isomorphism]]s between simple Lie algebras, which are reflected in [[Dynkin diagram#Isomorphisms|isomorphisms of Dynkin diagrams]]; the E<sub>n</sub> can also be extended down, but below E<sub>6</sub> are isomorphic to other, non-exceptional algebras.

Over a non-algebraically closed field, the classification is more complicated – one classifies simple Lie algebras over the algebraic closure, then for each of these, one classifies simple Lie algebras over the original field which have this form (over the closure). For example, to classify simple real Lie algebras, one classifies real Lie algebras with a given complexification, which are known as [[Real form (Lie theory)|real forms]] of the complex Lie algebra; this can be done by [[Satake diagram]]s, which are Dynkin diagrams with additional data ("decorations").<ref>{{harvnb|Knapp|2002}} Section VI.10</ref>