Semisimple Lie algebra

, if nonzero, the following conditions are equivalent: The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra.

Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their root system, which are in turn classified by Dynkin diagrams.

constitutes an important part of the representation theory for semisimple Lie algebras.

The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing (1888–90), though his proof lacked rigor.

His proof was made rigorous by Élie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras.

Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as (Humphreys 1972).

Each endomorphism x of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over the algebraic closure) and nilpotent part such that s and n commute with each other.

, The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the

The implication of the axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence, "all possible" semisimple Lie algebras (finite-dimensional over an algebraically closed field of characteristic zero).

The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of

the decomposition is and the associated root system is given by As noted in #Structure, semisimple Lie algebras over

(or more generally an algebraically closed field of characteristic zero) are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams.

Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank.

These four families, together with five exceptions (E6, E7, E8, F4, and G2), are in fact the only simple Lie algebras over the complex numbers.

The classification proceeds by considering a Cartan subalgebra (see below) and its 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.

For example, to classify simple real Lie algebras, one classifies real Lie algebras with a given complexification, which are known as real forms of the complex Lie algebra; this can be done by Satake diagrams, which are Dynkin diagrams with additional data ("decorations").

if it is a linear combination of the simple roots with non-negative integer coefficients.

, is called the highest weight of V. The basic yet nontrivial facts[14] then are (1) to each linear functional

satisfying the above equivalent condition is called a dominant integral weight.

Hence, in summary, there exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple

The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra

Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra.

consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in

This is the approach followed in (Bourbaki 2005), for instance, which classifies representations of split semisimple/reductive Lie algebras.

Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics.

of symmetries of an n-dimensional real vector space (equivalently, the group of invertible matrices) is reductive.