Semisimple Lie algebra


In mathematics,[] a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.
Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a 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 and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple.
Semisimple Lie algebras have a very elegant classification, in stark contrast to solvable Lie algebras. 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. Semisimple algebras over non-algebraically closed fields can be understood in terms of those over the algebraic closure, though the classification is somewhat more intricate; see real form for the case of real semisimple Lie algebras, which were classified by Élie Cartan.
Further, the representation theory of semisimple Lie algebras is much cleaner than that for general Lie algebras. For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general.
If is semisimple, then. In particular, every linear semisimple Lie algebra is a subalgebra of, the special linear Lie algebra. The study of the structure of constitutes an important part of the representation theory for semisimple Lie algebras.

History

The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing, though his proof lacked rigor. His proof was made rigorous by Élie Cartan in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made, but the proof is unchanged in its essentials and can be found in any standard reference, such as.

Basic properties

  • Every ideal, quotient and product of semisimple Lie algebras is again semisimple.
  • The center of a semisimple Lie algebra is trivial. In other words, the adjoint representation is injective. Moreover, the image turns out to be of derivations on. Hence, is an isomorphism.
  • As the adjoint representation is injective, a semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space, although not necessarily via the adjoint representation. But in practice, such ambiguity rarely occurs.
  • If is a semisimple Lie algebra, then .
  • A finite-dimensional Lie algebra over a field k of characteristic zero is semisimple if and only if the base extension is semisimple for each field extension. Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple.

    Jordan decomposition

Each endomorphism x of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple and nilpotent part
such that s and n commute with each other. Moreover, each of s and n is a polynomial in x. This is the Jordan decomposition of x.
The above applies to the adjoint representation of a semisimple Lie algebra. An element x of is said to be semisimple if is a semisimple operator. If, then the abstract Jordan decomposition states that x can be written uniquely as:
where is semisimple, is nilpotent and. Moreover, if commutes with x, then it commutes with both as well.
The abstract Jordan decomposition factors through any representation of in the sense that given any representation ρ,
is the Jordan decomposition of ρ in the endomorphism algebra of the representation space.

Structure

Let be a semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition, a Cartan subalgebra of is a maximal subalgebra such that, for each, is diagonalizable. As it turns out, is abelian and so all the operators in are simultaneously diagonalizable. For each linear functional of, let
Then
Let with the commutation relations ; i.e., the correspond to the standard basis of.
The linear functionals in are called the roots of relative to. The roots span Moreover, from the representation theory of, one deduces the following symmetry and integral properties of : for each,
Note that has the properties and the fixed-point set is, which means that is the reflection with respect to the hyperplane corresponding to. The above then says that is a root system.
It follows from the general theory of a root system that contains a basis of such that each root is a linear combination of with integer coefficients of the same sign; the roots are called simple roots. Let, etc. Then the elements generate as a Lie algebra. Moreover, they satisfy the relations : with,
The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a semisimple Lie algebra that has the root space decomposition as above. This is a theorem of Serre. In particular, two semisimple Lie algebras are isomorphic if they have the same root system.
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.
The Weyl group is the group of linear transformations of generated by the 's. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of are invariant under the Weyl group.

Example root space decomposition in sln(C)

For and the Cartan subalgebra of diagonal matrices, define by
where denotes the diagonal matrix with on the diagonal. Then the decomposition is given by
where
for the vector in with the standard basis, meaning represents the basis vector in the -th row and -th column. This decomposition of has an associated root system:

sl2(C)

For example, in the decomposition is
and the associated root system is

sl3(C)

In the decomposition is
and the associated root system is given by

Examples

As noted in #Structure, semisimple Lie algebras over are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams.
Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are:
The restriction in the family is needed because is one-dimensional and commutative and therefore not semisimple.
These Lie algebras are numbered so that n is the rank. 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. For example and. These four families, together with five exceptions, are in fact the only simple Lie algebras over the complex numbers.

Classification

Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras, and the finite-dimensional simple Lie algebras fall in four families – An, Bn, Cn, and Dn – with five exceptions
E6, E7, E8, F4, and G2. Simple Lie algebras are classified by the connected Dynkin diagrams, 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 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.
The enumeration of the four families is non-redundant and consists only of simple algebras if for An, for Bn, for Cn, and for Dn. If one starts numbering lower, the enumeration is redundant, and one has exceptional isomorphisms between simple Lie algebras, which are reflected in isomorphisms of Dynkin diagrams; the En can also be extended down, but below E6 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. 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.