In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals.) 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 \mathfrak g, if nonzero, the following conditions are equivalent:

Significance

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. 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 \mathfrak g 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 (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. 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 (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as.

Basic properties

Jordan decomposition

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. 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 \mathfrak g. An element x of \mathfrak g is said to be semisimple (resp. nilpotent) if is a semisimple (resp. nilpotent) operator. If, then the abstract Jordan decomposition states that x can be written uniquely as: where s is semisimple, n is nilpotent and [s, n] = 0. Moreover, if commutes with x, then it commutes with both s, n as well. The abstract Jordan decomposition factors through any representation of \mathfrak g in the sense that given any representation ρ, is the Jordan decomposition of ρ(x) in the endomorphism algebra of the representation space. (This is proved as a consequence of Weyl's complete reducibility theorem; see Weyl's theorem on complete reducibility.)

Structure

Let \mathfrak g be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of \mathfrak g can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition, a Cartan subalgebra (also called a maximal toral subalgebra) \mathfrak h of \mathfrak g is a maximal subalgebra such that, for each, is diagonalizable. As it turns out, \mathfrak h is abelian and so all the operators in are simultaneously diagonalizable. For each linear functional \alpha of \mathfrak h, let (Note that is the centralizer of \mathfrak h.) Then (The most difficult item to show is . The standard proofs all use some facts in the representation theory of ; e.g., Serre uses the fact that an -module with a primitive element of negative weight is infinite-dimensional, contradicting .) Let with the commutation relations ; i.e., the correspond to the standard basis of. The linear functionals in \Phi are called the roots of \mathfrak g relative to \mathfrak h. The roots span (since if, then is the zero operator; i.e., h is in the center, which is zero.) Moreover, from the representation theory of , one deduces the following symmetry and integral properties of \Phi: for each , • The endomorphism leaves \Phi invariant (i.e., ). • is an integer. Note that s_{\alpha} has the properties (1) and (2) the fixed-point set is, which means that s_{\alpha} is the reflection with respect to the hyperplane corresponding to \alpha. The above then says that \Phi is a root system. It follows from the general theory of a root system that \Phi contains a basis of such that each root is a linear combination of with integer coefficients of the same sign; the roots \alpha_i are called simple roots. Let, etc. Then the 3l elements (called Chevalley generators) generate \mathfrak g as a Lie algebra. Moreover, they satisfy the relations (called Serre 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 (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the is a Cartan matrix). 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 (finite-dimensional over an algebraically closed field of characteristic zero). The Weyl group is the group of linear transformations of generated by the s_\alpha'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 e_{ij} in with the standard (matrix) basis, meaning e_{ij} represents the basis vector in the i-th row and j-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, semisimple Lie algebras over [](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)b[](https://bliptext.com/articles/)b[](https://bliptext.com/articles/){[](https://bliptext.com/articles/)C[](https://bliptext.com/articles/)}[](https://bliptext.com/articles/) (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. Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are: The restriction n>1 in the D_n 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 (E6, E7, E8, F4, and G2), 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 (by definition), 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 (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. The enumeration of the four families is non-redundant and consists only of simple algebras if n \geq 1 for An, n \geq 2 for Bn, n \geq 3 for Cn, and n \geq 4 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 (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 forms of the complex Lie algebra; this can be done by Satake diagrams, which are Dynkin diagrams with additional data ("decorations").

Representation theory of semisimple Lie algebras

Let \mathfrak g be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in, where [](https://bliptext.com/articles/)P[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/) is the root system. Choose the simple roots in \Phi; a root \alpha of \Phi is then called positive and is denoted by \alpha > 0 if it is a linear combination of the simple roots with non-negative integer coefficients. Let, which is a maximal solvable subalgebra of \mathfrak g, the Borel subalgebra. Let V be a (possibly-infinite-dimensional) simple \mathfrak g-module. If V happens to admit a \mathfrak b-weight vector v_0, then it is unique up to scaling and is called the highest weight vector of V. It is also an \mathfrak h-weight vector and the \mathfrak h-weight of v_0, a linear functional of \mathfrak h, is called the highest weight of V. The basic yet nontrivial facts then are (1) to each linear functional, there exists a simple \mathfrak g-module V^{\mu} having \mu as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between and the set of the equivalence classes of simple \mathfrak g-modules admitting a Borel-weight vector. For applications, one is often interested in a finite-dimensional simple \mathfrak g-module (a finite-dimensional irreducible representation). This is especially the case when \mathfrak g is the Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber, we mean the convex cone where is a unique vector such that. The criterion then reads: A linear functional \mu 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 \mathfrak g-modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character formula. The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra \mathfrak g is completely reducible; i.e., it is a direct sum of simple \mathfrak g-modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra.

Real semisimple Lie algebra

For a semisimple Lie algebra over a field that has characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic zero. But over the field of real numbers, there are still the structure results. Let \mathfrak g be a finite-dimensional real semisimple Lie algebra and the complexification of it (which is again semisimple). The real Lie algebra \mathfrak g is called a real form of. A real form is called a compact form if the Killing form on it is negative-definite; it is necessarily the Lie algebra of a compact Lie group (hence, the name).

Compact case

Suppose \mathfrak g is a compact form and a maximal abelian subspace. One can show (for example, from the fact \mathfrak g is the Lie algebra of a compact Lie group) that consists of skew-Hermitian matrices, diagonalizable over \mathbb{C} with imaginary eigenvalues. Hence, is a [Cartan subalgebra](https://bliptext.com/articles/cartan-subalgebra) of and there results in the root space decomposition (cf. ) where each is real-valued on ; thus, can be identified with a real-linear functional on the real vector space. For example, let and take the subspace of all diagonal matrices. Note. Let e_i be the linear functional on given by for. Then for each , where E_{ij} is the matrix that has 1 on the (i, j)-th spot and zero elsewhere. Hence, each root \alpha is of the form and the root space decomposition is the decomposition of matrices:

Noncompact case

Suppose \mathfrak g is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a Cartan involution \theta and let be the eigenspace decomposition of \theta, where are the eigenspaces for 1 and -1, respectively. For example, if and \theta the negative transpose, then. Let be a maximal abelian subspace. Now, consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the restricted root space decomposition): where Moreover, \Phi is a root system but not necessarily reduced one (i.e., it can happen are both roots).

The case of sl(n,C)

If, then may be taken to be the diagonal subalgebra of , consisting of diagonal matrices whose diagonal entries sum to zero. Since has dimension n-1, we see that has rank n-1. The root vectors X in this case may be taken to be the matrices E_{i,j} with i\neq j, where E_{i,j} is the matrix with a 1 in the (i,j) spot and zeros elsewhere. If H is a diagonal matrix with diagonal entries, then we have Thus, the roots for are the linear functionals given by After identifying with its dual, the roots become the vectors in the space of n-tuples that sum to zero. This is the root system known as A {n-1} in the conventional labeling. The reflection associated to the root acts on by transposing the i and j diagonal entries. The Weyl group is then just the permutation group on n elements, acting by permuting the diagonal entries of matrices in .<!-- This is the old version of the structure section; it may (or may not) be needed to incorporated into the article

Structure

Let \mathfrak g be a complex semisimple Lie algebra. Let further \mathfrak h be a Cartan subalgebra of \mathfrak g. Then \mathfrak h acts on \mathfrak g via simultaneously diagonalizable linear maps in the adjoint representation. For λ in define the subspace by We say that is a root if and the subspace is nonzero. In this case is called the root space of λ . For each root \lambda, the root space is one-dimensional. Meanwhile, the definition of Cartan subalgebra guarantees that. Let R be the set of all roots. Since the elements of \mathfrak h are simultaneously diagonalizable, we have The Cartan subalgebra \mathfrak h inherits a nondegenerate bilinear form from the Killing form on \mathfrak g. This form induces a form on and the restriction of that form to the real span of the roots is an inner product. One can show that with respect to this inner product R is a reduced crystallographic root system. Let moreover Δ be a choice of simple roots. Now, it is not particularly difficult to find the following set of generators satisfying the Chevalley–Serre relations Here is the coefficient of the Cartan matrix, given by Note that if \lambda and \mu are in \Delta with, then , so that is a non-positive integer and is a positive integer. . For each simple root, we can find X_\lambda in the root space , Y_\lambda in the root space and H_\lambda in the Cartan subalgebra satisfying the standard relations: , , and. These will be our generators. Now, the element H_\lambda is the coroot associated to \lambda, which means that after we identify \mathfrak h with its dual, we have Then we have, for example, This sort of reasoning verifies the first four relations above. The last two relations hold because belongs to the root space, and more generally, belongs to. But, as we shall see momentarily, if \lambda and \mu are simple roots, then is not a root if, so that must be zero. To see that is not a root, note that if \lambda and \mu are distinct elements of \Delta, then cannot be a root, for this would violate one of the defining properties of a base—that the expansion of a root in terms of the base cannot have both positive and negative coefficients. But then if s_\lambda is the reflection associated to \lambda, we can easily calculate that Then since is not a root, neither is. Note that the elements do not span \mathfrak g as a vector space, because \lambda does not range over all the positive roots, but only over the base. Nevertheless, these elements generate \mathfrak g as a Lie algebra.

Serre's theorem

Serre's theorem asserts the above generators and relations completely determine \mathfrak g. In fact Serre's theorem asserts that starting from an arbitrary root system—not assumed to come from a semisimple Lie algebra—we can use the above relations to define a Lie algebra, the Lie algebra is finite-dimensional and semisimple, and the root system of that Lie algebra is the root system R we started from. A consequence of Serre's theorem is this:

Generalizations

Semisimple Lie algebras admit certain generalizations. Firstly, many statements that are true for semisimple Lie algebras are true more generally for reductive Lie algebras. Abstractly, a reductive Lie algebra is one whose adjoint representation is completely reducible, while concretely, a reductive Lie algebra is a direct sum of a semisimple Lie algebra and an abelian Lie algebra; for example, is semisimple, and is reductive. Many properties of semisimple Lie algebras depend only on reducibility. Many properties of complex semisimple/reductive Lie algebras are true not only for semisimple/reductive Lie algebras over algebraically closed fields, but more generally for split semisimple/reductive Lie algebras over other fields: semisimple/reductive Lie algebras over algebraically closed fields are always split, but over other fields this is not always the case. Split Lie algebras have essentially the same representation theory as semisimple Lie algebras over algebraically closed fields, for instance, the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in, for instance, which classifies representations of split semisimple/reductive Lie algebras.

Semisimple and reductive groups

A connected Lie group is called semisimple if its Lie algebra is a semisimple Lie algebra, i.e. a direct sum of simple Lie algebras. It is called reductive if its Lie algebra is a direct sum of simple and trivial (one-dimensional) Lie algebras. Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics. For example, the group of symmetries of an n-dimensional real vector space (equivalently, the group of invertible matrices) is reductive.