Group theory → Lie groups Lie groups
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups List of simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Exponential map Adjoint representation (group algebra) Killing form Index Lie point symmetry Simple Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
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Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Table of Lie groups
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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras gdisplaystyle mathfrak g whose only ideals are 0 and gdisplaystyle mathfrak g itself. It is important to emphasize that a one-dimensional Lie algebra (which is necessarily abelian) is by definition not considered a simple Lie algebra, even though such an algebra certainly has no nontrivial ideals. Thus, one-dimensional algebras are not allowed as summands in a semisimple Lie algebra.

Throughout the article, unless otherwise stated, gdisplaystyle mathfrak g is a non-zero finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent:

• 
  gdisplaystyle mathfrak g is semisimple


• the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate,


• 
  gdisplaystyle mathfrak g has no non-zero abelian ideals,


• 
  gdisplaystyle mathfrak g has no non-zero solvable ideals,


• The radical (maximal solvable ideal) of gdisplaystyle mathfrak g is zero.

Examples[edit]

As explained in greater detail below, semisimple Lie algebras over Cdisplaystyle mathbb C 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, with notation coming from their Dynkin diagrams, are:

• 
  An:displaystyle A_n: sln+1displaystyle mathfrak sl_n+1, the special linear Lie algebra.


• 
  Bn:displaystyle B_n: so2n+1displaystyle mathfrak so_2n+1, the odd-dimensional special orthogonal Lie algebra.


• 
  Cn:displaystyle C_n: sp2ndisplaystyle mathfrak sp_2n, the symplectic Lie algebra.


• 
  Dn:displaystyle D_n: so2ndisplaystyle mathfrak so_2n, the even-dimensional special orthogonal Lie algebra (n>1displaystyle n>1).

The restriction n>1displaystyle n>1 in the Dndisplaystyle D_n family is needed because so2displaystyle mathfrak so_2 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. 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[edit]

The simple Lie algebras are classified by the connected Dynkin diagrams.

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 – A_n, B_n, C_n, and D_n – 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 the adjoint action of the Lie algebra on this subalgebra. 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≥1displaystyle ngeq 1 for A_n, n≥2displaystyle ngeq 2 for B_n, n≥3displaystyle ngeq 3 for C_n, and n≥4displaystyle ngeq 4 for D_n. 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 E_n 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").^[1]

Connection with compact Lie groups[edit]

Let Kdisplaystyle K be a compact Lie group with Lie algebra kdisplaystyle mathfrak k. Then the complexification g=k+ikdisplaystyle mathfrak g=mathfrak k+imathfrak k of kdisplaystyle mathfrak k is reductive, that is, the direct sum of a complex semisimple Lie algebra and a commutative algebra.^[2] If Kdisplaystyle K is simply connected, then gdisplaystyle mathfrak g is actually semisimple.^[3] Conversely, every complex semisimple Lie algebra gdisplaystyle mathfrak g has a compact real form kdisplaystyle mathfrak k, where kdisplaystyle mathfrak k is the Lie algebra of a simply connected compact Lie group.^[4] For example, the complex semisimple Lie algebra sl(n,C)displaystyle mathrm sl (n,mathbb C ) is the complexification of su(n)displaystyle mathrm su (n), the Lie algebra of the simply connected compact group SU(n). It is possible to develop the theory of complex semisimple Lie algebras from the compact group perspective,^[5] leading to a simpler way to develop the existence and properties of Cartan subalgebras.

The classification and representation theory of connected compact Lie groups is thus closely related to the associated theories for complex semisimple Lie algebras.

History[edit]

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 (Humphreys 1972).

Properties[edit]

Complete reducibility[edit]

A consequence of semisimplicity is a theorem due to Weyl: every finite-dimensional representation is completely reducible; that is for every invariant subspace of the representation there is an invariant complement.^[6] Infinite-dimensional representations of semisimple Lie algebras are not in general completely reducible.

Centerless[edit]

Since the center of a Lie algebra gdisplaystyle mathfrak g is an abelian ideal, if gdisplaystyle mathfrak g is semisimple, then its center is zero. (Note: since glndisplaystyle mathfrak gl_n has non-trivial center, it is not semisimple.) In other words, the adjoint representation addisplaystyle operatorname ad is injective. Moreover, it can be shown that the dimension of the Lie algebra Der⁡(g)displaystyle operatorname Der (mathfrak g) of derivations on gdisplaystyle mathfrak g is equal to the dimension of gdisplaystyle mathfrak g. Hence, gdisplaystyle mathfrak g is Lie algebra isomorphic to Der⁡(g)displaystyle operatorname Der (mathfrak g). (This is a special case of Whitehead's lemma.) Every ideal, quotient and product of semisimple Lie algebras is again semisimple.

Linear[edit]

The adjoint representation is injective, and so a semisimple Lie algebra is also 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 (Ado's theorem), although not necessarily via the adjoint representation. But in practice, such ambiguity rarely occurs.

Jordan decomposition[edit]

Any endomorphism x of a finite-dimensional vector space over an algebraically closed field can be decomposed uniquely into a diagonalizable (or semisimple) and nilpotent part

x=s+n displaystyle x=s+n

such that s and n commute with each other. Moreover, each of s and n is a polynomial in x. This is a consequence of the Jordan decomposition.

If x∈gdisplaystyle xin mathfrak g, then the image of x under the adjoint map decomposes as

ad⁡(x)=ad⁡(s)+ad⁡(n).displaystyle operatorname ad (x)=operatorname ad (s)+operatorname ad (n).

The elements s and n are unique elements of gdisplaystyle mathfrak g such that n is nilpotent, s is semisimple, n and s commute, and for which such a decomposition holds. This abstract Jordan decomposition factors through any representation of gdisplaystyle mathfrak g in the sense that given any representation ρ,

ρ(x)=ρ(s)+ρ(n)displaystyle rho (x)=rho (s)+rho (n),

is the Jordan decomposition of ρ(x) in the endomorphism ring of the representation space.

Rank[edit]

The rank of a complex semisimple Lie algebra is the dimension of any of its Cartan subalgebras.

Representation Theory[edit]

Weyl's theorem on complete reducibility states that every finite-dimensional representation of a semisimple Lie algebra decomposes as a direct sum of irreducible representations. The finite-dimensional, irreducible representations, meanwhile, are classified by a theorem of the highest weight. One remarkable aspect of this theory is the Weyl character formula.

Cartan subalgebras and root systems[edit]

Definition of a Cartan subalgebra[edit]

Although there is a theory of Cartan subalgebras for any Lie algebra, the concept has particular importance and a special form in the case of complex semisimple Lie algebras. If gdisplaystyle mathfrak g is a complex semisimple Lie algebra, we say that hdisplaystyle mathfrak h is a Cartan subalgebra if hdisplaystyle mathfrak h is a maximal commutative subalgebra of gdisplaystyle mathfrak g and if adHdisplaystyle mathrm ad _H is diagonalizable for each H∈hdisplaystyle Hin mathfrak h. An important first step in the study of semisimple Lie algebras is to prove the existence of Cartan subalgebras, and their uniqueness up to automorphism.^[7] (If one assumes the existence of a compact real form, the existence of a Cartan subalgebra is much simpler to establish.^[8] In that case, hdisplaystyle mathfrak h may be taken as the complexification of the Lie algebra of a maximal torus of the compact group.)

Since Cartan subalgebras of a semisimple Lie algebra gdisplaystyle mathfrak g are unique up to automorphisms of gdisplaystyle mathfrak g, all Cartan subalgebras have the same dimension. This common dimension is the rank of gdisplaystyle mathfrak g.

Root systems[edit]

Given a Cartan subalgebra hdisplaystyle mathfrak h of gdisplaystyle mathfrak g, one defines a root to be a nonzero element αdisplaystyle alpha of h∗displaystyle mathfrak h^* for which there exists a nonzero X∈gdisplaystyle Xin mathfrak g with

[H,X]=α(H)Xdisplaystyle [H,X]=alpha (H)X

for all H∈hdisplaystyle Hin mathfrak h. That is to say, the roots are the nonzero weights of the adjoint representation. The collection of roots forms a root system^[9] and much of the structure of gdisplaystyle mathfrak g derives from its root system. Indeed, the classification of complex semisimple Lie algebras described above comes from a classification of the associated root systems, which in turn are classified by their Dynkin diagrams, as we will explain below.

Cartan subalgebras and the associated root systems are a basic tool for understanding both the classification of semisimple Lie algebras and their representation theory.

Weyl group[edit]

Let gdisplaystyle mathfrak g be a semisimple Lie algebra, let hdisplaystyle mathfrak h be a Cartan subalgebra, and let Rdisplaystyle R be the associated root system. For each α∈Rdisplaystyle alpha in R, we can consider the reflection sαdisplaystyle s_alpha about the hyperplane perpendicular to αdisplaystyle alpha . One of the basic properties of root systems ensures that Rdisplaystyle R is invariant under each sαdisplaystyle s_alpha . The Weyl group is then the group of linear transformations of hdisplaystyle mathfrak h generated by the sαdisplaystyle s_alpha 's. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of gdisplaystyle mathfrak g are invariant under the Weyl group.^[10]

The case of sl(n;C)displaystyle mathrm sl (n;mathbb C )[edit]

If g=sl(n,C)displaystyle mathfrak g=mathrm sl (n,mathbb C ), then hdisplaystyle mathfrak h may be taken to be the diagonal subalgebra of gdisplaystyle mathfrak g, consisting of diagonal matrices whose diagonal entries sum to zero. Since hdisplaystyle mathfrak h has dimension n−1displaystyle n-1, we see that sl(n;C)displaystyle mathrm sl (n;mathbb C ) has rank n−1displaystyle n-1.

The root vectors Xdisplaystyle X in this case may be taken to be the matrices Ei,jdisplaystyle E_i,j with i≠jdisplaystyle ineq j, where Ei,jdisplaystyle E_i,j is the matrix with a 1 in the (i,j)displaystyle (i,j) spot and zeros elsewhere.^[11] If Hdisplaystyle H is a diagonal matrix with diagonal entries λ1,…,λndisplaystyle lambda _1,ldots ,lambda _n, then we have

[H,Ei,j]=(λi−λj)Ei,jdisplaystyle [H,E_i,j]=(lambda _i-lambda _j)E_i,j.

Thus, the roots for sl(n,C)displaystyle mathrm sl (n,mathbb C ) are the linear functionals αi,jdisplaystyle alpha _i,j given by

αi,j(H)=λi−λjdisplaystyle alpha _i,j(H)=lambda _i-lambda _j.

After identifying hdisplaystyle mathfrak h with its dual, the roots become the vectors αi,j:=ei−ejdisplaystyle alpha _i,j:=e_i-e_j in the space of ndisplaystyle n-tuples that sum to zero. This is the root system known as An−1displaystyle A_n-1 in the conventional labeling.

The reflection associated to the root αi,jdisplaystyle alpha _i,j acts on hdisplaystyle mathfrak h by transposing the idisplaystyle i and jdisplaystyle j diagonal entries. The Weyl group is then just the permutation group on ndisplaystyle n elements, acting by permuting the diagonal entries of matrices in hdisplaystyle mathfrak h.

Role in the classification[edit]

Semisimple Lie algebras are ultimately classified by root systems (which in turn are classified by their Dynkin diagrams). The argument is as follows.

• First, one proves that the Cartan subalgebra of a semisimple Lie algebra is unique up to isomorphism. It follows that the root system is independent (up to isomorphism) of the choice of Cartan subalgebra.


• Next, one shows that the root system determines the Lie algebra up to isomorphism. (Thus, for example, there cannot be two nonisomorphic Lie algebras both having the root system Andisplaystyle A_n.)


• Finally, one shows that every root system comes from a Lie algebra. This can be done in a case-by-case fashion, with the only hard part being the exceptional root systems of type E, F, and G. Alternatively, one can use a systematic procedure, using Serre's relations.

These results can be found, for example, in various parts of (Humphreys 1972), culminating in Section 19.

Significance[edit]

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 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 gdisplaystyle mathfrak g is semisimple, then g=[g,g]displaystyle mathfrak g=[mathfrak g,mathfrak g]. In particular, every linear semisimple Lie algebra is a subalgebra of sldisplaystyle mathfrak sl, the special linear Lie algebra. The study of the structure of sldisplaystyle mathfrak sl constitutes an important part of the representation theory for semisimple Lie algebras.

Generalizations[edit]

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, slndisplaystyle mathfrak sl_n is semisimple, and glndisplaystyle mathfrak gl_n 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 semsimple 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 (Bourbaki 2005), for instance, which classifies representations of split semisimple/reductive Lie algebras.

See also[edit]

• Lie algebra


• Root system


• Lie algebra representation


• Compact group


• Simple Lie group

References[edit]