Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras (non-abelian Lie algebras 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 , if nonzero, the following conditions are equivalent:

  • is semisimple;
  • the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate;
  • has no non-zero abelian ideals;
  • has no non-zero solvable ideals;
  • the radical (maximal solvable ideal) of is zero.

SignificanceEdit

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   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.

HistoryEdit

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).

Basic propertiesEdit

  • Every ideal, quotient and product of semisimple Lie algebras is again semisimple.[1]
  • The center of a semisimple Lie algebra   is trivial (since the center is an abelian ideal). In other words, the adjoint representation   is injective. Moreover, the image turns out[2] to be   of derivations on  . Hence,   is an isomorphism.[3] (This is a special case of Whitehead's lemma.)
  • 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 (Ado's theorem), although not necessarily via the adjoint representation. But in practice, such ambiguity rarely occurs.
  • If   is a semisimple Lie algebra, then   (because   is semisimple and abelian).[4]
  • 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  .[5] Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple.

Jordan decompositionEdit

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  . An element x of   is said to be semisimple (resp. nilpotent) if   is a semisimple (resp. nilpotent) operator.[6] If  , then the abstract Jordan decomposition states that x can be written uniquely as:

 

where   is semisimple,   is nilpotent and  .[7] 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(x) in the endomorphism algebra of the representation space.[8] (This is proved as a consequence of Weyl's complete reducibility theorem; see Weyl's theorem on complete reducibility#Application: preservation of Jordan decomposition.)

StructureEdit

Let   be a (finite-dimensional) 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,[9] a Cartan subalgebra (also called a maximal toral 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

 .

(Note that   is the centralizer of  .) Then

Root space decomposition — [10] Given a Cartan subalgebra  , it holds that   and there is a decomposition (as an  -module):

 

where   is the set of all nonzero linear functionals   of   such that  . Moreover, for each  ,

  •  , which is the equality if  .
  •   as a Lie algebra.
  •  ; in particular,  .
  •  ; in other words,  .
  • With respect to the Killing form B,   are orthogonal to each other if  ; the restriction of B to   is nondegenerate.

(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   are called the roots of   relative to  . The roots span   (since if  , then   is the zero operator; i.e.,   is in the center, which is zero.) Moreover, from the representation theory of  , one deduces the following symmetry and integral properties of  : for each  ,

  • The endomorphism
     
    leaves   invariant (i.e.,  ).
  •   is an integer.

Note that   has the properties (1)   and (2) 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   (called Chevalley generators) generate   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. 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.[11]

Example root space decomposition in sln(C)Edit

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 (matrix) basis, meaning   represents the basis vector in the  -th row and  -th column. This decomposition of   has an associated root system:

 

sl2(C)Edit

For example, in   the decomposition is

 

and the associated root system is

 

sl3(C)Edit

In   the decomposition is

 

and the associated root system is given by

 

ExamplesEdit

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. 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 (E6, E7, E8, F4, and G2), are in fact the only simple Lie algebras over the complex numbers.

ClassificationEdit

 
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 – 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 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   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 (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").[12]

Representation theory of semisimple Lie algebrasEdit

Let   be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in #Structure,   where   is the root system. Choose the simple roots in  ; a root   of   is then called positive and is denoted by   if it is a linear combination of the simple roots with non-negative integers coefficients. Let  , which is a maximal solvable subalgebra of  , the Borel subalgebra.

Let V be a (possibly-infinite-dimensional) simple  -module. If V happens to admit a  -weight vector  ,[13] then it is unique up to scaling and is called the highest weight vector of V. It is also an  -weight vector and the  -weight of  , a linear functional of  , is called the highest weight of V. The basic yet nontrivial facts[14] then are (1) to each linear functional  , there exists a simple  -module   having   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  -modules admitting a Borel-weight vector.

For applications, one is often interested in a finite-dimensional simple  -module (a finite-dimensional irreducible representation). This is especially the case when   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:[15]

  •   if and only if, for each positive root  , (1)   is an integer and (2)   lies in  .

A 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  -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   is completely reducible; i.e., it is a direct sum of simple  -modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra.

Real semisimple Lie algebraEdit

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   be a finite-dimensional real semisimple Lie algebra and   the complexification of it (which is again semisimple). The real Lie algebra   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 caseEdit

Suppose   is a compact form and   a maximal abelian subspace. One can show (for example, from the fact   is the Lie algebra of a compact Lie group) that   consists of skew-Hermitian matrices, diagonalizable over   with imaginary eigenvalues. Hence,   is a Cartan subalgebra of   and there results in the root space decomposition (cf. #Structure)

 

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   be the linear functional on   given by   for  . Then for each  ,

 

where   is the matrix that has 1 on the  -th spot and zero elsewhere. Hence, each root   is of the form   and the root space decomposition is the decomposition of matrices:[16]

 

Noncompact caseEdit

Suppose   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   and let   be the eigenspace decomposition of  , where   are the eigenspaces for 1 and -1, respectively. For example, if   and   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):[17]

 

where

  • the elements in   are called the restricted roots,
  •   for any linear functional  ; in particular,  ,
  •  .

Moreover,   is a root system but not necessarily reduced one (i.e., it can happen   are both roots).

The case of Edit

If  , then   may be taken to be the diagonal subalgebra of  , consisting of diagonal matrices whose diagonal entries sum to zero. Since   has dimension  , we see that   has rank  .

The root vectors   in this case may be taken to be the matrices   with  , where   is the matrix with a 1 in the   spot and zeros elsewhere.[18] If   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  -tuples that sum to zero. This is the root system known as   in the conventional labeling.

The reflection associated to the root   acts on   by transposing the   and   diagonal entries. The Weyl group is then just the permutation group on   elements, acting by permuting the diagonal entries of matrices in  .

GeneralizationsEdit

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 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.

Semisimple and reductive groupsEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ Serre 2000, Ch. II, § 2, Corollary to Theorem 3.
  2. ^ Since the Killing form B is non-degenerate, given a derivation D, there is an x such that   for all y and then, by an easy computation,  .
  3. ^ Serre 2000, Ch. II, § 4, Theorem 5.
  4. ^ Serre 2000, Ch. II, § 3, Corollary to Theorem 4.
  5. ^ Jacobson 1979, Corollary at the end of Ch. III, § 4.
  6. ^ Serre 2000, Ch. II, § 5. Definition 3.
  7. ^ Serre 2000, Ch. II, § 5. Theorem 6.
  8. ^ Serre 2000, Ch. II, § 5. Theorem 7.
  9. ^ This is a definition of a Cartan subalgebra of a semisimple Lie algebra and coincides with the general one.
  10. ^ Serre 2000, Ch. VI, § 1.
  11. ^ Hall 2015 Theorem 9.3
  12. ^ Knapp 2002 Section VI.10
  13. ^ A  -weight vector is also called a primitive element, especially in older textbooks.
  14. ^ In textbooks, these facts is usually established by the theory of Verma modules.
  15. ^ Serre 2000, Ch. VII, § 4, Theorem 3.
  16. ^ Knapp, Ch. IV, § 1, Example 1.
  17. ^ Knapp, Ch. V, § 2, Proposition 5.9.
  18. ^ Hall 2015 Section 7.7.1
  • Bourbaki, Nicolas (2005), "VIII: Split Semi-simple Lie Algebras", Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9
  • Erdmann, Karin; Wildon, Mark (2006), Introduction to Lie Algebras (1st ed.), Springer, ISBN 1-84628-040-0.
  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666
  • Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7.
  • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
  • Knapp, Anthony W. (2002), Lie groups beyond an introduction (2nd ed.), Birkhäuser
  • Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.
  • Varadarajan, V. S. (2004), Lie Groups, Lie Algebras, and Their Representations (1st ed.), Springer, ISBN 0-387-90969-9.