Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras ${\mathfrak {g}}$ whose only ideals are {0} and ${\mathfrak {g}}$ 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, ${\mathfrak {g}}$ is a non-zero finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent:

${\mathfrak {g}}$ is semisimple
the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate,
${\mathfrak {g}}$ has no non-zero abelian ideals,
${\mathfrak {g}}$ has no non-zero solvable ideals,
The radical (maximal solvable ideal) of ${\mathfrak {g}}$ is zero.

Examples

As explained in greater detail below, semisimple Lie algebras over $\mathbb {C}$ 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:

$A_{n}:$ ${\mathfrak {sl}}_{{n+1}}$ , the special linear Lie algebra.
$B_{n}:$ ${\mathfrak {so}}_{{2n+1}}$ , the odd-dimensional special orthogonal Lie algebra.
$C_{n}:$ ${\mathfrak {sp}}_{{2n}}$ , the symplectic Lie algebra.
$D_{n}:$ ${\mathfrak {so}}_{{2n}}$ , the even-dimensional special orthogonal Lie algebra ( $n>1$ ).

The restriction $n>1$ in the $D_{n}$ family is needed because ${\mathfrak {so}}_{2}$ 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

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 $n\geq 1$ for A_n, $n\geq 2$ for B_n, $n\geq 3$ for C_n, and $n\geq 4$ 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

Let $K$ be a compact Lie group with Lie algebra $\mathfrak{k}$ . Then the complexification ${\mathfrak {g}}={\mathfrak {k}}+i{\mathfrak {k}}$ of $\mathfrak{k}$ is reductive, that is, the direct sum of a complex semisimple Lie algebra and a commutative algebra.^[2] If $K$ is simply connected, then ${\mathfrak {g}}$ is actually semisimple.^[3] Conversely, every complex semisimple Lie algebra ${\mathfrak {g}}$ has a compact real form $\mathfrak{k}$ , where $\mathfrak{k}$ is the Lie algebra of a simply connected compact Lie group.^[4] For example, the complex semisimple Lie algebra $\mathrm {sl} (n,\mathbb {C} )$ is the complexification of $\mathrm {su} (n)$ , 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

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

Complete reducibility

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

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

Linear

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

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\

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 $x\in {\mathfrak {g}}$ , then the image of x under the adjoint map decomposes as

\operatorname {ad}(x)=\operatorname {ad}(s)+\operatorname {ad}(n).

The elements s and n are unique elements of ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ in the sense that given any representation ρ,

\rho (x)=\rho (s)+\rho (n)\,

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

Rank

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

Representation Theory

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

Definition of a Cartan subalgebra

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 ${\mathfrak {g}}$ is a complex semisimple Lie algebra, we say that ${\mathfrak {h}}$ is a Cartan subalgebra if ${\mathfrak {h}}$ is a maximal commutative subalgebra of ${\mathfrak {g}}$ and if $\mathrm {ad} _{H}$ is diagonalizable for each $H\in {\mathfrak {h}}$ . 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, ${\mathfrak {h}}$ 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 ${\mathfrak {g}}$ are unique up to automorphisms of ${\mathfrak {g}}$ , all Cartan subalgebras have the same dimension. This common dimension is the rank of ${\mathfrak {g}}$ .

Root systems

Given a Cartan subalgebra ${\mathfrak {h}}$ of ${\mathfrak {g}}$ , one defines a root to be a nonzero element $\alpha$ of ${\mathfrak {h}}^{*}$ for which there exists a nonzero $X\in {\mathfrak {g}}$ with

[H,X]=\alpha (H)X

for all $H\in {\mathfrak {h}}$ . 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 ${\mathfrak {g}}$ 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

Let ${\mathfrak {g}}$ be a semisimple Lie algebra, let ${\mathfrak {h}}$ be a Cartan subalgebra, and let $R$ be the associated root system. For each $\alpha \in R$ , we can consider the reflection $s_{\alpha }$ about the hyperplane perpendicular to $\alpha$ . One of the basic properties of root systems ensures that $R$ is invariant under each $s_{\alpha }$ . The Weyl group is then the group of linear transformations of ${\mathfrak {h}}$ 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 ${\mathfrak {g}}$ are invariant under the Weyl group.^[10]

The case of $\mathrm {sl} (n;\mathbb {C} )$

If ${\mathfrak {g}}=\mathrm {sl} (n,\mathbb {C} )$ , then ${\mathfrak {h}}$ may be taken to be the diagonal subalgebra of ${\mathfrak {g}}$ , consisting of diagonal matrices whose diagonal entries sum to zero. Since ${\mathfrak {h}}$ has dimension $n-1$ , we see that $\mathrm {sl} (n;\mathbb {C} )$ 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.^[11] If $H$ is a diagonal matrix with diagonal entries $\lambda _{1},\ldots ,\lambda _{n}$ , then we have

[H,E_{i,j}]=(\lambda _{i}-\lambda _{j})E_{i,j}

.

Thus, the roots for $\mathrm {sl} (n,\mathbb {C} )$ are the linear functionals $\alpha _{i,j}$ given by

\alpha _{i,j}(H)=\lambda _{i}-\lambda _{j}

.

After identifying ${\mathfrak {h}}$ with its dual, the roots become the vectors $\alpha _{i,j}:=e_{i}-e_{j}$ 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 $\alpha _{i,j}$ acts on ${\mathfrak {h}}$ 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 ${\mathfrak {h}}$ .

Role in the classification

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 $A_{n}$ .)
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

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

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, ${\mathfrak {sl}}_{n}$ is semisimple, and ${\mathfrak {gl}}_{n}$ 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.

References

↑ Knapp 2002 Section VI.10
↑ Hall 2015 Proposition 7.6
↑ Hall 2015 Proposition 7.7
↑ Knapp 2002 Section VI.1
↑ Hall 2015 Chapter 7
↑ Hall 2015 Theorem 10.9
↑ Knapp 2002 Sections II.2 and II.3
↑ Hall 2015 Chapter 7
↑ Hall 2015 Section 7.5
↑ Hall 2015 Theorem 9.3
↑ 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 .
Knapp, Anthony W. (2002), Lie groups beyond an introduction (2nd ed.), Birkhäuser
Varadarajan, V. S. (2004), Lie Groups, Lie Algebras, and Their Representations (1st ed.), Springer, ISBN 0-387-90969-9 .

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[1] Knapp 2002 Section VI.10

[2] Hall 2015 Proposition 7.6

[3] Hall 2015 Proposition 7.7

[4] Knapp 2002 Section VI.1

[5] Hall 2015 Chapter 7

[6] Hall 2015 Theorem 10.9

[7] Knapp 2002 Sections II.2 and II.3

[8] Hall 2015 Chapter 7

[9] Hall 2015 Section 7.5

[10] Hall 2015 Theorem 9.3

[11] Hall 2015 Section 7.7.1