Semisimple Lie algebra: Information from Answers.com

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.

Throughout the article, unless otherwise stated, $\mathfrak g$ is a 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 of $\mathfrak g$ is zero.

1 Examples
2 Properties
3 Significance
4 Generalizations
5 References

Examples

Examples of semisimple Lie algebras, with notation coming from classification by Dynkin diagrams, are:

$A n :$ $\mathfrak {sl}_n$ , the special linear Lie algebra.
$C n :$ $\mathfrak {sp}_{2n}$ , the symplectic Lie algebra.
$B n, D n :$ $\mathfrak {so}_n,\quad n\ge 3$ , the special orthogonal Lie algebra. This family is divided into two families: $B_n = \mathfrak{so}_{2n+1}$ and $D_n = \mathfrak{so}_{2n}.$

Except certain exceptions in low dimensions, many of these are simple Lie algebras, which are a fortiori semisimple. These three (or more correctly four) families, together with five exceptions, are in fact the only simple Lie algebras over the complex numbers.

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. While in other contexts, complete reducibility is equivalent to being semisimple, for Lie algebras the two notions are different: Lie algebras whose finite-dimensional representations are all completely reducible are called reductive Lie algebras. For example, $\mathfrak{gl}_n$ is not semisimple, but its finite-dimensional representations are completely reducible. Here "semisimple" means that the Lie algebra is semisimple (a sum of simple Lie algebras), not that its representations are semisimple (sums of simple representations). Furthermore, 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 $\operatorname{Der}(\mathfrak g)$ . Every ideal, quotient and product of a semisimple Lie algebra is again semisimple.

Linear

The adjoint representation $\operatorname{ad}\colon \mathfrak g \to \operatorname{End}(\mathfrak{g})$ 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 and s is semisimple 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 complex semisimple Lie algebra is the dimension of any of its Cartan subalgebras.

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

Main articles: Reductive Lie algebra and Split Lie algebra

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

Bourbaki, Nicolas (2005), "VIII: Split Semi-simple Lie Algebras", Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9, http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA69
Erdmann, Karin; Wildon, Mark (2006), Introduction to Lie Algebras (1st ed.), Springer, ISBN 1-84628-040-0 .
Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7 .
Varadarajan, V. S. (2004), Lie Groups, Lie Algebras, and Their Representations (1st ed.), Springer .

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Semisimple Lie algebra

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