Semisimple_Lie

Semisimple Lie group

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 called reductive if it is the sum of a semisimple and an abelian Lie algebra.

Let $\mathfrak g$ be 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,
every representation is completely reducible; that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).
The radical of $\mathfrak g$ is zero.

The significance of semisimplicity is due to 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.) Moreover, the representation theory can be carried out more nicely on semisimple ones than on 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.

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, assuming $\mathfrak g$ is finite-dimensional, the dimension of $\operatorname{Der}(\mathfrak g)$ equals 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.

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

References

Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9

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References