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 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 and a semisimple algebra. In particular, the zero space is the only Lie algebra that is solvable and semisimple.

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.

ee also

*semisimple
*simple Lie algebra
*reductive group

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