Semisimple Lie algebra

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


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