Semisimple Lie Algebra

Let

\mathfrak g

be a finite dimensional Lie algebra. The following conditions are equivalent:

Additionally, when

\mathfrak g

is defined over a Field of Characteristic 0 we have:

$\mathfrak g$ is semisimple if and only if every Representation is completely reducible, that is for every invariant subspace of the representation there is an invariant complement ( Weyl's Theorem ).

\mathfrak g

is semisimple, then every element can be expressed as the bracket of two other elements, i.e.

\mathfrak g = g, \mathfrak g

. The converse of this statement does not always hold.

SEE ALSO

	CATEGORIES ABOUT SEMISIMPLE LIE ALGEBRA

	lie algebras

Information About