Let be a semisimple Lie algebra over and a Cartan subalgebra.

Given , we can define a subspace of

The nonzero that occur with are called **roots**, and they form a set . Because acts on by commuting diagonalizable operators (by semisimplicity of the elements of ), it follows by simultaneous diagonalization, that

Recall that , because a Cartan subalgebra is maximal abelian.

This is called the root space decomposition. A simple but important property is that ; this is checked because the are derivations.

The root space decomposition is highly useful in studying simple representations of .

I shall collect here a few facts about it.

Proposition 1are orthogonal under the Killing form unless .

This follows by a familiar argument, in view of . (more…)