Root System of a Semi- Simple Lie Algebra (Paperback)

High Quality Content by WIKIPEDIA articles In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. We show the construction of a root system from a semi-simple Lie algebra and conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic root system. Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g, i.e. a maximal abelian subalgebra. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For in h* define mathfrak{g}_lambda: = {ainmathfrak{g}: h, a]=lambda(h)atext{ for all }hinmathfrak{h}}.