root system underlying a semi-simple Lie algebra
Crystallographic, reduced root systems are in one-to-one correspondence with semi-simple, complex Lie algebras. First, let us describe how one passes from a Lie algebra to a root system. Let be a semi-simple, complex Lie algebra and let be a Cartan subalgebra. Since is semi-simple, is abelian. Moreover, acts on (via the adjoint representation) by commuting, simultaneously diagonalizable linear maps. The simultaneous eigenspaces of this action are called root spaces, and the decomposition of into and the root spaces is called a root decomposition of . To be more precise, for , set
We call a non-zero a root if is non-trivial, in which case is called a root space. It is possible to show that that is just the Cartan subalgebra , and that for each root . Letting denote the set of all roots, we have
The Cartan subalgebra has a natural inner product, called the Killing form, which in turn induces an inner product on . It is possible to show that, with respect to this inner product, is a reduced, crystallographic root system.
subject to the following relations:
Thanks to the above isomorphism, to the difficult task of classifying complex semi-simple Lie algebras is transformed into the somewhat easier task of classifying crystallographic, reduced roots systems. Furthermore, a complex Lie algebra is simple if and only if the corresponding root system is indecomposable. Thus, we only need to classify indecomposable root systems, since all other root systems and semi-simple Lie algebras are built out of these.
|Title||root system underlying a semi-simple Lie algebra|
|Date of creation||2013-03-22 15:28:59|
|Last modified on||2013-03-22 15:28:59|
|Last modified by||rmilson (146)|