Serre's theorem on a semisimple Lie algebra


In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a root system, there exists a finite-dimensional semisimple Lie algebra whose root system is the given.

Statement

The theorem states that: given a root system in an Euclidean space with an inner product, and a base of, the Lie algebra defined by generators and the relations
is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by 's and with the root system.
The square matrix is called the Cartan matrix. Thus, with this notion, the theorem states that, give a Cartan matrix A, there exists a unique finite-dimensional semisimple Lie algebra associated to. The construction of a semisimple Lie algebra from a Cartan matrix can be generalized by weakening the definition of a Cartan matrix. The Lie algebra associated to a generalized Cartan matrix is called a Kac–Moody algebra.

Sketch of proof

The proof here is taken from and.
Let and then let be the Lie algebra generated by the generators and the relations:
Let be the free vector space spanned by, V the free vector space with a basis and the tensor algebra over it. Consider the following representation of a Lie algebra:
given by: for,
It is not trivial that this is indeed a well-defined representation and that has to be checked by hand. From this representation, one deduces the following properties: let the subalgebras of generated by the 's.
For each ideal of, one can easily show that is homogeneous with respect to the grading given by the root space decomposition; i.e.,. It follows that the sum of ideals intersecting trivially, it itself intersects trivially. Let be the sum of all ideals intersecting trivially. Then there is a vector space decomposition:. In fact, it is a -module decomposition. Let
Then it contains a copy of, which is identified with and
where are the subalgebras generated by the images of 's.
One then shows: the derived algebra here is the same as in the lead, it is finite-dimensional and semisimple and .