Serre's theorem on a semisimple Lie algebra

In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system , there exists a finite-dimensional semisimple Lie algebra whose root system is the given . It is named after Jean-Pierre Serre.

Statement

Given a root system in a Euclidean space with an inner product , and the usual bilinear form , with a fixed base , there exists a Lie algebra generated by the elements (for ) and relations:

We also have that is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra and that the root system of is .

The square matrix is called the Cartan matrix. Thus, with this notion, the theorem states that, given a Cartan matrix A, there exists a unique (up to an isomorphism) 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 (generally infinite-dimensional) 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 (1) the generators and (2) 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 ,

• inductively,
• inductively.

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 (resp. ) the subalgebras of generated by the 's (resp. the 's).

• (resp. ) is a free Lie algebra generated by the 's (resp. the 's).
• As a vector space, .
• where and, similarly, .
• (root space decomposition) .

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 (resp. ) are the subalgebras generated by the images of 's (resp. the images of 's).

One then shows: (1) the derived algebra here is the same as in the lead, (2) it is finite-dimensional and semisimple and (3) .

References

Further reading