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 .

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 (1) generators and (2) 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 (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 (Kac 1990, Theorem 1.2.) and (Serre 2000, Ch. VI, Appendix.).

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

  • Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
  • Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90053-7.
  • Serre, Jean-Pierre (2000). Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras]. Translated by Jones, G. A. Springer. ISBN 978-3-540-67827-4.
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