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
.
The theorem states that: given a root system
in a Euclidean space with an inner product
,
and a base
of
, the Lie algebra
defined by (1)
generators
and (2) the relations
![{\displaystyle [h_{i},h_{j}]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1609195c508362f0f7c9eafa6823994031bed0e6)
,
,
,
.
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.
The proof here is taken from (Serre 1966, Ch. VI, Appendix.) and (Kac 1990, Theorem 1.2.).
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:
![{\displaystyle \pi :{\widetilde {\mathfrak {g}}}\to {\mathfrak {gl}}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fadac805a6c34fcde8637966652d28d0a57c4317)
given by: for
,
![{\displaystyle \pi (f_{i})a=v_{i}\otimes a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60b3f959717c5355c18652a3887f6f8c1faaecdd)
, 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
![{\displaystyle {\mathfrak {g}}={\mathfrak {n}}_{+}\bigoplus {\mathfrak {h}}\bigoplus {\mathfrak {n}}_{-}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e5901cc9f01d0be775855dafdb57d7ac576921b)
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)
.