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
![{\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.
Sketch of proof
The proof here is taken from (Kac 1990, Theorem 1.2.) and (Serre 2000, Ch. VI, Appendix.) harv error: no target: CITEREFSerre2000 (help).
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)
.
References