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
⟨ β , α ⟩ = 2 ( α , β )
and (2) the relations is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by
's and with the root system
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
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.).
be the free vector space spanned by
, V the free vector space with a basis
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
, 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
be the sum of all ideals intersecting
Then there is a vector space decomposition:
) are the subalgebras generated by the images of
One then shows: (1) the derived algebra
in the lead, (2) it is finite-dimensional and semisimple and (3)
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