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 formula_1, there exists a finite-dimensional semisimple Lie algebra whose root system is the given formula_1. Statement. The theorem states that: given a root system formula_1 in a Euclidean space with an inner product formula_4, formula_5 and a base formula_6 of formula_1, the Lie algebra formula_8 defined by (1) formula_9 generators formula_10 and (2) the relations is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by formula_16's and with  the root system formula_1. The square matrix formula_18 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 formula_19 associated to formula_20. 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 formula_21 and then let formula_22 be  the Lie algebra generated by (1) the generators formula_10 and (2) the relations: Let formula_28 be the free vector space spanned by formula_16, "V" the free vector space with a basis formula_30 and formula_31 the tensor algebra over it. Consider the following representation of a Lie algebra: given by: for formula_33, 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 formula_37 (resp. formula_38) the subalgebras of formula_22 generated by the formula_40's (resp. the formula_41's). For each ideal formula_51  of formula_22, one can easily show that formula_51 is homogeneous with respect to the grading given by the root space decomposition; i.e., formula_54. It follows that the sum of ideals intersecting formula_55 trivially, it itself intersects formula_55 trivially. Let formula_57 be the sum of all ideals intersecting formula_55 trivially. Then there is a vector space decomposition: formula_59. In fact, it is a formula_22-module decomposition. Let Then it contains a copy of formula_55, which is identified with formula_55 and where formula_65 (resp. formula_66) are the subalgebras generated by the images of formula_40's (resp. the images of formula_41's). One  then shows: (1) the derived algebra formula_69 here is the same as formula_8 in the lead, (2) it is finite-dimensional and semisimple and (3) formula_71.