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 [math]\displaystyle{ \Phi }[/math], there exists a finite-dimensional semisimple Lie algebra whose root system is the given [math]\displaystyle{ \Phi }[/math].

Statement

The theorem states that: given a root system [math]\displaystyle{ \Phi }[/math] in a Euclidean space with an inner product [math]\displaystyle{ (, ) }[/math], [math]\displaystyle{ \langle \beta, \alpha \rangle = 2(\alpha, \beta)/(\alpha, \alpha), \beta, \alpha \in E }[/math] and a base [math]\displaystyle{ \{ \alpha_1, \dots, \alpha_n \} }[/math] of [math]\displaystyle{ \Phi }[/math], the Lie algebra [math]\displaystyle{ \mathfrak g }[/math] defined by (1) [math]\displaystyle{ 3n }[/math] generators [math]\displaystyle{ e_i, f_i, h_i }[/math] and (2) the relations

[math]\displaystyle{ [h_i, h_j] = 0, }[/math]

[math]\displaystyle{ [e_i, f_i] = h_i, \, [e_i, f_j] = 0, i \ne j }[/math],

[math]\displaystyle{ [h_i, e_j] = \langle \alpha_i, \alpha_j \rangle e_j, \, [h_i, f_j] = -\langle \alpha_i, \alpha_j \rangle f_j }[/math],

[math]\displaystyle{ \operatorname{ad}(e_i)^{-\langle \alpha_i, \alpha_j \rangle+1}(e_j) = 0, i \ne j }[/math],

[math]\displaystyle{ \operatorname{ad}(f_i)^{-\langle \alpha_i, \alpha_j \rangle+1}(f_j) = 0, i \ne j }[/math].

is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by [math]\displaystyle{ h_i }[/math]'s and with the root system [math]\displaystyle{ \Phi }[/math].

The square matrix [math]\displaystyle{ [\langle \alpha_i, \alpha_j \rangle]_{1 \le i, j \le n} }[/math] 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 [math]\displaystyle{ \mathfrak g(A) }[/math] associated to [math]\displaystyle{ A }[/math]. 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 (Serre 1966) and (Kac 1990). Let [math]\displaystyle{ a_{ij} = \langle \alpha_i, \alpha_j \rangle }[/math] and then let [math]\displaystyle{ \widetilde{\mathfrak g} }[/math] be the Lie algebra generated by (1) the generators [math]\displaystyle{ e_i, f_i, h_i }[/math] and (2) the relations:

• [math]\displaystyle{ [h_i, h_j] = 0 }[/math],
• [math]\displaystyle{ [e_i, f_i] = h_i }[/math], [math]\displaystyle{ [e_i, f_j] = 0, i \ne j }[/math],
• [math]\displaystyle{ [h_i, e_j] = a_{ij} e_j, [h_i, f_j] = -a_{ij} f_j }[/math].

Let [math]\displaystyle{ \mathfrak{h} }[/math] be the free vector space spanned by [math]\displaystyle{ h_i }[/math], V the free vector space with a basis [math]\displaystyle{ v_1, \dots, v_n }[/math] and [math]\displaystyle{ T = \bigoplus_{l=0}^{\infty} V^{\otimes l} }[/math] the tensor algebra over it. Consider the following representation of a Lie algebra:

[math]\displaystyle{ \pi : \widetilde{\mathfrak g} \to \mathfrak{gl}(T) }[/math]

given by: for [math]\displaystyle{ a \in T, h \in \mathfrak{h}, \lambda \in \mathfrak{h}^* }[/math],

• [math]\displaystyle{ \pi(f_i)a = v_i \otimes a, }[/math]
• [math]\displaystyle{ \pi(h)1 = \langle \lambda, \, h \rangle 1, \pi(h)(v_j \otimes a) = -\langle \alpha_j, h \rangle v_j \otimes a + v_j \otimes \pi(h)a }[/math], inductively,
• [math]\displaystyle{ \pi(e_i)1 = 0, \, \pi(e_i)(v_j \otimes a) = \delta_{ij} \alpha_i(a) + v_j \otimes \pi(e_i)a }[/math], 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 [math]\displaystyle{ \widetilde{\mathfrak{n}}_+ }[/math] (resp. [math]\displaystyle{ \widetilde{\mathfrak{n}}_- }[/math]) the subalgebras of [math]\displaystyle{ \widetilde{\mathfrak g} }[/math] generated by the [math]\displaystyle{ e_i }[/math]'s (resp. the [math]\displaystyle{ f_i }[/math]'s).

• [math]\displaystyle{ \widetilde{\mathfrak{n}}_+ }[/math] (resp. [math]\displaystyle{ \widetilde{\mathfrak{n}}_- }[/math]) is a free Lie algebra generated by the [math]\displaystyle{ e_i }[/math]'s (resp. the [math]\displaystyle{ f_i }[/math]'s).
• As a vector space, [math]\displaystyle{ \widetilde{\mathfrak g} = \widetilde{\mathfrak{n}}_+ \bigoplus \mathfrak{h} \bigoplus \widetilde{\mathfrak{n}}_- }[/math].
• [math]\displaystyle{ \widetilde{\mathfrak{n}}_+ = \bigoplus_{0 \ne \alpha \in Q_+} \widetilde{\mathfrak g}_{\alpha} }[/math] where [math]\displaystyle{ \widetilde{\mathfrak g}_{\alpha} = \{ x \in \widetilde{\mathfrak g}|[h, x] = \alpha(h) x, h \in \mathfrak h \} }[/math] and, similarly, [math]\displaystyle{ \widetilde{\mathfrak{n}}_- = \bigoplus_{0 \ne \alpha \in Q_+} \widetilde{\mathfrak g}_{-\alpha} }[/math].
• (root space decomposition) [math]\displaystyle{ \widetilde{\mathfrak g} = \left( \bigoplus_{0 \ne \alpha \in Q_+} \widetilde{\mathfrak g}_{-\alpha} \right) \bigoplus \mathfrak h \bigoplus \left( \bigoplus_{0 \ne \alpha \in Q_+} \widetilde{\mathfrak g}_{\alpha} \right) }[/math].

For each ideal [math]\displaystyle{ \mathfrak i }[/math] of [math]\displaystyle{ \widetilde{\mathfrak g} }[/math], one can easily show that [math]\displaystyle{ \mathfrak i }[/math] is homogeneous with respect to the grading given by the root space decomposition; i.e., [math]\displaystyle{ \mathfrak i = \bigoplus_{\alpha} (\widetilde{\mathfrak g}_{\alpha} \cap \mathfrak i) }[/math]. It follows that the sum of ideals intersecting [math]\displaystyle{ \mathfrak h }[/math] trivially, it itself intersects [math]\displaystyle{ \mathfrak h }[/math] trivially. Let [math]\displaystyle{ \mathfrak r }[/math] be the sum of all ideals intersecting [math]\displaystyle{ \mathfrak h }[/math] trivially. Then there is a vector space decomposition: [math]\displaystyle{ \mathfrak r = (\mathfrak r \cap \widetilde{\mathfrak n}_-) \oplus (\mathfrak r \cap \widetilde{\mathfrak n}_+) }[/math]. In fact, it is a [math]\displaystyle{ \widetilde{\mathfrak g} }[/math]-module decomposition. Let

[math]\displaystyle{ \mathfrak g = \widetilde{\mathfrak g}/\mathfrak r }[/math].

Then it contains a copy of [math]\displaystyle{ \mathfrak h }[/math], which is identified with [math]\displaystyle{ \mathfrak h }[/math] and

[math]\displaystyle{ \mathfrak g = \mathfrak{n}_+ \bigoplus \mathfrak{h} \bigoplus \mathfrak{n}_- }[/math]

where [math]\displaystyle{ \mathfrak{n}_+ }[/math] (resp. [math]\displaystyle{ \mathfrak{n}_- }[/math]) are the subalgebras generated by the images of [math]\displaystyle{ e_i }[/math]'s (resp. the images of [math]\displaystyle{ f_i }[/math]'s).

One then shows: (1) the derived algebra [math]\displaystyle{ [\mathfrak g, \mathfrak g] }[/math] here is the same as [math]\displaystyle{ \mathfrak g }[/math] in the lead, (2) it is finite-dimensional and semisimple and (3) [math]\displaystyle{ [\mathfrak g, \mathfrak g] = \mathfrak g }[/math].

References

• Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8. https://books.google.com/books?id=kuEjSb9teJwC&dq=Victor%20G.%20Kac&pg=PP1. 
• Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90053-7. https://archive.org/details/introductiontoli00jame. 
• Serre, Jean-Pierre (1966) (in en). Algèbres de Lie semi-simples complexes. Benjamin. ISBN 978-3-540-67827-4. https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3.