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 $\mathrm{\Phi }$, there exists a finite-dimensional semisimple Lie algebra whose root system is the given $\mathrm{\Phi }$.

Statement

The theorem states that: given a root system $\mathrm{\Phi }$ in a Euclidean space with an inner product $(,)$, $⟨\beta ,\alpha ⟩=2(\alpha ,\beta )/(\alpha ,\alpha ),\beta ,\alpha \in E$ and a base $\{{\alpha }_1,\dots ,{\alpha }_n\}$ of $\mathrm{\Phi }$, the Lie algebra $\mathfrak{𝔤}$ defined by (1) $3n$ generators $e_i,f_i,h_i$ and (2) the relations

$[h_i,h_j]=0,$

$[e_i,f_i]=h_i,[e_i,f_j]=0,i\ne j$,

$[h_i,e_j]=⟨{\alpha }_i,{\alpha }_j⟩e_j,[h_i,f_j]=-⟨{\alpha }_i,{\alpha }_j⟩f_j$,

$\mathrm{ad}(e_i{)}^{-⟨{\alpha }_i,{\alpha }_j⟩+1}(e_j)=0,i\ne j$,

$\mathrm{ad}(f_i{)}^{-⟨{\alpha }_i,{\alpha }_j⟩+1}(f_j)=0,i\ne j$.

is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by $h_i$'s and with the root system $\mathrm{\Phi }$.

The square matrix $[⟨{\alpha }_i,{\alpha }_j⟩{]}_{1\le i,j\le n}$ 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 $\mathfrak{𝔤}(A)$ associated to $A$. 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 $a_{ij}=⟨{\alpha }_i,{\alpha }_j⟩$ and then let $\widetilde{\mathfrak{𝔤}}$ be the Lie algebra generated by (1) the generators $e_i,f_i,h_i$ and (2) the relations:

• $[h_i,h_j]=0$,
• $[e_i,f_i]=h_i$, $[e_i,f_j]=0,i\ne j$,
• $[h_i,e_j]=a_{ij}{e}_j,[h_i,f_j]=-a_{ij}{f}_j$.

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

$\pi :\widetilde{\mathfrak{𝔤}}\to \mathfrak{𝔤}\mathfrak{𝔩}(T)$

given by: for $a\in T,h\in \mathfrak{𝔥},\lambda \in {\mathfrak{𝔥}}^{*}$,

• $\pi (f_i)a=v_i\otimes a,$
• $\pi (h)1=⟨\lambda ,h⟩1,\pi (h)(v_j\otimes a)=-⟨{\alpha }_j,h⟩v_j\otimes a+v_j\otimes \pi (h)a$, inductively,
• $\pi (e_i)1=0,\pi (e_i)(v_j\otimes a)={\delta }_{ij}{\alpha }_i(a)+v_j\otimes \pi (e_i)a$, 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 ${\widetilde{\mathfrak{𝔫}}}_{+}$ (resp. ${\widetilde{\mathfrak{𝔫}}}_{-}$) the subalgebras of $\widetilde{\mathfrak{𝔤}}$ generated by the $e_i$'s (resp. the $f_i$'s).

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

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

$\mathfrak{𝔤}=\widetilde{\mathfrak{𝔤}}/\mathfrak{𝔯}$.

Then it contains a copy of $\mathfrak{𝔥}$, which is identified with $\mathfrak{𝔥}$ and

$\mathfrak{𝔤}={\mathfrak{𝔫}}_{+}⨁\mathfrak{𝔥}⨁{\mathfrak{𝔫}}_{-}$

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

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

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.