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

Statement

The theorem states that: given a root system $\Phi$ in a Euclidean space with an inner product $(,)$ , $\langle \beta ,\alpha \rangle =2(\alpha ,\beta )/(\alpha ,\alpha ),\beta ,\alpha \in E$ and a base $\{\alpha _{1},\dots ,\alpha _{n}\}$ of $\Phi$ , 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\neq j

,

[h_{i},e_{j}]=\langle \alpha _{i},\alpha _{j}\rangle e_{j},\,[h_{i},f_{j}]=-\langle \alpha _{i},\alpha _{j}\rangle f_{j}

,

\operatorname {ad} (e_{i})^{-\langle \alpha _{i},\alpha _{j}\rangle +1}(e_{j})=0,i\neq j

,

\operatorname {ad} (f_{i})^{-\langle \alpha _{i},\alpha _{j}\rangle +1}(f_{j})=0,i\neq j

.

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

The square matrix $[\langle \alpha _{i},\alpha _{j}\rangle ]_{1\leq i,j\leq n}$ 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 ${\mathfrak {g}}(A)$ 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, Ch. VI, Appendix.) and (Kac 1990, Theorem 1.2.). Let $a_{ij}=\langle \alpha _{i},\alpha _{j}\rangle$ 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:

$[h_{i},h_{j}]=0$ ,
$[e_{i},f_{i}]=h_{i}$ , $[e_{i},f_{j}]=0,i\neq j$ ,
$[h_{i},e_{j}]=a_{ij}e_{j},[h_{i},f_{j}]=-a_{ij}f_{j}$ .

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 ${\textstyle T=\bigoplus _{l=0}^{\infty }V^{\otimes l}}$ the tensor algebra over it. Consider the following representation of a Lie algebra:

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

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

$\pi (f_{i})a=v_{i}\otimes a,$
$\pi (h)1=\langle \lambda ,\,h\rangle 1,\pi (h)(v_{j}\otimes a)=-\langle \alpha _{j},h\rangle v_{j}\otimes a+v_{j}\otimes \pi (h)a$ , 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, ${\widetilde {\mathfrak {g}}}={\widetilde {\mathfrak {n}}}_{+}\bigoplus {\mathfrak {h}}\bigoplus {\widetilde {\mathfrak {n}}}_{-}$ .
${\widetilde {\mathfrak {n}}}_{+}=\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{\alpha }$ where ${\widetilde {\mathfrak {g}}}_{\alpha }=\{x\in {\widetilde {\mathfrak {g}}}|[h,x]=\alpha (h)x,h\in {\mathfrak {h}}\}$ and, similarly, ${\widetilde {\mathfrak {n}}}_{-}=\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{-\alpha }$ .
(root space decomposition) ${\widetilde {\mathfrak {g}}}=\left(\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{-\alpha }\right)\bigoplus {\mathfrak {h}}\bigoplus \left(\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{\alpha }\right)$ .

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., ${\mathfrak {i}}=\bigoplus _{\alpha }({\widetilde {\mathfrak {g}}}_{\alpha }\cap {\mathfrak {i}})$ . 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}}_{+}\bigoplus {\mathfrak {h}}\bigoplus {\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) $[{\mathfrak {g}},{\mathfrak {g}}]={\mathfrak {g}}$ .