Serre's theorem on a semisimple Lie algebra

Template:Format footnotes 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 $𝔤$ 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, given a Cartan matrix A, there exists a unique (up to an isomorphism) finite-dimensional semisimple Lie algebra $𝔤(A)$ 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 Template:Harv and Template:Harv. Let $a_{ij}=⟨{\alpha }_i,{\alpha }_j⟩$ and then let $\widetilde{𝔤}$ 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 $𝔥$ 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:

$\pi :\widetilde{𝔤}\to 𝔤𝔩(T)$

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

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

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

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

$𝔤=\widetilde{𝔤}/𝔯$.

Then it contains a copy of $𝔥$, which is identified with $𝔥$ and

$𝔤={𝔫}_{+}⨁𝔥⨁{𝔫}_{-}$

where ${𝔫}_{+}$ (resp. ${𝔫}_{-}$) 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 $[𝔤,𝔤]$ here is the same as $𝔤$ in the lead, (2) it is finite-dimensional and semisimple and (3) $[𝔤,𝔤]=𝔤$.

References

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