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

Statement

Given a root system ${\displaystyle \Phi }$ in a Euclidean space with an inner product ${\displaystyle (\cdot ,\cdot )}$, and the usual bilinear form ${\displaystyle \langle \beta ,\alpha \rangle =2(\alpha ,\beta )/(\alpha ,\alpha )}$, with a fixed base ${\displaystyle \{\alpha _{1},\dots ,\alpha _{n}\}}$, there exists a Lie algebra ${\displaystyle {\mathfrak {g}}}$ generated by the ${\displaystyle 3n}$ elements ${\displaystyle e_{i},f_{i},h_{i}}$ (for ${\displaystyle 1\leq i\leq n}$) and relations:

${\displaystyle [h_{i},h_{j}]=0,}$

${\displaystyle [e_{i},f_{j}]=\delta _{ij}h_{i}}$,

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

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

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

We also have that ${\displaystyle {\mathfrak {g}}}$ is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra ${\displaystyle {\mathfrak {h}}=\bigoplus _{i}h_{i}}$ and that the root system of ${\displaystyle {\mathfrak {g}}}$ is ${\displaystyle \Phi }$.

The square matrix ${\displaystyle [\langle \alpha _{i},\alpha _{j}\rangle ]_{1\leq i,j\leq n}}$ 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 ${\displaystyle {\mathfrak {g}}(A)}$ associated to ${\displaystyle A}$. 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.).^[1][2] Let ${\displaystyle a_{ij}=\langle \alpha _{i},\alpha _{j}\rangle }$ and then let ${\displaystyle {\widetilde {\mathfrak {g}}}}$ be the Lie algebra generated by (1) the generators ${\displaystyle e_{i},f_{i},h_{i}}$ and (2) the relations:

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

Let ${\displaystyle {\mathfrak {h}}}$ be the free vector space spanned by ${\displaystyle h_{i}}$, V the free vector space with a basis ${\displaystyle v_{1},\dots ,v_{n}}$ and ${\textstyle T=\bigoplus _{l=0}^{\infty }V^{\otimes l}}$ the tensor algebra over it. Consider the following representation of a Lie algebra:

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

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

• ${\displaystyle \pi (f_{i})a=v_{i}\otimes a,}$
• ${\displaystyle \pi (h)1=\langle \lambda ,\,h\rangle 1,\pi (h)(v_{j}\otimes a)=-\langle \alpha _{j},h\rangle v_{j}\otimes a+v_{j}\otimes \pi (h)a}$, inductively,
• ${\displaystyle \pi (e_{i})1=0,\,\pi (e_{i})(v_{j}\otimes a)=\delta _{ij}\alpha _{i}(a)+v_{j}\otimes \pi (e_{i})a}$, 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 ${\displaystyle {\widetilde {\mathfrak {n}}}_{+}}$ (resp. ${\displaystyle {\widetilde {\mathfrak {n}}}_{-}}$) the subalgebras of ${\displaystyle {\widetilde {\mathfrak {g}}}}$ generated by the ${\displaystyle e_{i}}$'s (resp. the ${\displaystyle f_{i}}$'s).

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

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

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

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

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

where ${\displaystyle {\mathfrak {n}}_{+}}$ (resp. ${\displaystyle {\mathfrak {n}}_{-}}$) are the subalgebras generated by the images of ${\displaystyle e_{i}}$'s (resp. the images of ${\displaystyle f_{i}}$'s).

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

References

1. Serre, Jean-Pierre (1966). Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras]. Translated by Jones, G. A. Benjamin. ISBN 978-3-540-67827-4.
2. Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.

Further reading

• Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90053-7.