root system underlying a semi-simple Lie algebra

Crystallographic, reduced root systems are in one-to-onecorrespondence with semi-simple, complex Lie algebras. First, let usdescribe how one passes from a Lie algebra to a root system. Let $𝔤$be a semi-simple, complex Lie algebra and let $𝔥$ be a Cartansubalgebra. Since $𝔤$ is semi-simple, $𝔥$ is abelian. Moreover,$𝔥$ acts on $𝔤$ (via the adjoint representation) by commuting,simultaneously diagonalizable linear maps. The simultaneouseigenspaces of this $𝔥$ action are called root spaces, and thedecomposition of $𝔤$ into $𝔥$ and the root spaces is called a root decomposition of $𝔤$. To be more precise, for$\lambda \in {𝔥}^{*}$, set

$${𝔤}_{\lambda }=\{a\in 𝔤:[h,a]=\lambda (h)a\text{ for all }h\in 𝔥\}.$$

We call a non-zero$\lambda \in {𝔥}^{*}$ a root if ${𝔤}_{\lambda }$ is non-trivial, in whichcase ${𝔤}_{\lambda }$ is called a root space. It is possible to showthat that ${𝔤}_0$ is just the Cartan subalgebra $𝔥$, and that$dim{𝔤}_{\lambda }=1$ for each root $\lambda$. Letting $R\subset {𝔥}^{*}$denote the set of all roots, we have

$$𝔤=𝔥\oplus \underset{\lambda \in R}{\oplus }{𝔤}_{\lambda }.$$

The Cartan subalgebra $𝔥$ has a natural inner product, called theKilling form, which in turn induces an inner product on ${𝔥}^{*}$. It ispossible to show that, with respect to this inner product, $R$ is areduced, crystallographic root system.

Conversely, let $R\subset E$ be a reduced, crystallographic rootsystem. Let $\mathrm{\Delta }$ be a base of positive roots. We define a Liealgebra by taking generators

$$H_{\lambda },X_{\lambda },Y_{\lambda },\lambda \in \mathrm{\Delta },$$

subject to the following relations:

$[H_{\lambda },H_{\mu }]$$=0,$

$=(\lambda ,\mu )X_{\lambda },$

$=-(\lambda ,\mu )Y_{\lambda },$

$=H_{\lambda },$

$=0,\lambda \ne \mu ;$

${(\mathrm{ad}{X}_{\lambda })}^{-(\lambda ,\mu )+1}(X_{\mu })$$=0,\lambda \ne \mu ,$

${(\mathrm{ad}{Y}_{\lambda })}^{-(\lambda ,\mu )+1}(Y_{\mu })$$=0,\lambda \ne \mu ,$

The above are known as the Chevalley-Serre relations The resulting Liealgebra turns out to be semi-simple, with a root system isomorphic tothe given $R$.

Thanks to the above isomorphism, to the difficult task of classifyingcomplex semi-simple Lie algebras is transformed into the somewhateasier task of classifying crystallographic, reduced roots systems.Furthermore, a complex Lie algebra is simple if and only if thecorresponding root system is indecomposable. Thus, we only need toclassify indecomposable root systems, since all other root systems andsemi-simple Lie algebras are built out of these.