# Root system of a semi-simple Lie algebra

In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semisimple Lie algebras. Here the construction of a root system of a semisimple Lie algebra – and, conversely, the construction of a semisimple Lie algebra from a reduced crystallographic root system – are shown.

## Associated root system

Let g be a complex semisimple Lie algebra. Let further h be a Cartan subalgebra of g. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For λ in h*, define the subspace gλg by

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

One calls a non-zero λ in h* a root if the subspace gλ is nontrivial. In this case gλ is called the root space of λ. The definition of Cartan subalgebra guarantees that g0 = h. One can show that each non-trivial gλ (i.e. for λ ≠ 0) is one-dimensional.[1] Let R be the set of all roots. Since the elements of h are simultaneously diagonalizable, we have

${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\lambda \in R}{\mathfrak {g}}_{\lambda }.}$

The Cartan subalgebra h inherits an inner product from the Killing form on g. This induces an inner product on h*. One can show that with respect to this inner product R is a reduced crystallographic root lattice.[2]

## Associated semisimple Lie algebra

Let E be a Euclidean space and R a reduced crystallographic root system in E. Let moreover Δ be a choice of simple roots. We define a complex Lie algebra over the generators

${\displaystyle H_{\lambda },X_{\lambda },Y_{\lambda }{\text{ for }}\lambda \in \Delta }$

with the Chevalley–Serre relations

{\displaystyle {\begin{aligned}[][H_{\lambda },H_{\mu }]&=0{\text{ for all }}\lambda ,\mu \in \Delta ,\\\left[H_{\lambda },X_{\mu }\right]&=(\lambda ,\mu )X_{\mu },\\\left[H_{\lambda },Y_{\mu }\right]&=-(\lambda ,\mu )Y_{\mu },\\\left[X_{\mu },Y_{\lambda }\right]&=\delta _{\mu \lambda }H_{\mu },\\\mathrm {ad} _{X_{\lambda }}^{-(\mu ,\lambda )+1}X_{\mu }&=0{\text{ for }}\lambda \neq \mu ,\\\mathrm {ad} _{Y_{\lambda }}^{-(\mu ,\lambda )+1}Y_{\mu }&=0{\text{ for }}\lambda \neq \mu .\end{aligned}}}

(Here the coefficients denoted by (λ, μ) should be replaced by the coefficients of the Cartan matrix.) It turns out that the generated Lie algebra is semisimple and has root system isomorphic to the given R.

## Application

Due to the isomorphism, classification of finite-dimensional representations of semi-simple Lie algebras is reduced to the somewhat easier task of classifying reduced crystallographic root systems.

## Notes

1. ^ Hall 2015 Theorem 7.23
2. ^ Hall 2015 Theorem 7.30