# Root system of a semi-simple Lie algebra

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

## Associated root system

Let g be a semi-simple complex 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

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

We call 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. Let R be the set of all roots. Since the elements of h are simultaneously diagonalizable, we have

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

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.

## Associated semi-simple Lie algebra

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

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

with the Chevalley-Serre relations

$[H_\lambda,H_\mu] =0 \text{ for all }\lambda,\mu\in\Delta$
$[H_\lambda,X_\mu] = (\lambda,\mu)X_\mu,$
$[H_\lambda,Y_\mu] = -(\lambda,\mu)Y_\mu,$
$[X_\mu,Y_\lambda] = \delta_{\mu\lambda}H_\mu,$
$\mathrm{ad}_{X_\lambda}^{-(\mu,\lambda)+1}X_\mu = 0\text{ for }\lambda\ne\mu,$
$\mathrm{ad}_{Y_\lambda}^{-(\mu,\lambda)+1}Y_\mu = 0\text{ for }\lambda\ne\mu.$

[Here the coefficients denoted by $(\lambda,\mu)$ should be replaced by the coefficients of the Cartan matrix.]

It turns out that the generated Lie algebra is semi-simple 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.