Root system of a semi-simple Lie algebra

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

1 Associated root system
2 Associated semi-simple Lie algebra
3 Application
4 References

Associated root system

Let g be a semi-simple complex Lie algebra. Let further h be a Cartan subalgebra of g, i.e. a maximal abelian subalgebra. Then h acts on g via simultaneously diagonalizable linear maps in the adjoint representation. For λ in h^* define

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

We call a non-zero λ in h^* a root if g_λ is nontrivial. g_λ is called the root space of λ. Obviously g₀ = h. One can show that each non-trivial g_λ 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

$[H_\lambda,H_\mu] =0 \text{ for all }\lambda,\mu\in\Delta$

$[H_\lambda,X_\mu] = (\lambda,\mu)X_\mu,$

$[H_\lambda,Y_\mu] = -(\lambda,\mu)Y_\mu,$

$[X_\mu,Y_\lambda] = \delta_{\mu\lambda}H_\mu,$

$\mathrm{ad}_{X_\lambda}^{-(\mu,\lambda)%2B1}X_\mu = 0\text{ for }\lambda\ne\mu,$

$\mathrm{ad}_{Y_\lambda}^{-(\mu,\lambda)%2B1}Y_\mu = 0\text{ for }\lambda\ne\mu.$

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 semi-simple Lie algebras is reduced to the somewhat easier task of classifying reduced crystallographic root systems.

References

This article incorporates material from Root system underlying a semi-simple Lie algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

V.S. Varadarajan: Lie groups, Lie algebras, and their representations, GTM, Springer 1984.

Root system of a semi-simple Lie algebra

Contents

Associated root system

Associated semi-simple Lie algebra

Application

References