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. 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 g_λ ⊂ g by

We call a non-zero λ in h^* a root if the subspace g_λ is nontrivial. In this case g_λ is called the root space of λ. The definition of Cartan subalgebra guarantees that g₀ = h. One can show that each non-trivial g_λ (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

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

with the Chevalley-Serre relations

[Here the coefficients denoted by 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.

Notes

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.