root system underlying a semi-simple Lie algebra

Crystallographic, reduced root systems are in one-to-one correspondence with semi-simple, complex Lie algebras. First, let us describe how one passes from a Lie algebraMathworldPlanetmath to a root system. Let 𝔤 be a semi-simple, complex Lie algebra and let 𝔥 be a Cartan subalgebraMathworldPlanetmath. Since 𝔤 is semi-simple, 𝔥 is abelian. Moreover, 𝔥 acts on 𝔤 (via the adjoint representationMathworldPlanetmath) by commuting, simultaneously diagonalizablePlanetmathPlanetmath linear maps. The simultaneous eigenspacesMathworldPlanetmath of this 𝔥 action are called root spaces, and the decomposition of 𝔤 into 𝔥 and the root spaces is called a root decomposition of 𝔤. To be more precise, for λ𝔥*, set

𝔤λ={a𝔤:[h,a]=λ(h)a for all h𝔥}.

We call a non-zero λ𝔥* a root if 𝔤λ is non-trivial, in which case 𝔤λ is called a root space. It is possible to show that that 𝔤0 is just the Cartan subalgebra 𝔥, and that dim𝔤λ=1 for each root λ. Letting R𝔥* denote the set of all roots, we have


The Cartan subalgebra 𝔥 has a natural inner product, called the Killing formPlanetmathPlanetmath, which in turn induces an inner product on 𝔥*. It is possible to show that, with respect to this inner product, R is a reduced, crystallographic root system.

Conversely, let RE be a reduced, crystallographic root system. Let Δ be a base of positive roots. We define a Lie algebra by taking generatorsPlanetmathPlanetmath


subject to the following relationsMathworldPlanetmathPlanetmath:

[Hλ,Hμ] =0,
(adXλ)-(λ,μ)+1(Xμ) =0,λμ,
(adYλ)-(λ,μ)+1(Yμ) =0,λμ,

The above are known as the Chevalley-Serre relationsMathworldPlanetmath The resulting Lie algebra turns out to be semi-simple, with a root system isomorphic to the given R.

Thanks to the above isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, to the difficult task of classifying complex semi-simple Lie algebras is transformed into the somewhat easier task of classifying crystallographic, reduced roots systems. Furthermore, a complex Lie algebra is simple if and only if the corresponding root system is indecomposable. Thus, we only need to classify indecomposable root systems, since all other root systems and semi-simple Lie algebras are built out of these.

Title root system underlying a semi-simple Lie algebra
Canonical name RootSystemUnderlyingASemisimpleLieAlgebra
Date of creation 2013-03-22 15:28:59
Last modified on 2013-03-22 15:28:59
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 7
Author rmilson (146)
Entry type Result
Classification msc 17B20
Related topic SimpleAndSemiSimpleLieAlgebras2
Defines Serre relations
Defines Chevalley-Serre relations