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 g_λ ⊂ g 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 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

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

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 semisimple 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

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

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

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

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

$\mathrm {ad} _{X_{\lambda }}^{-(\mu ,\lambda )+1}X_{\mu }=0{\text{ for }}\lambda \neq \mu ,$

$\mathrm {ad} _{Y_{\lambda }}^{-(\mu ,\lambda )+1}Y_{\mu }=0{\text{ for }}\lambda \neq \mu .$

[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 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.

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.

• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
• V.S. Varadarajan, Lie groups, Lie algebras, and their representations, GTM, Springer 1984.

External links

• "Coxeter group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Coxeter group". MathWorld.
• Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators
• Popov, V.L.; Fedenko, A.S. (2001), "Weyl group", Encyclopaedia of Mathematics, SpringerLink