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[edit]

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[edit]

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[edit]

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[edit]

References[edit]

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.