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 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 $g λ \subset g$ by

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

One calls 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 $g 0 = h$ . One can show that each non-trivial $g λ$ (i.e. for $λ \neq 0$ ) is one-dimensional.^[1] 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.^[2]

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

{\begin{aligned}[][H_{\lambda },H_{\mu }]&=0{\text{ for all }}\lambda ,\mu \in \Delta ,\\\left[H_{\lambda },X_{\mu }\right]&=(\lambda ,\mu )X_{\mu },\\\left[H_{\lambda },Y_{\mu }\right]&=-(\lambda ,\mu )Y_{\mu },\\\left[X_{\mu },Y_{\lambda }\right]&=\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 .\end{aligned}}

(Here the coefficients denoted by $(λ, μ)$ 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.

Notes

↑ Hall 2015 Theorem 7.23
↑ Hall 2015 Theorem 7.30

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, 222 (2nd ed.), Springer
V.S. Varadarajan, Lie groups, Lie algebras, and their representations, GTM, Springer 1984.

External links

Hazewinkel, Michiel, ed. (2001), "Coxeter group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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

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