Math 2.0

Nonassociative rings and algebras Semisimple Lie algebras Theorem

Root system of a semisimple Lie algebra

Builds on Root space decompositionSemisimple Lie algebraCartan subalgebraKilling formRoot systemExistence and conjugacy of Cartan subalgebras

Statement

Let 𝔤\mathfrak{g} be a finite-dimensional semisimple Lie algebra over ℂ\C, and let 𝔥\mathfrak{h} be a Cartan subalgebra. Let Φ⊆𝔥∗\Phi \subseteq \mathfrak{h}^* be the set of roots from the root space decomposition. For λ∈𝔥∗\lambda \in \mathfrak{h}^*, let tλ∈𝔥t_\lambda \in \mathfrak{h} be the element with B(tλ,h)=λ(h)B(t_\lambda, h) = \lambda(h) for all h∈𝔥h \in \mathfrak{h}, where BB is the Killing form. Put (λ,μ)=B(tλ,tμ)(\lambda, \mu) = B(t_\lambda, t_\mu), and let EℚE_\Q be the ℚ\Q-span of Φ\Phi. Then:

  1. the form (⋅,⋅)(\cdot, \cdot) takes rational values on EℚE_\Q and is positive definite there;
  2. Φ\Phi is a reduced root system in the Euclidean space E=ℝ⊗ℚEℚE = \R \otimes_\Q E_\Q, and dim⁡ℝE=dim⁡ℂ𝔥\dim_\R E = \dim_\C \mathfrak{h};
  3. up to isomorphism, this root system does not depend on the choice of 𝔥\mathfrak{h};
  4. 𝔤\mathfrak{g} is simple if and only if Φ\Phi is irreducible.

Remarks

The proof studies, for each root α\alpha, the subalgebra 𝔤α⊕𝔤−α⊕[𝔤α,𝔤−α]\mathfrak{g}_\alpha \oplus \mathfrak{g}_{-\alpha} \oplus [\mathfrak{g}_\alpha, \mathfrak{g}_{-\alpha}], which is isomorphic to 𝔰𝔩2(ℂ)\mathfrak{sl}_2(\C). The weights of 𝔤\mathfrak{g} under this copy of 𝔰𝔩2(ℂ)\mathfrak{sl}_2(\C) are integers, which gives the integrality of ⟨β,α⟩\langle \beta, \alpha \rangle, and the symmetry of the weights of 𝔰𝔩2\mathfrak{sl}_2-modules gives sα(Φ)=Φs_\alpha(\Phi) = \Phi. Item 3 follows from the conjugacy of Cartan subalgebras. For 𝔰𝔩n+1(ℂ)\mathfrak{sl}_{n+1}(\C), the roots εi−εj\varepsilon_i - \varepsilon_j form the root system AnA_n. Two semisimple Lie algebras over ℂ\C are isomorphic if and only if their root systems are isomorphic, so the root system is a complete invariant.

Going deeper

For a student

This is the bridge between Lie algebras and combinatorics. Together with the classification of root systems by Dynkin diagrams and the existence result of Serre, it gives the classification of the simple Lie algebras over ℂ\C.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Cartan subalgebra, concept, no evidence levelCartan subalgebraconceptDual space, concept, no evidence levelDual spaceconceptEngel subalgebra, concept, no evidence levelEngel subalgebraconceptKilling form, concept, no evidence levelKilling formconceptLie algebra, concept, no evidence levelLie algebraconceptNilpotent Lie algebra, concept, no evidence levelNilpotent Lie algebraconceptRoot system, concept, no evidence levelRoot systemconceptSemisimple Lie algebra, concept, no evidence levelSemisimple Lie algebraconceptWeight space, concept, no evidence levelWeight spaceconceptExistence and conjugacy of Cartan subalgebras, statement, no evidence levelExistence and conjuga…statementRoot space decomposition, statement, no evidence levelRoot space decomposit…statementRoot system of a semisimple Lie algebra, statement, no evidence levelRoot system of a semi…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cartan subalgebra concept
  2. Dual space concept
  3. Engel subalgebra concept
  4. Killing form concept
  5. Lie algebra concept
  6. Nilpotent Lie algebra concept
  7. Root system concept
  8. Semisimple Lie algebra concept
  9. Weight space concept
  10. Existence and conjugacy of Cartan subalgebras statement
  11. Root space decomposition statement
  12. Root system of a semisimple Lie algebra statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Semisimple_Lie_algebra
About this page
TagAR0D7V1J8T
URIm2://AR0D7V1J8T
Kindstatement, theorem
WritingW0 stub
MSC17B20, 17B22
StatusThis copy of the site has no build data. Run m2 status AR0D7V1J8T.
AboutSemisimple Lie algebra, Root system
UsesRoot space decomposition, Semisimple Lie algebra, Cartan subalgebra, Killing form, Root system, Existence and conjugacy of Cartan subalgebras
Changed3 November 2026

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