Math 2.0

Nonassociative rings and algebras Semisimple Lie algebras Theorem

Root space decomposition

Builds on Semisimple Lie algebraCartan subalgebraKilling formWeight spaceDual space

Statement

Let FF be an algebraically closed field of characteristic 00. Let 𝔤\mathfrak{g} be a finite-dimensional semisimple Lie algebra over FF, with a Cartan subalgebra 𝔥\mathfrak{h}. Let BB be the Killing form of 𝔤\mathfrak{g}. For a linear form α\alpha in the dual space 𝔥∗\mathfrak{h}^*, let 𝔤α={x∈𝔤:[h,x]=α(h)x for all h∈𝔥}\mathfrak{g}_\alpha = \set{x \in \mathfrak{g} : [h, x] = \alpha(h) x \text{ for all } h \in \mathfrak{h}} be the weight space of α\alpha for the adjoint action of 𝔥\mathfrak{h}. Let Φ\Phi be the set of the nonzero α∈𝔥∗\alpha \in \mathfrak{h}^* with 𝔤α≠0\mathfrak{g}_\alpha \ne 0; its elements are the roots. Then:

  1. 𝔥\mathfrak{h} is abelian and 𝔤0=𝔥\mathfrak{g}_0 = \mathfrak{h};
  2. Φ\Phi is finite and 𝔤=𝔥⊕⨁α∈Φ𝔤α\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha;
  3. dim⁡𝔤α=1\dim \mathfrak{g}_\alpha = 1 for each α∈Φ\alpha \in \Phi;
  4. [𝔤α,𝔤β]⊆𝔤α+β[\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha + \beta} for all α,β∈𝔥∗\alpha, \beta \in \mathfrak{h}^*;
  5. B(𝔤α,𝔤β)=0B(\mathfrak{g}_\alpha, \mathfrak{g}_\beta) = 0 unless α+β=0\alpha + \beta = 0, and the restriction of BB to 𝔥\mathfrak{h} is nondegenerate;
  6. Φ\Phi spans 𝔥∗\mathfrak{h}^*, and −α∈Φ-\alpha \in \Phi whenever α∈Φ\alpha \in \Phi.

Remarks

The elements ad⁡h\operatorname{ad} h, h∈𝔥h \in \mathfrak{h}, are semisimple and commute, so they can be diagonalized at the same time; this gives the decomposition in 2. Item 4 follows from the Jacobi identity, and item 5 from the invariance of BB. Item 3 and the rest come from the copy of 𝔰𝔩2\mathfrak{sl}_2 spanned by 𝔤α\mathfrak{g}_\alpha, 𝔤−α\mathfrak{g}_{-\alpha} and [𝔤α,𝔤−α][\mathfrak{g}_\alpha, \mathfrak{g}_{-\alpha}], and from the representation theory of 𝔰𝔩2\mathfrak{sl}_2. For 𝔤=𝔰𝔩n(F)\mathfrak{g} = \mathfrak{sl}_n(F) with 𝔥\mathfrak{h} the diagonal matrices of trace 00, the root spaces are the lines FEijF E_{ij} with i≠ji \ne j. The root of EijE_{ij} is εi−εj\varepsilon_i - \varepsilon_j, where εi\varepsilon_i reads off the ii-th diagonal entry. The hypothesis of semisimplicity is needed: for a nilpotent Lie algebra 𝔤\mathfrak{g}, the whole of 𝔤\mathfrak{g} is a Cartan subalgebra and there are no roots.

Going deeper

For a student

The root space decomposition is the first step of the structure theory of semisimple Lie algebras. From it one shows that the roots form a root system, which leads to the classification by Dynkin diagrams. The same method with an arbitrary finite-dimensional module in place of 𝔤\mathfrak{g} gives the weight space decomposition of representations.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Cartan subalgebra, concept, no evidence levelCartan subalgebraconceptDual space, concept, no evidence levelDual spaceconceptKilling form, concept, no evidence levelKilling formconceptSemisimple Lie algebra, concept, no evidence levelSemisimple Lie algebraconceptWeight space, concept, no evidence levelWeight spaceconceptRoot space decomposition, statement, no evidence levelRoot space decomposit…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.
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  1. Cartan subalgebra concept
  2. Dual space concept
  3. Killing form concept
  4. Semisimple Lie algebra concept
  5. Weight space concept
  6. Root space decomposition statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Semisimple_Lie_algebra
About this page
TagKJKH7NJGGP
URIm2://KJKH7NJGGP
Kindstatement, theorem
WritingW0 stub
MSC17B20, 17B22
StatusThis copy of the site has no build data. Run m2 status KJKH7NJGGP.
AboutSemisimple Lie algebra, Cartan subalgebra
UsesSemisimple Lie algebra, Cartan subalgebra, Killing form, Weight space, Dual space
Changed3 November 2026

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