Lectures on Chevalley groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 18 July 2013

§1. A basis for

We start with some basic properties of semisimple Lie algebras over , and establish some notation to be used throughout. The assertions not proved here are proved in the standard books on Lie algebras, e.g., those of Dynkin, Jacobson or Sophus Lie (Séminaire).

Let be a semisimple Lie algebra over , and a Cartan subalgebra of . Then is necessarily Abelian and =Σα0α where α* and α={X|[H,X]=α(H)X for all H}. Note that =0. The α's are linear functions on , called roots. We adopt the convention that γ=0 if γ is not a root. Then [α,β]α+β. The rank of =dim=, say. The roots generate * as a vector space over .

Write V for Q*, the vector space over Q generated by the roots. Then dimQV=. Let γV. Since the Killing form is nondegenerate there exists an Hγ such that (H,Hγ)=γ(H) for all H. Define (γ,δ)=(Hγ,Hδ) for all γ,δV. This is a symmetric, nondegenerate, positive definite bilinear form on V.

Denote the collection of all roots by Σ. Then Σ is a subset of the nonzero elements of V satisfying:

(0) Σ generates V as a vector space over Q.
(1) αΣ-αΣ and kαΣ for k an integer ±1.
(2) 2(α,β)/ (β,β) for all α,βΣ. (Write α,β=2(α,β)/(β,β). These are called Cartan integers).
(3) Σ is invariant under all reflections wα (αΣ) (where wα is the reflection in the hyperplane orthogonal to α, i.e., wαv=v-2(v,α)/(α,α) α).

Thus Σ is a root system in the sense of Appendix I. Conversely, if Σ is any root system satisfying condition (2), then Σ is the root system of some Lie algebra.

The group W generated by all wα is a finite group (Appendix 1.6) called the Weyl group. If {α1,,α} is a simple system of roots (Appendix I.8), then W is generated by the wαi (i=1,,n) (Appendix I.16) and every root is congruent under W to a simple root (Appendix I.15).

Lemma 1: For each root α, let Hα be such that (H,Hα)=α(H) for all H. Define Hα=2/(α,α)Hα and Hi=Hαi (i=1,,). Then each Hα is an integral linear combination of the Hi.


Write wi for wαiW. Define an action of W on by wiHj=Hj- αj,αiHi.


wiHj = 2(αj,αj) wiHj = 2(αj,αj) Hj- 2(αj,αj) · 2(αi,αj) (αi,αi) Hi = Hj- 2(αi,αi)· 2(αi,αj) (αj,αj) Hi = Hj-αi,αj Hi = Hwiαj

Then since the wi generate W, wHj is an integral linear combination of the Hi for all wW. Now if α is an arbitrary root then α=wαj for some wW and some j. Then Hα=Hwαj= wHαj=wHj = an integral linear combination of the Hi.

For every root α choose Xαα, Xα0. If α+β0 define Nα,β by [Xα,Xβ]=Nα,βXα+β. Set Nα,β=0 if α+β is not a root.

If α and β are roots the α-string of roots through β is the sequence β-rα,,β,,β+qα where β+iα is a root for -riq but β-(r+1)α and β+(q+1)α are not roots.

Lemma 2: The Xα can be chosen so that:

(a) [Xα,X-α]=Hα.
(b) If α and β are roots, β±α, and β-rα,,β,β+qα is the α-string of roots through β then Nα,β2= q(r+1) |α+β|2 /|β|2.


See the first part of the proof of Theorem 10, p. 147 in Jacobson, Lie Algebras.

Lemma 3: If α,β and α+β are roots, then q(r+1) |α+β|2 /|β|2 =(r+1)2.


We use two facts:

(*) r-q=β,α.
(For wα maps β-rα to β+qα so β+qα= wα(β-rα)= β-rα- 2(β-rα,α)/ (α,α)α =β-β,αα +rα).
(**) In the α-string of roots through β at most two root lengths occur.
(For if V is the vector space over Q generated by α and β and Σ=ΣV, then Σ is a root system and every root in the α-string of roots through β belongs to Σ. Now V is two dimensional; so a system of simple roots for Σ has at most two elements. Since every root in Σ is conjugate under the Weyl group of Σ to a simple root, Σ and hence the α-string of roots through β has at most two root lengths). We must show that q|α+β|2/ |β|2 =r+1. Now by (*):

r+1-q |α+β|2 /|β|2 = q+β,α+ 1-q (α+β,α+β) /(β,β) = β,α+1- q|α|2/ |β|2-q α,β = (β,α+1) ( 1-q|α|2 /|β|2 ) .

Set A=β,α+1 and B=1-q|α|2/|β|2. We must show A=0 or B=0.

If |α||β| then |β,α|= 2|(β,α)| /|α|2 2|(β,α)| /|β|2 =|α,β|. By Schwarz's inequality β,α α,β = 4(α,β)2/ |α||β| 4 with equality if and only if α=kβ. Since α and β are roots and α±β we have αkβ so β,αα,β<4. Then since |β,α| |α,β| we have β,α=-1, 0, or 1. If β,α=-1 then A=0. If β,α0 then |β+2α|> |β+α|> |β|. Since there are only two root lengths β+2α is not a root and hence q=1. Since |β+α|>|α| and |β+α|>|β| and at most two root lengths occur |α|=|β|. Hence B=0.

If |α|<|β|, then |α+β||β| (since otherwise three root lengths would occur). Hence (α,β)<0 so α,β<0. Then |β-α|>|β|>|α| so β-α is not a root and hence r=0. As above α,ββ,α<4 and |α,β|< |β,α| so α,β=-1,0, or 1. Hence α,β=-1. Then by (*) q=-β,α= β,α/ α,β = |β|2/ |α|2. Hence B=0.

We collect these results in:

Theorem 1: The Hi (i=1,2,,) chosen as in Lemma 1 together with the Xα chosen as in Lemma 2 form a basis for relative to which the equations of structure are as follows (and, in particular, fare integral):

(a) [Hi,Hj]=0
(b) [Hi,Xα]= α,αiXα
(c) [Xα,X-α]=Hα= an integral linear combination of the Hi.
(d) [Xα,Xβ]= ±(r+1)Xα+β if α+β is a root.
(e) [Xα,Xβ]=0 if α+β0 and α+β is not a root.


(a) holds since is abelian. (b) holds since [Hβ,Xα]= a(Hβ)Xα= α,βXα. (c) follows from the choice of the Xα and the Hi and from Lemma 1. (d) follows from Lemma 2(b) and Lemma 3. (e) holds since [α,β]=0 if α+β is not a root.


(a) Such a basis is called a Chevalley basis. It is unique up to sign changes and automorphisms of .
(b) Xα,X-α and Hα span a 3-dimensional subalgebra isomorphic to s𝓁2 (2×2 matrices of trace 0). Xα[0100] X-α [0010] Hα [100-1]
(c) As an example let =s𝓁+1.
Then = {diagonal matrices} is a Cartan subalgebra. For i,j=1,, +1,ij, def±ne α=α(i,j) by α(i,j): diag(a1,,a+1) ai-aj. Then the α(i,j) are the roots. Let Ei,j be the matrix unit with 1 in the (i,j) position and 0 elsewhere. Then Xα=Ei,j, X-α=Ej,i, Hα=Ei,i-Ej,j and Hi=Ei,i-Ei+1,i+1.

Exercise: If only one root length occurs then all coefficients in (d) of Theorem 1 are ±1. Otherwise ±2 and ±3 can occur.

Notes and References

This is a typed excerpt of Lectures on Chevalley groups by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson. This work was partially supported by Contract ARO-D-336-8230-31-43033.

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