Lectures on Chevalley groups

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

Last update: 1 August 2013

§5. Chevalley groups and algebraic groups

The significance of the results so far to the theory of semisimple algebraic groups will now be indicated.

Let k be an algebraically closed field. A subset Vkn is said to be algebraic if there exists a subset 𝒫k[x1,,xn] such that V= {v= (v1,,vn) kn| p(v1,,vn) =0 for all p𝒫}. The algebraic subsets of kn are the closed sets of the Zariski topology on kn. For Vkn set Ik(V)= {p k[x1,,xn] | p(v1,,vn) =0 for all (v1,,vn) V}.

Let r=n2+1. Define D(x)k[x0;xij]1i,jn by D(x)=1-x0 det(xij). Then GLn(k)={vkr|D(v)=0} is an algebraic subset of kr. G is a matric algebraic group if G is a subgroup of GLn(k) for some n and some algebraically closed field k, and G is an algebraic subset of kn2+1. If k0 is a subfield of k, G is defined over k0 if Ik(G) has a basis of polynomials with coefficients in k0.


(a) SLn(k),
(b) Superdiagonal subgroup,
(c) Diagonal subgroup,
(d) {[1t01]} =Ga=additive group,
(e) {[t00t-1]} =Gm=multiplicative group,
(f) Sp2n,
(g) SOn,
(h) any finite subgroup.
The groups in (a) - (e) are defined over the prime field. Whether Sp2n, SOn are or not depends on the coefficients of the defining forms. The groups in (h) are not connected in the Zariski topology, the others are.

A map of algebraic groups φ:GH is a homomorphism if it is a group homomorphism and each of the matric coefficients φ(g)ij is a rational function of the gij. A homomorphism φ:GH is an isomorphism if there exists a homomorphism ψ:HG such that φψ=idH and ψφ=idG. A homomorphism φ:GH is defined over k0 if each of the rational functions above has its coefficients in k0.

Except for the last assertion, the following results are proved in Séminaire Chevalley (1956-8), Exposé 3.

(i) Let G be a matric algebraic group. Then the following are equivalent:
(a) G is connected (in the Zariski topology).
(b) G is irreducible (as an algebraic variety).
(c) Ik(G) is a prime ideal.
(ii) The image of an algebraic group under a rational homomorphism is algebraic.
(iii) A group generated by connected algebraic subgroups is algebraic and connected (e.g. (a) - (g) are connected). It is defined over the perfect field k0 if each of the subgroups is.

If G is an algebraic group, the radical of G (radG) is the maximal connected solvable normal subgroup. G is semisimple if (1) radG={1} and (2) G is connected.

Example: { [ 1** 0 A 0 ] |A SLn-1 } has radical { [ 1** 01 ] }

For the remainder of this section we assume that k is algebraically closed, k0 is the prime field, G is a Chevalley group based on k and M the lattice. (Since a change of basis in M is given by polynomials with integral coefficients we may speak of a basis over M.)

Theorem 6: With the preceding notations:

(a) G is a semisimple algebraic group relative to M.
(b) B is a maximal connected solvable subgroup (Borel subgroup).
(c) H is a maximal connected diagonalizable subgroup (maximal torus).
(d) N is the normalizer of H and N/HW.
(e) G,B,H, and N are all defined over k0 relative to M.

Remark: B and H are determined by the abstract group G:

(a) B is maximal solvable and has no subgroups of finite index.
(b) H is maximal nilpotent and every subgroup of finite index is of finite index in its normalizer.

Proof of Theorem 6.

(a) MapGa𝔛α by xα:txα(t). This is a rational homomorphism. So since Ga is a connected algebraic group so is 𝔛α. Hence G is algebraic and connected. Let R=radG. Since R is solvable and normal it is finite by the Corollary to Theorem 5. Since R is also connected R=1, and hence G is semisimple.

(b and c) H is the image of Gm under (t1,,t) Πi=1hi(ti) and hence is algebraic and connected; so B=UH is connected, algebraic, and solvable. Let G1B. Then G1BwαB (some simple root α), so G1𝔛α,𝔛-α, and hence by Corollary 6 of Theorem 4' G1 is not solvable and hence (b) holds. H is a maximal connected diagonalizable subgroup of B (for any larger subgroup must intersect U nontrivially). Hence H is a maximal connected diagonalizable subgroup of G (by a theorem in Chevalley's Séminaire); so (c) holds.

(d) is clear. To prove (e) it suffices by (iii) to prove:

Lemma 34: Let 𝔛α={xα(t)|tk} and 𝔥α={hα(t)|tk*}. Then:

(a) 𝔛α is defined over k0 and xα:Ga𝔛α is an isomorphism over k0.
(b) 𝔥α is defined over k0 and hα:Gm𝔥α is a homomorphism over k0.


Let {vi} be a basis of M formed of weight vectors. Choose vi so that Xαvi0, then write Xαci=Σcijvj, and choose vj so that cij0. If vi is of weight μ, then vj is of weight μ+α. Since xα(t)=1+tXα+t2Xα2/2+ it follows that if aij is the (i,j) matric coordinate (ij) function then aij(xα(t))=cijt. All other coefficients of xα(t) are polynomials over k0 in t, hence also in aij. This set of polynomial relations defines 𝔛α as a group over k0. Now 1cijaij:xα(t)t is an inverse of xa, so the map xα is an isomorphism over k0. The proof of (b) is left as an excercise.

We can recover the lattices L0 and L from the group G as follows. Let μL. Define μˆ:HGm by μˆ(Πhi(ti))= Πtiμ(Hi). This is a character defined over k0. {μˆ} generates a lattice Lˆ, the character group of H. The 𝔛α's are determined by H as the unique minimal unipotent subgroups normalized by H. If h=Πhi(ti) then hxα(t)h-1= xα(αˆ(h)t) where αˆ(h)=Πtiα(Hi). αˆ is called a global root. Define Lˆ0= the lattice generated by all αˆ. Then Lˆ0Lˆ.

Exercise: There exists a W-isomorphism: LLˆ such that L0Lˆ0, μμˆ, and ααˆ. (The action of W on Lˆ is given by the action of N/H on the character group).

We summarize our results in:

Existence Theorem: Given a root system Σ, a lattice L with L0LL1 (where L0 and L1 are the root and weight lattices, respectively), and an algebraically closed field k, then there exists a semisimple algebraic group G defined over k such that L0 and L are realized as the lattices of global roots and characters, respectively, relative to a maximal torus. Furthermore G,𝔛α, can be taken over the prime field.

The classification theorem, that up to k-isomorphism every semisimple algebraic group over k has been obtained above, is much more difficult, (See Séminair Chevalley, 1956-8).

We recall that == { H| μ(H) for allμL } .

Lemma 35: Let k be algebraically closed, G a Chevalley group over k,H1,,H a basis for . Define hi by hi(v)=tμ(Hi)v for vVμ. Then the map φ:GmH given by (t1,,t) Πj=1hj(tj) is an isomorphism over k0 of algebraic groups.


Write Hi=ΣnijHj, nij. Given {tj} we can find {ti} such that tj=Πitinij (for det(nij)0 and k* is divisible). Then Πjhj(tj) acts on Vμ as multiplication by Πjtjμ(Hj)= Πitiμ(Hj), i.e. as Πhi(ti). This shows that φ maps Gm onto H. Clearly φ is a rational mapping defined over k0. Let {μi} be the basis of L dual to {Hj} (i.e. μi(Hj)=δij). Write μi=ΣμLnμμ. Then Πμ(Πjtjμ(Hj))nμ=ti, so φ-1 exists and is defined over k0.

Theorem 7: Let k be an algebraically closed field and k0 the prime subfield. Let G be a Chevalley group parametrized by k and viewed as an algebraic group defined over k0 as above. Then:

(a) U-HU is an open subvariety of G defined over k0.
(b) If n is the number of positive roots, then the map φ:kn×k*×knU-Hu defined by φ ( (tα)α<0, (ti)1i, (tα)α>0 ) = Πα<0 xα(tα)Π hi(ti) Πα>0 xα(tα) is an isomorphism of of varieties over k0.


(a) We consider the natural action of G on Λn relative to a basis {Y1,Y2,,Yr} over k0 made up of products of Hi's and Xα's such that Y1=ΛXα (α>0). For xG we set xYi=Σaij(x)Yj and then d=a11, a function on G over k0. We claim that xU-HU=U-B if and only if d(x)0. Assume xU-B. Since B fixes Y1 up to a nonzero multiple and if uU- then uXαXα++ht(β)<ht(α)kXβ, it follows that d(x)0. If xU-wB with wW,w1, the same considerations show that d(x)=0. If w0W makes all positive roots negative then by the equation w0U-wB=Bw0wB and Theorem 4' the two cases above are exclusive and exhaustive, whence (a).

The map φ is composed of the two maps ψ=(ψ1,ψ2,ψ3): (tα)α>0× (ti)× (tα)α>0 U-×H×U, and θ:U-×H×UU-HU. We will show that these are isomorphisms over k0. For ψ2 this follows from Lemma 35. Consider ψ3 Let {vi} be a basis for V, the underlying vector space, made up of weight vectors in the lattice M, and fij the corresponding coordinate functions on EndV. For each root α choose i=i(α), j=j(α), nij=n(α) as in the proof of Lemma 34. Set x=Πβ>0xβ(tb). Choosing an ordering of the positive roots consistent with addition^,we see at once that fi(α),j(α)(x)=n(α)tα+ an integral polynomial in the earlier t's and that fij(x) is an integral polynomial in the t's for all i,j. Thus ψ3 is an isomorphism over k0, and similarly for ψ1. To prove θ is an isomorphism we order the vi so that U-,H,U consist respectively of subdiagonal unipotent, diagonal, super-diagonal unipotent matrices (see Lemma 18, Cor. 3), and then we may assume that they consist of all of the invertible matrices of these types. Let x=u-hu be in U-HU and let the subdiagonal entries of u-, the diagonal entries of h, the superdiagonal entries of u be labelled tij with i>j,i=j,i<j respectively. We order the indices so that ij precedes k in case ik, j and ijk. Then in the three cases above fij(x)=tijtjj, resp. tij, resp. tiitij, increased by an integral polynomial in t's preceeding tij. We may now inductively solve for the t's as rational forms over in the f's, the division by the forms representing the tjj's being justified by the fact that they are nonzero on U-HU. Thus θ is an isomorphism over k0 and (b) follows.

Example: In SLn U-HU consists of all (aij) such that the minors [a11], [a11a12a21a22], are nonsingular.

Remark: It easily follows that the Lie algebra of G is k.

We can now easily prove the following important fact (but will refer the reader to Séminaire Bourbaki, Exp. 219 instead). Let G be a Chevalley group over , viewed as above as an algebraic matric group over , the prime field, and I the corresponding ideal over (consisting of all polynomials over which vanish on G). Then the set of zeros of I in any algebraically closed field k is just the Chevalley group over k of the same type (same root system and same weight lattice) as G. Thus we have a functorial definition in terms of equations of all of the semisimple algebraic groups of any given type.

Corollary 1: Let k,k0,G,V be as above. Let G be a Chevalley group constructed using V instead of V but with the same . Assume that LVLV. Then the homomorphism φ:GG taking xα(t)xα(t) for all α and t is a homomorphism of algebraic groups over k0.


Consider first φ|U-HU. By Theorem 7 we need only show that φ|H is rational over k0. The nonzero coordinates of Πhi(ti) are Πtiμ(Hi) (μLV). The nonzero coordinates of Πhi(ti) are Πtiμ(Hi) (μLV). Each of the former is a monomial in the latter (because LVLV), and hence is rational over k0. Now for wW,ωW (resp. ωW) can be chosen with coefficients in k0 (for wα(1)=xα(1)x-α(-1)xα(1)), so that φ|ωw-1U-B is rational over k0. Since BωwBωw-1U-B, we conclude that φ is rational over k0.

Corollary 2: The homomorphism φα:SL2𝔛α,𝔛-α (of Corollary 6 to Theorem 4') is a homomorphism of algebraic groups over k0.


This is a special case of Corollary 1.

Corollary 3: Assume ,V, and M are fixed, that V is universal, kK are fields and Gk and GK are the corresponding Chevalley groups. Then Gk=GkGLM,k.


Clearly GkGkGLM,k. Suppose xGkGLM,k. Then x=uhωwv (see Theorem 4') with ωw defined over the prime field. We must show that xωw-1Gk, i.e. uhu-Gk where u-=ωwvωw-1. Write uhu-= Πα>0xα(t) Πhi(ti) Πα<0xα(t) with tα,tiK. Applying φ-1 of Theorem 7, we get (tα)α>0× (ti)× (tα)α<0. Since uhu- is defined over k and φ-1 is defined over k0, all tα,tik. Hence uhu-Gk.

Remark: Suppose k= and G is a Chevalley group over k. Then G has the structure of a complex Lie group, and all the preceding statements have obvious modifications in the language of Lie groups, all of which are true. For example, all complex semisimple Lie groups are included in the construction, and φ in Theorem 7 is an isomorphism of complex analytic manifolds.

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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