Linear Algebraic Groups - Department of Mathematics · 8 Commutative Linear Algebraic Groups II (10/07) 46 9 Derivations and Di erentials (10/14) 51 10 The Lie Algebra of a Linear

Linear Algebraic Groups

Fall 2015

These are notes for the graduate course Math 6690 (Linear Algebraic Groups) taughtby Dr. Mahdi Asgari at the Oklahoma State University in Fall 2015. The notes are takenby Pan Yan ([email protected]), who is responsible for any mistakes. If you noticeany mistakes or have any comments, please let me know.

Contents

1 Root Systems (08/19) 3

2 Review of Algebraic Geometry I (08/26) 14

3 Review of Algebraic Geometry II, Introduction to Linear AlgebraicGroups I (09/02) 18

4 Introduction to Linear Algebraic Groups II (09/09) 24

5 Introduction to Linear Algebraic Groups III (09/16) 30

6 Jordan Decomposition (09/23) 34

7 Commutative Linear Algebraic Groups I (09/30) 40

8 Commutative Linear Algebraic Groups II (10/07) 46

9 Derivations and Differentials (10/14) 51

10 The Lie Algebra of a Linear Algebraic Group (10/21) 56

11 Homogeneous Spaces, Quotients of Linear Algebraic Groups (10/28) 61

12 Parabolic and Borel Subgroups (11/4) 66

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13 Weyl Group, Roots, and Root Datum (11/11) 72

14 More on Roots, and Reductive Groups (11/18) 79

15 Bruhat Decomposition, Parabolic Subgroups, the Isomorphism Theo-rem, and the Existence Theorem (12/2) 86

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1 Root Systems (08/19)

Root Systems

Reference for this part is Lie Groups and Lie Algebras, Chapters 4-6 by N. Bourbaki.Let V be a finite dimensional vector space over R. An endomorphism s : V → V is

called a reflection if there exists 0 6= a ∈ V such that s(a) = −a and s fixes pointwise ahyperplane (i.e., a subspace of codimension 1) in V . Then

V = ker(s− 1)⊕ ker(s+ 1)

and s2 = 1. We denote V +s = ker(s− 1) which is a hyperplane in V , and V −s = ker(s+ 1)

which is just Ra.Let D = im(1− s), then dim(D) = 1. This implies that given 0 6= a ∈ D, there exists

a nonzero linear form a∗ : V → R such that

x− s(x) = 〈x, a∗〉 a,∀x ∈ V

where 〈x, a∗〉 = a∗(x). Conversely, given some 0 6= a ∈ V and a linear form a∗ 6= 0 on V ,set

sa,a∗(x) = x− 〈x, a∗〉 a,∀x ∈ V

this gives an endomorphism of V such that 1− sa,a∗ is of rank 1. Note that

s2a,a∗(x) = sa,a∗(x− 〈x, a∗〉 a)

= x− 〈x, a∗〉 a− 〈x− 〈x, a∗〉 a, a∗〉 a= x− 2 〈x, a∗〉 a+ 〈x, a∗〉 〈a, a∗〉 a= x+ (〈a, a∗〉 a− 2) 〈x, a∗〉 a.

So sa,a∗ is a reflection if and only if 〈a, a∗〉 = 2, i.e., sa,a∗(a) = −a.WARNING: 〈x, a∗〉 is only linear in the first variable, but not the second.

Remark 1.1. (i) When V is equipped with a scalar product (i.e., a non-degenerate sym-metric bilinear form B), then we can consider the so called orthogonal reflections, i.e., thefollowing equivalent conditions hold:

V +s and V −s are perpendicular w.r.t. B ⇔ B is invariant under s.

In that case,

s(x) = x− 2B(x, a)

B(a, a)a.

(ii) A reflection s determines the hyperplane uniquely, but not the choice of the nonzeroa (but it does in a root system, which we will talk about later).

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Definition 1.2. Let V be a finite dimensional vector space over R, and let R be a subsetof V . Then R is called a root system in V if(i) R is finite, 0 6∈ R, and R spans V ;(ii) For any α ∈ R, there is an α∨ ∈ V ∗ where V ∗ = f : V → R linear is the dual of V ;(iii) For any α ∈ R, α∨(R) ⊂ Z.

Lemma 1.3. Let V be a vector space over R and let R be a finite subset of V generatingV . For any α ∈ R such that α 6= 0, there exists at most one reflection s of V such thats(α) = −α and s(R) = R.

Proof. Suppose there are two reflections s, s′ such that s(α) = s′(α) = −α and s(R) =s′(R) = R. Then s(x) = x−f(x)α, s′(x) = x− g(x)α for some linear functions f(x), g(x).Since s(α) = s′(α) = −α, we have f(α) = g(α) = 2. Then

s(s′(x)) = x− g(x)α− f (x− g(x)α)α

= x− g(x)α− f(x)α+ f(α)g(x)α

= x− g(x)α− f(x)α+ 2g(x)α

= x− (g(x)− f(x))α

is a linear function, and s(s′(R)) = R. Since R is finite, s s′ is of finite order, i.e.,(s s′)n = (s s′) (s s′) · · · (s s′) is identity for some n ≥ 1. Moreover,

(s s′)2(x) = x− (g(x)− f(x))α− (g(x− (g(x)− f(x))α)− f(x− (g(x)− f(x))α))α

= x− 2(g(x)− f(x))α

and by applying the composition repeatedly, we have

(s s′)n(x) = x− n(g(x)− f(x))α.

But (s s′)n(x) = x for all x ∈ V , therefore, g(x) = f(x). Hence s(x) = s′(x).

Lemma 1.3 shows that given α ∈ R, there is a unique reflection s of V such thats(α) = −α and s(R) = R. That implies α determines sα,α∨ and α∨ uniquely, and hence(iii) in the definition makes sense.

We can write sα,α∨ = sα. Then

sα(x) = x−⟨x, α∨

⟩α,∀x ∈ V.

The elements of R are called roots (of this system). The rank of the root system is thedimension of V . We define

A(R) = finite group of automorphisms of V leaving R stable

and the Weyl group of the root system R to be

W = W (R) = the subgroup of A(R) generated by the sα, α ∈ R.

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Remark 1.4. Let R be a root system in V . Let (x|y) be a symmetric bilinear form on V ,non-degenerate and invariant under W (R). We can use this form to identify V with V ∗.Now if α ∈ R, then α is non-isotropic (i.e., (α|α) 6= 0) and

α∨ =2α

(α|α).

This is because we saw that (x|y) invariant under sα implies

sα(x) = x− 2(x|α)

(α|α)α.

Proposition 1.5. R∨ = α∨ : α ∈ R is a root system in V ∗ and α∨∨ = α, ∀α ∈ R.

Proof. (Sketch). For (i) in Definition 1.2, R∨ is finite and does not contain 0. To see thatR∨ spans V ∗, we need to use the canonical bilinear form on V × V ∗ to identify

VQ = Q− vector space of V generated by the α

andV ∗Q = Q− vector space of V ∗ generated by the α∨

with the dual of the other. This way, the α∨ generate V ∗.For (ii) in Definition 1.2, sα,α∨ is an automorphism of V equipped with the root system

R and t(sα,α∨)−1 leaved R∨ stable, but one can check that t(sα,α∨)−1 = sα,α∨ and α∨∨ = α.For (iii) in Definition 1.2, note that 〈β, α∨〉 ∈ Z ∀β ∈ R,∀α∨ ∈ R∨, so R∨ satisfies

(iii).

Remark 1.6. R∨ is called the dual root system of R. The map α 7→ α∨ is a bijectionfrom R to R∨ and is called the canonical bijection from R to R∨.

WARNING: If α, β ∈ R and α+ β ∈ R, then (α+ β)∨ 6= α∨ + β∨ in general.

Remark 1.7. (i) The facts sα(α) = −α and sα(R) ⊂ R imply R = −R.(ii) It is also clear that (−α)∨ = −α∨. −1 ∈ A(R), but -1 is not always an element of

W (R).(iii) The equality t(sα,α∨)−1 = sα∨,α implies the map u 7→ tu−1 is an isomorphism from

W (R) to W (R∨), so we can identify these two via this isomorphism, and simply considerW (R) as acting on both V and V ∗. It is similar for A(R).

First Examples

Now we give a few examples of root systems.

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Example 1.8. (A1): V = Re. The root system is

R = α = e,−e.

The reflection is sα(x) = −x. V +s = 0, V −s = V . A(R) = W (R) = Z/2Z. The usual scalar

product (x|y) = xy is W (R)-invariant. The dual space is V ∗ = Re∗ where e∗ : V → Rsuch that e∗(e) = 1. Then α∨ = 2e∗ and 〈α, α∨〉 = (2e∗)(e) = 2. R∨ = α∨ = 2e∗,−2e∗is a root system in V ∗, which is the dual root system of R. Observe that if we identify Vand V ∗ via e↔ e∗, then α∨ = 2α

(α|α) . See Figure 1.

−e e

Figure 1: Root system for A1, Example 1.8

Example 1.9. (A1-non-reduced): V = Re. The root system is

R = e, 2e,−e,−2e.

The dual space is R∗ = Re∗, and the dual root system is R∨ = ±e∗,±2e∗. E∨ =2e∗, (2e)∨ = e∗. See Figure 2

Remark 1.10. Example 1.8 and Example 1.9 are the only dimension 1 root systems forV = R.

Example 1.11. (A1 ×A1): V = R2 = Re1 ⊕ Re2. The root system is

R = α = e1,−α, β = e2,−β.

The dual space is V ∗ = Re∗1 ⊕ Re∗2. We have α∨ = 2e∗1, β∨ = 2e∗2. The dual root systemis R∨ = ±2e∗1,±2e∗2. This root system will be called reducible. See Figure 3.

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−e−2e e 2e

Figure 2: Root system for A1-non-reduced, Example 1.9

−e1 α = e1

β = e2

−e2

Figure 3: Root system for A1 ×A1, Example 1.11

Example 1.12. (A2): E = R3, V = (x1, x2, x3) ∈ E : x1 + x2 + x3 = 0. The rootsystem is

R = ±(e1 − e2),±(e1 − e3),±(e2 − e3).

Moreover,W (R) = S3 = permutations on e1, e2, e3,

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A(R) = S3 × 1,−1where −1 maps ei to −ei. See Figure 4.

−α α = e1 − e2

β = e2 − e3 α+ β

−α− β −β

Figure 4: Root system for A2, Example 1.12

Example 1.13. (B2): V = R2 = Re1 ⊕ Re2. The root system is

R = ±e1,±e2,±e1 ± e2.

Moreover,A(R) = W (R) = (Z/2Z)2 o S2.

See Figure 5.

Example 1.14. (C2) – the dual of (B2): The root system is

R = ±2e1,±2e2,±e1 ± e2

AndA(R) = W (R) = (Z/2Z)2 o S2.

See Figure 6.

Example 1.15. (BC2) – this is non-reduced (also the unique irreducible non-reduced rootsystem of rank 2): V = R2. The root system is

R = ±e1,±e2,±2e1,±2e2,±e1 ± e2

andA(R) = W (R) = (Z/2Z)2 o S2.

See Figure 7.

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−e1 e1

β = e2

α = e1 − e2

Figure 5: Root system for B2, Example 1.13

α = e1 − e2

β = 2e2

Figure 6: Root system for C2, Example 1.14

Example 1.16. (G2): E = R3, V = (x1, x2, x3) ∈ E : x1 + x2 + x3 = 0. The rootsystem is

R = ±(e1−e2),±(e1−e3),±(e2−e3),±(2e1−e2−e3),±(2e2−e1−e3),±(2e3−e1−e2)

andA(R) = W (R) = dihedral group of order 12.

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e1 − e2

2e1

2e2

Figure 7: Non-reduced root system for BC2, Example 1.15

See Figure 8.

α = e1 − e2

β = −2e1 + e2 + e3

Figure 8: Root system for G2, Example 1.16

Remark 1.17. The above eight examples comprise of all rank 1 and rank 2 root systems(up to isomorphism). The rank 1 root systems are A1, and non-reduced A1. The rank 2root systems are A1 ×A1, A2, B2

∼= C2, G2, BC2.

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Irreducible Root Systems

Let V be the direct sum of Vi, 1 ≤ i ≤ r. Identify V ∗ with the direct sum of V ∗i , and foreach i, let Ri be a root system in Vi. Then R =

∐iRi is a root system in V whose dual

system is R∨ =∐iR∨i . The canonical bijection R↔ R∨ extends each canonical bijection

Ri ↔ R∨i for each i. We say R is the direct sum of root systems Ri.Let α ∈ Ri. If j 6= i, then ker(α∨) ⊃ Vj . So sα induces identity on Vj , j 6= i. On

the other hand, Rα ⊂ Vi, so sα leaves Vi stable. Then W (R) can be identified withW (R1)× · · · ×W (Rr).

Definition 1.18. A root system R is irreducible if R 6= ∅ and R is not the direct sum oftwo nonempty root systems.

It is easy to check that every root system R in V is the direct sum of a family of (Ri)i∈Iof irreducible root systems. The direct sum is unique up to permutation of the index setI. The Ri are called irreducible components of R.

Definition 1.19. A root system R is reduced if α ∈ R implies 12α 6∈ R. α is called

indivisible root .

Here is the complete list of irreducible, reduced root systems (up to isomorphism).

(I) (Al), l ≥ 1 :E = Rl+1, V = (α1, · · · , αl+1) :l+1∑i=1

αi = 0,

R = ±(ei − ej) : 1 ≤ i < j ≤ l + 1,#R = l(l + 1),

W (R) = Sl+1, A(R) =

W (R), if l = 1,

W (R)× Z/2Z, if l ≥ 2.

(II) (Bl), l ≥ 2 :E = V = Rl,R = ±ei, 1 ≤ i ≤ l;±ei ± ej , 1 ≤ i < j ≤ l,#R = 2l2,

A(R) = W (R) = (Z/2Z)l o Sl.

(III) (Cl), l ≥ 2 :E = V = Rl,R = ±2ei, 1 ≤ i ≤ l;±ei ± ej , 1 ≤ i < j ≤ l,#R = 2l2,

A(R) = W (R) = (Z/2Z)l o Sl.

(IV ) (Dl), l ≥ 3 :E = V = Rl,R = ±ei ± ej , 1 ≤ i < j ≤ l,#R = 2l(l − 1),

W (R) = (Z/2Z)l−1 o Sl,

A(R)/W (R) ∼=

Z/2Z, if l 6= 4,

S3, if l = 4,

(V ) Exceptional root systems: E6, E7, E8, F4, G2.

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Remark 1.20. The above list will classify split, connected, semisimple linear algebraicgroups over an algebraically closed field (up to isogeny).

Angles between Roots

Let α, β ∈ R. Put 〈α, β∨〉 = n(α, β). Then we have

n(α, α) = 2,

n(−α, β) = n(α,−β) = −n(α, β),

n(α, β) ∈ Z,sβ(α) = α− n(α, β)β,

n(α, β) = n(β∨, α∨).

Let (x|y) be a symmetric bilinear form on V , non-degenerate, invariant under W (R). Then

n(α, β) =2(α|β)

(β|β).

So

n(α, β) = 0

⇔ n(β, α) = 0

⇔ (α, β) = 0

⇔ sα and sβ commute ,

and

(α|β) 6= 0⇒ n(β, α)

n(α, β)=

(β|β)

(α|α).

We can determine possible angles between α and β. Let (x|y) be scalar product,W (R)-invariant and α, β ∈ R. Then

n(α, β)n(β, α) =2(α|β)

(β|β)· 2(β|α)

(α|α)= 4 cos2(α, β) ≤ 4.

We list all the possibilities in Table 1.

Corollary 1.21. Let α, β ∈ R. If α = cβ, then c ∈ ±1,±2,±12.

Corollary 1.22. Let α, β be non-proportional roots. If (α|β) > 0 (i.e., if the angle betweenα and β is strictly acute), then α− β is a root. If (α|β) < 0, then α+ β is a root.

Proof. Without loss of generality we may assume ||α|| ≤ ||β||. If (α|β) > 0, then sβ(α) =α − n(α, β)β ∈ R must be α − β by Table 1 (case 1 is the only possibility). Similarly, if(α|β) < 0, then sβ(α) = α−n(α, β)β ∈ R must be α+β (case 2 is the only possibility).

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case angle between α and β order of sαsβ1 n(α, β) = n(β, α) = 0 π/2 2

2 n(α, β) = n(β, α) = 1 π/3 and ||α|| = ||β|| 3

3 n(α, β) = n(β, α) = −1 2π/3 and ||α|| = ||β|| 3

4 n(α, β) = 1, n(β, α) = 2 π/4 and ||α|| =√

2||β|| 4

5 n(α, β) = −1, n(β, α) = −2 3π/4 and ||α|| =√

2||β|| 4

6 n(α, β) = 1, n(β, α) = 3 π/6 and ||α|| =√

3||β|| 6

7 n(α, β) = −1, n(β, α) = −3 5π/6 and ||α|| =√

3||β|| 6

8 n(α, β) = 2, n(β, α) = 2 α = β

9 n(α, β) = −2, n(β, α) = −2 α = −β10 n(α, β) = 1, n(β, α) = 4 β = 2α

11 n(α, β) = −1, n(β, α) = −4 β = −2α

Table 1: Possible Angels Between Two Roots

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2 Review of Algebraic Geometry I (08/26)

The Zariski Topology

Let k be an algebraically closed field (of any characteristic, occasionally char(k) 6= 2, 3).Let V = kn, S = k[T ] := k[T1, T2, · · · , Tn]. f ∈ S can be thought of as a functionf : V → k, via evaluation. We say v ∈ V is a zero of f ∈ k[T ] if f(v) = 0. Wesay v ∈ V is a zero of an ideal I of S if f(v) = 0,∀f ∈ I. Given an ideal I, writeν(I) = set of zeros of I. In the opposite direction, if X ⊂ V , define I(X) ⊂ S = k[T ] tobe the ideal consisting of polynomials f ∈ S with f(v) = 0, ∀v ∈ X.

Example 2.1. Let S = k[T ] = [T1], consider I = (T 2), then ν(I) = 0 and I(0) = (T ).

Definition 2.2. The radical or nilradical√I of an ideal I is

√I = f ∈ S : fm ∈ I for some m ≥ 1.

Theorem 2.3 (Hilbert’s Nullstellensatz). (i) If I is a proper ideal in S, then ν(I) 6= ∅.(ii) For any ideal I of S we have I(ν(I)) =

√I.

Definition 2.4. Observe that(i) ν(0) = V , ν(S) = ∅;(ii) I ⊂ J ⇒ ν(J) ⊂ ν(I);(iii) ν(I ∩ J) = ν(I) ∪ ν(J);(iv) If (Iα)α∈A is a family of ideals and I =

∑α∈A Iα, then ν(I) =

⋂α∈A ν(Iα).

Note that (i), (ii), (iv) imply that there is a topology on V = kn whose closed setsare the ν(I) where I is an ideal in S – we call it the Zariski topology . A closed subset inthe Zariski topology is called an algebraic set . Also, for any X ⊂ V , we have a Zariskisubspace topology on X.

Proposition 2.5. Let X ⊂ V be an algebraic set.(i) The Zariski topology on X is T1, i.e., points are closed.(ii) The topology space X is noetherian, i.e., it satisfies the following two equivalent

properties: any family of closed subsets of X contains a minimal one , or equivalently ifX1 ⊃ X2 ⊃ X3 ⊃ · · · is a decreasing sequence of closed subsets of X, then there existssome index h such that Xi = Xh for i ≥ h.

(iii) X is quasi-compact, i.e., any open covering of X has a finite subcover.

Note that in algebraic geometry, compact means quasi-compact and Hausdorff.

Review of Reducibility of Topological Spaces

Definition 2.6. A non-empty topological space X is called reducible if it is the union oftwo proper, closed subsets. Otherwise, it is called irreducible.

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Remark 2.7. If X is irreducible, then any two non-empty open subsets of X have anon-empty intersection.

This is mostly interesting only in non-Hausdorff space. In fact, any irreducible Haus-dorff space is simply a point.

If X,Y are two topological spaces. Then

A ⊂ X irreducible ⇔ A is irreducible,

f : X → Y continuous and X irreducible ⇒ f(X) is irreducible.

If X is noetherian topological space, then X has finitely many maximal irreduciblesubsets, called the (irreducible) components of X. The components are closed and theycover X.

Now, we consider the Zariski topology on V = kn.

Proposition 2.8. A closed subset X of V is irreducible if and only if I(X) is prime.

Proof. Let f, g ∈ S with fg ∈ I(X). Then

X = (X ∩ ν(fS)) ∪ (X ∩ ν(gS))

where both X ∩ ν(fS) and X ∩ ν(gS) are closed subsets of V . Since X is irreducible,X ⊂ ν(fS) or X ⊂ ν(gS). Hence f ∈ I(X) or g ∈ I(X). So I(X) is prime.

Conversely, assume I(X) is a prime ideal. If X = ν(I1) ∪ ν(I2) = ν(I1 ∩ I2) andX 6= ν(I1), then there exists f ∈ I1 such that f 6∈ I(X). But fg ∈ I(X) for all g ∈ I2.By primeness, g ∈ I(X) implies I2 ⊂ I(X). Hence X = ν(I2). So X is irreducible.

Recall that a topological space is connected if it is not the union of two disjoint properclosed subsets. So if a topological space is irreducible, then it must be connected (but theinverse direction is not true, see Example 2.9).

A noetherian topological space X is a disjoint union of finitely many connected closedsubsets – its connected components. A connected component is a union of irreduciblecomponents. A closed subset X of V = kn is not connected if and only if there exists twoideals I1, I2 of S with I1 + I2 = S and I1 ∩ I2 = I(X).

Example 2.9. X = (x, y) ∈ k2 : xy = 0 is closed in k2 which is connected, but notirreducible.

Here is a dictionary between algebraic objects and geometric objects.

Algebra Geometryk[T1, · · · , Tn] V = kn

radical ideals closed subsetsmaximal ideals points

prime ideals irreducible closed subsets

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Review of Affine Algebras

A k-algebra is a vector space A over k together with a bilinear operation A×A→ A suchthat for all f, g, h ∈ A and scalars c1, c2 ∈ k, we have (f+g)h = fh+gh, f(g+h) = fg+fh,(c1f)(c2g) = (c1c2)fg. A k-algebra homomorphism F : A → B is a ring homomorphismwhich is k-linear.

Let X ⊂ V = kn be an algebraic set. Define

k[X] := f |X : f ∈ S = k[T ].

Then k[X] ∼= k[T ]/I(X) (this is an isomorphism of k-algebra). k[X] is called an affinek-algebra, i.e., it has the following two properties: (i) k[X] is an algebra of finite type, i.e.,there exists a finite subset f1, · · · , fr of k[X] such that k[X] = k[f1, · · · , fr]; (ii) k[X] isreduced, i.e., 0 is the only nilpotent element of k[X].

An affine k-algebra A also determines an algebraic subset X of some kr such that

A ∼= k[X]. If A ∼= k[T1, · · · , Tr]/I where I = ker(Ti1≤i≤r−−−−→ fi), then

A is reduced ⇔ I is a radical ideal.

The affine k-algebra k[X] determines both the algebraic set X and its Zariski topology.We have the following one-to-one correspondence

points of X ↔ Max(k[X]) = maximal ideals of S containing I(X)x 7→Mx = IX(x),

where for Y ⊂ X, IX(Y ) = f ∈ k[X] : f(y) = 0,∀y ∈ Y . Note that k[X]/Mx∼= k, so

Mx is a maximal ideal. It is easy to check that(i) x 7→Mx is a bijection;(ii) x ∈ νX(I)⇔ I ⊂Mx;(iii) The closed sets of X are the νX(I), where I is an ideal in k[X];Hence the algebra k[X] determines X and its Zariski topology.

For f ∈ k[X], setDX(f) = D(f) := x ∈ X : f(x) 6= 0.

This is an open set of X and we call it a principal open subset of X. It is easy to checkthat the principal opens form a basis for the Zariski topology.

Review of Field of Definitions and F -structures

Definition 2.10. Let F be a subfield of k. We say F is a field of definition of the closedsubset X of V = kn if the ideal I(X) is generated by polynomials with coefficients in F .

SetF [X] := F [T ]/(I(X) ∩ F [T ]).

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Then F [T ] → k[T ] = S induces an isomorphism of F -algebras

F [X] ∼= (an F − subalgebra of S)

and an isomorphism of k-algebras

k ⊗F F [X] ∼= k[X]

(F [X] will be called an F -structure on X). However, this definition of field of definitionand F -structure is not intrinsic.

Definition 2.11. Let A = k[X] be an affine algebra. An F -structure on X is an F -subalgebra A0 of A which is of finite type over F such that the homomorphism

k ⊗F A0 → A = k[X]

induced by multiplication is an isomorphism. We then write A0 = F [X] and X(F ) :=F − homomorphism : F [X] → F which is called the F -rational points for the givenF -structure.

Example 2.12. Let k = C and F = R. Let X = (z, w) ∈ C2 : z2 +w2 = 1, A = k[X] =C[T,U ]/(T 2 + U2 − 1). Let a = T mod (T 2 + U2 − 1), b = U mod (T 2 + U2 − 1). Hereare two R-structure on X:

A1 = R[a, b],

A2 = R[ia, ib].

These are two different R-structures. To see this, consider the R-rational points for A1

and A2. The R-rational points for A1 is

X(R) = R− homomorphism R[a, b]→ R = S1

while the R-rational points for A2 is

X(R) = R− homomorphism R[ia, ib]→ R = ∅.

17

3 Review of Algebraic Geometry II, Introduction to LinearAlgebraic Groups I (09/02)

Review of Regular Functions

Let x ∈ X ⊂ V = kn.

Definition 3.1. A function f : U → k with U a neighborhood of x in X is regular at x if

f(y) =g(y)

h(y), g, h ∈ k[X]

on a neighborhood V ⊂ U ∩D(h) of x (i.e., h 6= 0 in V ). As usual, we say f is regular ina non-empty, open subset U if it is regular at each x ∈ U . We define

OX(U) = O(U) := the k − algebra of regular functions in U.

Observe that if U, V are non-empty, open sets and U ⊂ V , then the restriction O(V )→O(U) is a k-algebra homomorphism.

Let U =⋃α∈A Uα be an open cover of the open set U . Assume that for each α, we

have fα ∈ O(Uα) such that if Uα∩Uβ 6= ∅, then fα and fβ restrict to the same function inO(Uα ∩ Uβ). Then there exists f ∈ O(U) such that f |Uα = fα for any α ∈ A (patching).(X,O) is called a ringed space and O is called a sheaf of k-valued functions on X.

Definition 3.2. The ringed space (X,OX) (or simply X) as above is called an affinealgebraic variety over k or an affine k-variety or simply an affine algebraic variety.

Lemma 3.3. Let (X,OX) be an affine algebraic variety. Then the homomorphism

ϕ : k[X]→ O(X)

f 7→ f/1

is an isomorphism of k-algebras.

If (X,OX) and (Y,OY ) are two ringed space or affine algebraic varieties, and φ : X → Yis a continuous map, and f is a function on an open set V ⊂ Y , then define

φ∗V (f) := f φ|φ−1(V ),

a function on an open subset φ−1(V ) ⊂ X.

Definition 3.4. φ is called a morphism of ringed space or of affine algebraic varieties iffor each V ⊂ Y , φ∗V maps OY (V ) into OX(φ−1V ).

18

If X ⊂ Y , φ : X → Y is injection and OX = OY |X , then φ : X → Y is a morphism ofringed spaces. This is the notion of ringed subspace.

A morphism ϕ : X → Y of affine algebraic varieties induces an algebraic homomor-phism OY (Y )→ OX(X) by composition with ϕ. Then we get an algebraic homomorphismϕ∗ : k[Y ]→ k[X] by Lemma 3.3. Conversely, an algebraic homomorphism ψ : k[Y ]→ k[X]also gives a continuous map (ψ) : X → Y such that (ψ)∗ = ψ. Hence there is an equiva-lence of categories

affine k-varieties and their morphisms←→ affine k-algebras and their homomorphisms

affine k-variety X 7→ affine k-algebra k[X]

morphism ϕ : X → Y 7→ ϕ∗ : k[Y ]→ k[X] defined by ϕ∗(f) = f ϕ

Let F be a subfield of k. Similar remarks apply to affine F -varieties and F -subalgebras.Hence affine F -varieties can also be described algebraically. An example is that the affinen-space An, n ≥ 0 with algebra k[T1, T2, · · · , Tn].

Review on Products

Given two affine algebraic varieties X and Y over k, we would like to define a productaffine algebraic variety X × Y .

Definition 3.5 (Universal Property of Product (in any category)). A product of X and Yis defined as an affine algebraic variety Z together with morphisms p : Z → X, q : Z → Ysuch that the following holds: for any triaple (Z ′, p′, q′) as above, there exists a uniquemorphism r : Z ′ → Z such that the diagram

Z ′

X <p

p′

<Z

r

∨

..........q> Y

q′

>

commutes.

Equivalently, we can do this in the category of affine k-algebras. Put A = k[X],B = k[Y ], and C = k[Z]. Then using the equivalence of categories we can expressthe universal property algebraically: there exists k-algebra homomorphisms a : A → C,b : B → C such that for any triple (C ′, a′, b′) of affine k-algebras, there is a unique k-algebrahomomorphism c : C → C ′ such that the diagram

C

Aa′>

a>

C ′

c∨

.........

<b′

B

b

<

commutes.

19

Having this property just for the k-algebras (forgetting that C is an affine k-algebra)we already know from abstract algebra that C = A⊗k B with

a(x) = x⊗ 1, x ∈ A,b(y) = 1⊗ y, y ∈ A,

satisfies all the requirements.

Lemma 3.6. Let A, B be k-algebras of finite type.(i) If A,B are reduced, then A⊗k B is reduced.(ii) If A,B are integral domains, then A⊗k B is an integral domain.

Therefore, for X,Y affine k-varieties, a product variety X × Y exists (as an affinek-variety). It is unique up to isomorphism. If X and Y are irreducible, then so is X × Y .In fact, it is easy to see the set underlying X × Y can be identified with the product ofthe sets underlying X and Y . With this identification, the Zariski topology on X × Y isfiner than the product topology. If F is a subfield of k, a product of two affine F -varietiesexists and is unique up to F -isomorphism.

Prevarieties and Varieties

Definition 3.7. A prevariety over k is a quasi-compact ringed space (X,O) such thatany point of X has an open neighborhood U such that

(U,O|U ) ∼= an affine k-variety

is an isomorphism in the category of affine k-algebras or affine k-varieties.

Definition 3.8. A morphism of prevarieties is a morphism of ringed spaces.

Definition 3.9. A sub prevariety of a prevariety is a ringed subspace which is also aprevariety.

A product of two prevarieties exists and is unique up to isomorphism. This allows usto consider the diagonal subset ∆X = (x, x) : x ∈ X of X×X equipped with its reducedtopology. Denote by

i : X → ∆X

x 7→ (x, x).

Then i : X → ∆X is a homeomorphism of topological spaces for any prevariety X.

Definition 3.10. A prevariety X is called a variety or an algebraic variety over k ork-variety if it satisfies the Separation Axiom, i.e.,

(Separation Axiom): ∆X is closed in X ×X.

20

Morphisms of varieties are now defined in the usual way.

Example 3.11. Let X be an affine k-variety. Then ∆X = νX×X(I) where I is the kernelof the map defined from universal property

k[X ×X] = k[X]⊗k k[X]→ k[X].

In fact, I is generated by f ⊗1−1⊗f , f ∈ k[X]. Hence X satisfies the Separation Axiom,i.e., it is a variety over k. Also note that

k[X ×X]/I ∼= k[X],

which implies that i gives a homeomorphism of topological spaces X → ∆X .

Lemma 3.12. A topological space X is Hausdorff if and only if ∆X is closed in X ×Xfor the product topology.

Lemma 3.13. The product of two varieties is a variety.

Lemma 3.14. For X a variety, Y a prevariety, if ϕ : Y → X is a morphism of prevari-eties, then its graph Γφ = (y, φ(y)) : y ∈ Y is closed in Y ×X.

Lemma 3.15. Again, for X a variety, Y a prevariety, if two morphisms ϕ : Y → X,ψ : Y → X coincide on a dense subset, then ϕ = ψ.

Lemma 3.16 (Criterion for a prevariety to be a variety). (i) Let X be a variety, U, V beaffine open sets in X. Then U ∩ V is an affine open set and the images under restrictionof OX(U) and OX(V ) in OX(U ∩ V ) generate it.

(ii) Let X be a prevariety and let X = ∪mi=1Ui be a covering by affine open sets. ThenX is a variety if and only if for each pair (i, j), the intersection Ui ∩ Uj is an affine openset and the images under restriction of OX(Ui) and OX(Uj) in OX(Ui ∩Uj) generate it.

Remark 3.17. There are more examples of varieties, for example, projective varieties,which are not affine.

Definition of Linear Algebraic Groups

Now we introduce the notion of linear algebraic groups.

Definition 3.18. Let k be an algebraically closed field, and let F be a subfield. Analgebraic group G is an algebraic variety over k which is also a group such that the maps

µ : G×G→ G

(x, y) 7→ xy

and

i : G→ G

x 7→ x−1

are morphisms of varieties. An algebraic group G is called a linear algebraic group if it isaffine as an algebraic variety.

21

Definition 3.19. Let G, G′ be algebraic groups. A homomorphism of algebraic groupsϕ : G → G′ is a group homomorphism and a morphism of varieties. (Hence we have thenotion of isomorphism and automorphism of algebraic groups).

Note that G × G′ is automatically an algebraic group – called the direct product ofG×G′.

A closed subgroup H of an algebraic group G (with respect to the Zariski topology)can be made into an algebraic group such that H → G is a homomorphism of algebraicgroups.

Definition 3.20. The algebraic group G is called an F -group where F ⊂ k is a subfield if(i) G is an F -variety;(ii) the morphisms µ and i are defined over F ;(iii) the identity element e is an F -rational point.

Similarly, we get F -homomorphisms. For G an F -group, set

G(F ) := the set of F -rational points, which come with a canonical group structure.

Let G be a linear algebraic group. Put A = k[G]. Recall that there is an equivalenceof categories

affine k-varieties and their morphisms←→ affine k-algebras and their homomorphisms.

So the morphisms µ and i can be described as algebraic homomorphisms. µ is definedby ∆ : A → A ⊗k A, called “multiplication”. i can be defined by ι : A → A, called“amtipode”. Moreover, the identity element e is a homomorphism A → k. With this inhand, we can write the group axioms algebraically. We denote

m : A⊗k A→ A

f ⊗ g 7→ fg

and

ε : Aε> A

k.∪

∧e >

Then associativity in Group Axioms is the same as the diagram

A∆

> A⊗k A

A⊗k A

∆

∨∆⊗kid

> A⊗k A⊗k A

id⊗∆

∨

22

commutes. The existence of the inverse in Group Axioms is the same as the diagram

A⊗k Aι⊗id> A⊗k A

A

∆

∧

ε> A

m

∨

A⊗k A

∆

∨id⊗ι> A⊗k A

m

∧

commutes. The existence of identity in Group Axioms is the same as the diagram

A <e⊗id

A⊗k A

A⊗k A

id⊗e∧

<∆

A

∆

∧id

<

commutes.

23

4 Introduction to Linear Algebraic Groups II (09/09)

Examples of Algebraic Groups

We first give several examples of algebraic groups.Recall that k is algebraically closed, and F ⊂ k is a subfield.

Example 4.1. G = k = A1. Another notation is Ga – “the additive group”. A = k[G] =k[T ]. Multiplication and inversion are

∆ : k[T ]→ k[T ]⊗k k[T ] ∼= k[T,U ]

T 7→ T + U

and

ι : k[T ]→ k[T ]

T 7→ −T.

Note that G is a variety because we have the separation axiom: ∆G = (g, g) : g ∈ G isclosed in G×G. Therefore, ∆ and ι are k-algebra homomorphism. This implies µ, i givenby

µ(x, y) = x+ y, i(x) = −x,are morphisms of varieties. For any F ⊂ k, F [T ] defines an F -structure on Ga:

Ga(F ) ∼= F.

Example 4.2. G = k∗ = A1\0. Other notation for this group is Gm – “the multiplica-tive group”, or GL1. A = k[G] = k[T, T−1]. Multiplication and inversion are

∆ : k[T, T−1]→ k[T, T−1]⊗k k[T, T−1] ∼= k[T, T−1, U, U−1]

T 7→ TU

and

ι : k[T, T−1]→ k[T, T−1]

T 7→ T−1.

Also,

e : k[T, T−1]→ k

T 7→ 1

Again, F [T, T−1] defines an F -structure, Gm(F ) ∼= F ∗. Observe that for any n ∈ Z\0,x 7→ xn defines a homomorphism of algebraic groups Gm → Gm. When is this anisomorphism?

Gm → Gm is an isomorphism⇔ φ∗ : A = k[T, T−1]→ A is an isomorphism.

Hence Aut(Gm) ∼= ±1.

24

Example 4.3. G = An, n ≥ 1. µ and i are given by

µ(x, y) = xy, i(x) = −x,

and e = 0. In particular, G = Mn = all n× n matrices ∼= kn2.

Example 4.4. G = GLn = x ∈Mn : D(x) 6= 0 where D is the determinant. Note thatD is a regular function on Mn, and GLn is the principal open set given by D 6= 0. µ andi are given by

µ(x, y) = xy, i(x) = x−1,

and e = In. The k-algebra is A = k[GLn] = k[Tij , D−1]1≤i,j≤n,D=det(Tij) with homomor-

phisms

∆ : A→ A⊗k A

Tij 7→n∑h=1

TihThj

ι : A→ A

Tij 7→ (i, j)− entry of the

inverse of [Tkl]1≤k,l≤n,

and

e : A→ k

Tij 7→ δij .

For any F ⊂ k, F [Tij , D−1] defines an F -structure on G = GLn and G(F ) = GLn(F ).

Note that any Zariski closed subgroup of GLn defines a linear algebraic group.

Example 4.5. Any finite closed subgroup of GLn is a linear algebraic group.

Example 4.6. Dn, the diagonal matrices in GLn, is a linear algebraic group.

Example 4.7. Tn, the upper triangulars in GLn, is a linear algebraic group.

Example 4.8. Un, the unipotent upper triangular matrices in GLn, is a linear algebraicgroup.

Example 4.9. SLn = X ∈ GLn : det(X) = 1, the special linear group, is a linearalgebraic group.

Example 4.10. On = X ∈ GLn : tXX = 1, the orthogonal group, is a linear algebraicgroup. Let

J =

1...

1

.

Then On = On(J) = X ∈ GLn : tXJX = J.

25

Example 4.11. SOn = On ∩ SLn, the special orthogonal group, is a linear algebraicgroup. Sometimes we distinguish the odd and the even indices as SO2n+1 and SO2n.

Example 4.12. The special orthogonal group, Sp2n = X ∈ GL2n : tXJX = J where

J =

1...

1−1

...

−1

2n×2n

or J =

(0 In−In 0

),

is a linear algebraic group.

Review of Projective Varieties

Definition 4.13. The projective space Pn is the set 1−dim subspace of kn+1 or equiv-alently

kn+1\0/∼

wherex ∼ y ⇐⇒ y = ax for some a ∈ k∗ = k\0.

If x = (x0, x1, · · · , xn) ∈ kn+1\0, we write x∗ or [x0 : x1 : · · · : xn] for the equivalenceclass of x. The xi’s are called the homogeneous coordinates of x∗.

We cover the set Pn by U0, U1, · · · , Un where

Ui := (x0, x1, · · · , xn)∗ : xi 6= 0.

Each Ui can be given an affine variety structure of An via

ϕi : Ui → An

(x0, x1, · · · , xn)∗ 7→(x0

xi,x1

xi, · · · , xi

xi, · · · , xn

xi

).

Then ϕi(Ui ∩ Uj) is a principal open D(f) in An because we may take

f =

Tj , j > i

1, j = i

Tj+1, j < i.

Declare a subset U of Pn open if U∩Ui is open in the affine variety Ui for any i = 0, 1, · · · , n.For x ∈ Pn, assume x ∈ Ui for some i. Then a function f in a neighborhood of x is declared

26

regular at x if f |Ui is regular in the affine structure of Ui and we get a sheaf OPn and aringed space (Pn,OPn) that makes Pn into a prevariety.

In fact, Pn is a variety. We can check this by using the criterion we had before inLemma 3.16.

Definition 4.14. A projective variety is a closed subvariety of some Pn. A quasi-projectivevariety is an open subvariety of a projective variety.

Closed sets in Pn are of the form

ν∗(I) = x∗ ∈ Pn : x ∈ νkn+1(I)

where I is a homogeneous ideal. Recall that a homogeneous ideal means an ideal I ∈ S =k[T0, T1, · · · , Tn] generated by homogeneous polynomials.

Example 4.15. We assume char(k) 6= 2, 3. Define

G = (x0, x1, x2)∗ ∈ P2 : x0x22 = x3

1 + ax1x20 + bx3

0

where a, b ∈ k such that the polynomial T 3+aT+b has no multiple roots. Let e = (0, 0, 1)∗

be the point at “∞′′. Define the sum of three corlinear points in P2 to be e. It is easy tocheck that if x = (x0, x1, x2)∗ ∈ G, then −x = (x0, x1,−x2)∗. It is a bit of work to writeaddition explicitly. We can also check the associativity. Then G is an algebraic group,which is non-linear.

Review of Dimension

Let X be an irreducible variety. First, assume X to be affine. Since X is irreducible,k[X] is an integral domain. Then we get its fraction field k(X). It is an easy fact (bylocalization) that if U is any open affine subset of X, then

k(U) ∼= k(X).

If X is any variety, then the above and the criterion for a prevariety to be a variety inLemma 3.16 imply that if U, V are any two affine open sets, then k(U), k(V ) can becanonically identified. Hence we can speak of the fraction field k(X).

Definition 4.16. We define the dimension of an irreducible variety X to be

dimX = transcendence degree of k(X) over k.

If X is reducible and (Xi)1≤i≤m are its irreducible components, then

dimX = max1≤i≤m

dimXi.

27

Lemma 4.17. If X is affine and k[X] = k[x1, x2, · · · , xr], then

dimX = maximal number of elements among x1, · · · , xr that are algebraically independent over k.

Lemma 4.18. If X is irreducible and Y is proper irreducible closed subvariety of X, then

dimY < dimX.

Lemma 4.19. If X,Y are irreducible varieties, then

dim(X × Y ) = dimX + dimY.

Lemma 4.20. If ϕ : X → Y is a morphism of affine varieties and X is irreducible, thenϕ(X) is irreducible, and dimϕ(X) ≤ dimX.

Example 4.21. dim An = n, and dim Pn = n.

Remark 4.22. If U is an open set in X, then dimU = dimX. If dimX = 0, then Xis finite. If f ∈ k[T1, · · · , Tn] is irreducible, then ν(f) is (n − 1)-dimensional irreduciblesubvariety of An. Dimension respects field of definition. In other words, if X is an F -variety, then

dimX = transcendence degree of F (X) over F .

Basic Results on Algebraic Groups

Let k be an algebraically closed field, G an algebraic group. For g ∈ G, the maps

Lg : G→ G

x 7→ gx

and

Rg : G→ G

x 7→ xg

define isomorphisms of the varieties G.

Proposition 4.23. (i) There is a unique irreducible component G0 of G that contains e.It is closed, normal subgroup of finite index.

(ii) G0 is the unique connected component of G containing e.(iii) Any closed subgroup of G of finite index contains G0.

Proof. (i) Let X,Y be two irreducible components of G containing e. Then XY = µ(X ×Y ) is irreducible (because X × Y is irreducible, and µ is continuous), and its closure XYis irreducible, closed. But irreducible components are maximal irreducible closed subsets,so X ⊂ XY ⊂ X, so X = XY = Y . This implies X is closed under multiplication.

28

Now, i is a homeomorphism, hence X−1 is an irreducible component of G containing e. SoX−1 = X, i.e., X is a closed subgroup. Now for g ∈ G, gXg−1 is an irreducible componentcontaining e. This implies gXg−1 = X for any g ∈ G. So X is a normal subgroup ofG. So gX must be the irreducible components of G and there are finitely many of them.Hence G0 = X satisfies (i).

(ii) The cosets gG0 are mutually disjoint, and each connected component is a union ofthem. So the irreducible and connected components of G must coincide. This proves (ii).

(iii) Let H be a closed subgroup of G of finite index, then H0 is a closed subgroupof finite index in G0. Now H0 is both open and closed in G0, but G0 is connected, soH0 = G0.

Convention: we talk about “connected algebraic groups” and not “irreducible algebraicgroups”.

We need the following two lemmas about morphisms of varieties.

Lemma 4.24. If φ : X → Y is a morphism of varieties, then φ(X) contains a nonemptyopen subset of its closure φ(X).

Lemma 4.25. If X,Y are F -varieties, and φ is defined over F , then φ(X) is an F -subvariety of Y .

Proposition 4.26. Let φ : G→ G′ be a homomorphism of algebraic groups. Then(i) kerφ is a closed normal subgroup of G.(ii) φ(G) is a closed subgroup of G.(iii) If G and G′ are F -groups and φ is defined over F , then φ(G) is an F -subgroup of G′.(iv) φ(G0) = φ(G)0.

We need the following two lemmas to prove it.

Lemma 4.27. If U and V are dense open subgroups of G, then G = UV .

Lemma 4.28. If H is a subgroup of G, then(i) The closure H is also a subgroup of G.(ii) If H contains a non-empty open subset of H, then H = H.

Proposition 4.29 (Chevalley). Let (Xi, φi)i∈I be a family of irreducible varieties andmorphisms φi : Xi → G. Denote by H the smallest closed subgroup of G containingYi = φi(Xi). Assume that all Yi contain e. Then(i) H is connected.(ii) H = Y ±1

i1Y ±1i2· · ·Y ±1

infor some n ≥ 0, i1, · · · , in ∈ I.

(iii) If G is an F -group, and for all i ∈ I, Xi is an F -variety, and φi is defined over F ,then H is an F -subgroup of G.

Corollary 4.30. (i) If H and K are closed subgroups of G, one of which is connected,then the commutator subgroup (H,K) is connected.

(ii) If G is an F -group and H,K are F -subgroups, then (H,K) is a connected F -subgroup. In particular, (G,G) is a connected F -subgroup.

29

5 Introduction to Linear Algebraic Groups III (09/16)

G-spaces

Let k be an algebraically closed field, X an variety over k, G an algebraic group over k.

Definition 5.1. Let a : G×X → X defined by a(g, x) = g · x be a morphism of varietiessuch that

g · (h · x) = (gh) · x, ∀g, h ∈ G,e · x = x.

Then X is called a G-space or G-variety .

Definition 5.2. Let F ⊂ k be a subfield. If G is an F -group and X is an F -variety, anda is defined over F , then we say X is a G-space over F .

Definition 5.3. If F acts trivially on the G-space X, we say X is a homogeneous spacefor G. For x ∈ X, define the orbit of X to be

G · x = g · x : g ∈ G

and the isotropy group of x to be

Gx = g ∈ G : g · x = x.

Lemma 5.4. Gx is a closed subgroup of G.

Proof. Fix x ∈ X.

G→ G×X → X

g 7→ (g, x) 7→ g · xis continuous and Gx is the inverse image of x, and x is closed in the Zariski topology,so Gx is closed.

Definition 5.5. Let X and Y be G-spaces. A morphism ϕ : X → Y is called a G-morphism or G-equivalent if

ϕ(g · x) = g · ϕ(x), ∀g ∈ G, x ∈ X.

Lemma 5.6. (i) An orbit G · x is open in G · x.(ii) There exists closed orbits.

Proof. (i) Fix x ∈ X and consider the morphism ϕ : G → X given by ϕ(g) = g · x. Bya general fact from algebraic geometry, we know ϕ(G) = G · x contains a nonempty opensubset U in its closure ϕ(G) = G · x. Now G · x =

⋃g∈G g · U , so G · x is open in G · x.

(ii) Let Sx = G · x−G · x, which is closed in X. It is a union of orbits. Consider thefamily Sxx∈X of closed subsets in X. It has a minimal subset Sx0 . By (i), Sx0 must beempty. Then G · x = G · x is closed.

Corollary 5.7. G · x is locally closed in X, i.e., an open subset of a closed set in X. Ithas an algebraic variety structure, and is automatically a homogeneous space for G.

30

Examples of G-spaces

Example 5.8 (Inner automorphisms). X = G, a : G × G → G is defined by a(g, x) =gxg−1. The orbits are conjugacy classes G · x = gxg−1 : g ∈ G. The isotropy group isGx = CG(x) = g ∈ G : gx = xg.

Example 5.9 (Left and right actions). X = G, a : G×G→ G is defined by (g, x) 7→ gx or(g, x) 7→ xg−1. G acts simply-transitively, i.e., Gx = 1 ∀x ∈ G, and G is a homogeneousspace. Then G is called a principal homogeneous space.

Example 5.10. Let V be a finite dimensional vector space over k of dimension n. Arational representation of G in V is a homomorphism of algebraic groups r : G→ GL(V ).V is also called a G-module, via g · v = r(g)v.

Remark 5.11. Let F ⊂ k be a subfield. View V as a finite dimensional vector space withan F -structure and view GL(V ) as an F -group and r is defined over F , then we call r arational map over F .

Example 5.12. With the same notation, any closed subgroup G of GLn acts on X = An

(left action) so An is a G-space. The orbits of X are 0 and An\0. For example, forG = SLn, the orbit is 0.

Now assume G is affine. X is an affine G-space with action a : G×X → X. We havek[G×X] = k[G]⊗k k[X] and a is given by a∗ : k[X]→ k[G]⊗k k[X]. For g ∈ G, x ∈ X,f ∈ k[X], define

s(g) : k[X]→ k[X]

(s(g)f)(x) = f(g−1x).

Then s(g) is an invertible linear map from (often infinite-dimensional) vector space k[X]to itself. This way, we get a representation of abstract groups s : G→ GL(k[X]).

Proposition 5.13. Let V be a finite dimensional subspace of k[X].(i) There is a finite dimensional subspace W of k[X] containing V such that s(g)W ⊂

W , ∀g ∈ G.(ii) V is stable under all s(g) if and only if a∗(V ) ⊂ k[G]⊗k V . In this case, we get a

map sV : G× V → V which is a rational representation of G in V .(iii) If G is an F -group, X is an F -variety, V is defined over F , and a is an F -

morphism, then W in part (i) can be taken to be defined over F .

Proof. (i) Without loss of generality, we may assume that V = kf is one dimensional.Write

a∗(f) =n∑i=1

ui ⊗ fi, ui ∈ k[G], fi ∈ k[X].

31

Then (s(g)f)(x) = f(g−1x) =∑n

i=1 ui(g−1)fi(x). Now W ′ = 〈fi〉i=1,··· ,n is finite dimen-

sional and let W be its subspace spanned by all s(g)f , g ∈ G. Then W satisfies (i).(ii) (⇐) is just as in (i).(⇒) Assume V is s(G)-stable. Let (fi) be a basis for V and extend it to a basis

(fi) ∪ (gi) for k[X]. Take f ∈ V , and write

a∗(f) =∑i

ui ⊗ fi +∑j

vj ⊗ gj , ui, vj ∈ k[G].

Nows(g)f =

∑i

ui(g−1)fi +

∑j

vj(g−1)gj .

By assumption, vj(g−1) = 0 for all g ∈ G. Hence vj = 0 for all j. So a∗f ∈ k[G]⊗k V .

(iii) In the argument for (i), check that if all data is defined over F , then so is W .

Observe that there exists an increasing sequence of finite dimensional subspaces (Vi)of k[X] such that (i) each Vi is stable under s(G) and s defines a rational map of G in Vi,and (ii) k[X] =

⋃i Vi.

Now we still assume that G is affine. Consider the left and right action of G on itself.For g, x ∈ G, f ∈ k[G], define

(λ(g)f)(x) = f(g−1x),

(ρ(g)f)(x) = f(xg).

They both define representations of abstract group G in GL(k[G]). If ι : k[G] → k[G] isthe automorphism of k[G] defined by inversion in G, then we have

ρ = ι λ ι−1.

Lemma 5.14. Both λ and ρ have trivial kernels, i.e., they are “faithful” representations.

Proof. If λ(g) = id, then f(g−1) = f(e) for all f ∈ k[G]. Hence g−1 = e. So g = e. Thisproves that kerλ is trivial. The proof for ρ is similar.

Theorem 5.15. Let G be a linear algebraic group.(i) There is an isomorphism of G onto a closed subgroup of some GLn.(ii) If G is an F -group, the isomorphism may be taken to be defined over F .

Proof. (i) By part (i) of Proposition 5.13, we may assume k[G] = k[f1, · · · , fn] where (fi)is a basis of ρ(G)-stable subspace V of k[G]. By part (ii) of Proposition 5.13, we can write

ρ(g)fi =n∑j=1

mji(g)fj , mji ∈ k[G],∀g ∈ G, i, j = 1, · · · , n.

32

Define

φ : G→ GLn

g 7→ (mij(g))n×n.

Then φ is a group homomorphism and a morphism of affine varieties.We claim that φ is injective. If φ(g) = e, then ρ(g)fi = fi, ∀i. But ρ(g) is an algebraic

homomorphism and k[G] is generated by the fi, so

ρ(g)f = f, ∀f ∈ k[G].

Hence g = e.We claim that φ∗ is surjective. Note that φ∗ : k[GLn] = k[Tij , D

−1]→ k[G] is given by

φ∗(Tij) = mij ,

φ∗(D−1) = det(mij)−1.

But fi(g) =∑

jmji(e)fj(e), so each fi is in im(φ∗), hence φ∗ is surjective. This impliesthat φ(G) is a closed subgroup of GLn. Its algebra is isomorphic to k[GLn]/kerφ∗ ∼= k[G].Therefore, φ is an isomorphism of algebraic groups G ∼= φ(G). So we have proved (i).

For (ii), we check that the maps above can be taken to be defined over F .

Lemma 5.16. Let H be a closed subgroup of G. Then

H = g ∈ G : λ(g)IG(H) = IG(H) = g ∈ G : ρ(g)IG(H) = IG(H).

Proof. We consider λ. The proof for ρ is similar. For g, h ∈ H, f ∈ IG(H), we have(ρ(g)f)(h) = f(g−1h) = 0, so ρ(g)f ∈ IG(H). This proves H ⊂ g ∈ G : λ(g)IG(H) =IG(H). Now assume that g ∈ G and ρ(g)IG(H) = IG(H). Then for all f ∈ IG(H) wehave f(g−1) = (λ(g)f)(e) = 0. So g−1 ∈ H, and hence g ∈ H. This proves H ⊃ g ∈ G :λ(g)IG(H) = IG(H).

33

6 Jordan Decomposition (09/23)

Jordan Decomposition

Definition 6.1. Let V be a finite dimensional vector space over an algebraically closedfield k. Let x ∈ End(V ). x is called nilpotent if xn = 0 for some n ≥ 1 ( ⇐⇒ 0 is theonly eigenvalue of x). x is semisimple if the minimal polynomial of x has distinct roots(⇐⇒ x is diagonalizable over k). x is unipotent if x = 1 + n where n is nilpotent.

Remark 6.2. 0 is the only endomorphism of V that is both nilpotent and semisimple.

Remark 6.3. Suppose x, y ∈ End(V ) commute. Then(i) x, y nilpotent ⇒ x+ y nilpotent.(ii) x, y unipotent ⇒ xy unipotent.(iii) x, y semisimple ⇒ x+ y and xy are semisimple.

Proposition 6.4 (Additive Jordan Decomposition). Let x ∈ End(V ).(i) There exists unique xs, xn ∈ End(V ) such that x = xs + xn and xs is semisimple,

xn is nilpotent, and xs · xn = xn · xs.(ii) There exists polynomials p(T ), q(T ) ∈ k[T ] satisfying p(0) = q(0) = 0 such that

xs = p(x) and xn = q(x). In particular, xs and xn commute with x and in fact, theycommute with any endomorphism of V that commutes with x.

(iii) If A ⊂ B ⊂ V are subspaces and x(B) ⊂ A, then xs(B) ⊂ A, xn(B) ⊂ A.(iv) If xy = yx for some y ∈ End(V ), then

(x+ y)s = xs + ys,

(x+ y)n = xn + yn.

Proof. (i) Let det(T ·I−x) =∏ri=1(T −αi)mi be the characteristic polynomial of x, where

αi are distinct engenvalues of x. Let

Vi = ker(x− αiI)mi = v ∈ V : (x− αiI)miv = 0

be the generalized eigenspaces. Note that if v ∈ Vi, then (x−αiI)mixv = x(x−αiI)miv = 0,so Vi is x-stable. By the Chinese Remainder Theorem for polynomials, there is somep(T ) ∈ k(T ) such that

p(T ) ≡ 0 (mod T ), p(T ) ≡ αi (mod (T − αi)mi) for all i.

Let xs = p(x). Note that p(αi) = αi, so the eigenvalues of xs = p(x) are the same as thoseof x. Since Vi is x-invariant and p is a polynomial, Vi is xs-invariant. Also, xs|Vi = αiI|Vi .It follows that the Vi are the eigenspaces of xs. Moreover, V = ⊕ri=1Vi (which can be

34

proved by induction on r). Thus, xs is semisimple (because the sum of dimensions of itseigenspaces is equal to dim(V ), namely r), and xn = x− xs is nilpotent.

Since xs = p(x) where p is a polynomial, then xsx = xxs, so

xsxn = xs(x− xs) = xsx− xsxs = xxs − xsxs = xnxs.

To prove the uniqueness, suppose x = ys + yn is another such decomposition. Thenxs − ys = yn − xn, hence xs − ys and yn − xn are both semisimple and nilpotent, hencexs = ys, yn = xn.

(ii) Take p(T ) ∈ k[T ] as in part (i), q(T ) = T − p(T ).(iii) Since x(B) ⊂ A and p is a polynomial, p(x)(B) ⊂ A, thus xs(B) ⊂ A by part (ii).

Similarly xn(B) ⊂ A.(iv) This follows from 6.3 and uniqueness in part (i).

If we let xu = 1 + x−1s xn, then we get the Multiplicative Jordan Decomposition.

Corollary 6.5 (Multiplicative Jordan Decomposition). Let x ∈ GL(V ). There existsunique elements xs, xu ∈ GL(V ) such that x = xsxu = xuxs and xs is semisimple, xu isunipotent.

Remark 6.6. Suppose V is a finite dimensional vector space over an algebraically closedfield k. Let a ∈ End(V ). Let W ⊂ V be a a-stable space. Then W is stable under asand an and a|W = as|W + an|W and a = as + au where¯means the linear transformationinduced on V/W . Similarly, if a ∈ GL(V ), then a|W = as|W · au|W , and similarly forV/W .

Remark 6.7. Suppose V,W are two finite dimensional vector space over k. Let ϕ : V →W be linear. Let a ∈ End(W ), b ∈ End(W ). If ϕ a = b ϕ, i.e., the diagram

Va> V

W

ϕ

∨b> W

ϕ

∨

is commutative, thenϕ as = bs ϕ

andϕ an = bn ϕ.

Let V be a not necessarily finite dimensional vector space over k. Again

End(V ) := algebra of endomorphisms of V,

35

GL(V ) := group of invertible endomorphisms of V.

We say a ∈ End(V ) is locally finite if V is a union of finite dimensional a-stable subspaces.We say a ∈ End(V ) is semisimple if its restriction to any finite dimensional a-stablesubspace is semisimple. We say a ∈ End(V ) is locally nilpotent if its restriction to anyfinite dimensional a-stable subspace is nilpotent. We say a ∈ End(V ) is locally unipotentif its restriction to any finite dimensional a-stable subspace is unipotent. For a locallyfinite a ∈ End(V ), we have a = as + an with as locally finite and semisimple, an locallyfinite and locally nilpotent. For x ∈ V , take a finite dimensional a-stable subspace Wcontaining x, and put

asx := (a|W )s,

anx := (a|W )n.

It follows from the uniqueness statement of the finite dimensional Additive Jordan decom-position that asx and anx are independent of the choice of W . If a ∈ GL(V ), we havea similar multiplicative Jordan decomposition a = as · au where as is semisimple, au islocally unipotent.

Remark 6.8. There is an infinite-dimensional generalization of Remark 6.7.

Jordan Decomposition in Linear Algebraic Groups

We now come to the Jordan decomposition in linear algebraic groups.Let G be a linear algebraic group and A = k[G]. From our discussion of G-actions, we

can conclude that the right translation ρ(g), g ∈ G, is a locally finite element of GL(A),i.e., ρ(g) = ρ(g)sρ(g)u.

Theorem 6.9. (i) There exists unique elements gs and gu in G such that ρ(g)s = ρ(gs),ρ(g)u = ρ(gu), and g = gsgu = gugs.

(ii) If φ : G → G′ is a homomorphism of linear algebraic groups, then φ(g)s = φ(gs)and φ(g)u = φ(gu).

(iii) If G = GLn, then gs and gu are the semisimple and unipotent parts of g ∈ GL(V ),where V = kn as before.

Remark 6.10. gs is called the semisimple part of g, and gu is called the unipotent partof g.

Proof of Theorem 6.9. (i) Let m : A ⊗ A → A be the k-algebra homomorphism corre-sponding to multiplication in G. ρ(g) is an algebra automorphism of A. That means

m (ρ(g)⊗ ρ(g)) = ρ(g) m.

By Remark 6.7, we have

m (ρ(g)s ⊗ ρ(g)s) = ρ(g)s m.

36

So ρ(g)s is also an automorphism of A. So f 7→ (ρ(g)sf)(e) defines a homomorphismA → k, i.e., a point in G, and we call it gs. Now ρ(g) commutes with all left translationλ(x), x ∈ G, and the λ(x) are locally finite, so ρ(g)s also commutes with all λ(x). In otherwords, for f ∈ A,

(ρ(g)sf)(x) = (λ(x−1)ρ(g)sf)(e)

= (ρ(g)sλ(x−1)f)(e)

= (λ(x−1)f)(gs) (by definiton of gs)

= f(xgs)

= (ρ(gs)f)(x).

Hence ρ(g)s = ρ(gs). A similar argument also gives ρ(g)u = ρ(gu). So

ρ(g) = ρ(g)sρ(g)u = ρ(gs)ρ(gu) = ρ(gsgu).

But ρ is a faithful representation of G (i.e., ker ρ is trivial), so g = gsgu. Similarly g = gugs.(ii) Recall that for homomorphism of algebraic groups φ : G → G′, we saw that

Im(φ) = φ(G) is closed in G′. So φ can be factored into

G→ Im(φ)→ G′.

This reduces the proof to two cases: case (a) the inclusion Imφ → G′, and case (b) thesurjection G→ Imφ.

For case (a), G is a closed subgroup of G′ and φ is the inclusion. Let k[G] = k[G′]/I.By Lemma 5.16,

G = g ∈ G′ : ρ(g)I = I.

Now W = I is a subspace of V = k[G′] and it is stable under ρ(g), so by Remark 6.6, wehave a Jordan decomposition on V/W = k[G]. So

φ(g)s = φ(gs),

φ(g)u = φ(gu),

as φ is just inclusion.For case (b), if φ is surjective, then k[G′] can be viewed as a subspace of k[G], which

is stable under all ρ(g), g ∈ G. Again, the result follows from Remark 6.6.(iii) Let G = GL(V ) with V = kn. Let 0 6= f ∈ V ∨ = dual of V and define f(v) ∈ k[G]

via f(v)(g) = f(gv). Then f is an injective linear map V → k[G], and ∀x ∈ G, we have

f(gv)(x) = f(xgv) = f(v)(xg) = [ρ(g)f(v)](x).

Hence f(gv) = ρ(g)f(v). By Remark 6.8, we have

f(gsv) = ρ(g)sf(v),

37

f(guv) = ρ(g)uf(v),

which implies (iii).

Corollary 6.11. x ∈ G is semisimple ⇐⇒ for any homomorphism φ from G onto aclosed subgroup of some GLn, φ(x) is semisimple. Similarly for unipotent elements.

Jordan Decomposition and F -structures

Let F ⊂ k be a subfield. Assume G is an F -group. Note that if x ∈ G(F ), then xs andxu need not lie in G(F ). Here is an example.

Example 6.12. Assume that char(k) = 2 and F 6= F 2 (i.e., F is non-perfect). Let

G = GL2. Let a ∈ F\F 2 and x =

(0 1a 0

). Then the Jordan decomposition of x in

GL2(k) is

x =

(0 1a 0

)= xsxu =

(√a 0

0√a

)(0 1√

a√a 0

).

But xs, xu 6∈ GL2(F ). Moreover, it is the case that if F is perfect, then the semisimpleand unipotent parts of an element in G(F ) are again in G(F ).

Unipotent Groups

Definition 6.13. A linear algebraic group G is unipotent if all its elements are unipotent.

Example 6.14. The linear group Un =

1 ∗ ∗ · · · ∗0 1 ∗ · · · ∗0 0 1 · · · ∗

0 0 0. . . ∗

0 0 0 · · · 1

is unipotent. Actually

it turns out that this is essentially the only example.

Proposition 6.15. Let G be a subgroup of GLn consisting of unipotent matrices. Thenthere exists x ∈ GLn such that xGx−1 ⊂ Un.

Before proving Proposition 6.15, we need the Burnside’s Theorem.

Theorem 6.16 (Burnside’s Theorem). Let E be a finite dimensional vector space overan algebraically closed field k, R be a subalgebra of End(E). If E is a simple R-module(i.e., the action is irreducible), then R = End(E).

38

Proof of Proposition 6.15. We prove this by induction on n. Suppose this is true form < n, let V = kn. Suppose that there is a non-trivial G-invariant subspace 0 (W1 ( V .Let W2 be the complementary to W1 so that V = W1 ⊕W2. Since n > dimW1,dimW2,there are xi ∈ GL(Wi) so that xiGx

−1i consists of unipotent elements for i = 1, 2. Let

x = x1 ⊕ x2. Then xGx−1 consists of unipotent elements as well.Next, suppose no non-trivial G-invariant subspace exists, i.e., G acts irreducibly in V .

Let g ∈ G. Then Tr(g) = n. Then for any h ∈ G, Tr((1−g)h) = Tr(h)−Tr(gh) = n−n = 0.By Burnside’s Theorem, the elements in G span the vector space End(V ). This meansthat Tr(h) = Tr(gh) for all h ∈ End(V ). Now choosing h = Eij , we see that this is onlypossible when g = 1, i.e., G = 1.

Remark 6.17. By Proposition 6.15, if G is unipotent linear algebraic group and G →GL(V ) is a rational representation of G, then there is a nonzero vector v ∈ V which isfixed by all of G (consider the first basis element after conjugating into Un).

Proposition 6.18 (Kostant-Rosenlicht). Let G be a unipotent linear algebraic group andlet X be an affine G-space. Then all orbits of G in X are closed.

Proof. Let O be an orbit. Without loss of generality we may assume that X = O andhence O is dense in X. Recall that an orbit is open in its closure by Lemma 5.6, so O isopen in O. Let Y = O\O. Then G acts locally finitely on the ideal IX(Y ). Because Gis unipotent, we may apply Remark 6.17 to the rational representation G→ GL(IX(Y )).So there is a non-zero function f ∈ IX(Y ) fixed by elements of G, i.e., ρ(g)f = f, ∀g ∈ G.Now for any o ∈ O, (ρ(g)f)(o) = f(o). So f(og) = f(o),∀o ∈ O,∀g ∈ G. Hencef(o) = f(e), ∀o ∈ O, i.e., f is constant on O. Since O is dense in X, f is constant on X.Thus IX(Y ) = k[X], i.e., Y = ∅, and hence O = O.

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7 Commutative Linear Algebraic Groups I (09/30)

Structure of Commutative Algebraic Groups

Theorem 7.1 (Kolchin). Let G be a commutative linear algebraic group. Then(i) The sets Gs and Gu of semisimple and unipotent elements are closed subgroups.(ii) The product map π : Gs ×Gu → G is an isomorphism of algebraic groups.

Proof. (i) We may assume that G is a closed subgroup of some GLn, by Theorem 5.15.Recall that if x, y ∈ End(V ), then xy = yx implies that (xy)s = xsys and (xy)u = xuyu.This implies that both Gs and Gu are subgroups.

Gu is a closed subset for general (not necessarily commutative) linear algebraic groupG because the set of all unipotent matrices in GLn(k) is the zero set of polynomials impliedby (x− 1)n = 0.

To see Gs is closed, recall that without loss of generality we may assume G ⊂ Tn =upper triangular matrices in GLn and Gs ⊂ Dn. This forces Gs = G ∩Dn which showsthat Gs is closed. (Note that for general G, it is rare that Gs is closed).

(ii) π is an isomorphism of abstract groups by the uniqueness of Jordan decompositionin G. Also, π is a morphism of varieties and the map G → Gs defined by x 7→ xs isa morphism of algebraic varieties because it maps x to some of its matrix entries, so itgives polynomials. Hence π−1 : x 7→ (xs, x

−1s x) is a morphism of varieties. Hence π is an

isomorphism of algebraic groups.

Corollary 7.2. If G is connected, then so are Gs and Gu.

Proof. Gs and Gu are imagies of the connected group G under continuous maps, so theyare connected.

Proposition 7.3. Let G be a connected linear algebraic group of dimension 1. Then(i) G is commutative.(ii) Either G is Gs or Gu.(iii) If G is unipotent and p = char(k) > 0, then the elements of G have order dividing

p.

Proof. (i) Fix g ∈ G and consider the morphism φ : G → G defined by x 7→ xgx−1.Because G is connected (i.e., irreducible topological group), its image φ(G) is also anirreducible topological group, which implies φ(G) is an irreducible closed subset of G. Ifφ(G) is a proper irreducible closed subset of G, it must have dimension less than dimG = 1.So either φ(G) = g (i.e., G is commutative) or φ(G) = G.

Let’s assume φ(G) = G. Because φ(G) contains a nonempty open subset U of φ(G), wewould have G− φ(G) is finite (it suffices to show G−U is finite since G−U ⊃ G− φ(G),but G − U is closed and a variety, and dim(G − U) = 0, so G − U is finite). ViewingG as a closed subgroup of some GLn, there are only finitely many possibilities (becausechar(x) = char(yx−1y)) for the characteristic polynomial det(T · 1 − x), x ∈ G. But G is

40

connected, so the characteristic polynomial is constant. Taking x to be identity, it mustbe (T − 1)n. This means G is unipotent. Hence G is solvable. Now G′ = (G,G) thecommutator subgroup of G is a connected, closed subgroup and can only be e. Nowg−1φ(G) ⊂ G′, which is a contradiction.

(ii) Because G is connected, both Gs and Gu are irreducible, closed subvarieties of G.If G 6= Gs, then Gs is a proper subvariety so dim(Gs) < dim(G) = 1, i.e., Gs = e. ThusG = Gu.

(iii) Assume that G is unipotent and p = char(k) > 0. Let

Gph

= ph − power of elements of G.

Then it is easy to check that Gph

is a connected, closed subgroup of G, so it must be Gor e. Viewing G as an upper triangular matrices in some GLn, Gp

h= e if ph ≥ n,

which in characteristic p implies Gp = e.

Algebraic Tori

Definition 7.4. Let G be a linear algebraic group. A rational character (or just a char-acter) of G is a homomorphism of algebraic groups χ : G→ Gm. We denote

X∗(G) =abelian group of rational characters with additive notation,

i.e., (χ1 + χ2)(g) = χ1(g)χ2(g).

Note that characters are regular functions on G, so X∗(G) ⊂ k[G]. Also, charactersare linearly independent in k[G] (this is the Dedekind’s Lemma).

Lemma 7.5 (Dedekind’s Lemma). Let G be any group, E be any field. X(G) = Homgroup(G,E∗)is a linearly independent subset of the vector space over E of functions G→ E.

Proof. If there is a nontrivial linear independence relation among the elements in X(G),take one of minimal length:

a1χ1 + · · ·+ anχn = 0, 0 6= ai ∈ Z, χi distinct.

For g, h ∈ G,n∑i=1

aiχi(g)χi(h) = 0 = χ1(g)

n∑i=1

aiχi(h).

Son∑i=2

ai(χi(g)− χ1(g))χi(h) = 0.

Since χ2 6= χ1, there exists some g ∈ G such that χ2(g)− χ1(g) 6= 0. This contradicts theminimality of the length.

41

Definition 7.6. A cocharacter (or a multiplicative one parameter subgroup) of G is ahomomorphism of algebraic groups Gm → G. We denote

X∗(G) = the set of cocharacters.

Note that cocharacters may not necessarily be abelian. However, if G is commutative,then X∗(G) is an abelian group. Even G is not commutative, we still have an action of Zon X∗(G) by

(n · λ)(a) = λ(a)n.

We write −λ = (−1) · λ.

Definition 7.7. A linear algebraic group G is called diagonalizable if it is isomorphicto a closed subgroup of some Dn. G is called an algebraic torus (or just torus) if it isisomorphic to some Dn.

Example 7.8. G = Dn is a torus, while G = Dn × ±1 is diagonalizable.

Example 7.9. G = Dn =

x1

. . .

xn

: xi 6= 0

. Set χi(x) = xi. Then each χi is

a character of Dn and in fact k[Dn] = k[χ1, · · · , χn, χ−11 , · · · , χ−1

n ]. χa11 χa22 · · ·χann , where

(a1, · · · , an) ∈ Zn, are all the characters of Dn, and they form a basis for k[Dn]. Moreover,

X∗(Dn) ∼= Zn

as abelian groups. Also, any cocharacter Gm → Dn is given by

x 7→

xa1

xa2

. . .

xan

where (a1, · · · , an) ∈ Zn. In other words,

X∗(Dn) ∼= Zn

as abelian groups.

Theorem 7.10. The following are equivalent for a linear algebraic group G.(i) G is diagonalizable.(ii) X∗(G) is an abelian group of finite type. X∗(G) is a k-basis for k[G].(iii) Any rational representation of G is a direct sum of one dimensional representa-

tions.

42

Proof. (i) ⇒ (ii). Assume G is diagonalizable. Then G is a closed subgroup of someDn. Hence k[G] is a quotient of k[Dn]. Restriction of characters from Dn to G reducescharacters of G and they span k[G]. By Dedekind’s Lemma (Lemma 7.5), they form a basisand any character of G is a linear combination of these restrictions. Hence X∗(Dn) →X∗(G) is a surjective homomorphism of abelian groups. Recall that X∗(Dn) ∼= Zn. SoX∗(G) is of finite type.

(ii)⇒ (iii). Let φ : G→ GL(V ) be a rational representation of G in a finite dimensionalvector space V . Then (ii) implies that we can define linear maps Aχ : V → V, χ ∈ X∗(G)via φ(x) =

∑χ∈X∗(G) χ(x)Aχ with Aχ = 0 for all but finitely many χ’s. To see this, fix

a basis for V and write φ(x) = [φij(x)]n×n. Then φij ∈ k[G] and by (ii), we can writeφij =

∑χ∈X∗(G) αijχχ. Then φ(x) =

∑χ∈X∗(G)Aχχ(x) where Aχ has the matrix [αijχ]n×n

with respect to the fixed basis. For x, y ∈ G,

φ(xy) =∑

χ∈X∗(G)

χ(xy)Aχ = φ(x)φ(y) =

∑χ∈X∗(G)

χ(x)Aχ

∑χ∈X∗(G)

χ(y)Aχ

.

By Dedekind’s Lemma (Lemma 7.5),

AχAψ =

0 if χ 6= ψ

Aχ if χ = ψ.

Also,∑

χ∈X∗(G)Aχ = φ(e) = id. Put Vχ = im(Aχ). Then it follows that V is a direct sumof Vχ and x ∈ G acts on Vχ via mapping by χ(x).

(iii) ⇒ (i). This direction is clear.

Corollary 7.11. If a linear algebraic group G is diagonalizable, then X∗(G) is an abeliangroup of finite type without p-torsion if p = char(k) > 0.

In fact, if G is diagonalizable, the algebra k[G] is isomorphic to the group algebra ofX∗(G).

Group Algebras of Abelian Groups

Let M be an abelian group of finite type. The group algebra of M is

k[M ] := the algebra with basis (em)m∈M with mapping defined by em · en = em+n.

Observe that if M1,M2 are two abelian groups of finite type, then

k[M1 ⊕M2] = k[M1]⊗k k[M2].

Define

∆ : k[M ]→ k[M ]⊗k k[M ]

em → em ⊗ em,

43

ι : k[M ]→ k[M ]

em → e−m,

and

e : k[M ]→ k

em → 1.

Recall that if M is of finite type, then M ∼= Zr ⊕ (direct sum of finite groups). Ifp ·m = 0 for a prime p, then m is called a p-torsion element.

Proposition 7.12. Assume that p = char(k) > 0, and M has no p-torsion.(i) k[M ] is an affine algebra, and there is a diagonalizable linear algebraic group G(M)

with k[G(M)] = k[M ] such that ∆, ι, e are comultiplication, antipode and the identityelements of G(M).

(ii) There is a canonical isomorphism M ∼= X∗(G(M)).(iii) If G is diagonalizable, then there is a canonical algebraic group isomorphism

G(X∗(G)) ∼= G.

Example 7.13. Let M = Z ⊕ Z/12Z = M1 ⊕ M2. Then G(M) = G1 × G2 whereG1 = G(M1) and G2 = G(M2), and k[M1] ∼= k[T, T−1], k[M2] ∼= k[T ]/(T 12 − 1). Byassumption, if p > 0, p 6 |12 (i.e., p 6= 2, 3), then k[M2] is a reduced algebraic group. ThenG(M) ∼= D1 × (finite group).

Example 7.14. G(Zn) ∼= Dn.

Corollary 7.15. Let G be a diagonalizable group.(i) G is a direct product of a torus and a finite abelian group of order prime to p, if

p = char(k) > 0.(ii) G is a torus ⇐⇒ G is connected.(iii) G is a torus ⇐⇒ X∗(G) is a free abelian group.

Proposition 7.16 (Rigidity of Diagonalizable Groups). Let G and H be diagonalizablegroups and let V be a connected affine variety. Assume that ϕ : V ×G→ H is a morphismof varieties such that for each v ∈ V , the map G → H defined by x → ϕ(v, x) is ahomomorphism of algebraic groups. Then ϕ(v, x) is independent of v.

For G an arbitrary linear algebraic group and H a closed subgroup, set

ZG(H) = g ∈ G : ghg−1 = h,∀h ∈ H, − centralizer of H in G,

NG(H) = g ∈ G : gHg−1 ∈ H, − normalizer of H in G,

The defining conditions can be expressed as polynomial conditions, so these are closedsubgroups of G and ZG(H) / NG(H).

Corollary 7.17. If H is a diagonalizable subgroup of G, then NG(H)0 = ZG(H)0 andNG(H)/ZG(H) is finite.

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Proof. Let V = NG(H)0. Apply rigidity (Proposition 7.16) to

ϕ : NG(H)0 ×H → H

(x, y) 7→ xyx−1

to conclude that xyx−1 is independent of x, i.e., xyx−1 = y, ∀x ∈ NG(H)0. ThusNG(H)0 ⊂ ZG(H). This proves NG(H)0 = ZG(H)0.

45

8 Commutative Linear Algebraic Groups II (10/07)

Review of Pairings

Let R be a commutative ring with 1 and let M,N be two (left) R-modules. The setHomR(M,N) = R-linear maps from M to N is an R-module.

Example 8.1. M is any R-module, N = R, then M∨ = HomR(R,N) is called the dualmodule, or dual space, or R-module of M .

Example 8.2. R∨ = HomR(R,R) ∼= R.

Example 8.3. R = F is a field, M,N are vector spaces over F . This comes from linearalgebra.

Definition 8.4. A pairing between M and N is a bilinear map 〈·, ·〉 : M ×N → R, i.e.,R-linear in each component when the other is fixed.

Example 8.5. A dot product Rn ×Rn → R is a pairing.

Example 8.6. There are two natural pairings,

Mn(R)×Mn(R)→ R

〈A,B〉 = Tr(AB)

and

Mn(R)×Mn(R)→ R

〈A,B〉 = Tr(ABT).

Example 8.7. The map

M ×M∨ → R

〈m,ϕ〉 = ϕ(m)

is called the standard pairing between a module and its dual.

Example 8.8. The map

R[x]×R[x]→ R

〈f, g〉 = f(0)g(0)

is a pairing. Then 〈x, g〉 = 0 for all g ∈ R[x] even though x 6= 0. In fact, 〈f, g〉 = 0 for allg ∈ R[x] if x|f .

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We can use a pairing to think of M and N as part of the dual of the other module.For m ∈ M , n 7→ 〈m,n〉 is a functional on N and for n ∈ N , m 7→ 〈m,n〉 is a functionalon M . However, if the pairing behaves badly, we may have 〈m,n〉 = 0 ∀n with m 6= 0.

For R-modules M and N , HomR(M,N∨), HomR(N,M∨) and

BilR(M,N ;R) = bilinear maps from M to N

are all isomorphic as R-modules. The point is that a bilinear map allows us to use M toparametrize a piece of N∨ and similarly for N . However, some pairings may make differentelements of M behave like the same element of N∨. For example, a nonzero element of Mmight pair with every element of N to have the value 0, as behavior we expect if m = 0.

The pairings that allow us to identify M and N with each other’s full dual module arethe “perfect” pairings.

Definition 8.9. A pairing 〈·, ·〉 : M × N → R is called a perfect pairing if the inducedlinear maps M → N∨ and N →M∨ are both isomorphisms.

Note that when R is a field and M , N are finite dimensional vector spaces of the samedimension, then a pairing 〈·, ·〉 : M × N → R is perfect if and only if the induced mapM → N∨ is injective, i.e., 〈m,n〉 = 0 for all n ∈ N implies m = 0. (Then N → M∨

is also automatically an isomorphism.) However, an injective linear map of free moduleswith the same rank need not be an isomorphism, for example, the map Z→ Z defined byx 7→ 2x. So in the case of non-field commutative ring R with M and N free of the samefinite rank, it is not enough to just check that M → N∨ is injective.

Perfect pairing of modules M and N is stronger than just identification of one of themwith the dual of the other. It identifies each module as the dual of the other M ∼= N∨ andN ∼= M∨, both coming from the perfect pairing 〈·, ·〉 : M ×N → R.

Characters and Cocharacters of Tori

Let T be a torus. Denote the character group

X = X∗(T ) = χ : T → Gm

and the cocharacter group

Y = X∗(T ) = λ : Gm → T.

For χ ∈ X,λ ∈ Y , a ∈ k∗, consider the character

Gm → Gm

a 7→ χ(λ(a)).

Recall X∗(Gm) ∼= Z so χ(λ(a)) = a 〈χ, λ〉 for some 〈χ, λ〉 ∈ Z.

47

Lemma 8.10. (i) 〈·, ·〉 : X × Y → Z defines a perfect pairing, i.e., any homomorphismX → Z is of the form χ 7→ 〈χ, λ〉 for some λ ∈ Y and any homomorphism Y → Z is ofthe form λ 7→ 〈χ, λ〉 for some χ ∈ X. In particular, Y is a free Z-module.

(ii) The map a⊗λ 7→ λ(a) defines a canonical isomorphism of abelian groups k∗⊗ZY ∼=T .

Proof. (i) Since T is a torus, it is isomorphic to some Dn. Then X = χa11 χa22 · · ·χann :

(a1, a2, · · · , an) ∈ Zn ∼= Zn and Y = x 7→ diag(xb1xb2 · · ·xbn : (b1, b2, · · · , bn) ∈ Zn ∼=Zn. So the assertion is clear.

(ii) This follows from the freeness of Y .

Tori and F -structures

Let F ⊂ k be a subfield.

Definition 8.11. An F -torus is an F -group which is also a torus. An F -torus T whichis F -isomorphic to some Dn is called F -split .

Example 8.12. Let k = C, F = R. Then G =

(a −bb a

)∈ GL2

is an R-torus which

is not R-split.

Proposition 8.13. (i) An F -torus T is F -split ⇐⇒ all its characters are defined overF . In that case, the characters form a basis of F [T ].

(ii) Any rational representation over F of an F -split torus is a direct sum of one-dimensional representations over F .

Torus Action

Let X,Y, T as before. Let V be an affine T -space. This leads to locally finite representations of T in k[V ] as before.

For χ ∈ X, put

k[V ]χ = f ∈ k[V ] : s(t) · f = χ(t)f,∀t ∈ T.

We saw that any rational representation of a diagonalizable group was a direst sum of1-dimensional rational representations of the subspaces. k[V ]χ define an X-grading of thealgebra k[V ], i.e., k[V ] = ⊕χ∈Xk[V ]χ and k[V ]χk[V ]ψ ⊂ k[V ]χ+ψ for χ, ψ ∈ X.

Example 8.14. If T = Gm, then X = Z and the grading structure on k[V ] is the usualone (given by degrees of monomials).

For Z a variety and ϕ : Gm → Z a morphism of varieties, write lima→0 ϕ(a) = z if ϕextends to a morphism ϕ : A1 → Z such that ϕ(0) = z. Put ϕ′(a) = ϕ(a−1) and definelima→∞ ϕ(a) = lima→0 ϕ

′(a).

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If V is a T -space and λ ∈ T , we write

V (λ) = v ∈ V : lima→0

λ(a) · v exists.

ThenV (−λ) = v ∈ V : lim

a→∞λ(a) · v exists.

Lemma 8.15. Assume V is affine.(i) V (λ) is a closed subset of V .(ii) V (λ) ∩ V (−λ) is the set of fixed points in Im(λ), i.e.,

V (λ) ∩ V (−λ) = v ∈ V : λ(k∗) · v = v.

Proof. (i) An element f ∈ V (λ) can be written as f =∑

χ fχ, fχ ∈ k[V ]χ. Then

s(λ(a)) · f =∑χ

a〈χ,λ〉fχ.

So lima→0 λ(a) · v exists ⇐⇒ v annihilates all functions in Vχ with 〈χ, λ〉 < 0. Thisproves (i).

(ii) Now, V (λ) ∩ V (−λ) is the set of v, annihilating all Vχ with 〈χ, λ〉 6= 0. Then

V (λ) ∩ V (−λ) = v ∈ V : f(λ(a) · v) = f(v), ∀f ∈ k[V ], a ∈ k∗.

This is just the set of fixed points.

Example 8.16. Let G is a linear algebra group, λ : Gm → G be a cocharacter. Considerthe action of T = Gm on G by a · x = λ(a)xλ(a)−1. Write P (λ) = x ∈ G : lima→0 a ·x exists. This is a subgroup of G. By Lemma 8.15 (i), it is closed and by Lemma 8.15(ii), P (λ) ∩ P (−λ) is the centralizer of Im(λ).

Additive Functions and Elementary Unipotent Groups

The next goal is to classify connected 1-dimensional groups. It requires a study of “additivefunctions”.

Definition 8.17. An additive function on a linear algebraic group G is a homomorphismof algebraic groups f : G→ Ga. Denote

A = A(G) = the set of additive functions on G,

which is a subspace of the algebra k[G]. Let F ⊂ k be a subfield and G be an F -group,then write

A(F ) = A(G)(F ) = F -vector space of additive functions defined over F .

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Note that if p = char(k) > 0, then p-th power of an additive function is again anadditive function. This will allow us to define a ring R over which A is a module.

If p = char(k) > 0, then ϕ : x 7→ xp defines an isomorphism of F onto a subfield of F p

(recall F is perfect if F = F p). We define a ring R = R(F ) as follows. The underlyingadditive group is F [T ] – polynomials in T , and multiplication is defined by

(∑

aiTi)(∑

bjTj) =

∑ai(ϕ

i(bj))Ti+j .

Then R is an associative but non-commutative ring. Observe that the subfield F of Rdoes not lie in the center of R, and degree has its usual property (i.e., R has no non-zerodivisors). If p = char(k) = 0, then define R(F ) = F .

Now, if p > 0, we define a left R-module structure on A(F ) by (∑aiT

i) · f =∑aif

pi .If p = 0, then R = F and A(F ) is trivially an R-module.

Example 8.18. Consider G = Gna . Then F [G] = F [T1, · · · , Tn] and any additive function

in F [G] is an additive polynomial, i.e., f ∈ F [T1, · · · , Tn] satisfying f(T1 + u1, · · · , Tn +un) = f(T1, · · · , Tn) + f(u1, · · · , un). The set of additive polynomials is a left R-moduledenoted by A(Gn

a)(F ).

Definition 8.19. A unipotent linear algebraic group G is called elementary if it is abelianand when p = char(k) > 0, its elements have order dividing p. G is called a vector groupif G ∼= Gn

a for some n.

Theorem 8.20. The followings are equivalent for a linear algebraic group G:(i) G is an elementary group;(ii) A(G) is an R-module of finite type and its elements generate the algebra k[G];(iii) G is a vector group when p = 0, and a product of a vector group and a finite elementaryabelian p-group if p > 0 (note that elementary abelian p-group is a product of cyclic groupsof order p).

Corollary 8.21. Let G be an F -group. Then G is elementary unipotent if and only ifone of the following equivalent conditions hold:(i) A(G)(F ) generate F [G];(ii) G is F -isomorphic to a closed subgroup of some Gn

a .

Corollary 8.22 (Classification of 1-dimensional linear algebraic groups). Let G be aconnected linear algebraic group of dimension 1. Then G ∼= Gm or G ∼= Ga.

Proof. We already know that G must be commutative, and G = Gs or G = Gu.If G = Gs, then G is diagonizable, and by connectedness, it is a torus of dimension 1,

i.e., G ∼= Gm.If G = Gu, then we have an elementary unipotent group, and by (iii) of Theorem 8.20,

G ∼= Ga because it is connected.

50

9 Derivations and Differentials (10/14)

Derivations and Tangent Spaces of Varieties

Definition 9.1. Let R be a commutative ring with unit and let A be an R-algebra. Also,let M be a left A-module. An R-derivation of A in M is an R-linear map D : A → Msuch that

D(ab) = a ·D(b) + b ·D(a), ∀a, b ∈ A.

Note that D(1) = 0 because D(1) = D(1 · 1) = 1 · D(1) + 1 · D(1) = 2D(1). SoD(r) = rD(1) = 0 for all r ∈ R. Also, DerR(A,M) becomes a left A-module via

(D +D′)(a) = D(a) +D′(a),

(bD)(a) = bD(a), ∀b ∈ A.

DerR(A,A) is the derivations of the R-algebra A. If ϕ : A → B is a homomorphism ofR-algebras and N is a left B-module, then N becomes an A-module via a ·n = ϕ(a)n andwe get a homomorphism of A-modules

ϕ0 : DerR(B,N)→ DerR(A,N)

D 7→ D ϕ

because

(D ϕ)(a1a2) = D(ϕ(a1)ϕ(a2))

= ϕ(a1)D(ϕ(a2)) + ϕ(a2)D(ϕ(a1))

= a1 ·D(ϕ(a2)) + a2 ·D(ϕ(a1))

and so D ϕ ∈ DerR(A,N). The fact that ϕ0 is indeed a module homomorphism nowfollows formally.

Next, we claim that ker(ϕ0) = DerA(B,N). To see this, if D ∈ ker(ϕ0), then D(a ·b) =D(ϕ(a)b) = ϕ(a)D(b) + bD(ϕ(a)) = ϕ(a)D(b) + bϕ0(D)(a) = ϕ(a)D(b) = a · D(b), soD ∈ DerA(B,N). On the other hand, if D ∈ DerA(B,N), then D(a) = 0 for all a ∈ A, soD(ϕ(a)) = 0 for all a ∈ A, so D ϕ(a) = 0 for all a ∈ A, then ϕ0(D) = D ϕ = 0, thusD ∈ ker(ϕ0). Therefore, we get an exact sequence

1 > DerA(B,N) > DerR(B,N)ϕ0> DerR(A,N).

Review of Basics of Lie Algebras

Let k be an algebraically closed field (of any characteristic). For our purposes, a Liealgebra g over k is a subspace of an associative k-algebra which is closed under the bracketoperation

[x, y] := xy − yx.

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The adjoint representation (of Lie algebras) is defined via

ad : g→ End(g)

(ad x)(y) = [x, y].

Example 9.2. g = gl(n, k) = Mn(k) = n× n matrices with entries in k.

Example 9.3. If A is an arbitrary commutative k-algebra, then the space D = Derk(A,A)has a Lie algebra structure with the Lie bracket given by

[D,D′] = D D′ −D′ D, ∀D,D′ ∈ D.

We need to check that [D,D′] is again a derivation in D. Note

[D,D′](ab) = D D′(ab)−D′ D(ab)

= D(aD′(b) + bD′(a))−D′(aD(b) + bD(a))

= aD D′(b) +D(a)D′(b) + bD D′(a) +D′(a)D(b)

− aD′ D(b)−D′(a)D(b)− bD′ D(a)−D(a)D′(b)

= a(D D′(b)−D′ D(b)) + b(D D′(a)−D′ D(a))

= a[D,D′](b) + b[D,D′](a),

so [D,D′] ∈ D.

Example 9.4. Fix a prime p. We say a Lie algebra g is a p-Lie algebra (or a restrictedLie algebra) if g has a p-operation X 7→ X [p], X ∈ g, such that for X,X ′ ∈ g, a ∈ k, wehave(a) (aD)[p] = apD[p];(b) ad (D[p]) = (ad D)[p];

(c) (Jacobson’s formula) (D +D′)[p] = D[p] +D′[p] +∑p−1

i=1si(D,D

′)i where si(D,D

′) is thecoefficient of ai in ad (aD +D′)p−1(D′).

Example 9.5. If p = char(k) > 0, D = Derk(A,A) is a p-Lie algebra with the operation

D[p] := Dp = D D · · · D.

One checks that (a)-(c) holds for D. Using the fact that if p = char(k) > 0, pD = 0 forall D ∈ D, they reduce to straightforward, but tedious calculations.

The main example for us will be the “tangent space” at e to an algebraic group. Thereis a general algebraic construction of tangent spaces of algebraic varieties and for linearalgebraic groups we have a way to identify TeG with “left invariant derivations”.

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Tangent Spaces of an Algebraic Group

(Heuristic)LetX ⊂ A2 be an irreducible curve defined by a single polynomial equation f(T1, T2) =

0. Tangent at x = (x1, x2) is defined as the solutions to the linear equation

∂f

∂T1(x)(T1 − x1) +

∂f

∂T2(x)(T2 − x2) = 0.

Unless both partials vanish, the solution set is a line through x = (x1, x2).More generally, let X ⊂ An be a closed subvariety of An. Write k[X] = k[T1, · · · , Tn]/I

where I is the ideal of polynomial functions vanishing on X. Let I = (f1, · · · , fs). Forx ∈ X, let L be a line in An through x. Then L = x + tv|t ∈ k for a direction vectorv = (v1, · · · , vn). L∩X is the solution of the system of equations fi(x+ tv) = 0, 1 ≤ i ≤ s(and of course, t = 0 is a solution). Writing Di for the partial derivation with respect toTi in k[T1, · · · , Tn], we have

fi(x+ tv) = t

n∑j=1

vj(Djfi)(x) + terms of degree in t higher than 1.

Now t = 0 is a “multiple root” of the system of equations if and only if∑n

j=1 vj(Djfi)(x) =0, 1 ≤ i ≤ s. If this holds, we call L a tangent line and v a tangent vector of X in x.

Write D′ =∑n

j=1 vjDj . Then D′ is a k-derivation of k[T1, · · · , Tn], and the system ofequations simply says D′fi(x) = 0 for 1 ≤ i ≤ s. Let Mx be the maximal ideal in k[T ] offunctions vanishing at x. Then D′T ⊂Mx. Consider the diagram

k[T ] > k ∼= k[X]/Mx∼= kx

k[X] = k[T ]/I∨ D

>

Viewing k as a k[X] −module kx via the above homomorphism f 7→ f(x), it turns outthat D ∈ Derk(k[X], kx). Conversely, any D ∈ Derk(k[X], kx) can be obtained this wayfrom a derivation D′ of k[T ] with D′T ⊂Mx. We conclude that

tangent vectors v such that the systemfi(x+ tv) = 0, 1 ≤ i ≤ s, has a “multiple root” t=0

bijection←→ Derk(k[X], kx).

Here is a summary of the formal definition of tangent space. First, let X be an affinealgebraic variety, x ∈ X, and define

TxX := the k-vector space Derk(k[X], kx).

We observe that:

53

(1) A morphism ϕ : X → Y of affine algebraic varieties with ϕ∗ : k[Y ] → k[X],the corresponding homomorphism of k-algebras, gives rise to ϕ∗0 : Derk(k[X], kx) →Derk(k[Y ], kϕ(x)), i.e., we get a “differential of ϕ at x” dϕx : TxX → Tϕ(x)Y .

(2) If Xϕ−→ Y

ψ−→ Z are morphisms of affine varieties, then d(ψ ϕ)x = dψϕ(x) dϕx .(3) If ϕ is an isomorphism of affine varieties, then dϕx is also an isomorphism of k-

vector spaces. In fact, if ϕ is an isomorphism of X onto an affine open subvariety of Y ,then dϕx is an isomorphism of TxX onto Tϕ(x)Y .

(4) d(id)x = id.(5) Here are two more equivalent definitions of TxX:(5)-i Let Mx ⊂ k[X] be the maximal ideal of functions vanishing at x. If D ∈ TxX,

then D maps M2x to 0, so it defines a linear map

λ(D) : Mx/M2x → k.

In fact, λ : TxX → dual of Mx/M2x is an isomorphism.

(5)-ii Recall that Ox = ring of regular functions at x. It is a k-algebra with a maximalideal Mx := regular functions vanishing at x. Then Ox/Mx

∼= k and we may view k asan Ox-module. Recall that there is an algebraic homomorphism

k[X]→ Oxf 7→ f/1

which induces a linear map α0 : Derk(Ox, k)→ Derk(k[X], kx). In fact, α0 is a bijection.(6) Let X be an affine F -variety and x ∈ X(F ). Then the point x defines an algebraic

homomorphism F [X] → F which makes F into an F [X]-module Fx. We then define (avector space over F )

TxX(F ) := DerF (F [X], Fx).

We have a canonical isomorphism k ⊗F TxX(F ) ∼= TxX.(7) Both notions in (5) and (6) generalize to arbitrary (not necessarily affine) varieties.

Definition 9.6. Let X be an algebraic variety. We say x ∈ X is a simple point or X issmooth at x, or X is non-singular at x if dimTxX = dimX.

(8) (algebraic differential) Let R be a commutative ring with 1, A a commutative R-algebra. Denote by m : A ⊗R A → A the product homomorphism (i.e, m(a ⊗ b) = ab)and let I = kerm. Then I is an ideal of A ⊗R A generated by elements a ⊗ 1 − 1 ⊗ a,a ∈ A. The quotient algebra (A ⊗ A)/I is isomorphic to A (as R-algebras). The moduleof differentials of the R-algebra A is defined as

ΩA/R := I/I2.

A priori, this is an A⊗A-module, but it is annihilated by I, so we can and will view ΩA/R

as an A-module. Define

da = dA/Ra := image of (a⊗ 1− 1⊗ a) in ΩA/R.

54

It’s easy to check that d is in fact an R-derivation of A in ΩA/R, i.e., d ∈ DerR(A,ΩA/R),and da’s generate the A-module ΩA/R.

There are two basic results here.Let X be an irreducible affine algebraic variety over k. Put ΩX = Ωk[X]/k. If x ∈ X,

then TxX ∼= Hom(ΩX , kx). But for any k[X]-module M , we have

Homk[X](M,kx) ∼= Homk(M(x), k),

where M(x) := M/MxM . Hence TxX ∼= Hom(ΩX , k). This allows for the following:

(TxX)∗ := “cotangent space” = dual vector space of the tangent space

can be identified with ΩX(x) = ΩX/MxΩx.

Theorem 9.7. Let X be an irreducible variety of dimention e.(i) If x is a simple point of X, there is an affine open neighborhood U of x such that ΩU

is a free k[U ]-module with a bases (dg1, · · · , dge) for some suitable gi ∈ k[U ].(ii) The simple points of X form a non-empty open subset of X.(iii) For any x ∈ X, we have dimTxX ≥ e.

Definition 9.8. A morphism ϕ : X → Y of irreducible varieties is called dominant ifϕ(X) is dense in Y .

Recall from algebraic geometry that ϕ∗ is injective ⇔ ϕ(X) is dense in Y . So if ϕ isdominant, then there is an injection of the quotient fields k(Y )→ k(X). So we may viewk(X) as a field extension of k(Y ). We say ϕ is separable if this extension is separablygenerated.

Theorem 9.9. Let ϕ : X → Y be a homomorphism of irreducible varieties.(i) Assume there is a simple point x ∈ X such that ϕ(x) is a simple point of Y and dϕxis surjective. Then ϕ is dominant and separable.(ii) Assume that ϕ is dominant and separable, then the points x ∈ X as in (i) form anon-empty open subset of X.

We may apply previous results to homogeneous spaces.

Theorem 9.10. Let G be a connected algebraic group.(i) Let X be a homogeneous space for G. Then X is irreducible and smooth. In particular,G is smooth.(ii) Let ϕ : X → Y to be a G-morphism of homogeneous spaces. Then ϕ is separable ⇐⇒dϕx is surjective for some x ∈ X. If this is the case, then dϕx is surjective for all x ∈ X.(iii) Let ϕ : G→ G′ be a surjective homomorphism of algebraic groups. Then ϕ is separable⇐⇒ dϕe is surjective.

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10 The Lie Algebra of a Linear Algebraic Group (10/21)

Let G be a linear algebraic group. Let λ and ρ be left and right translation on A = k[G],so they are both representations of G. A ⊗k A can be viewed as the algebra of regularfunctions k[G×G]. If m : A⊗k A→ A is the multiplication map, then for F ∈ k[G×G],we have

(mF )(x) = F (x, x).

Let I = kerm, the ideal of functions in k[G×G] vanishing on the diagonal. For x ∈ G, bothautomorphisms λ(x)⊗λ(x) and ρ(x)⊗ρ(x) stabilize I and I2. This induces automorphismof ΩG = I/I2, again denoted by λ(x) and ρ(x). Hence, we have two representations λ, ρof G in ΩG which are locally finite. (Recall we had a k-derivation d : A → ΩG = I/I2

with da = image of (a⊗ 1− 1⊗ a)). Now, the derivation d : A→ ΩG commutes with allλ(x) and ρ(x).

Fix x ∈ G. Consider Int : G→ G defined by y 7→ xyx−1 which is an automorphism ofalgebraic groups, fixing e ∈ G. Therefore, this induces two linear automorphisms

Ad(x) : TeG→ TeG

and(Ad(x))∗ : (TeG)∗ → (TeG)∗,

and we have

((Ad(x)∗u)(X) = u(Ad(x−1X)), u ∈ (TeG)∗, X ∈ TeG.

Recall that if Me ⊂ A is the maximal ideal of functions vanishing at e, then weidentify (TeG)∗ with Me/M

2e . (We saw that λ : TxX ∼= dual of Mx/M

2x , where λ(D) :

Mx/M2x → k). If f ∈ A, denote by δf the element f − f(e) + M2

e in (TeG)∗ = Me/M2e .

For X ∈ Derk(A, ke), we have (δf)(X) = Xf .

Proposition 10.1. There is an isomorphism of k[G]-modules

Φ : ΩG = I/I2 → k[G]⊗k (TeG)∗

such that(a) the diagram

ΩGΦ> k[G]⊗k (TeG)∗

ΩG

λ(x)

∨Φ> k[G]⊗k (TeG)∗

λ(x)⊗id∨

ΩG

ρ(x)

∨Φ> k[G]⊗k (TeG)∗

ρ(x)⊗(Ad(x))∗

∨

56

is commutative, so

Φ λ(x) Φ−1 = λ(x)⊗ id, Φ ρ(x) Φ−1 = λ(x)⊗ (Ad(x))∗;

(b) If f ∈ k[G] and ∆f =∑

i fi ⊗ gi (recall ∆ is comultiplication given by (∆f)(x, y) =f(xy)), then Φ(df) = −

∑i fi ⊗ δgi.

Proof. Consider the automorphism G×G→ G×G given by (x, y) 7→ (x, xy). This definesan algebra automorphism ψ : A ⊗ A → A ⊗ A with (ψF )(x, y) = F (x, xy). So ψI is theideal of functions vanishing on G × e, which is k[G] ⊗k Me. Then ψI2 = k[G] ⊗k M2

e .So ψ induces a bijection

ΩG = I/I2 → k[G]⊗kMe/M2e .

Also, recall that we have an isomorphism Me/M2e∼= (TeG)∗. Now, let Φ be the composite

of these two maps:

ΩG > k[G]⊗kMe/M2e

k[G]⊗k (TeG)∗∨Φ

.........................>

Then, for x ∈ G, we have

(λ(x)⊗ id)(ψF )(x′, y′) = (ψF )(xx′, y′) = F (xx′, xx′y′) = ψ (λ(x)⊗ λ(x))(F )(x′, y′),

and

((ρ(x)⊗ Int(x)) ψ)(F )(x′, y′) = (ψF )(x′x−1, xy′x−1) = F (x′x−1, x′x−1xy′x−1)

= ψ (ρ(x)⊗ ρ(x))(F )(x′, y′).

Thus

(λ(x)⊗ id) ψ = ψ (λ(x)⊗ λ(x)), (ρ(x)⊗ Int(x)) ψ = ψ (ρ(x)⊗ ρ(x)).

These formulas for ψ now give the requirements of (a) for Φ, so (a) is true.For (b),

ψ(f ⊗ 1− 1⊗ f)(x, y) = (f ⊗ 1− 1⊗ f)(x, xy) = f(x)− f(xy)

= (∆f)(x, e)− (∆f)(x, y)

=∑i

fi(x)gi(e)− fi(x)gi(y)

=∑i

fi(x)(gi(e)− gi(y))

= −∑i

fi(x)(δgi)(y).

Thus, ψ(df) = −∑

i fi ⊗ δgi.

57

Let G be a linearly algebraic group and A = k[G]. Write D = DG = Derk(A,A) as lasttime. Recall that D had a Lie algebra structure with bracket [D,D′] = D D′ −D′ D.Then λ and ρ define representations of G in D, and we denote by λ and ρ again. So

λ(x)D = λ(x) D λ(x)−1, ∀D ∈ D,∀x ∈ G,

ρ(x)D = ρ(x) D ρ(x)−1, ∀D ∈ D,∀x ∈ G.

Definition 10.2. The Lie algebra of G is defined as

L(G) := D ∈ D : D commutes with all λ(x), x ∈ G.

Here are a few remarks.(1) L(G) is a subalgebra of the Lie algebra D.(2) If p = char(k) > 0, L(G) is stable under the p-operation D 7→ Dp.(3) Let and right translations commute, so all ρ(x) stabilize L(G). We denote the linearmap induced again by ρ(x).(4) There is an isomorphism of k[G]-modules

Ψ : DG → k[G]⊗k TeG

such thatΨ λ(x) Ψ−1 = λ(x)⊗ id,

Ψ ρ(x) Ψ−1 = ρ(x)⊗Ad(x),

Ψ−1(1⊗X)(f) = −∑i

fi(Xgi), ∀X ∈ TeG.

(5) Let α = αG : DG → TeG be the linear map with (αGD)(f) = (Df)(e). Then α inducesan isomorphism of vector spaces L(G) ∼= TeG, and α ρ(x) α−1 = Adx, and Ad is arational representation of G in TeG called the “adjoint representation”.(6) As a corollary of (5), dimk L(G) = dimG.(7) (closed subgroup) Let H be a closed subgroup of G. Let J ⊂ K[G] be the ideal offunctions vanishing on H, so k[H] ∼= k[G]/J . Put DG,H := D ∈ DG : DJ ⊂ J. ThenDG,H is a subalgebra of the Lie algebra DG and there is an obvious homomorphism of Liealgebra

ψ : DG,H → DH .

In fact, TeH = X ∈ TeG : XJ = 0.

Lemma 10.3. ψ gives an isomorphism of DG,H ∩ L(G) onto L(H).

From now on, we identify the Lie algebra L(G) and the tangent space TeG via αG soTeG gets a Lie algebra structure. We usually write

g = Lie(G), h = Lie(H).

58

(8) If ψ : G → G′ is a homomorphism of linear algebraic groups, we write dψ for thetangent map

dψ : g→ g′,

called the differential of ψ. dψ is a homomorphism of Lie algebras, compatible with thep-operation if p = char(k) > 0.

We now add some differential formulas for later use.(9) Let µ : G × G → G, µ(x, y) = xy, and i : G → G, i(x) = x−1 as before. Identify

L(G×G) with g⊕ g (this is easy to check). Then (dµ)(e,e) : g⊕ g→ g is given by

(X,Y ) 7→ X + Y,

and (di)e : g→ g is given byX 7→ −X.

(10) (i) Let σ : G→ G be a morphism of varieties and put ψ(x) = (σ(x))x−1. Then

(dψ)e = (dσ)e − 1.

(ii) Let a ∈ G. If ϕ(x) = axa−1x−1, then

(dϕ)e = Ad(a)− 1.

(iii) (direct sums and tensor products) Start with a finite dimensional vector space Vover k and write gl(V ) = Lie algebra of endomorphisms of V . Then gl(V ) ∼= gldimV . Ifφ : G → GL(V ) is a rational representation of G, its differential dφ is a Lie algebrahomomorphism g→ gl(V ), i.e., a representation of g in V .

Let G1, G2 be two linear algebraic groups and let φi : Gi → GL(Vi), i = 1, 2, betwo rational representations. Let φ1 ⊕ φ2 be the direct sum representation of G1 × G2

in V1 ⊕ V2 and let φ1 ⊗ φ2 be the tensor product representation of G1 × G2 in V1 ⊗ V2.Identify L(G1 ×G2) = g1 ⊕ g2. Then

d(φ1 ⊕ φ2) = dφ1 ⊕ dφ2,

and(d(φ1 ⊗ φ2))(X1, X2)(v1, v2) = (dφ1)(X1)(v1)⊗ v2 + v1 ⊗ (dφ2)(X2)(v2).

Example 10.4. Let G = Ga, A = k[G] = K[T ]. Derivations of k[G] that commute withall translations T → T + a, a ∈ k are multiplies of X = d

dT . If p = char(k) > 0, we haveXp = 0. So g = k ·X with [X,X] = 0.

Example 10.5. Let G = Gm, A = k[G] = K[T, T−1]. First, for all a ∈ k×, we have(Td

dT

)(λ(a)f) (x) = x

d

dTf(ax) = x · df

dT(ax) · a.

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Also,

λ(a)

(Td

dTf

)(x) =

(Td

dTf

)(ax) = ax · df

dT(ax).

So T ddT commutes with all (left and right) translations T → aT , a ∈ k×. (Note that in

this case, a left translation is the same as a right translation). In fact, derivations of k[G]commutating with translations T → aT , a ∈ k×, are all multiplies of T d

dT . If p > 0, we

have Xp = X. So g = multiplies of T ddT . So g is the same as in Example 10.4, but the

p-operation is different when p > 0.

Example 10.6. Let G = GLn, k[G] = k[Tij , D−1] where D = det(Tij). Recall the

notation gln; [X,Y ] = XY − Y X, and the usual p-th power p-operation when p > 0. IfX = (xij) ∈ gln, then

DXTij = −n∑h=1

Tihxhj

defines a derivation of k[G] that commutes with all left-translations. So it lies in L(G).Also, the map X 7→ DX is injective. By equality of dimensions of Lie algebra and group,L(G) consists of all of the DX ’s. So we can identify g and gln (with p-th power operation).For x ∈ GLn, X ∈ gln, we have Ad(x)X = xXx−1.

Example 10.7. If H is a closed subgroup of GLn, we can view h as a subalgebra ofgln. Using Remark (7) on closed subgroups, h = DG,H ∩ gln, where DG,H = D ∈ DG :DJ ⊂ J, J ⊂ k[G] the ideal of functions vanishing on H. For example, H = SLn,h = X ∈ gln : tr(X) = 0.

60

11 Homogeneous Spaces, Quotients of Linear Algebraic Groups(10/28)

Review of Homogeneous Spaces

Let G be an algebraic group (not necessarily linear) and let X be a homogeneous spacefor G. Recall that a homogeneous space for G is a G-space in which G acts transitively.Let G0 be the identity component of G.

Theorem 11.1. Let G be an algebraic group and let φ : X → Y be an equivarianthomomorphism of homogeneous spaces for G. Put r = dimX − dimY .

(i) For any variety Z, the morphism (φ, id) : X × Z → Y × Z is open.(ii) If Y ′ is an irreducible closed subvariety of Y and X ′ is an irreducible component of

φ−1Y ′, then dimX ′ = dimY ′+ r. In particular, if y ∈ Y , then all irreducible componentsof φ−1(y) have dimension r.

(iii) φ is an isomorphism ⇐⇒ it is bijective and for some x ∈ X, the tangent mapdφx : TxX → Tφ(x)Y is bijective.

Corollary 11.2. Let φ : G→ G′ be a surjective homomorphism of algebraic groups.(i) dimG = dimG′ + dim ker(φ).(ii) φ is an isomorphism ⇐⇒ φ and the tangent map dφe are bijective.

Proof. View G and G′ as homogeneous spaces for G (via left translation for G and viag · g′ := φ(g)g′ for G′). Apply Theorem 11.1.

Remark 11.3. Note that if G and G′ are both linear algebraic groups, then the condi-tion on the tangent map in (iii) in Theorem 11.1 can be rephrased as: the Lie algebrahomomorphism dφ : g→ g′ is bijective.

Remark 11.4. In (iii) in Theorem 11.1, it is not enough to check that φ is bijective, wealso need that dφe is bijejctive. For example, consider φ : Gm → Gm defined by x 7→ xn

with char(k) = p > 0, and n = pf , then φ is a bijection (in fact, an isomorphism ofabstract groups), but not an isomorphism of algebraic groups.

Quotients of Linear Algebraic Groups

Let G be a linear algebraic group, H a closed subgroup with respective Lie algebras g andh. Let F ⊂ k be a subfield. Assume that G is an F -group, and H is an F -subgroup. Ourgoal is to (1) construct a quotient variety G/H; (2) when H is closed and normal, G/Hwill be an affine variety, hence a linear algebraic group.

Theorem 11.5 (Chevalley). There exists a rational representation φ : G → GL(V ) overF , where V is a finite dimensional subspace of k[G] and there is a nonzero v ∈ V (F ) suchthat

H = x ∈ G : (φx)v ∈ kv, h = X ∈ g : (dφX)v ∈ kv.

61

Corollary 11.6. There is a quasi-projective homogeneous space X for G together with apoint x ∈ X such that(a) The isotropy group of x in G is H;(b) The morphism ϕ : G → X defined by g 7→ g · x defines an separable morphismG0 → ϕG0;(c) The fibres of ϕ are cosets gH, g ∈ G.

Recall that a quasi-projective variety is an open subvariety of a projective variety.

Proof of Corollary 11.6. We skip proof for (b), and prove (a) and (c). Take V and v asin Theorem 11.5 and consider the projective space P(V ). Denote by x the point in P(V )determined by the line kv. Consider π : V −0 → P(V ) sending each non-zero vector tothe line through it. Now, G acts on P(V ) by g · π(v) = π(φ(g) · v), with φ as in Theorem11.5. Let X be the G-orbit of x. Recall that X = G · x is open in its closure, so X isquasi-projective. Now

Gx = g ∈ G : g · x = x= g ∈ G : π(φ(g) · v) = π(v)= g ∈ G : φ(g)v ∈ kv = H (by Theorem 11.5).

This proves (a).For y = g · x ∈ X, g ∈ G, its fibre is

ϕ−1(y) = γ ∈ G : γ · x = y= γ ∈ G : γ · x = g · x= γ ∈ G : g−1γ ∈ H= γ ∈ G : γ ∈ gH= gH.

This proves (c).

Now we prove Theorem 11.5.

Proof of Chevalley’s Theorem, Theorem 11.5. It follows from combining the following twolemmas.

Lemma 11.7. There exists a finite dimensional subspace V of k[G] together with a sub-space W of V such that(a) V is stable under all right translations ρ(x), x ∈ G;(b) We have

H = x ∈ G : ρ(x)W = Wh = X ∈ g : X ·W ⊂W.

(c) V is defined over F and W is an F -subspace of V .

62

Proof of Lemma 11.7. Let I ∈ k[G] be the ideal of functions vanishing on H and let V bea finite dimensional ρ(G)-stable subspace of k[G] that is defined over F and contains a setof generators (f1, · · · , fr) of I which lie in F [G]. Set W = V ∩ I. Then (a) is automatic,and (c) is clear.

For (b), if x ∈ H, then ρ(x)W = W (recall by Lemma 5.16, we had H = g ∈G : λ(g)IG(H) = IG(H) = g ∈ G : ρ(g)IG(H) = IG(H)). On the other hand, ifρ(W ) = W , then ρ(x)fi ∈ I for all 1 ≤ i ≤ r. So ρ(x)I ⊂ I. By the same reason (Lemma5.16) we must have x ∈ H. For 〈, similar argument works (recall we had DG,H = D ∈DG : DI ⊂ I ∩ L(G) = L(H)).

Now let V be an arbitrary finite dimension vector space, W subspace of dimension d.Then the d-exterior power of V ∧dV contains the one dimension subspace L = ∧dW . Letφ be the canonical representation of GL(V ) in ∧dV . Then

(dφ)(X)(v1 ∧ v2 · · · ∧ vd) =

d∑i=1

v1 ∧ · · · ∧ (dφ)(x)vi ∧ · · · ∧ vd.

Lemma 11.8. (i) Let x ∈ GL(V ). We have x ·W = W ⇐⇒ (φx)L = L.(ii) Let X ∈ gl(V ). We have X ·W ⊂W ⇐⇒ (dφ)(X)L ⊂ L.

Proof of Lemma 11.8. The direction (=⇒) is clear for both (i) and (ii).(i) (⇐=) Choose a basis (v1, · · · , vn) of V such that (v1, · · · , vd) is a basis of W . Then

vi1 ∧ · · · ∧ vid with i1 < i2 < · · · < id form a basis for ∧dV and v1 ∧ · · · ∧ vd is a basis forL. Let x ∈ GL(V ). We may also assume that vl+1, · · · , vl+d is a basis for x ·W for somel. Put e = v1 ∧ · · · ∧ vd and f = vl+1 ∧ · · · ∧ vl+d. Then (φ(x))e is a multiple of f . Ifl > 0, then e and f are linearly independent and φ(x) does not stabilize L. So l = 0, i.e.,x ·W = W .

(ii) (⇐=) If X ∈ gl(V ), then

(dφ)(X)e =d∑i=1

v1 ∧ · · · ∧X · vi ∧ · · · ∧ vd.

Write Xvi =∑n

j=1 aijvj . Then

(dφ)(X)e =

d∑i=1

n∑j=1

aijv1 ∧ · · · ∧ vj ∧ · · · ∧ vd

If aij 6= 0 for i ≤ d and j > d, then L is not mapped into itself. So aij = 0 for i ≤ d andj > d. So X ·W ⊂W .

Thus, we have finished the proof of Theorem 11.5.

63

Construction of Quotient

A quotient of G by H over F is a pair (G/H, a) of a homogeneous space G/H of G overF together with a point a ∈ G/H(F ) such that the following universal property holds:for any pair (Y, b) of a G-space Y for G over F and a point b ∈ Y (F ) whose isotropygroup contains H, there exists a unique equivariant F -morphism φ : G/H → Y such thatφ(a) = b.

G > G/H

Y

φ

∨

.........>

Theorem 11.9. A quotient (G/H, a) over F exists and is unique up to a G-isomorphism.In fact, if X and x are as earlier, then (X,x) is such a quotient.

We prove this in the case F = k. The proof over general F is similar, but uses someinformation about the ground fields, which is done in Chapter 11 and 12 of Springer’sLinear Algebraic Groups.

Proof of Theorem 11.9 over k, i.e., F = k. The uniqueness is trivial given the universalproperty. So we prove existence. The proof of existence has two steps: in step 1, we define(G/H, a) in the category of ringed spaces, and in step 2, we show that it is isomorphic asa ringed space, to the pair (X,x) as before.

Step 1. G/H has its points the cosets gH, and a = H. Let π : G → G/H be thecanonical map. Declare U ⊂ G/H open if π−1(U) is open in G (called the quotienttopology). Then we get a topology space G/H such that the map π is an open map. Wedefine a sheaf O of k-valued functions on G/H as follows: if U ⊂ G/H is open, then set

O(U) = f : U → k : f π is a regular function on π−1(U).

It’s easy to check thatO(U) is a ring of functions, andO defines a sheaf. G acts transitivelyon G/H by left translations and for x ∈ G, the map gH 7→ xgH defines an isomorphism ofringed spaces (G/H,O). Now, if (Y, b) is as in the universal property, then there exists aunique G-morphism of ringed space φ : G/H → Y with φ(a) = b. Just take φ(gH) = g · b.

Step 2. Let X,x and ψ : G→ X be as before. In particular, we have a G-morphism ofringed spaces φ : G/H → X defined by gH 7→ g · x. We prove that this is an isomorphismof ringed spaces, which will imply that as an algebraic variety, the ringed space G/Hsatisfies the theorem.

First, by Corollary 11.6, we know the fibres of ϕ are the cosets gH, so φ is a continuousbijection. If U ⊂ G/H is open, then φ(U) = ψ(π−1U) is open, so by Theorem 11.1 weconclude that φ is a homeomorphism of topological spaces.

Next, we show that φ is an isomorphism of ringed spaces. We need to show that ifU ⊂ X is open, then the homomorphism of k-algebras OX(U) → O(φ−1(U)) defined by

64

φ is an isomorphism of k-algebras. By definition of O, this means that for any regularfunction f on V = ψ−1(U) such that f(gh) = f(g), g ∈ V, h ∈ H, there is a unique regularfunction F on U such that F (ψg) = f(g) for all g ∈ V . Without loss of generality, wemay assume G is connected (because of the following lemma).

Lemma 11.10. Let G be an algebraic group and let X be a homogeneous space for G.Then(i) each irreducible component of X is a homogeneous space for G0;(ii) the components of X are open and closed and X is their disjoint union.

Let Γ = (g, f(g)) : g ∈ G ⊂ V × A1 be the graph of f and put Γ′ = (ψ, id)Γ, soΓ′ ⊂ U ×A1. Since Γ is closed in V ×A1, by Theorem 11.1, we have

(ψ, id)(V ×A1 − Γ) = U ×A1 − Γ′ is open in U ×A1.

So Γ′ is closed in U ×A1. Let λ : Γ′ → U be the morphism induced by projection ontothe first component. Then it follows from definitions that λ is bijective and by Corollary11.6, λ is separable. In fact, by results from algebraic geometry, λ is an isomorphism.This implies that there is a regular function F on U such that Γ′ = (u, F (u)) : u ∈ Uwhich is what we want.

Corollary 11.11. (i) G/H is a quasi-projective variety of dimension dim(G)− dim(H).(ii) If G is connected, the morphism G→ G/H via g 7→ g · a is separable.

Proof. This follows from Corollary 11.6.

Proposition 11.12 (normal subgroups). Let G be a linear algebraic group and H a normalclosed subgroup of G. Then(i) G/H is an affine variety;(ii) with the usual group structure, G/H is a linear algebraic group.

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12 Parabolic and Borel Subgroups (11/4)

Review of Complete Varieties

Definition 12.1. An algebraic variety X is called complete if for any variety Y , theprojection morphism X × Y → Y is closed, i.e., it maps closed sets to closed sets.

This is analogous to the notion of “compactness” in topology.

Example 12.2. X = A1 is not complete. Take Y = A1 and consider φ : A1 ×A1 → A1

given by (x, y) 7→ y. Now C = (x, y) ∈ A1 ×A1 : xy = 1 is closed in A2, but ϕ(C) isopen in A1.

Example 12.3. X = P1 is complete.

Theorem 12.4. A projectivve variety is complete.

Proposition 12.5. Let X be complete.(i) A closed subvariety of X is complete.(ii) If Y is complete, then so is X × Y .(iii) If φ : X → Y is a morphism, then φ(X) is closed and complete.(iv) If X is a subvariety of Y , then X is closed in Y .(v) If X is irreducible, then any regular function on X is constant.(vi) If X is affine, then X is finite.

Definition 12.6. A closed subgroup P of G is called parabolic if the quotient variety G/Pis complete.

Facts on Parabolics

Here are a few facts on parabolics.(1) If X,Y are homogeneous spaces for G and φ : X → Y is a bijection of G-morphism,

then X is complete ⇐⇒ Y is complete.

Proof. Recall that given X and Y as above, for any variety Z, the map (φ, id) : X ×Z →Y × Z is a homeomorphism of topological spaces. Now X × Z → Z is closed ⇐⇒Y × Z → Z is closed.

(2) P is parabolic in G ⇒ G/P is a projective variety.

Proof. Recall that G/P is a quasi-projective variety, i.e., an open subvariety of a projectivevariety. Now by part (iv) of Proposition 12.5, G/P is closed, so it is a projective variety.

(3) Let P be a parabolic in G, Q be a parabolic in P , then Q is a parabolic in G.

66

Proof. We need to show that for any variety X, the projection map G/Q × X → X isclosed. Consider

P ×G×X α> G×X β

> G/Q×X pr2> X

(p, g, x) > (gp, x)

If C is closed in G/Q ×X, then A = β−1(C) ⊂ G ×X is closed with the property thatif (g, x) ∈ A, then (gQ, x) ⊂ A. We need to show A′ = pr2(β(A)) is closed in X. Nowα−1(A) = (p, g, x) ∈ P × G × X : (gp, x) ∈ A is closed in P × G × X. Since P/Q iscomplete, P/Q× (G×X)→ G×X is closed, so the image of α−1(A) under this map, i.e.,⋃

(g,x)∈A(gp, x) is closed in G×X. Now completeness of G/P implies that the projection

of this set is closed in X, but that is just A′.

(4) (i) If P is parabolic in G and Q is a closed subgroup of G containing P , then Q isparabolic in G. (ii) P is parabolic in G ⇐⇒ P 0 is parabolic in G0.

(5) A connected linear algebraic group G contains no proper parabolic subgroups ⇐⇒G is solvable.

Theorem 12.7 (Borel’s Fixed Point Theorem). Let G be a connected solvable linearalgebraic group, and let X be a complete G-variety. Then there exists a point x ∈ X thatis fixed by all elements of G.

Proof. Recall that G has a closed orbit in X. The isotropy group of a point in that orbitis parabolic. By Fact (5) above, this group must be all of G, i.e., we get a fixed point.

Borel Subgroups

Definition 12.8. A Borel subgroup of G is a closed, connected, solvable subgroup of G,which is maximal for these properties.

Note that Borel subgroups always exist, as we can see by taking the maximal dimension.

Theorem 12.9. (i) A closed subgroup of G is parabolic iff it contains a Borel.(ii) A Borel subgroup is parabolic.(iii) Any two Borel subgroups in G are conjugate.

Proof. (i) By Fact (4)(ii) above, we may assume G is connected. Let B be a Borel andlet P be any parabolic. Now B acts on the complete variety G/P and by Borel’s FixedPoint Theorem, there exists gP ∈ G/P such that bgP = gP for all b ∈ B. This impliesthat g−1bg ∈ P for all b ∈ B. So P contains a conjugate of B, which is also a Borel. Thisproves ⇒. ⇐ will follow if we prove (ii) by Fact (4)(1) above.

(ii) If G is solvable, then there is no proper parabolic and (ii) is obvious. Now assumeG is non-solvable, so there is a proper parabolic P . By what we already saw, we mayassume B ⊂ P . By induction on dimG, we may assume B is parabolic in P . Now Fact(3) implies B is parabolic in G.

67

(iii) If B,B′ are Borel, then B′ is conjugate to a subgroup of B and B is conjugate toa subgroup of B′. Hence dimB = dimB′. So B is conjugate to B′.

Corollary 12.10. If G is connected, then we have C(G)0 ⊂ C(B) ⊂ C(G) for the centers.

Proof. C(G)0 is closed, connected, and commutative, so it lies in a Borel subgroup. Henceit lies in all Borels since Borels are conjugate. So C(G)0 ⊂ C(B).

Next, if g ∈ C(B), then the morphism G → G defined by x 7→ gxg−1x−1 induces amorphism G/B → G which must be constant because G/B is an irreducible completevariety. So gxg−1x−1 = e for all x ∈ G, so g ∈ C(G).

Corollary 12.11. Let φ : G → G′ be a surjective homomorphism of linear algebraicgroups. Let P be a parabolic subgroup, respectively a Borel subgroup of G. Then φ(P ) isa parabolic subgroup, respectively a Borel subgroup of G.

Proof. By part (i) of Theorem 12.9, it is enough to prove it for Borels, B = P . Thenφ(B) is closed, connected, solvable. Moreover, the homomorphism G/B → G′/φ(B) issurjective. By Proposition 12.5 (iii), G′/φ(B) is complete, so φ(B) is a parabolic, henceit contains a Borel of G′. Then φ(B) is a Borel by maximality.

Connected Solvable Groups

Theorem 12.12 (Lie-Kolchin). Let G be a connected solvable closed subgroup of GLn.Then there exists x ∈ GLn such that xGx−1 ⊂ Tn where Tn is the upper triangularmatrices.

Proof. Apply Borel’s Fixed Point Theorem (Theorem 12.7) to G acting on Pn−1 to con-clude that the elements of G have a non-zero eigenvector. Now induction on n gives thetheorem.

Corollary 12.13. Assume G is a nilpotent group. Then(i) the sets Gs and Gu of semisimple and unipotent elements are closed, connected sub-groups and Gs is a central torus;(ii) The product map Gs ×Gu → G is an isomorphism of algebraic groups.

Corollary 12.14. For G connected, solvable closed as above, we have(i) the commutator subgroup (G,G) is closed, connected, nilpotent, normal subgroup;(ii) the set Gu of unipotent elements is a closed, connected, nilpotent, normal subgroup ofG and the quotient G/Gu is a torus.

Lemma 12.15. Assume G is connected, solvable, linear algebraic group which is not atorus. Then there exists a closed normal subgroup N of G that is isomorphic to Ga andit lies in the center of Gu.

68

Definition 12.16. Let G be a connected solvable linear algebraic group. A maximal torusof G is a subtorus that has the same dimension as S = G/Gu.

Theorem 12.17. Let G be a connected solvable linear algebraic group.(i) Let s ∈ G be semisimple. Then s lies in a maximal torus. In particular, maximal toriexist.(ii) The centralizer ZG(s) of a semisimple element s ∈ G is connected.(iii) Two maximal tori in G are conjugate.(iv) If T is a maximal torus, the product map π : T × Gu → G is an isomorphism ofvarieties.

Corollary 12.18. Let H ⊂ G be a subgroup whose elements are semisimple.(i) H is contained in a maximal torus of G. In particular, a subtorus of G is containedin a maximal torus of G.(ii) ZG(H) is connected and it is equal to NG(H).

Proof. The restriction to H of the canonical homomorphism G→ G/Gu is bijective, so His commutative. If H ⊂ C(G), then Corollary 12.18 is clear. Otherwise, take a non-centralelement s ∈ H. By Theorem 12.17 (ii) , ZG(s) is connected. Also, it contains H. Now (i)and connectedness of ZG(H) in (ii) follows by induction on dimG. Finally, if x ∈ NG(H),then for h ∈ H, xhx−1h−1 ∈ H ∩ (G,G) ⊂ H ∩ Gu = e, so x ∈ ZG(H), and henceZG(H) = NG(H).

Definition 12.19. Now let G be a connected linear algebraic group over k. A maximaltorus of G is a subtorus of G that is not strictly contained in another subtorus. A Cartansubgroup of G is the identity component of the centralizer of a maximal torus. (We willsee such centralizers are always connected.)

Example 12.20. Let G = GL2. T1 =

(a

b

): ab 6= 0

is a maximal torus of G,

ZG(T1) = T1 is the Cartan subgroup of G.

Example 12.21. Let G = GL2. Let T2 =

(p qq p

): p2 − q2 6= 0

. Note

T2∼= D2 = GL2

1(p qq p

)7→ (p+ q, p− q).

So T2 is a torus. It is actually a maximal torus, and ZG(T2) = T2 is the Cartan subgroupof G. In fact,

T2 =

(1√2− 1√

21√2

1√2

)T1

(1√2− 1√

21√2

1√2

)−1

.

69

Theorem 12.22. Any two maximal tori in G are conjugate.

Proof. Fix a Borel B in G. A maximal torus T , being connected and solvable, lies in someBorel. By conjugacy of Borels, T must be conjugate to some subtorus of B, which in turnmust be a maximal torus in B. But for connected solvable groups, any two maximal toriare conjugate. So we get the theorem.

Proposition 12.23. Let T be a maximal torus of G and let C = ZG(T )0 be the corre-sponding subgroup.(i) C is nilpotent and T is its maximal torus.(ii) There exists elements t ∈ T lying in only finitely many conjugates of C.

Proof. We skip the proof, which uses the following Lemma 12.24.

Lemma 12.24. Let S be a subtorus in G. Then there exists s ∈ S such that ZG(s) =ZG(S).

We will state three more theorems.

Theorem 12.25. (i) Every element of G lies in a Borel.(ii) Every semisimple element of G lies in a maximal torus.(iii) The union of Cartan subgroups of G contains a dense open subset.

The proof of Theorem 12.25 uses the following Lemma 12.26.

Lemma 12.26. Let H be a closed subgroup of G and let X = ∪x∈GxHx−1. Then(i) X contains a non-empty open subset of X. If X is parabolic, then X = X is closed.(ii) Assume H has finite index in its normalizer N and that there exist elements in Hthat lie in only finitely many conjugates of H. Then X = G.

Corollary 12.27. If B is a Borel in G, then C(B) = C(G).

Theorem 12.28. Let S be a subtorus of G.(i) ZG(S) is connected.(ii) If B is a Borel containing S, then ZG(S) ∩B is a Borel in ZG(B).

Remark 12.29. All Borels of ZG(S) are obtained in this way.

We apply Theorem 12.28 to the case S = T , to get the following corollary.

Corollary 12.30. Let T be a maximal torus in G.(i) C = ZG(T ) is Cartan in G.(ii) If B is a Borel containing T , then B ⊃ C = ZG(T ).

Theorem 12.31. Let B be a Borel in G. Then NG(B) = B.

Corollary 12.32. If P is a parabolic in G, then P is connected and NG(P ) = P .

70

Proof. Since P is a parabolic, P contains a Borel B, which lies in P 0. By conjugacy ofBorels, there is some y ∈ P 0 such that xBx−1 = yBy−1. So y−1xB(y−1x)−1 = B. Soy−1x ∈ B ⊂ P 0 by Theorem 12.31. Thus x ∈ P 0.

Corollary 12.33. If P , Q are two conjugate parabolics in G such that P ∩Q contains aBorel B, then P = Q.

Proof. Let P = xQx−1. Then B and xBx−1 are two Borels in P , which must be conjugatein P , i.e., xBx−1 = yBy−1 for some y ∈ P . Arguing as above, we conclude that x ∈ Q,so P = Q.

Corollary 12.34. Let T be a maximum torus in G, and let B ⊃ T be a Borel. Then thereis a bijection

NG(T )/ZG(T )←→ Borel subgroups containing Tx 7→ xBx−1.

Reductive Groups

Observe that if N and N ′ are normal subgroups of G, then so is N · N ′. Also, recall if(G′i)i∈I was a family of closed connected subgroups of G, then the subgroup generatedby them was also closed and connected. Using these, it follows that there is a (unique)maximal, closed, connected, normal, solvable subgroup of G, namely a group with theseproperties of maximal dimension.

Definition 12.35. The radical of G, R(G), is the unique maximal, closed, connected,normal, solvable subgroup of G. Similarly, the unipotent radical of G, Ru(G), is themaximal, closed, connected, unipotent, normal subgroup of G.

Definition 12.36. G is called semisimple if R(G) = e. G is called reductive if Ru(G) =e.

For example, SLn is semisimple, and GLn is reductive.

71

13 Weyl Group, Roots, and Root Datum (11/11)

Weyl Group and Roots

Let G be a connected linear algebraic group, let T be a maximal torus in G, and letX = X∗(T ) be the character group of T = χ : T → Gm.

Remark 13.1. Let S be a torus and r : S → GL(V ) be a rational representation. V is adirect sum of 1-dimensional S spaces. In each of these, S acts via a character. Set

Vχ = v ∈ V : r(s)v = χ(s)v,∀s ∈ S.

The character χ for which Vχ is nonzero are called the weights of S in V . Vχ is the weightspace. Nonzero weights in a weight space are called weight vectors.

If S = T , V = g, r = Ad. Then let P be the set of nonzero weights. Then P is a finitesubset of X. (X viewed as a free abelian group in the additive notation.) We now do afew examples.

Example 13.2. LetG = GL2, T =

t =

(a1

a2

): a1, a2 6= 0

, g =

X =

(x11 x12

x21 x22

)=

M2(k). X ∼= Z2 = Z〈e1, e2〉 where e1(t) = a1, e2(t) = a2. We have

Ad(t) ·X = tXt−1 =

(a1

a2

)(x11 x12

x21 x22

)(a−1

1

a−12

)=

(x11

a1a2x12

a2a1x21 x22

),

V = k

(1 00 0

)⊕ k

(0 00 1

)⊕ k

(0 10 0

)⊕ k

(0 01 0

),

where the first two factors correspond to χ = 0, the third factor corresponds to χ = e1−e2,and the last factor corresponds to χ = −(e1 − e2). Moreover,

P = ±(e1 − e2).

Example 13.3. Let

G = GSp4 =g ∈ GL4 : tgJg = µ(g)J

where J =

1

1−1

−1

. Then B = upper triangular matrices in G is a Borel,

and

T =

t =

a1

a2

b2b1

: a1b1 = a2b2 = µ(t)

72

is a maximal torus in B. We have

X ∼= Z3 = Ze1 ⊕ Ze2 ⊕ Ze0

where e1(t) = a1, e2(t) = a2, e0(t) = µ(t). Also,

V = g =k

⟨x1

x2

y2

y1

: x1 + y1 = x2 + y2

⟩⊕ k

0 1

00 −1

0

⊕ k

01 0

0−1 0

⊕ k

0

0 10

0

⊕ k

001 0

0

⊕ k

0 10 −1

00

⊕ k

00

1 0−1 0

⊕ k

0 1

00

0

⊕ k

00

01 0

,

where the first factor is 3-dimensional, corresponding to χ = 0, and the second factorcorresponds to χ = e1−e2, the third factor corresponds to χ = −(e1−e2), the forth factorcorresponds to χ = 2e2−e0, the fifth factor corresponds to χ = −(2e2−e0), the sixth factorcorresponds to χ = e1 + e2− e0, the seventh factor corresponds to χ = −(e1 + e2− e0), theeighth factor corresponds to χ = 2e1−e0, the the last factor corresponds to χ = −(2e1−e0).Moreover,

P = ±(e1 − e2),±(2e2 − e0),±(e1 + e2 − e0),±(2e1 − e0).

Example 13.4. Consider S =

a1

11

a−11

⊂ T ⊂ GSp4. Let V = g =

Lie(GSp4), r = Ad|S . ThenP = ±e1,±2e1.

Remark 13.5. Let S ⊂ T be a subtorus. Then ZG(S) ⊂ ZG(T ) ⇐⇒ S is not containedin any of the subgroups kerα of T , α ∈ P .

Here is an example. Let G = GL3, P = ±(e1 − e2),±(e1 − e3),±(e2 − e3). If

S = ker(e1 − e2) =

a a

c

=

(aI2

c

), then ZG(S) =

(A 00 c

): A is 2× 2

,

T = ZG(T ).

73

For α ∈ P , let Gα := ZG((kerα)0), which is a closed connected subgroup. ThenRemark 13.5 means that if S is a subtorus of T with ZG(S) 6= ZG(T ), then there is α ∈ Psuch that ZG(S) ⊃ Gα. In fact, we have the following lemma.

Lemma 13.6. (i) The Gα, α ∈ P generate G.(ii) If all Gα are solvable, then G in solvable.

We skip the proof of Lemma 13.6. Here is an example:

G =

a d f

b ec

∈ GL3

, T =

a b

c

∈ G , P = e1 − e2, e1 − e3, e2 − e3,

Ge1−e2 =

a d

bc

, Ge2−e3 =

a b e

c

, Ge1−e3 =

a f

bc

.

Recall that for H diagonalizable, we saw NG(H)0 = ZG(H)0 and NG(H)/ZG(H) isfinite. Now if T is a maximal torus in G, we define

W = W (G,T ) := NG(T )/ZG(T ),

which is a finite group, called the Weyl group of G relative to T . W acts faithfully asa group of automorphisms of X = X×(T ), which is a free abelian group of finite rank.We identify W with this subgroup of automorphisms of X. Let P be as before andP ′ = α ∈ P : Gα is non-solvable. Note that for S ⊂ T a subtorus, the Weyl groupW (ZG(S), T ) is a subgroup of W (G,T ). Also note that if S ⊂ C is a subset of the centerof G, then G 7→ G/S induces an isomorphism W (G,T ) ∼= W (G/S, T/S). Recall that wehad a bijection

NG(T )/ZG(T )→ Borel subgroups containing Tx 7→ xBx−1.

This implies that for fixed B, we have bijections

W ∼= Borel subgroups of G containing T ∼= set of fixed points of T in G/B(bylefttranslation).

Fix α ∈ P ′. Then the torus S = (kerα)0 ⊂ C(Gα) and the Weyl group Wα :=W (Gα, T ) ∼= W (Gα/S, T/S). Now T/S ∼= Gm, this implies that Wα has order ≤ 2(because Wα can be identified with a subgroup of Aut(Gm) = ±1.

Proposition 13.7. Assume G is non-solvable and dimT = 1. Then(i) W has order 2.(ii) If B is a Borel in G, then dimG/B = 1.

74

Example 13.8. G = SL2, T =

(a

a−1

), NG(T ) =

(a

a−1

),

(b

−b−1

).

Then

Wα = ±1, B =

(a b

a−1

), dimG/B = 1.

Fix α ∈ P ′. By Proposition 13.7 (i), Wα has order 2, so we can choose nα ∈NG(T )\ZG(T ). Let Sα = (nα (mod ZG(T ))) ∈ W . Recall that X = X∗(T ), so wecan identify it as a subgroup of V = R ⊗Z X. Similarly, X∨ = Hom(X,Z) viewed as agruop of cocharacters of T , so we can identify it with a subgroup of V ∨ = R⊗ZX

∨. Thenwe obtain the induced pairing 〈 , 〉 between V and V ∨. Let ( , ) be a positive definitesymmetric bilinear form on V , invariant under the action of W (such form always exist, forexample, via averaging over W ). Then sα, α ∈ P ′, is a Euclidean reflection with respect

to the metric defined by ( , ) and sα(x) = x− 2(x,α)(α,α) α.

Lemma 13.9. (i) There exists a unique α∨ ∈ V ∨ with 〈α, α∨〉 = 2 such that

sα(x) = x− 〈x, α∨〉α,∀x ∈ X.

(ii) If β ∈ P ′ and Gβ = Gα, then sβ = sα.

Theorem 13.10. W is generated by the sα, α ∈ P ′.

Theorem 13.10 can be proved by using induction on dimG. We skip the proof.

Semisimple Groups of Rank One

We definerank G := dimT,

s.s.rank G := rank(G/R(G)).

Theorem 13.11. Assume that G is connected, semisimple of rank 1. Then G ∼= SL2 orPSL2.

Example 13.12. Consider G = SL2, T =

t =

(a

a−1

), B =

(a b

a−1

), U =(

1 b1

). Let n =

(1

−1

). Then

ntn−1 =

(1

−1

)(a

a−1

)(1

−1

)=

(a−1

a

)= t−1,

B =

(a bc d

)∈ SL2 : c = 0

, UnB =

(a bc d

)∈ SL2 : c 6= 0

, nUn−1 =

(1b 1

),

U ∩ nUn−1 =

(1

1

).

75

Reductive Groups of Semisimple Rank 1

Here is a general fact.

Proposition 13.13. Let G be a connected reductive linear algebraic group.(i) R(G) is a central torus. In fact, R(G) = C(G)0.(ii) R(G) ∩ (G,G) is finite.

Example 13.14. Consider G = GLn, then C(G) =

a

a. . .

a

is connected

and C(G) = R(G). (G,G) = SLn. R(G) ∩ (G,G) = µn.

Example 13.15. Consider G = (g1, g2) ∈ GL2×GL2 : det(g1) = det(g2). Then C(G) =((a

a

),

(b

b

)): a2 = b2, a 6= 0

is not connected, and C(G)0 =

((a

a

),

(a

a

)): a 6= 0

=

R(G). (G,G) = SL2 × SL2. The center of (G,G) is (±I2,±I2) ∼= Z/2Z × Z/2Z.R(G) ∩ (G,G) = (I, I), (−I,−I).

Now we assume that G is connected, reductive, and s.s. rank G = 1.Let C = R(G). Then G/C is s.s. of rank 1, so it is isomorphic to SL2 or PSL2. By

Proposition 13.13, (G,G) is connected s.s. of rank 1. Let T1 ⊂ (G,G) be a maximal toruscontained in T , so T1 ⊂ T ⊂ G. Then P = ±α and g = t⊕ gα⊕ g−α where both gα andg−α are 1-dimensional.

Lemma 13.16. (i) There exists a homomorphism of algebraic groups uα : Ga → G suchthat

tuα(x)t−1 = uα(α(t)x) ∀t ∈ T, x ∈ X and Im duα = gα.

If u′α is another homomorphism of this type, then u′α(x) = uα(a · x), a ∈ k×.(ii) T and Im uα generate a Borel in G with Lie algebra t⊕ gα.

Example 13.17. Consider G = GL2, α = e1 − e2, uα(x) =

(1 x0 1

)or u′α(x) =(

1 ax0 1

), a 6= 0. Let λ : Gm → T be an isomorphism viewed as an element of the

cocharacter group X∨. It turns out that

±〈α, λ〉 =

1 if (G,G) = PSL2

2 if (G,G) = SL2.

The Weyl group W ((G,G), T1) has order 2. Let n ∈ N(G,G)(T1) be a representative of thenontrivial element of W and let sα be the corresponding reflection of V = R⊗ZX. Let α∨

be the corresponding element of R⊗ZX∨ such that 〈α, α∨〉 = 2. Then sα(x) = x−〈x, α∨〉α

for all x ∈ X, α∨ ∈ X∨ and Im α∨ = T1, and n2 = α∨(−1).

76

Example 13.18. ConsiderG = GL2, (G,G) = SL2, T1 =

(a

a−1

)⊂ T =

(a

b

),

α = e1 − e2. Then

α∨(λ) =

(λ

λ−1

), n = nα =

(1

−1

), n2 =

(−1

−1

)= α∨(−1).

Let B be the Borel of G containing T whose Lie algebra is t⊕gα as before. Let χ ∈ X.Consider χ : B → Gm via the composition

B → B/Bu → Tχ→ Gm.

Proposition 13.19. Let f ∈ k[G] be a regular function on G whose restriction to (G,G)is non-constant. Assume that for g ∈ G, b ∈ B, we have f(gb) = χ(b)f(g). Then〈χ, α∨〉 > 0.

Root Data

Definition 13.20. A root datum is a quadruple Ψ = (X,R,X∨, R∨) where X,X∨ arefree abelian groups of finite rank, in duality by a pairing 〈 , 〉 : X ×X∨ → Z and R andR∨ are finite subsets of X and X∨ respectively, and we are given a bijection α 7→ α∨ of Ronto R∨, and with

sα(x) := x− 〈x, α∨〉α, ∀x ∈ X,

sα∨(y) := y − 〈α, y〉α∨, ∀y ∈ X∨.

We have:

(RD1) :α ∈ R⇒ 〈α, α∨〉 = 2,

(RD2) :α ∈ R, sαR = R, sα∨R∨ = R∨.

R is called the set of roots. R∨ is called the set of coroots. W = W (Ψ) := 〈sα : α ∈ R〉 isthe Weyl group.

Remark 13.21. Oberve that:(i) s2

α = 1 and sα(α) = −α.(ii) Ψ∨ = (X∨, R∨, X,R) is also a root datum, called the dual root datum.(iii) Let Q be the subgroup of X generated by R and put V ′ = R⊗Z Q. If R 6= ∅, then Ris a root system in V ′. Similarly, R∨ is a root system in the dual of V ′.

77

Root Datum of Linear Algebraic Groups

Let G be a connected linear algebraic group. Consider β ∈ P ′ and the group Gβ =ZG((kerβ)0). Then H = Gβ/Ru(Gβ) is connected of s.s. rank 1. By earlier resultstoday, there are two nontrivial characters ±α′ of the image of T in H, and their imageis isomorphic to T , so we have two corresponding characters ±α of T . Now we have(kerα)0 = (kerβ)0, so α is a rational multiple of β, so α ∈ P ′. α : β ∈ P ′ are calledroots of G relative to T , or roots of (G,T ).

Notation: let R = R(G,T ) be the set of roots. R = ∅ ⇐⇒ G is solvable.Next, we have a map α 7→ α∨ of R onto a subset R′ of X∨, which turns out to be

bijective. The elements of R∨ are the coroots of G relative to T . These are the ingredientsof a root datum and (RD1), (RD2) hold. So if G is a connected linear algebraic groupwith T a maximal torus in G, then we get Ψ = Ψ(G,T ) a root datum. Since maximal toriare conjugate, G determines Ψ up to isomorphism and also the root system R = R(G,T )up to isomorphism. Also, if c ∈ Q and cα ∈ R, then Gα = Gcα and Gα determines a pairof roots ±α uniquely. So c = ±1, i.e., R(G,T ) is a reduced root system.

78

14 More on Roots, and Reductive Groups (11/18)

Recall that last time we discussed that if G is a connected linear algebraic group, T amaximal torus, then we get Ψ = Ψ(G,T ) = (X,R,X∨, R∨) a root datum, where (X,R)and (X∨, R∨) turned out to be reduced root systems.

Positive Roots

Let (X,R,X∨, R∨) be a root datum with Weyl group W and fix a W -invariant positivedefinite symmetric bilinear form on V = R⊗Z X.

Definition 14.1. A subset R+ of R is called a system of positive roots if there existsx ∈ V with (α, x) 6= 0 for all α ∈ R such that R+ = α ∈ R : (α, x) > 0. Equivalently, ifthere exists λ ∈ X∨ with 〈α, λ〉 6= 0,∀α ∈ R such that

R+ = α ∈ R : 〈α, λ〉 > 0.

Observe that convex hull of R+ in V does not contain 0; R = R+ ∪ (−R+), so ifα, β ∈ R+ and α+ β ∈ R then α+ β ∈ R+; (R+)∨ is a system of positive roots in R∨.

Positive Roots and Borels

Recall that if B is a Borel containing R and α ∈ R(G,T ), then Gα ∩ B is a Borel in Gα.(Recall Gα := ZG((kerα)0). Then B′ = Gα ∩ B/Ru(Gα) ∩ B is a Borel subgroup of thereductive group G′ = Gα/Ru(Gα) containing T ′ = ImT . Let ±α′ be the characters of T ′

corresponding to ±α. As we saw

B(B′) = L(T ′)⊕ (1-dimensional weight space)

where the last one is either weight space of α′ or weight space of −α′. Hence, B picks outone root out of each pair ±α. Set

R+(B) = roots obtained this way as α ranges through R(G,T ).

Proposition 14.2. R+(B) is a system of positive roots.

Example 14.3. G = GL3, B=upper triangulars, R+(B) = e1 − e2, e1 − e3, e2 − e3.

Example 14.4. G = GL3, B=lower triangulars, R+(B) = −e1 + e2,−e1 + e3,−e2 + e3.

79

Unipotent Radical

Theorem 14.5. Let G be a connected linear algebraic group and T a maximal torus. ThenRu(G) = C where

C =

( ⋂B⊃T Borel

Bu

)0

.

Corollary 14.6. Assume G is reductive, then(i) if S is a subtorus of G, then ZG(S) is connected, reductive;(ii) ZG(T ) = T , i.e., Cartans are maximal tori in reductive linear algebraic group;(iii) the center C(G) ⊂ T .

Example 14.7. G =

a1 ∗ ∗

. . . ∗0 an

∈ GLn

, then B = G is a Borel in G. But all

Borels are conjugate, so this is the only Borel. T =

a1 0

. . .

0 an

∈ GLn

, Bu =

1 ∗ ∗

. . . ∗0 1

. So

Ru(G) =

1 ∗ ∗

. . . ∗0 1

.

Example 14.8. G = GLn, T =

a1 0

. . .

0 an

∈ GLn

, B =

a1 ∗

. . .

0 an

∈ GLn

is a Borel, Bu =

1 ∗

. . .

0 1

∈ GLn

. B′ =

a1 0

. . .

∗ an

∈ GLn

is also a Borel,

B′u =

1 0

. . .

∗ 1

∈ GLn

. There are other Borels. So Ru(G) = 1. So GLn is re-

ductive.

Example 14.9. Let G =

a b

0 dA

:

(a b0 d

), A ∈ GL2

, T =

a

dp

q

.

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ThenB =

a b

dp x

q

is a Borel, withBu =

1 b1

1 x1

. B′ =

a b

dpy q

is also a Borel, with B′u =

1 b1

1y 1

. So

Ru(G) =

1 b1

11

,

therefore G is not reductive.

Review of Structure Theory of Reductive Groups

Let G be a connected, reductive, linear algebraic group, and let T be a maximal torus ofG. So we have a root datum (X,R,X∨, R∨) of (G,T ).

Proposition 14.10. (i) For α ∈ R, there is an isomorphism uα : Ga → Uα onto a closedsubgroup of G such that

tuα(x)t−1 = uα(α(t)x) ∀t ∈ T, x ∈ R.

Moreover, Imduα = gα where gα is the weight space for the α.(ii) T and the Uα, α ∈ R, generate G.

As a result, the roots in R are the nonzero weights of T in g. For each α ∈ R,dim gα = 1. If B is a Borel containing T , α ∈ R, then α ∈ R+(B) ⇐⇒ Uα ⊂ B ⇐⇒gα ⊂ b = Lie(B). dimB = dimT + 1

2 |R| and dimG = dimT + |R|.

Example 14.11. G = GL3, T =

t =

a1

a2

a3

, R = ±(e1 − e2),±(e1 −

e3),±(e2 − e3). Let α = e1 − e2, then uα(x) =

1 x1

1

,

tuα(x)t−1 =

a1

a2

a3

1 x1

1

a−11

a−12

a−13

=

1 a1a2x

11

= uα(a1

a2x) = uα(α(t)x),

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ge1−e2 =

1 x

11

.

Take B =

a ∗ ∗

b ∗c

. R+(B) = e1 − e2, e1 − e3, e2 − e3. |R| = 6, dim(T ) = 3,

dim(B) = 6. So dim(G) = 3 + 6 = 9.

Example 14.12. G = Sp4 =g ∈ GL4 : tgJg = J

where J =

1

1−1

−1

, and

T =

t =

a1

a2

a−12

a−11

.

R = ±(e1±e2),±2e1,±2e2. Ue1−e2(x) =

t =

1 x

11 −x

1

, U2e2(x) =

t =

1

1 x1

1

.

If B =

a1 ∗ ∗ ∗

a2 ∗ ∗a−1

2 ∗a−1

1

, then R+(B) = e1 − e2, 2e1, e1 + e2, 2e2, dim(B) =

2 + 4 = 6, dim(B) = 2 + 8 = 10.

The Weyl group of (G,T ) is W := NG(T )\T . We identify it with the Weyl group ofthe root datum of (G,T ). For α ∈ R, sα ∈W and sα = s−α, we have the following:

• We may choose uα such that for all α ∈ R, we have nα = uα(1)u−α(1)uα(1) ∈ NG(T )and its image is sα ∈W . For x ∈ k×, uα(x)u−α(−x−1)uα(x) = α∨(x)nα.

• n2α = α∨(−1) and n−α = n−1

α .

• For u ∈ Uα − 1, there exists a unique u′ ∈ U−α − 1 such that uu′u ∈ NG(T ).

• If (u′α)α∈R is another such family, then u′α(x) = uα(cαx) with cαc−α = 1 for allα ∈ R, x ∈ k.

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Example 14.13. G = SL2, R = ±(e1 − e2), T =

t =

(a

a−1

). ue1−e2(x) =(

1 x1

), u−(e1−e2)(x) =

(1x 1

). Let α = e1 − e2, then

uα(x)u−α(−x−1)uα(x) =

(1 x

1

)(1−x−1 1

)(1 x

1

)=

(x

−x−1

)=

(x

x−1

)(1

−1

)

where α∨(x) =

(x

−x−1

), nα =

(1

−1

), and

n2α =

(1

−1

)2

=

(−1

−1

)= α∨(−1).

Definition 14.14. We call a family (uα)α∈R with these properties a realization of theroot system R in G.

Theorem 14.15. Assume that G is semisimple.(i) The Uα, α ∈ R, generate G.(ii) G = (G,G).(iii) Let G1 6= G be a nontrivial connected closed normal subgroup of G. Then G1 issemisimple, and there is a similar group G2 such that (G1, G2) = e, G1 ∩G2 finite andG = G1G2.(iv) The number of minimal non-trivial connected, closed normal subgroups of G is finite,say G1, · · · , Gr. Then (Gi, Gj) = e if i 6= j, and Gi ∩

∏j 6=iGj is finite and G =

G1G2 · · ·Gr and the Gi’s have no closed, normal subgroups of dimension > 0.

If G is reductive, then(i) G = R(G) · (G,G).(ii) (G,G) is semisimple.

Example 14.16. G = GLn, (G,G) = SLn, and R(G) =

a

a. . .

a

.

Recall that given (G,T ), we have the root datum (X,R,X∨, R∨).Notation: For A a subgroup of X, we denote

A⊥ := y ∈ X∨ : 〈A, y〉 = 0, annihilator of A,

A := x ∈ X : Z · x ∩A 6= 0, rational closure of A.

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A/A is the torsion subgroup of X/A. We have similar notions for X∨. It’s easy to checkthat A = (A⊥)⊥.

We define Q to be the subgroup of X generated by R, and Q∨ to be the subgroup ofX∨ generated by R∨.

Here are some facts:

• C(G) = ∩α∈R kerα.

• R(G) = the subgroup of G generated by Imy, y ∈ Q∨ ⊂ T and

X∗(R(G)) ∼= X\Q,

X∗(R(G)) ∼= Q⊥.

• The subtorus T1 = 〈Imα∨ : α ∈ R〉 ⊂ T is a maximal subtorus of (G,G). Also

X∗(T1) ∼= X/(Q∨)⊥,

X∗(T1) ∼= Q⊥.

• The root datum of ((G,G), T1) is (X/(Q∨)⊥, R, Q⊥, R∨).

• Assume G is semisimple, then Q∨ = 0, X = Q. Q has finite index in X. Thefinite group C∗ = C∗(G) := X/Q is called the cocenter of G. Define P := x ∈ V =R ⊗Z X : 〈x,R∨〉 ⊂ Z. Then P is a lattice in V and Q ⊂ P . Given a root systemR in V , there are finitely many possibilities for x. Q is the root lattice of R, and Pis the weight lattice of R. For G semisimple as above, G is called adjoint if X = Q.G is called simply-connected if X = P . The finite abelian group P/Q is called thefundamental group of R.

Borels and Positive Systems

Fix a Borel B ⊃ T in G. Write U : Bu the unipotent radical of B, R+ = R+(B) positivesystem of roots determined by B. Fix a total order on R.

Bu is generated by groups Uα, α ∈ R+.For α, β ∈ R, α = ±β, there exist “structure constants” cα,β;i,j ∈ R such that

(uα(x), uβ(y)) =∏

iα+jβ∈Ri,j>0

uiα+jβ(cα,β;i,jxiyj) ∀x, y ∈ R

(order according to the fixed total ordering).

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Let R+ be an arbitrary system of positive roots. Then(i) T and the Uα, α ∈ R+ generate a Borel in G.(ii) there exists a unique w ∈W with R+ = w ·R+.

Let W be the set of Borel subgroups of G containing T . We saw that W acts simplytransitively onW. Two Borels B,B′ ⊂ W are called adjacent if dim(B∩B′) = dimB−1 =dimB′ − 1.

We say two systems of positive roots R+ and R+ are adjacent if

|R+ ∩ R+| = |R+| − 1 = |R+| − 1.

We have:(i) B,B′ ∈ W ⇒ there exists a family B = B0, B1, B2, · · · , Bn = B′ in W such that Biand Bi+1 are adjacent.(ii) If R+ and R+ are two systems of positive roots, then there exists a family of systemof positive roots R+ = R+

0 , R+1 , · · · , R+

n = R+ such that R+i and R+

i+1 are adjacent.

(iii) If R+ and R+ are adjacent, then there exists a unique α ∈ R+ such that R+ = sαR+.

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15 Bruhat Decomposition, Parabolic Subgroups, the Iso-morphism Theorem, and the Existence Theorem (12/2)

Recall that the pair of a reductive group and a maximal torus (G,T ) determines a rootdatum Ψ = Ψ(G,T ) = (X,R,X∨, R∨). Given α ∈ R, we have uα : Ga → Uα satisfying

tuα(x)t−1 = uα(α(t)x), ∀x ∈ k, t ∈ T,

and gα = Imduα. We know that T and Uα generate G. Let W = NG(T )/T , identifiedwith W (R(G,T )). Take B ⊃ T a Borel in G, then we have R+(B) a system of positiveroots. Then R = R+(B) ∪ (−R+(B)).

Since G is reductive, G = R(G) · (G,G) where R(G) is a central torus, (G,G) issemisimple, generated by the Uα’s. Last time, we described the root data for both R(G)and (G,G). We continue the study on Borel subgroups.

Let R+ be a system of positive roots. Let

D = D(R+) := α ∈ R+ : sα ·R+ and R+ are adjacent.

If R+ = R+(B) with R ⊃ T , we also write D = D(B). We call D the basis of R definedby R+. Its elements are called the simple roots in R+.

Let S = S(R+) = S(B) := sα : α ∈ D be simple reflections defined by B or R+. Wehave

• D(w ·R+) = wD(R+).

• S(w ·R+) = wS(R+)w−1.

• For α ∈ D, sα permutes the elements of R+ − 0.

• For α, β ∈ D, α 6= β, 〈α, β〉 ≤ 0.

Theorem 15.1. (i) S generate W .(ii) R = W ·D.(iii) The roots in D are linearly independent. A root in R+ is a linear combination∑

α∈D nαα, nα ∈ Z≥0. These two properties characterize the subsets D of R+.

Corollary 15.2. G is generated by T and the groups U±α, α ∈ D.

Bruhat Decomposition

For w ∈W , defineR(w) := α ∈ R+ : wα ∈ −R+.

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For example, for α ∈ D, R(sα) = α, we have

R(wsα) =

sαR(w) ∪ α if wα ∈ R+,

sα(R(w) ∪ α) if wα ∈ −R+.

Define l(w) := smallest integer h ≥ 0 such that w is a product of heights in S. A re-duced decomposition of w is a sequence s = (s1, · · · , sh) in S with w = s1s2 · · · sh andh = l(w). Note that l(w) = l(w−1).

Let (w)w∈W be a set of representatives in NG(T ) of elements in W . Then C(w) = BwBis the Bruhat cell, which is a locally closed subvariety of G.

Theorem 15.3 (Bruhat Decomposition).

G =∐w∈W

C(w).

In other words, an element of G can be written uniquely as uwb, w ∈W , u ∈ Uw−1 =∏α∈R(w−1) Uα.

Corollary 15.4. The intersection of two Borels of G contains a maximal torus.

Proof. Let B and B′ = gBg−1 be two Borels. Write g = bwb′, then bTb−1 ∈ B ∩B′.

Corollary 15.5. There is a unique open double coset, namely C(w0), where w0 is thelongest element of W .

Example 15.6. G = GL2, T =

(a

b

), B =

(a ∗

b

), W =

1,

(1

−1

).

Parabolic Subgroups

Let G,T,B be as before. Let

I ⊂ D = simple roots

WI : = subgroup of W generated by reflections sα, α ∈ I;

RI : = set of roots in R that are linear combinations of the roots in I;

SI : =

(⋂α∈I

kerα

)0

, LI := ZG(SI).

We saw that LI is a connected reductive subgroup of G with maximal torus T and Borelsubgroup BI = B ∩ LI .

Here are some facts:

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• The root system R(LI , T ) = RI , and W (LI , T ) = WI .

• R+I (BI) = R+

I = R+ ∩RI and the corresponding simple roots in I.

• PI :=⋃w∈WI

C(w) is a parabolic subgroup of G containing B and LI .

• Ru(PI) is generated by the Uα, α ∈ R+ −RI .

• The product map LI ×Ru(PI)→ PI is an isomorphism of varieties.

• If P is a parabolic subgroup of G containing B, then there exists a unique subset Iof D such that P = PI .

Definition 15.7. Let P be a parabolic subgroup of G. A Levi subgroup of P is a closedsubgroup of P such that the product map L×Ru(P )→ P is bijective.

Note that a maximal torus of P lies in a unique Levi subgroup.

Example 15.8. G = GLn, T =

a1 0

. . .

0 an

∈ GLn

, B =

a1 ∗

. . .

0 an

∈ GLn

.

The root is of type An−1:

R(G,T ) = ±(ei − ej), 1 ≤ i, j ≤ n, i 6= j,

R+(B) = (ei − ej), i < j,

D(B) = e1 − e2, e2 − e3, · · · , en−1 − en.

Dynkin Diagrams

Let R be a root system in the Euclidean space V . Fix a metric ( , ) in V that is W (R)-

invariant such that 〈α, β∨〉 = 2(α,β)(β,β) as before. Let D be a basis of R.

Definition 15.9. The Dynkin diagram D defined by D is a graph with vertex set D andtwo vertices α, β joined by 〈α, β∨〉〈β, α∨〉 bonds. When α and β have different lengths,we add an arrow pointing towards the shortest root.

Observe that if α ⊥ β, then 〈α, β∨〉 = 0, so no bonds (no edges between them).

The Cartan matrix is(〈αi, α∨j

).

If R is an irreducible root system, then it’s easy to check that:

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• D is a connected graph, in fact, a tree.

• A multiple bond is either double or triple.

• Triple bond only occurs in the case .

• At most one double bond can occur and if it does, D is a chain.

• If multiple bonds occur, we have only two root lengths (long and short). In fact,here is a classification of irreducible Dynkin diagrams.

ABC

D

E6

E7

E8

F4

G2

Recall that given an irreducible root system R in the Euclidean space V , we defineP = x ∈ V : 〈x,R∨〉 ⊂ Z and Q ⊂ P by Q = subgroup in V generated by R. They areboth free Z-modules. When R = R(G,T ), then Q ⊂ X = X∗(T ) ⊂ P .

Example 15.10. One-dimensional irreducible root systems:

A1 : . α = e1− e2, −α = e2− e1, α∨ = e∗1− e∗2, and 〈α, α∨〉 = (1)(1) + (−1)(−1) = 2.V = (x, y) ∈ R2 : x+y = 0. The Cartan matrix is (2). P = Z· e1−e22 , and Q = Z·(e1−e2).We have two possibilities: if X = P , then we get simplely-connected G = SL2; if X = Q,then we get an adjoint group G = PSL2 or PGL2.

Example 15.11. Two-dimensional irreducible root systems: (two-dimensional reducible:A1 ×A1 = D2)

(a) A2 : . α = e1−e2, β = e2−e3, V = (x, y, z) ∈ R3 : x+y+z = 0. The Cartan

matrix is

(2 −1−1 2

). We have P = Z〈13α+ 2

3β,23α+ 1

3β〉, and Q = Z〈e1−e2, e2−e3〉 . The

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index is [P : Q] = 2. We have two possibilities: if X = P , then we get simplely-connectedG = SL3; if X = Q, then we get an adjoint group G = PSL3 or PGL3.

(b) B2 : . α = e1−e2, β = e2, α∨ = e∗1−e∗2, β∨ = 2e∗2, V = R2. The Cartan matrix

is

(2 −2−1 2

). We have P = (m + n

2 ,n2 ),m, n ∈ Z = Z〈e1,

e1+e22 〉, and Q = Z〈e1, e2〉.

If X = P , then we get simplely-connected G = Sp4∼= Spin5; if X = Q, then we get an

adjoint group G = SO5∼= PSp4

∼= PGSp4.(c) G2 : . α = e1 − e2, β = −2e1 + e2 + e3. V = (x, y, z) : x+ y + z = 0, and the

Cartan matrix is

(2 −1−3 2

). X = P = Q. So we get simply-connected and adjoint G2.

Here are identical isomorphisms:

A1 = B1 = C1,

SL2∼= Spin3

∼= Sp2,

PSL2∼= SO3

∼= PSp2,

B2∼= C2,

Spin5∼= Sp4,

SO5∼= PSp4.

The Isomorphism Theorem

Definition 15.12. Let G and G1 be two connected, reductive algebraic groups over k,with maximal torus T and T1 respectively, and root data Ψ = (X,R,X∨, R∨) and Ψ1 =(X1, R1, X

∨1 , R

∨1 ). An isogeny ϕ : G → G1 is a surjective homomorphism of algebraic

groups with finite kernel.

We have the following:

• kerϕ is a central subgroup of G lying in T .

• dimG = dimG1.

• Assume ϕ(T ) = T1. Then ϕ defines homomorphisms f = f(ϕ) : X1 → X andf∨ = f∨(ϕ) : X∨ → X∨1 such that 〈x1, f

∨(λ)〉 = 〈f(x1), λ〉 for all x1 ∈ X1, λ ∈ X∨.

• Also, there is a bijection b : R→ R1 with ϕUα = Rbα for all α ∈ R.

• If ϕ is an isomorphism of algebraic groups, then f(bα) = α for all α ∈ R and fdefines an isomorphism of root data Ψ1 → Ψ, i.e., isomorphisms X1 → X mappingR1 onto R such that its dual maps R∨ onto R∨1 .

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Theorem 15.13 (Isomorphism Theorem). Let f : Ψ1 → Ψ be an isomorphism of rootdata. Then there is an isomorphism of algebraic groups ϕ : G → G1 with ϕ(T ) = T1 andf = f(ϕ). If ϕ′ is another isomorphism, then there exists t ∈ T such that ϕ′(t) = ϕ(tgt−1).

Let ϕ be an arbitrary isogeny. Then f is an isomorphism of X1 onto a subgroup of Xof finite index and we have

f(bα) = q(α)α,

f∨(α∨) = q(α)(bα)∨,

for some q(α) =

1 if char k = 0

some power of p if char k = p.

If q(α) ≡ 1, then we say ϕ is a central isogeny. We say a triple µ = (f, b, q) withf : X1 a subgroup of X of finite index, b : R → R1 bijective, q : R → pnn>0, definesa p-morphism of ϕ1 to ϕ if the above properties hold.

Theorem 15.14. Let µ = (f, b, q) be a p-morphism of ϕ1 to ϕ. There is an isogenyϕ : G → G1 with ϕT = T1 and µ = µ(ϕ). If ϕ′ is another such isogeny with thesesproperties, then there exists t ∈ T such that ϕ′(g) = ϕ(tgt−1).

The Existence Theorem

Theorem 15.15. Let Ψ = (X,R,X∨, R∨) be a root datum. There exists a connected,reductive, linear algebraic group G over k with a maximal torus T , such that the rootdatum Ψ(G,T ) is isomorphic to Ψ.

We outline the ideas of proof. Step (a): Reduce to the case that R spans X andis irreducible. Then the group is quasi-simple (i.e., no proper, connected, closed normalsubgroup) and adjoint. Step (b): In that case, G is constructed as a group of automor-phisms of its Lie algebra in the case R is simply-faced. In this case, one constructs the Liealgebra first, which requires explicit description of the structure of constants. Step (c):For arbitrary (non-simply-faced) root system R, the construction of G is reduced to thesimply-faced case via “automorphisms folding”.

Example 15.16. Consider Ψn = (X,R,X∨, R∨), X = Ze0 ⊕ Ze1 ⊕ · · · ⊕ Zen, X∨ =Ze∗0 ⊕ Ze∗1 ⊕ · · · ⊕ Ze∗n.

R =

±(ei − ej),±(ei + ej), 1 ≤ i < j ≤ n,±ei 1 ≤ i ≤ n,

R∨ =

±(e∗i − e∗j ),±(e∗i + e∗j − e∗0), 1 ≤ i < j ≤ n,±2e∗i − e∗0 1 ≤ i ≤ n.

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〈 , 〉 : X ×X∨ → Z is the standard pairing. It’s easy to check that this is a root datum.(X,R) is a root system of type Bn, and (X∨, R∨) is a root system of type Cn. Choose aBorel B such that the simple roots are D = e1 − e2, e2 − e3, · · · , en−1 − en, en. Then

D∨ = e∗1 − e∗2, e∗2 − e∗3, · · · , e∗n−1 − e∗n, e∗n.

The group determined by Ψn is usually called GSpin2n+1.When n = 2, one can construct the Uα’s abstractly. The Uα’s corresponding to positive

roots are Ue1−e2 , Ue1+e2 , Ue1 , Ue2 , and they generate Ru(B). The Uα’s corresponding tonegative roots are U−e1+e2 , U−e1−e2 , U−e1 , U−e2 , and they generate Ru(B). To generateT , choose e∗o(x) and γ∨(x) where γ = α, α + β, α + 2β. This group (n = 2) is calledGSpin5, where root system is of type B2 and coroot system is of type C2. But the tworoot systems are isomorphic, so Ψ∨2 = (X∨, R∨, x, r) is a root datum isomorphic to Ψ2. Sowhen n = 2, GSpin5

∼= GSp4.

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