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Linear Algebraic Groups: a Crash Course

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Linear Algebraic Groups: a Crash Course Linear Algebraic Groups: a Crash Course Dave Anderson January 24, 2011 This is a collection of notes for three lectures, designed to introduce linear algebraic groups quickly in a course on Geometric Invariant Theory. There are several good introductory textbooks; in particular, the books by Humphreys [H], Springer [S], and Borel [B]. Here I merely distill some of the material from Humphreys and Springer. 1 Definitions We’ll work over a fixed algebraically closed base field k. Definition 1.1 An algebraic group G is a group object in the category of varieties over k. That is, G is a group and a variety, and the maps G × G → G and G → G (g, h) 7→ gh g 7→ g−1 are morphisms of varieties. (And there is a distinguished k-point e ∈ G, the identity.) A homomorphism of algebraic groups is a group homomorphism that is also a map of varieties. In schemey language, another way to say this is that the functor hG : Schemes → Sets factors through Groups. Definition 1.2 A linear algebraic group is an affine variety that is an algebraic group. ∗ −1 Example 1.3 The multiplicative group Gm = k = Spec k[x,x ] is an algebraic group. 1 The coordinate ring k[G] of a linear algebraic group is a (commutative) Hopf algebra: it comes with maps δ : k[G] → k[G] ⊗ k[G]= k[G × G] (comultiplication), c : k[G] → k[G](antipode), ǫ : k[G] → k(counit), corresponding to the multiplication, inverse, and unit maps, respectively. −1 Example 1.4 For Gm, we have δ(x)= x ⊗ x, c(x)= x , and ǫ(x) = 1. Exercise 1.5 Work out the maps for the additive group Ga = Spec k[x]. A pleasant feature of the theory is that the most important examples (for now) are also the most familiar ones. Example 1.6 The general linear group is a LAG, with GLn = Spec k[xij]det , as is any Zariski-closed subgroup of GLn. Example 1.7 Particular examples of closed subgroups that come up: B ⊂ GLn, upper-triangular matrices (“Borel”). U ⊂ B ⊂ GLn, strictly upper-triangular matrices, with 1’s on diagonal (unipotent). D ⊂ B ⊂ GLn, diagonal matrices (maximal torus). In fact, although we’ve defined “linear” to mean “affine”, it turns out that all such groups are closed subgroups of GLn. (This justifies the termi- nology.) Proposition 1.8 Every linear algebraic group can be embedded as a closed subgroup in some GLn. To prove this, we’ll need a couple more basic notions. Definition 1.9 A (rational) representation of G on a k-vector space V is a homomorphism G → GL(V ). A representation is irreducible if there is no nontrivial proper G-stable subspace; that is, no W such that 0 6= W ( V with G · W ⊆ W . 2 One can talk about representations for infinite-dimensional V , but we’ll always assume they’re locally finite: for all v ∈ W , there is a G-stable, finite-dimensional subspace W with v ∈ W ⊆ V . Example 1.10 The main (and for us, essentially only) example of this is the action of G on k[G]. (Or k[X], when G acts on a variety X.) Here, for f ∈ k[G], we have g · f defined by (g · f)(x)= f(g−1x) for all x ∈ G. This is sometimes called the action by left translation on functions, and it works whenever G acts (on the left) on a variety. There is also an action by right translation which is sometimes useful. (But note that it is still a left action!) Denoting this by rg, we have rg · f given by (rg · f)(x)= f(xg). The key thing here is that k[G] is locally finite: Lemma 1.11 If V ⊂ k[G] is a finite-dimensional subspace, then there is a finite-dimensional G-stable subspace W with V ⊆ W ⊆ k[G]. (In particular, k[G] is locally finite.) Proof. It clearly suffices to treat the case where V is one-dimensional, say spanned by f. Write δ(f)= mi ⊗ fi Xi e in k[G × G], corresponding to the map (g, h) 7→ g−1h. Only finitely many terms appear, say i = 1,...,n. Then −1 (g · f)(x)= f(g x)= mi(g)fi(x), Xi so g · f = mi(g)fi Xi lies in the span of f1,...,fn. Therefore the space W spanned by {g · f | g ∈ G}, which is manifestly G-stable, is also finite-dimensional. We now prove the Proposition. Proof. Take generators f1,...,fn for k[G]. By the Lemma, we may assume →֒ they’re a basis for a G-stable subspace. We’ll produce an embedding G GLn. 3 In fact, we have a map k[GLn]= k[xij]det → k[G], as follows. Consider the right translation action. As in the lemma, there are elements mij ∈ k[G] with rg · fi = fj mij(g). Xj Define the map by xij 7→ mij. Since fi(g)= fi(eg)= fj(e)mij(g), Xj we see that fi = j fj(e)mij , and therefore the mij also generate k[G]. It follows that the mapP we defined is surjective, so it corresponds to a closed embedding of varieties. This is all good culture, but many of the groups you encounter come automatically linearized. A major example is that of diagonalizable groups. 2 Diagonalizable groups and characters The group of diagonal matrices Dn ⊂ GLn is special in several ways. First, observe that ∼ ±1 ±1 ∼ n k[Dn] = k[x1 , · · · ,xn ] = k[Z ]. Definition 2.1 A character of an algebraic group G is a homomorphism ∗ χ : G → Gm = k . The set of all characters forms an abelian group under pointwise multiplication, the character group of G, denoted X(G)= Homalg. gp.(G, Gm). (Warning: the group operation in X(G) is often written additively, so you may see the character g 7→ χ1(g)χ2(g) written as either χ1 · χ2 or χ1 + χ2.) Example 2.2 For Gm, we have X(Gm) = Z canonically (up to choice of generator 1 ∈ Z), by sending the identity in X(G) = Homalg. gp.(Gm, Gm) to 1 ∈ Z. The character χ : z 7→ zn then corresponds to the integer n. n From the example, we see X(Dn) =∼ Z , and k[Dn] =∼ k[X(Dn)]. In other words, the group of diagonal matrices has lots of characters, enough to form a linear basis for all functions. Contrast with this with the case of a simple group like P GLn, which has no nontrivial characters (since ker(χ) would be a nontrivial normal subgroup). 4 Definition 2.3 A linear algebraic group is diagonalizable if it is isomor- phic to a closed subgroup of some Dn. A connected diagonalizable group is called a torus. The key fact about diagonalizable groups is the following structure the- orem: Proposition 2.4 For a linear algebraic group D, the following are equiva- lent: (1). D is diagonalizable ∼ (2). X(D) is finitely generated, and k[D] = k[X(D)] := χ∈X(D) k · χ. L (3). Every rational representation of D is isomorphic to a direct sum of one-dimensional representations. (4). D is isomorphic to (k∗)r × A, for some finite abelian group A. Remark 2.5 In (3), the claim is that a representation V breaks up as V = χ Vχ, where Vχ = {v ∈ V | g · v = χ(g)v for all g ∈ D}. Characters with LVχ 6= 0 are called weights, and Vχ are called weight spaces. ∗ 2 Example 2.6 Take T = (k ) = D2 ⊂ GL2, acting on 2 × 2 matrices M2,2 by conjugation: −1 −1 a b z1 0 a b z1 0 a z1z2 b g· = = −1 . c d 0 z2 c d 0 z2 z1 z2c d The group D2 has a basis of characters χ1,χ2, with χi(g) = zi, and the four-dimensional vector space M2,2 breaks up as 0 1 0 0 ∗ 0 M = k · ⊕ k · ⊕ 2,2 0 0 1 0 0 ∗ −1 −1 weight χ1χ2 weight χ1 χ2 weight 0 | {z } | {z } | {z } Remark 2.7 Actually, there’s some delicacy about which finite abelian groups A can occur in (4). The condition is that A should have no p-torsion if char(k)= p. Example 2.8 The diagonalizable group D with character group X(D) = −1 2 Z ⊕ Z/2Z has k[D] =∼ k[x,x ,y]/(y − 1). So D =∼ Gm × µ2. Note that if char(k) = 2, this is a non-reduced group scheme (so not an algebraic group). 5 We now prove the proposition (see [S, §3]). Proof. The implications (2)⇒(4)⇒(1)⇒(2) are easily verified. We’ll show (2)⇒(3). Let V be a finite-dimensional (rational) representation of D, correspond- ing to a homomorphism ϕ : D → GL(V ). Choosing a basis for V , the map ϕ is given by ϕ(g) = (aij(g)), χ for some functions aij ∈ k[D]. By (2), we can write aij = χ cijχ (with finitely many nonzero terms). Grouping these by characters,Pwe can define χ matrices Aχ = (cij), and then we have ϕ(g)= χ(g)Aχ. Xχ It’s easy to see that the endomorphisms Aχ do not depend on the choice of basis. We claim that Aχ is actually the projection on the weight space Vχ. To see this, we first show that Aχ · Aψ = δχ,ψAχ. Using ϕ(gh)= ϕ(g)ϕ(h), we obtain η(gh)Aη = χ(g)ψ(h) B. Xη XB AχXA =B ψ We’ll write this entrywise, for ϕ(gh)= ϕ(g)ϕ(h), and using η(gh)= η(g)η(h): this is an equality of coefficients η χ,ψ cijη(g)η(h) = bij χ(g)ψ(h).
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