Lie Algebra of an Algebraic Group

1.4 Linear Algebraic Groups

1.4.3 Lie Algebra of an Algebraic Group

1.4 Linear Algebraic Groups 39

Using the identificationO[G×G] =O[G]⊗O[G], we can write (1.31) as µ^∗(f) =

∑

i

f_i⁰⊗f_i⁰⁰.

This shows thatµis a regular map. Furthermore, L^∗_g(f) =∑if_i⁰(g)f_i⁰⁰ and R^∗_g(f) =

∑if_i⁰⁰(g)f_i⁰, which proves thatL_gandR_gare regular maps. ut

Examples

1.LetD_nbe the subgroup of diagonal matrices inGL(n,C). The map (x₁, . . . ,x_n)7→diag[x₁, . . . ,x_n]

from(C^×)ⁿtoD_nis obviously an isomorphism of algebraic groups. SinceO[C^×] = C[x,x⁻¹]consists of the Laurent polynomials in one variable, it follows that

O[Dn]∼=C[x1,x⁻₁¹, . . . ,xn,x⁻_n¹]

is the algebra of the Laurent polynomials innvariables. We call an algebraic group Hthat is isomorphic toD_nanalgebraic torusofrank n.

2.LetN_n⁺⊂GL(n,C)be the subgroup of upper-triangular matrices with unit diag- onal. It is easy to show that the functionsx_{i j}fori>jandx_ii−1 generateI_Nn⁺, and that

O[N_n⁺]∼=C[x₁₂,x₁₃, . . . ,x_n−1,n]

is the algebra of polynomials in then(n−1)/2 variables{x_{i j}: i<j}.

Remark 1.4.5.In the examples of algebraic groupsGjust given, the determination of generators for the idealIGand the structure ofO[G]is straightforward because IGis generated by linear functions of the matrix entries. In general, it is a difficult problem to find generators forIGand to determine the structure of the algebraO[G].

lowing notion of a derivation (infinitesimal transformation) plays an important role in Lie theory.

Definition 1.4.6.LetAbe an algebra (not assumed to be associative) over a fieldF. Then Der(A)⊂End(A)is the set of all linear transformationsD:A //Athat satisfyD(ab) = (Da)b+a(Db)for alla,b∈A(callDaderivationofA).

We leave it as an exercise to verify that Der(A)is a Lie subalgebra of End(A), called the algebra ofderivationsofA.

We begin with the caseG=GL(n,C), which we view as a linear algebraic group with the algebra of regular functionsO[G] =C[x₁₁,x₁₂, . . . ,x_nn,det⁻¹]. Aregular vector fieldonGis a complex linear transformationX:O[G] //O[G]of the form

X f(g) =

∑

ⁿ

i,j=1

c_{i j}(g)∂f

∂x_{i j}(g) (1.32)

for f ∈O[G]andg∈G, where we assume that the coefficientsc_{i j}are inO[G]. In addition to being a linear transformation ofO[G], the operatorXsatisfies

X(f₁f₂)(g) = (X f₁)(g)f₂(g) +f₁(g)(X f₂)(g) (1.33) for f1,f2∈O[G]andg∈G, by the product rule for differentiation. Any linear trans- formationXofO[G]that satisfies (1.33) is called aderivationof the algebraO[G]. If X1andX2are derivations, then so is the linear transformation[X1,X2], and we write Der(O[G])for the Lie algebra of all derivations ofO[G].

We will show that every derivation ofO[G]is given by a regular vector field on G. For this purpose it is useful to consider equation (1.33) withgfixed. We say that a complex linear mapv:O[G] //Cis atangent vector to G at gif

v(f₁f₂) =v(f₁)f₂(g) +f₁(g)v(f₂). (1.34) The set of all tangent vectors atgis a vector subspace of the complex dual vector spaceO[G]^∗, since equation (1.34) is linear inv. We call this vector space thetangent spacetoGatg(in the sense of algebraic groups), and denote it byT(G)_g. For any A= [a_{i j}]∈M_n(C)we can define a tangent vectorv_Aatgby

v_A(f) =

∑

ⁿ

i,j=1

a_{i j} ∂f

∂x_{i j}(g) for f∈O[G]. (1.35) Lemma 1.4.7.Let G=GL(n,C)and let v∈T(G)_g. Set a_{i j}=v(x_{i j})and A= [a_{i j}]∈ M_n(C). Then v=v_A. Hence the map A7→v_Ais a linear isomorphism from M_n(C) to T(G)_g.

Proof. By (1.34) we havev(1) =v(1·1) =2v(1). Hencev(1) =0. In particular, if f =det^kfor some positive integerk, then

0=v(f·f⁻¹) =v(f)f(g)⁻¹+f(g)v(f⁻¹),

1.4 Linear Algebraic Groups 41

and so v(1/f) =−v(f)/f(g)². Hence vis uniquely determined by its restriction to the polynomial functions on G. Furthermore,v(f1f2) =0 whenever f1 and f2

are polynomials on Mn(C)with f1(g) =0 and f2(g) =0. Let f be a polynomial function onMn(C). Whenvis evaluated on the Taylor polynomial of f centered at g, one obtains zero for the constant term and for all terms of degree greater than one.

Alsov(xi j−x_{i j}(g)) =v(xi j). This implies thatv=v_A, wherea_{i j}=v(xi j). ut Corollary 1.4.8.(G=GL(n,C)) If X∈Der(O[G])then X is given by(1.32), where c_{i j}=X(xi j).

Proof. For fixedg∈G, the linear functional f 7→X f(g)is a tangent vector atg.

HenceX f(g) =v_A(f), wherea_{i j}=X(xi j)(g). Now definec_{i j}(g) =X(xi j)(g)for all g∈G. Thenc_{i j}∈O[G]by assumption, and equation (1.32) holds. ut We continue to study the groupG=GL(n,C)as an algebraic group. Just as in the Lie group case, we say that a regular vector fieldX onGisleft invariantif it commutes with the left translation operatorsL(y)for ally∈G(where now these operators are understood to act onO[G]).

LetA∈M_n(C). Define a derivationX_AofO[G]by XAf(u) = d

dtf u(I+tA) _t=0

foru∈Gand f ∈O[G], where the derivative is defined algebraically as usual for rational functions of the complex variablet. WhenA=ei jis an elementary matrix, we write X_{ei j} =E_{i j}, as in Section 1.3.7 (but now understood as acting onO[G]).

Then the mapA7→X_Ais complex linear, and whenA= [ai j]we have X_A=

∑

i,j

a_{i j}E_{i j}, with E_{i j}=

∑

ⁿ

k=1

x_ki ∂

∂x_{k j} ,

by the same proof as for (1.22). The commutator correspondence (1.25) holds as an equality of regular vector fields onGL(n,C)(with the same proof). Thus the map A7→X_Ais a complex Lie algebra isomorphism fromM_n(n,C)onto the Lie algebra of left-invariant regular vector fields onGL(n,C). Furthermore,

X_Af_B(u) = d

dttr u(I+tA)B

_t=0=tr(uAB) =f_AB(u) (1.36) for allA,B∈M_n(C), where the trace function f_Bis defined by (1.28).

Now letG⊂GL(n,C)be a linear algebraic group. We define its Lie algebragas a complex Lie subalgebra ofM_n(C)as follows: Recall thatIG⊂O[GL(n,C)]is the ideal of regular functions that vanish onG. Define

g={A∈M_n(C): X_Af ∈IG for all f ∈IG}. (1.37) WhenG=GL(n,C), we haveIG=0, sog=M_n(C)in this case, in agreement with the previous definition of Lie(G). An arbitrary algebraic subgroupGofGL(n,C)is

closed, and hence a Lie group by Theorem 1.3.11. After developing some algebraic tools, we shall show (in Section 1.4.4) thatg=Lie(G)is the same set of matrices, whether we considerGas an algebraic group or as a Lie group.

LetA∈g. Then the left-invariant vector fieldXAonGL(n,C)induces a linear transformation of the quotient algebraO[G] =O[GL(n,C)]/IG:

X_A(f+IG) =X_A(f) +IG

(sinceX_A(IG)⊂IG). For simplicity of notation we will also denote this transfor- mation by X_A when the domain is clear. Clearly X_A is a derivation of O[G] that commutes with left translations by elements ofG.

Proposition 1.4.9.Let G be an algebraic subgroup ofGL(n,C). Then gis a Lie subalgebra of M_n(C)(viewed as a Lie algebra overC). Furthermore, the map A7→

X_Ais an injective complex linear Lie algebra homomorphism fromgtoDer(O[G]).

Proof. Since the map A7→X_A is complex linear, it follows that A+λB∈g if A,B∈gandλ∈C. The differential operatorsX_AX_BandX_BX_AonO[GL(V)]leave the subspace IG invariant. Hence [X_A,XB] also leaves this space invariant. But [XA,XB] =X_[A,B]by (1.25), so we have[A,B]∈g.

SupposeA∈Lie(G)andX_Aacts by zero onO[G]. ThenX_Af|^G=0 for all f ∈ O[GL(n,C)]. Since I∈Gand X_A commutes with left translations by GL(n,C), it follows that X_Af =0 for all regular functions f onGL(n,C). HenceA=0 by

Corollary 1.4.8. ut

To calculategit is convenient to use the following property: IfG⊂GL(n,C) andA∈M_n(C), thenAis ingif and only if

X_Af|G=0 for all f ∈P(Mn(C))∩IG. (1.38) This is an easy consequence of the definition of gand (1.29), and we leave the proof as an exercise. Another basic relation between algebraic groups and their Lie algebras is the following:

IfG⊂Hare linear algebraic groups with Lie algebrasgandh,

respectively, theng⊂h. (1.39)

This is clear from the definition of the Lie algebras, sinceIH⊂IG. Examples

1.LetD_nbe the group of invertible diagonaln×nmatrices. Then the Lie algebrad_n ofD_n(in the sense of algebraic groups) consists of the diagonal matrices inM_n(C).

To prove this, take any polynomial f onM_n(C)that vanishes onD_n. Then we can write

f =

∑

i6=j

x_{i j}f_{i j},

1.4 Linear Algebraic Groups 43

wheref_{i j}∈P(M_n(C))and 1≤i,j≤n. HenceA= [a_{i j}]∈d_nif and only ifX_Ax_{i j}|Dn= 0 for alli6=j. SetB=A e_{j i}. Then

X_Ax_{i j}=f_B=

∑

ⁿ

p=1

a_{p j}x_ip

by (1.36). Thus we see thatX_Ax_{i j}vanishes onD_nfor alli6=jif and only ifa_{i j}=0 for alli6= j.

2.LetN_n⁺be the group of upper-triangular matrices with diagonal entries 1. Then its Lie algebran⁺_n consists of the strictly upper-triangular matrices inM_n(C). To prove this, let f be any polynomial onM_n(C)that vanishes onN_n⁺. Then we can write

f=

∑

ⁿ

i=1

(x_ii−1)f_i+

∑

1≤j<i≤n

x_{i j}f_{i j},

wheref_iandf_{i j}are polynomials onM_n(C). HenceA∈n⁺_n if and only ifX_Ax_{i j}|Nn⁺=0 for all 1≤j≤i≤n. By the same calculation as in Example 1, we have

X_Ax_{i j}|Nn⁺=a_{i j}+

∑

ⁿ

p=i+1

a_{p j}x_ip.

HenceA∈n⁺_n if and only ifa_{i j}=0 for all 1≤j≤i≤n.

3.Let 1≤p≤nand letPbe the subgroup ofGL(n,C)consisting of all matrices in block upper-triangular form

g= a b 0d

, wherea∈GL(p,C),d∈GL(n−p,C), andb∈M_p,n−p(C).

The same arguments as in Example 2 show that the ideal IP is generated by the matrix entry functionsx_{i j}withp<i≤nand 1≤j≤pand that the Lie algebra of P(as an algebraic group) consists of all matricesX in block upper-triangular form

X= A B

0 D

, whereA∈M_p(C), D∈M_n−p(C), andB∈M_p,n−p(C).