Notes to Lie Algebras and Representation Theory

205 Pages·2019·1.299 MB·English

by  Zhengyao Wu (吴正尧)

NOTES TO LIE ALGEBRAS AND REPRESENTATION THEORY
ZHENGYAO WU
Abstract. • Main reference: [Hum78, Parts I, II, III].
• Lecture notes to the graduate course “Finite dimensional algebra” during Spring 2019 at
Shantou University taught by me.
• Targeted audience: Graduate students in pure mathematics.
Contents
1. February 26th, Introduction to Lie algebra 2
2. March 5th, Solvable and nilpotent Lie algebras 12
3. March 12th, Lie theorem, Jordan decomposition 26
4. March 19th, Cartan’s criterion, Killing form 36
5. March 26th, Semisimple decomposition and Lie modules 49
6. April 2nd, Casimir element, Weyl’s theorem 61
7. April 9th, Representation of sl(2,F), toral subalgebras 73
8. April 16th, Centralizer of H; Orthogonal, integral properties 83
9. April 23th, Rationality properties, reflections, root systems 98
10. April 30th, Bases and Weyl chambers 110
11. May 7th, Weyl group and its actions 122
12. May 14th, Irreducible root systems, two root lengths and Cartan matrix 133
13. May 21th, Coxeter graphs, Dynkin diagrams 147
14. May 28th, Classification, irreducible root systems of types A, B and C 157
15. June 4th, Irreducible root systems of types D,E,F,G 169
16. June 11th, Weyl group of each type, Automorphisms of the Dynkin diagram, Weights 185
References 205
1
2 ZHENGYAO WU
1. February 26th, Introduction to Lie algebra
Sophus Lie (1842-1899) established the theory in late 1880s in Oslo, Norway.
Definition 1.1
Let F be a field. Let L be a F-vector space. We say that L is a F-Lie algebra if there exists a
map L×L → L, (x,y) 7→ [x,y] such that
(L1) [x,y] is bilinear over F. (L2) [x,x] = 0 for all x ∈ L.
(L3) Jacobi identity [x,[y,z]]+[y,[z,x]]+[z,[x,y]] = 0 for all x,y,z ∈ L.
Remark 1.2
(L2’) [x,y] = −[y,x] for all x,y ∈ L.
(1) If L is a F-Lie algebra, then (L2’) holds.
(2) If char(F) 6= 2, then (L2’) implies (L2).
(3) Let H,K be subsets of L. We write [H,K] = Span {[x,y],x ∈ H,y ∈ K}. Then [H,K] =
F
[K,H].
Definition 1.3
Let K be a subset of L. We call K a subalgebra of L if
(1) If K is a sub-F-vector space of L, and
(2) [K,K] ⊂ K, then K is a F-Lie algebra.
Example 1.4
Let V be a F-vector space. Let End (V) be the space of endomorphisms (linear transformations)
F
of V. For all x,y ∈ End (V), define [x,y] = x◦y−y◦x, where ◦ means the composition of maps.
F
Show that gl(V) = (End (V),[•,•]) is a F-Lie algebra, called the general linear algebra.
F
If dim (V) = n, then we write gl(n,F) the F-Lie algebra of all n×n matrices such that [e ,e ] =
F ij kl
δ e −δ e where e is the n×n matrix whose (i,j)-entry is 1 and all other entries are 0; δ = 1
jk il li kj ij ij
if i = j, otherwise δ = 0.
ij
Subalgebras of gl(V) ’ gl(n,F) are called linear algebras.
Definition 1.5
Let L,L0 be two F-Lie algebras. A map f : L → L0 is an isomorphism if
(1) f is a linear isomorphism of vector F-spaces, and
(2) f([x,y]) = [f(x),f(y)] for all x,y ∈ L.
Theorem 1.6 Ado-Iwasawa
Every F-Lie algebra is isomorphic to a linear F-Lie algebra.
Proof. Omit. (cid:3)
Definition 1.7
Classical algebras are the following proper subalgebras of gl(V) of type A ,B ,C ,D , l ≥ 1.
l l l l
NOTES TO HUMPHREYS 3
Example 1.8
Type A (l ≥ 1): special linear algebras
l
sl(V) = {x ∈ gl(V) : Tr(x) = 0}, dim (V) = l+1.
F
sl(l+1,F) = {x ∈ gl(l+1,F) : Tr(x) = 0}.
dim (sl(l+1,F)) = l2 +2l since it has standard basis
F
{e −e : 1 ≤ i ≤ l}∪{e : 1 ≤ i 6= j ≤ l+1}.
ii i+1,i+1 ij
Example 1.9
Type B (l ≥ 1): orthogonal algebras of odd degree.
l
 
1 0 0
 
 
 
Let f be a non-degenerate symmertic bilinear form on V whose matrix is s = 0 0 I .
 l
 
 
 
0 I 0
l
o(V) = {x ∈ gl(V) : f(x(v),w) = −f(v,x(w)), ∀v,w ∈ V}, dim (V) = 2l+1,
F
o(2l+1,F) = {x ∈ gl(2l+1,F) : sx = −xts}
  
 0 b1 b2  
= −bt m n  ∈ gl(2l+1,F) : nt = −n, pt = −p
 2 
−bt1 p −mt 
dim (B ) = 2l2 +l since B has standard basis
F l l
{e −e : 1 ≤ i ≤ l}∪{e −e : 1 ≤ i 6= j ≤ l}
i+1,i+1 l+i+1,l+i+1 i+1,j+1 l+j+1,l+i+1
∪{e −e : 1 ≤ i ≤ l}∪{e −e : 1 ≤ i ≤ l}
1,l+i+1 i+1,1 1,i+1 l+i+1,1
∪{e −e : 1 ≤ i < j ≤ l}∪{e −e : 1 ≤ j < i ≤ l}.
i+1,l+j+1 j+1,l+i+1 l+i+1,j+1 l+j+1,i+1
Example 1.10
Type C (l ≥ 1): symplectic algebras
l
 
0 I
 l
Let f be a non-degenerate skew-symmertic bilinear form on V whose matrix is s =  .
 
−I 0
l
sp(V) = {x ∈ gl(V) : f(x(v),w) = −f(v,x(w)), ∀v,w ∈ V},dim (V) = 2l,
F
sp(2l,F) = {x ∈ gl(2l,F) : sx = −xts}
4 ZHENGYAO WU
  
= m n  ∈ gl(2l,F) : nt = n, pt = p.
p −mt 
dim (sp(2l,F)) = 2l2 +l since it has standard basis
F
{e −e : 1 ≤ i ≤ l}∪{e −e : 1 ≤ i 6= j ≤ l}
i,i l+i,l+i i,j l+j,l+i
∪{e : 1 ≤ i ≤ l}∪{e +e : 1 ≤ i < j ≤ l}
i,l+i i,l+j j,l+i
∪{e : 1 ≤ i ≤ l}∪{e +e : 1 ≤ i < j ≤ l}.
l+i,i l+i,j l+j,i
Example 1.11
Type D (l ≥ 1): orthogonal algebras of even degree:
l
 
0 I
 l
Let f be a non-degenerate symmertic bilinear form on V whose matrix is s =  .
 
I 0
l
o(V) = {x ∈ gl(V) : f(x(v),w) = −f(v,x(w)), ∀v,w ∈ V}, dim (V) = 2l.
F
o(2l,F) = {x ∈ gl(2l,F) : sx = −xts}
  
= m n  ∈ gl(2l,F) : nt = −n, pt = −p.
p −mt 
dim (o(2l,F)) = 2l2 −l since it has standard basis
F
{e −e : 1 ≤ i ≤ l}∪{e −e : 1 ≤ i 6= j ≤ l}
i,i l+i,l+i i,j l+j,l+i
∪{e −e : 1 ≤ i < j ≤ l}∪{e −e : 1 ≤ j < i ≤ l}.
i,l+j j,l+i l+i,j l+j,i
Definition 1.12
A derivation of an F-algebra A is an F-linear map δ : A → A such that δ(ab) = aδ(b)+δ(a)b for
all a,b ∈ A. Let Der(A) be the set of all derivations of A. We have Der(A) ⊂ gl(A).
Definition 1.13
Let L be a F-Lie algebra. The map ad : L → gl(L) such that (adx)(y) = [x,y] for all x,y ∈ L is
the adjoint representation of L.
Lemma 1.14
Let L be a F-Lie algebra. Then ad(L) ⊂ Der(L).
Proof. We need to show that (adx)([y,z]) = [(adx)(y),z]+[y,(adx)(z)] for all x,y,z ∈ L.
NOTES TO HUMPHREYS 5
Steps Statements Reasons
1. (adx)([y,z]) = [x,[y,z]] Definition 1.13
2. = −[z,[x,y]]−[y,[z,x]] Definition 1.1(L3)
3. = [[x,y],z]+[y,[x,z]] Definition 1.1(L2)
4. = [(adx)(y),z]+[y,(adx)(z)] Definition 1.13
5. (adx) is a derivation for all x ∈ L Definition 1.12
(cid:3)
Definition 1.15
Elements in ad(L) are called inner derivations, elements in Der(L) − ad(L) are called outer
derivations.
Example 1.16
Let K be a subalgebra of L. Then ad (x) and ad (x) are different in general.
K L
Let d(n,F) be the set of diagonal matrices. For x ∈ d(n,F) ⊂ gl(n,F), ad (x) = 0 and
d(n,F)
ad (x) 6= 0 for n ≥ 2.
gl(n,F)
Definition 1.17
A F-Lie algebra L is abelian if [x,y] = 0 for all x,y ∈ L.
Example 1.18
Every F-Lie algebra L of dimension 1 is abelian by Definition 1.1(L2).
Example 1.19
A two dimensional F-Lie algebras is either abelian or isomorphic to Fx+Fy such that [x,y] = x.
Proof. Suppose L is not abelian and L = Fa+Fb.
Steps Statements Reasons
1. [L,L] = Fx for x = [a,b]. Definition 1.1(L2)
2. There exists z ∈ L−Fx. dim L = 2
F
3. [x,z] = cx for some c ∈ F∗. step 1
4. Let y = c−1z. Then [x,y] = x Definition 1.1(L1)
6 ZHENGYAO WU
Steps Statements Reasons
and L = Fx+Fy. step 2
(cid:3)
Definition 1.20
A subset I of a F-Lie algebra L is an ideal if
(1) I is a sub-F-vector space of L. (2) [I,L] ⊂ I.
By (2), I is a subalgebra of L.
Example 1.21 Ideals
(1) 0 and L are ideals of L, called trivial ideals of L.
(2) The center Z(L) = {z ∈ L : [z,L] = 0} of L is an ideal of L.
(3) The derived algebra [L,L] = Span{[x,y] : x,y ∈ L} of L is an ideal of L. In particular,
[L,L] = 0 iff L is abelian. If L is a classical algebra, then L = [L,L].
(4) If I and J are ideals of L, then I +J = {x+y : x ∈ I, y ∈ J} is an ideal of L.
(5) If I and J are ideals of L, then [I,J] = {P[x ,y ] : x ∈ I, y ∈ J} is an ideal of L.
i i i i
Definition 1.22
We call L simple if .
(1) L only has ideals 0 and L;
(2) [L,L] 6= 0.
Lemma 1.23
If L is simple, then Z(L) = 0.
Proof.
Steps Statements Reasons
1. Z(L) = 0 or L. Definition 1.22(1)
2. Z(L) 6= L. Definition 1.22(2)
3. Z(L) = 0. steps 1,2
(cid:3)
NOTES TO HUMPHREYS 7
Lemma 1.24
If L is simple, then L = [L,L].
Proof.
Steps Statements Reasons
1. [L,L] = 0 or L. Definition 1.22(1)
2. [L,L] 6= 0. Definition 1.22(2)
3. [L,L] = L. steps 1,2
(cid:3)
Example 1.25
If char(F) 6= 2, then sl(2,F) is simple.
Proof. A standard basis for L = sl(2,F) is
     
0 1 0 0 1 0
     
x =  , y =  , h =  .
     
0 0 1 0 0 −1
such that [x,y] = h, [h,x] = 2x, [h,y] = −2y.
Let I 6= 0 be an ideal of L. We need to show that I = L. Suppose 0 6= ax+by +ch ∈ I.
The case a 6= 0:
Steps Statements Reasons
1. [ax+by +ch,y] = ah−2cy ∈ I. ax+by +ch ∈ I and y ∈ L
2. [ah−2cy,y] = −2ay ∈ I. ah−2cy ∈ I and y ∈ L
3. y ∈ I. charF 6= 2 and a 6= 0
4. [x,y] = h ∈ I. x ∈ L and y ∈ I
5. [h,x] = 2x ∈ I. h ∈ I and x ∈ L
6. x ∈ I. charF 6= 2
7. I = L. x,y,h ∈ I by steps 3,4,6.
8 ZHENGYAO WU
The case b 6= 0:
Steps Statements Reasons
1. [ax+by +ch,x] = −bh+2cx ∈ I. ax+by +ch ∈ I and x ∈ L
2. [−bh+2cx,x] = −2bx ∈ I. −bh+2cx ∈ I and x ∈ L
3. x ∈ I. charF 6= 2 and b 6= 0
4. [x,y] = h ∈ I. x ∈ I and y ∈ L
5. [h,y] = −2y ∈ I. h ∈ I and y ∈ L
6. y ∈ I. charF 6= 2
7. I = L. x,y,h ∈ I by steps 3,4,6.
The case a = b = 0:
Steps Statements Reasons
1. c 6= 0. ax+by +ch 6= 0
2. h ∈ I. ax+by +ch = ch ∈ I
3.1 x ∈ I. steps 5,6 of case a 6= 0
3.2 y ∈ I. steps 5,6 of case b 6= 0
4. I = L. x,y,h ∈ I by steps 3,4.
Hence L is simple. (cid:3)
Example 1.26
Let L be a F-Lie algebra.
(1) Let K be a sub-F-vector space of L. Its normalizer is N (K) = {x ∈ L : [x,K] ⊂ K}
L
• N (K) is a subalgebra of L.
L
• K is an ideal of N (K).
L
• If A is subalgebra of L and K is an ideal of A, then A ⊂ N (K).
L
(2) Let X be a subset of L. Its centralizer C (X) = {x ∈ L : [x,X] = 0} is a subalgebra of L.
L
In particular, C (L) = Z(L).
L
NOTES TO HUMPHREYS 9
Definition 1.27
A linear map φ : L → L0 between F-Lie algebras over F is a
• homomorphism if φ([x,y]) = [φ(x),φ(y)] for all x,y ∈ L;
• monomorphism if it is a homomorphism and ker(φ) = {0};
• epimorphism if it is a homomorphism and im(φ) = L0;
• isomorphism if it is both a monomorphism and an epimorphism;
• automorphism of L0 = L and it is an isomorphism. Let Aut(L) denote the group of
automorphisms of L.
Example 1.28
Let φ : L → L0 be a homomorphism between F-Lie algebras over F. Then
(1) ker(φ) is an ideal of L. (2) φ(L) is a subalgebra of L.
(3) If I is an ideal of L, then the quotient vector F-space L/I with [x+I,y +I] = [x,y]+I for
all x,y ∈ L is a F-Lie algebra, called the quotient algebra. There is a canonical homomorphism
π: L → L/I such that π(x) = x+I for all x ∈ L. We have ker(π) = I and im(π) = L/I.
Proposition 1.29
(1) Let φ : L → L0 be a homomorphism between F-Lie algebras over F. Then L/ker(φ) ’ im(φ).
(1’) If I ⊂ ker(φ) is an ideal of L, then there exists a unique homomorphism ψ : L/I → L0 such
that the following diagram commutes
L φ (cid:47)(cid:47) L0
(cid:79)(cid:79)
ψ
π (cid:32)(cid:32)
L/I
(2) If I ⊂ J are ideals of L, then J/I is an ideal of L/I and (L/I)/(J/I) ’ L/J.
(3) If I and J are ideals of L, then (I +J)/J ’ I/(I ∩J).
Definition 1.30
Let L be a F-Lie algebra.
(1) A representation of L is a homomorphism φ : L → gl(V) for some vector space V over F.
(2) We call φ faithful if it is a monomorphism.
Lemma 1.31
Let L be a F-Lie algebra.
(1) ad: L → gl(L) is a representation. (2) ker(ad) = Z(L).
(3) If L is simple, then ad is faithful. Thus L is isomorphic to a linear F-Lie algebra.
Proof. (1) We need to show that [(adx),(ady)](z) = ad([x,y])(z) for all x,y,z ∈ L.
10 ZHENGYAO WU
Steps Statements Reasons
1. [(adx),(ady)](z) = [x,[y,z]]−[y,[x,z]]. Definition 1.13
2. = [x,[y,z]]+[y,[z,x]] Definition 1.1(L2)
3. = −[z,[x,y]] Definition 1.1(L3)
4. = [[x,y],z] Definition 1.1(L2)
5. = ad([x,y])(z) Definition 1.13
6. [(adx),(ady)] = ad([x,y]) for all x,y ∈ L. steps 1-5
7. ad: L → gl(L) is a representation. Definition 1.30(1)
(2) x ∈ ker(ad) iff (adx) = 0 iff [x,L] = 0 iff x ∈ Z(L).
(3)
Steps Statements Reasons
1. Z(L) = 0. L is simple and Lemma 1.23
2. ker(ad) = 0. (2)
3. ad is faithful. Definition 1.30(2)
4. L ’ ad(L) ⊂ gl(L). Proposition 1.29(1)
(cid:3)
Example 1.32
k−1 δn
Suppose char(F) = 0 and δ ∈ Der(L) such that δk = 0 for some k > 0. Define expδ = P .
n!
n=0
Then (1)(expδ)([x,y]) = [(expδ)(x),(expδ)(y)]. (2) expδ ∈ Aut(L).
Proof. (1)
Steps Statements Reasons
n (cid:16) (cid:17)
1. δn([x,y]) = P n [δi(x),δn−i(y)]. Leibniz rule of Definition 1.12
i
i=0
δn n "δi δn−i # n! n!
2. ([x,y]) = P (x), (y) = and Definition 1.1(L1)
n! i! (n−i)! i i!(n−i)!
i=0