Representation Theory of Finite dimensional Lie algebras [Lecture notes]

111 Pages·2016·0.502 MB·English

by  Catharina Stroppel, J. Torres

Representation Theory of Finite dimensional
Lie algebras
Notes by Jacinta Torres, taken from the lecture
”Representation Theory II” given by Catharina Stroppel in
2011
Bonn, Germany
2
Contents
1 Introduction to Lie Algebras 5
1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Definition of the classical simple Lie algebras . . . . . . . . . . 10
1.3 Nilpotent and solvable Lie algebras . . . . . . . . . . . . . . . 11
1.4 g-modules, basic definitions . . . . . . . . . . . . . . . . . . . 14
1.5 Testing solvability and semisimplicity . . . . . . . . . . . . . . 17
1.6 Jordan Decomposition and
Proof of Cartan’s Criterion . . . . . . . . . . . . . . . . . . . . 20
1.6.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Theorems of Levi and Malcev . . . . . . . . . . . . . . . . . . 23
1.7.1 Weyl’s complete irreducibility theorem. . . . . . . . . . 26
1.7.2 Classification of irreducible finite dimensional sl(2,C)
modules . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8 Universal enveloping algebras . . . . . . . . . . . . . . . . . . 33
2 Representations of Lie algebras 45
2.1 Constructing new representations . . . . . . . . . . . . . . . . 45
2.1.1 Pull-back and restriction . . . . . . . . . . . . . . . . . 45
2.1.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Verma Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Abstract Jordan Decomposition . . . . . . . . . . . . . . . . . 52
3 Structure Theory of Semisimple Complex Lie Algebras 57
3.1 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . 58
3.2 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.1 Changing scalar . . . . . . . . . . . . . . . . . . . . . . 69
3.2.2 Bases of root system . . . . . . . . . . . . . . . . . . . 75
3.2.3 Weyl Chambers . . . . . . . . . . . . . . . . . . . . . . 77
3
4 CONTENTS
3.2.4 Subsets of roots . . . . . . . . . . . . . . . . . . . . . . 80
3.2.5 Classification of a parabolic subset over a fixed R+(B) . 82
3.3 Borel and Parabolic subalgebras of a complex semi simple Lie
algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Highest Weight Theory 85
4.1 Construction of highest weight modules . . . . . . . . . . . . . 91
4.2 Character formula . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Category O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Cartan matrices and Dynkin diagrams . . . . . . . . . . . . . 100
4.4.1 Classificationofirreducible,reducedrootsystems/Dynkin
Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 105
Literature
Humphreys:
• Introduction to Lie algebras and Representation Theory
• Complex Reflection Groups
• Representations of semi simple Lie Algebras
• Knapp: Lie groups: beyond an introduction
• V.S. Varadarajan: Lie Groups, Lie Algebras and their Representations
Chapter 1
Introduction to Lie Algebras
(Lecture 1)
1.1 Basic Definitions
Definition 1.1.1. ALiealgebraisavectorspacegoversomefieldk,together
with a bilinear map [−,−] : g×g → g such that:
[x,x] = 0∀x ∈ g(anti-symmetry) (1.1)
[x,[y,z]]+[y,[z,x]]+[z,[x,y]](Jacobi identity) (1.2)
Remarks.
• [−,−]is called a Lie bracket.
• ∀x,y ∈ g,[x,y] = −[y,x](because 0 = [x+y,x+y] = [x,x]+[x,y]+
[y,x]+[y,y])
• Often: Say a k-Lie algebra if we have a Lie algebra over k.
Examples.
V a k-vector space, [v,w] = 0∀v,w ∈ V defines a k-Lie algebra.
Any associative k-algebra is naturally a k-Lie algebra by
[a,b] := ab−ba∀a,b ∈ A
. Excercise: check 1.2.
5
6 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS
In particular: The vector space
gl(n,k) = {A ∈ M (k)}
n×n
is a lie algebra via [A,B] = AB−BA (usual commutator of matrices). More
generally:
gl(V) = {ρ : V → Vk −linear};
V a k-vector space, is a lie algebra via [f,g] = f ◦ g − g ◦ f ∀f,g ∈ gl(V).
These are the general linear Lie algebras.
Let g ,g be Lie algebras, then g ⊕g is a Lie algebra via
1 2 1 2
[(x,y),(x(cid:48),y(cid:48))] = [[x,x(cid:48)],[y,y(cid:48)]](take the Lie algebra component wise)
. The Lie algebra g ⊕g is called the direct sum of g and g .
1 2 1 2
Definition 1.1.2. Given g ,g k-Lie algebras, a morphism f : g → g of
1 2 1 2
k-Lie algebras is a k-linear map such that f([x,y]) = [f(x),f(y)].
Remarks. • id : g → g is a Lie algebra homomorphism.
• f : g → g ,g : g → g Lie algebra homomorphisms, then g◦f : g →
1 2 2 3 1
g is a Lie algebra homomorphism. (because
2
g ◦f([x,y]) = g([f(x),f(y)]) = [g ◦f(x),g ◦f(y)]).
Hence: k-Lie algebras with Lie algebra homomorphisms form a cate-
gory.
Example 1.1.3. Let g be a Lie algebra.
ad :g → gl(g)
x (cid:55)→ ad(x)
where ad(x)(y) = [x,y] ∀x,y ∈ g is a lie algebra homomorphism. It is called
the adjoint representation. (because: linear clear, since [−,−] is bilinear, and
[ad(x),ad(y)](z) = ad(x)ad(y)(z)−ad(y)ad(x)(z)
Jacobi−id
= [x,[y,z]]−[y,[x,z]] = [[x,y],z] = ad([x,y])(z).)
1.1. BASIC DEFINITIONS 7
Definition 1.1.4. Let g be a k-Lie algebra, l ⊂ g a vector subspace.
• l is called a sub-lie algebra if [x,y] ∈ l for all x,y ∈ l.
• l is called an ideal if [x,y] ∈ l for all x ∈ l, y ∈ g.(denoted l (cid:67) g
( ⇐⇒ [x,y] ∈ l for all x ∈ g, y ∈ l)).
Given a Lie algebra g, I (cid:67) g, then the vector space g/I becomes a Lie
algebra via [x+I,y +I] = [x,y]+I.
To check this is well defined:
Assumex+I = x(cid:48)+I,y+I = y(cid:48)+I =⇒ x(cid:48) = x+u,y(cid:48) = y+v,u,v ∈ I =⇒
[x(cid:48) +I,y(cid:48) +I] = [u+x+I,v +y +I] =
[u+x,v +y]+I = [u,v]+[u,y]+[x,v]+[x,y]+I = [x,y]+I =⇒
well defined.
Proposition 1.1.5. Let f : g → g be a Lie algebra homomorphism. Then
1 2
1 ker(f)(cid:67)g .
1
2 im(f) is a Lie subalgebra of g .
2
3 Universal Property: If I (cid:67) g ,ker(f) ⊂ I, the following diagram com-
1
mutes:
x g f (cid:45) g
1 2
(cid:64)
(cid:64)
(cid:64)
(cid:64)(cid:82) (cid:63)
x+I ∈ g /I
1
∼
In particular: g /ker(f) = im(f) is an isomorphism of Lie algebras.
1
Proof. : Standard.
Remark 1.1.6. There are the usual isomorphism Theorems:
a. I,J(cid:67)g,I ⊆ J, then J/I(cid:67)g/I and
∼
(g/I)/(J/I) = g/J
8 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS
b. I,J(cid:67)g. Then I+J(cid:67)g;I∩J(cid:67)g, and
∼
I/I∩J = I+J/J
Definition 1.1.7. A Lie algebra g is called simple if g contains no ideals
I(cid:67)g except for I = 0 or I = g, and if [g,g] (cid:54)= 0.
Def.
Remark 1.1.8. [g,g] = 0 ⇐⇒ [x,y] = 0∀x,y ∈ g (i.e g is abelian).
If g is simple, then:
a. Z(g) = {x ∈ g : [x,y] = 0∀y ∈ g}
b. [g,g] = g
(because: Z(g)(cid:67)g and g not abelian, hence Z(g) = 0); Z(g)(cid:67)g because
∀x,y ∈ g,z ∈ Z(g):
[[x,z],y] = [x,[z,y]]+[z,[y,x]]
[g,g] is obviously an ideal, because for x,y,z ∈ g, [x,[y,z]] ∈ [g,g].
[g,g] is called the derived lie algebra of g. Also, [g,g] is the smallest
ideal of g such that g/[g,g] is abelian.
Example 1.1.9.
sl(2,k) := {A ∈ Mat(2×2,k)|Tr(A) = 0}
Is a lie subalgebra of gl(2,k), because
Tr([A,B]) = Tr([A,B]) = Tr(AB−BA) = 0 ∀A,B ∈ sl(2,k).
Fact 1.1.10. sl(2,k) is simple ⇐⇒ char(k) (cid:54)= 2.
Proof. Choose a standard basis (sl ”triple”) as follows:
2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
e = 0 1 f = 0 0 h = 1 0
0 0 1 0 0 −1
Then we have (excercise!):
[h,e] = 2e
[h,f] = −2f
[e,f] = h
These relations define the lie bracket.
If char(k) = 2, then the vector subspace h spanned by h is a non-trivial
ideal because [e,h] = 0, [f,h] = 0. Since [h,h] = 0, then g = sl(2,k) is not
simple.
1.1. BASIC DEFINITIONS 9
Remark 1.1.11. gl(n,k) is not simple, because 1 spans a non-trivial ideal
(Note Z(gl(n,k)) (cid:54)= 0).
Lemma 1.1.12. ker(ad(g) → gl(g)) = {x ∈ g|ad(x) = 0} = {x ∈ g|[x,y] =
0∀x,y ∈ g} = Z(g).
Proof. Just definitions.
Corollary 1.1.13. LetgbeasimpleLiealgebra,thengisalinear Lie algebra
(i.e. g is a Lie subalgebra of some Lie algebra of matrices, i.e., of gl(V) for
some vector space V).
Proof. Since g is simple, then Z(g) = 0, which implies ad is injective, hence
∼
g = ad(g) is an isomorphism of Lie algebras, and ad(g) is a Linear lie algebra.
Theorem 1.1.14. (Theorem od Ado) Every finite dimensional Lie algebra
is linear.
Proof. Later.
Theorem1.1.15. (Cartan-KillingClassification)Everycomplexfinite-dimensional
simple Lie algebra is isomorphic to exactly one of the following list:
• Classical Lie Algebras:
A sl(n+1,C);n ≥ 1
n
B so(2n+1,C);n ≥ 2
n
C sp(2n,C);n ≥ 3
n
D so(2n,C);n ≥ 4
n
• Exceptional Lie Algebras:
E
6
E
7
E
8
F
4
10 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS
G
2
Remark 1.1.16. Every finite dimensional complex Lie algebra which is a
direct sum of simple Lie algebras is called semi-simple.
Remark 1.1.17. (Without further explanations)
(cid:8) (cid:9) 1:1 (cid:8) (cid:9)
connected, compact Lie group with trivial center ←→ s.s. complex f.d. Lie alg
G (cid:55)→ C⊗ Lie(G)
R
Where Lie(G) is the tangent space of G at the origin.
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
(Lecture 2- 6th April 2011)
1.2 Definition of the classical simple Lie al-
gebras
• sl(n+1,C) = {A ∈ gl(n+1,C) : Tr(A) = 0}
• sp(2n,C) = {A ∈ gl(2n,C) : f(Ax,y) = −f(x,Ay)∀x,y ∈ C2n} where
f is the bilinear form given by
f(x,y) := xtMy
(cid:0) (cid:1)
where M = 0 In .
−In 0
Explicitly:
(cid:18) (cid:19)t(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
A B 0 I 0 I A B
n + n = 0
C D −I 0 −I 0 C D
n n
(cid:18) (cid:19) (cid:18) (cid:19)
−Ct At C D
⇐⇒ =
−Dt Bt −A −B
Thus
(cid:18) (cid:19)
sp(2n,C) = (cid:8) A skew (cid:9) =⇒ dim(sp(2n,C)) = n2 +n(n−1) = 2n2 −n.
skew At