Lectures on sl2(C)-modules

254 Pages·2009·1.235 MB·English

by  Volodymyr Mazorchuk

Volodymyr Mazorchuk
C
Lectures on sl ( )-modules
2

Preface
The representation theory of Lie algebras is an important and intensively
studied area of modern mathematics with applications to, basically, all ma-
jor areas of mathematics and physics. There are several textbooks which
specialize in different aspects of the representation theory of Lie algebras
and its applications, but the usual topics covered in such books are finite-
dimensional, highest weight, or Harish-Chandra modules.
ThesmallestsimpleLiealgebrasl differsinmanyaspectsfromallother
2
semi-simple Lie algebras. One could, for example, mention that sl is the
2
only semi-simple Lie algebra for which all simple (not necessarily finite-
dimensional) modules are in some sense understood. The algebra sl is
2
generated by only two elements and hence is an invaluable source of com-
putable examples. Moreover, in many cases the ideas which one gets from
working with sl generalize relatively easily to other Lie algebras with a
2
minimum of extra knowledge required.
The aim of these lecture notes is to give a relatively short introduction
to the representation theory of Lie algebras, based on the Lie algebra sl ,
2
with a special emphasis on explicit examples. Due to the choice of this
Lie algebra, it is possible to mention and describe many more aspects of
the representation theory of Lie algebras than those covered in standard
textbooks.
Thenotesstartwithanabsolutelyclassicalintroductorysectiononfinite-
dimensional modules and the universal enveloping algebra. The third chap-
ter moves on to the study of weight modules, including a complete classifi-
cation and explicit construction of all weight modules and a description of
thecategoryofallweightmoduleswithfinite-dimensionalweightspaces, via
quiver algebras. This is followed by a description and study of the primitive
spectrum of the universal enveloping algebra and its primitive quotients.
The next step is a relatively complete description of the Bernstein-Gelfand-
Gelfand category O and its properties. The two last chapters contain a
description of all simple sl -modules and various categorifications of simple
2
finite-dimensional modules. The material presented in the last chapter is
based on papers which were published in the last two years.
Thenotesareprimarilydirectedtowardspost-graduatestudentswhoare
interested in learning the basics of the representation theory of Lie algebras.
v
vi PREFACE
I hope that these notes could serve as a textbook for both lecture courses
and reading courses on this subject. Originally, they were written and used
for reading courses which I gave in Uppsala in 2008.
The prerequisites needed to understand these notes depend on the chap-
ter. Forthefirsttwochapters,oneneedsonlysomebasicknowledgeinlinear
algebra and rings and modules. For the next two chapters, it is assumed
that the reader is familiar with the basics of the representation theory of
finite-dimensional associative algebras and basic homological algebra. The
last three chapters also require some basic experience with category theory.
Everychaptercontainsattheendsomecomments,includingsomehistor-
ical background, brief descriptions of more advanced results, and references
to some original papers. I tried to present these comments to the best of
my knowledge and I would like to apologize in advance if some of them are
not absolutely correct.
Therearemanyexercisesinthemaintextandattheendofeachchapter.
The exercises in the main text are usually relatively straightforward and
necessary to understand the material. It is strongly recommended that the
reader at least looks through them. There are some answers and hints at
the end of the notes.
I would like to thank Ekaterina Orekhova and Valentina Chapovalova
for their corrections and comments on the earlier version of the manuscript.
Contents
Preface v
1 Finite-dimensional modules 1
1.1 The Lie algebra sl and sl -modules . . . . . . . . . . . . . . 1
2 2
1.2 Classification of simple finite-dimensional modules . . . . . . 5
1.3 Semi-simplicity of finite-dimensional modules . . . . . . . . . 9
1.4 Tensor products of finite-dimensional modules . . . . . . . . . 13
1.5 Unitarizability of finite-dimensional modules . . . . . . . . . . 15
1.6 Bilinear forms on tensor products . . . . . . . . . . . . . . . . 19
1.7 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 21
1.8 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 24
2 The universal enveloping algebra of sl 29
2
2.1 Construction and the universal property . . . . . . . . . . . . 29
2.2 Poincar´e-Birkhoff-Witt Theorem . . . . . . . . . . . . . . . . 33
2.3 Filtration on U(g) and the associated graded algebra . . . . . 37
2.4 Centralizer of the Cartan subalgebra and center of U(sl ) . . 39
2
2.5 Harish-Chandra homomorphism . . . . . . . . . . . . . . . . . 43
2.6 Noetherian property . . . . . . . . . . . . . . . . . . . . . . . 45
2.7 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 47
2.8 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 50
3 Weight sl -modules 53
2
3.1 Weight modules . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Verma modules . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Dense modules . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Classification of simple weight modules . . . . . . . . . . . . . 65
3.5 Coherent families . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Categoryofallweightmoduleswithfinite-dimensionalweight
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
ξ,τ
3.7 Structure of W in the case of one simple object . . . . . . . 76
ξ,τ
3.8 Structure of W in the case of two simple objects . . . . . . 78
ξ,τ
3.9 Structure of W in the case of three simple objects . . . . . 81
vii
viii CONTENTS
3.10 Tensoring with a finite-dimensional module . . . . . . . . . . 83
3.11 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.12 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 93
3.13 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 97
4 The primitive spectrum of U(sl ) 103
2
4.1 Annihilators of Verma modules . . . . . . . . . . . . . . . . . 103
4.2 Simple modules and central characters . . . . . . . . . . . . . 105
4.3 Classification of primitive ideals . . . . . . . . . . . . . . . . . 106
4.4 Primitive quotients . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5 Centralizers of elements in primitive quotients . . . . . . . . . 111
4.6 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 115
4.7 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 118
5 Category O 121
5.1 Definition and basic properties . . . . . . . . . . . . . . . . . 121
5.2 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3 Blocks via quiver and relation . . . . . . . . . . . . . . . . . . 129
5.4 Structure of a highest weight category . . . . . . . . . . . . . 135
5.5 Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.6 Homological properties . . . . . . . . . . . . . . . . . . . . . . 144
5.7 Category of bounded linear complexes of
projective graded D-modules . . . . . . . . . . . . . . . . . . . 148
5.8 Projective functors on O . . . . . . . . . . . . . . . . . . . . 153
0
5.9 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 160
5.10 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 170
6 Description of all simple sl -modules 175
2
6.1 Weight and nonweight modules . . . . . . . . . . . . . . . . . 175
6.2 Embedding into an Euclidean algebra . . . . . . . . . . . . . 177
6.3 Description of simple nonweight modules . . . . . . . . . . . . 181
6.4 Finite-dimensionality of kernels and cokernels . . . . . . . . . 183
6.5 Finite-dimensionality of extensions . . . . . . . . . . . . . . . 188
6.6 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 191
6.7 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 193
7 Categorification of simple finite-dimensional sl -modules 195
2
7.1 Decategorification and categorification . . . . . . . . . . . . . 195
7.2 Naive categorification of V(n) . . . . . . . . . . . . . . . . . . 197
7.3 Weak categorification of V(n) . . . . . . . . . . . . . . . . . . 201
7.4 Categorification of V(n) via coinvariant algebras . . . . . . . 206
7.5 Addenda and comments . . . . . . . . . . . . . . . . . . . . . 209
7.6 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 211
CONTENTS ix
Answers and hints to exercises 215
Bibliography 223
List of Notation 233
Index 241
x CONTENTS
Chapter 1
Finite-dimensional modules
1.1 The Lie algebra sl and sl -modules
2 2
In what follows we will always work over the field C of complex numbers.
If not stated otherwise, all vector spaces, tensor products and spaces of
homomorphisms are taken over C. As usual, we denote by Z, Q and R the
sets of integer, rational and real numbers, respectively. We also denote by N
the set of all positive integers and by N the set of all non-negative integers.
0
The Lie algebra g = sl = sl (C) consists of the vectorspace
2 2
‰(cid:181) ¶ (cid:190)
a b
sl = : a,b,c,d ∈ C; a+d = 0
2 c d
of all complex 2 × 2 matrices with zero trace and the binary bilinear op-
eration [X,Y] = XY − YX of taking the commutant of two matrices on
this vectorspace. Here XY denotes the usual (associative) product of the
matrices X and Y. To simplify the notation we will usually denote the Lie
algebra sl simply by g.
2
Exercise 1.1.1. Prove that for any two square complex matrices X and Y
of the same size the matrix [X,Y] has zero trace.
FromExercise1.1.1itfollowsthattheoperation[·,·]ongiswell-defined.
The fact that g is a Lie algebra means that it has the following properties:
Lemma 1.1.2. (a) For any X ∈ g we have [X,X] = 0.
(b) For any X,Y,Z ∈ g we have [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] = 0.
Proof. We have [X,X] = XX −XX = 0, proving the statement (a). The
1
2 CHAPTER 1. FINITE-DIMENSIONAL MODULES
statement (b) is proved by the following computation:
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] =
X(YZ −ZY)−(YZ −ZY)X +Y(ZX −XZ)−
−(ZX −XZ)Y +Z(XY −YX)−(XY −YX)Z =
XYZ −XZY −YZX +ZYX +YZX −YXZ−
−ZXY +XZY +ZXY −ZYX −XYZ +YXZ = 0.
Exercise 1.1.3. Show that the condition in Lemma 1.1.2(a) is equivalent
to the following condition: [X,Y] = −[Y,X] for all X,Y ∈ g.
The condition in Lemma 1.1.2(b) is called the Jacobi identity. The as-
sertion of Exercise 1.1.3 is true over any field of characteristic different from
2 and basically says that the operation [·,·] is antisymmetric.
From the definition we have that elements of the algebra g are given by
four parameters and one non-trivial linear relation. This means that this
algebra has dimension three. We now fix the following natural basis of g:
(cid:181) ¶ (cid:181) ¶ (cid:181) ¶
0 1 0 0 1 0
e = , f = , h = .
0 0 1 0 0 −1
By a direct calculation one gets the following Cayley table for the oper-
ation [·,·] in the natural bases:
[·,·] e f h
e 0 h −2e
f −h 0 2f
h 2e −2f 0
Another way to write down the essential part of the information from the
above Cayley table (the diagonal of the table is fairly obvious and given by
Lemma 1.1.2(a)) is the following:
[e,f] = ef −fe = h,
[h,e] = he−eh = 2e, (1.1)
[h,f] = hf −fh = −2f.
A module over g (or, simply, a g-module) is a vector space V together
with three fixed linear operators E = E , F = F and H = H on V,
V V V
which satisfy the right-hand side equalities in (1.1), that is
EF −FE = H, HE −EH = 2E, HF −FH = −2F. (1.2)