Lectures on SL2(C)-Modules [Illustrated] 184816517X, 9781848165175

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Lectures on sC2(CC)-modules

Volodymyr Mazorchuk Uppsala University, Sweden

~f)r---~~~~-Im_pe_ria_lC_oll_eg_eP_re_ss_

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LECTURES ON SL_2(C)-MODULES Copyright© 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-1-84816-517-5 ISBN-10 1-84816-517-X

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Preface

The representation theory of Lie algebras is an important and intensively studied area of modern mathematics with applications in practically all major 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 finitedimensional, highest weight or Harish-Chandra modules. The smallest simple Lie algebra s[2 differs in many aspects from all other semi-simple Lie algebras. One could, for example, mention that s[2 is the only semi-simple Lie algebra for which all simple (not necessarily finite-dimensional) modules are in some sense understood. The algebra s[ 2 is generated by only two elements and hence is an invaluable source of computable examples. Moreover, in many cases the ideas which one gets from working with s[ 2 generalize relatively easily to other Lie algebras with a 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 s[ 2 , with a special emphasis on explicit examples. Using this Lie algebra, we can examine and describe many more aspects of the representation theory of Lie algebras than are covered in standard textbooks. The notes start with two conventional introductory chapters on finitedimensional modules and the universal enveloping algebra. The third chapter moves on to the study of weight modules, including a complete classification and explicit construction of all weight modules and a description of the category of all weight modules with finite-dimensional weight spaces, 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-

v

vi

Lectures on sl2(C)-modules

Gelfand category 0 and its properties. The two last chapters contain a description of all simple s[ 2 -modules and various categorifications of simple finite-dimensional modules. The material presented in the last chapter is based on papers which were published in the last two years. The notes are primarily directed towards postgraduate students interested in learning the basics of the representation theory of Lie algebras. 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 for understanding these notes depend on the chapter. For the first two chapters, one needs only some basic knowledge in linear 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. At the end of each chapter are comments including some historical 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 for any unforeseen errors or omissions. There are numerous exercises in the main text and at the end of each chapter. The exercises in the main text are usually relatively straightforward and required to understand the material. It is strongly recommended that the reader at least looks through them. Answers and hints are supplied 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.

Uppsala,. August 2009

Volodymyr Mazorchuk

Contents

Preface l.

Finite-dimensional modules 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

2.

3.

v

1

The Lie algebra s[ 2 and s[ 2 -modules Classification of simple finite-dimensional modules Semi-simplicity of finite-dimensional modules . Tensor products of finite-dimensional modules . Unitarizability of finite-dimensional modules. Bilinear forms on tensor products . Addenda and comments Additional exercises . . . .

1

5 10 15 17 21 23 26

The universal enveloping algebra

33

2.1 2.2 2.3 2.4 2.5 2.6 2. 7 2.8

33 37 41 44 48 50 52 56

Construction and the universal property . . . . . . . . Poincare-Birkhoff-Witt Theorem . . . . . . . . . . . . Filtration on U(g) and the associated graded algebra . Centralizer of h and center of U(sC 2 ) Harish-Chandra homomorphism. Noetherian property . . Addenda and comments Additional exercises

Weight modules

59

3.1 3.2 3.3

59 62 68

Weights and weight modules Verma modules Dense modules . . . . . vii

Lectures on s[2(C)-modules

viii

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4.

5.

72 75 82 85 86 90 92 101 103 108

The primitive spectrum

115

4.1 4.2 4.3 4.4 4.5 4.6 4.7

115 117 119 121 124 128 132

Annihilators of Verma modules Simple modules and central characters Classification of primitive ideals . Primitive quotients . Centralizers of elements in primitive quotients . Addenda and comments Additional exercises

Category 0

135

5.1 5.2 5.3 5.4 5.5 5.6 5.7

135 139 144 149 153 160

5.8 5.9 5.10 6.

Classification of simple weight modules . Coherent families . Category of all weight modules with finite-dimensional weight spaces Structure of wl;,T in the case of one simple object Structure of wl;,T in the case of two simple objects Structure of wl;,T in the case of three simple objects Tensoring with a finite-dimensional module Duality Addenda and comments Additional exercises

Definition and basic properties Projective modules . Blocks via quiver and relation . Structure of a highest weight category Grading Homological properties . Category of bounded linear complexes of projective graded D-modules . Projective functors on 0 0 Addenda and comments Additional exercises

164 170 178 189

Description of all simple modules

195

6.1 6.2 6.3

195 197 201

Weight and nonweight modules Embedding into a Euclidean algebra Description of simple nonweight modules .

Contents

6.4 6.5 6.6 6. 7 7.

Finite-dimensionality of kernels and cokernels Finite-dimensionality of extensions Addenda and comments Additional exercises . . . . . . . .

ix

204 210

213 215

Categorification of simple finite-dimensional modules

219

7.1 7.2 7.3 7.4 7.5 7.6

219 221

Decategorification and categorification Nai:ve categorification of y(n) . . . . . . . . . . Weak categorification of y(n) . . . . . . . . . . Categorification of y(n) via coinvariant algebras Addenda and comments Additional exercises . . . . . . . .

Appendix A

Answers and hints to exercises

226

232 234 237

241

Bibliography

249

Index of Notation

255

Index

259

Chapter 1

Finite-dimensional modules

1.1

The Lie algebra sb and sb-modules

In what follows we will always work over the field . E .w}

and the subspace { w E W : (A - >.)kw= 0 for some k E N}

are invariant with respect to B. Applying the Jordan Decomposition Theorem to the linear operator C on V we find that

where

V(C,T) = {v EV

(C -T)kv = 0 for some k EN}.

Lemma 1.32. For any TE . -1- n-1, -n-1. Indeed, for any v E V(>.)nKer(E) we have E(H(v))

= EHv c;;;l HEv - 2Ev = 0,

and hence V(>.) n Ker(E) is invariant under the action of H. If we have the inequality V(>.) n Ker(E) -1- 0, then VA n Ker(E) -1- 0 and for any v E VA n Ker( E) we have Cv = ((H + 1) 2 + 4FE)v = (H + 1) 2 v + 4FEv = (>. + 1) 2 v. At the same time Cv = n 2 v as V = V(C,n 2 ), which implies>.= n-1 or >. = -n-1. Analogously, one can show that Facts injectively on any V(>.) such that >. -1- 1 - n, n + 1. Since V is finite-dimensional, the previous paragraph implies that the inequality V(>.) -1- 0 is possible only if>. E {-n + 1, -n + 3, ... , n - 1} and that Ker(E) = V(n - 1), Ker(F) = V(l - n). In particular, dim V(>.) = dim V(µ) for any>.,µ E {-n + 1, -n + 3, ... , n - l}. Furthermore, for any i E {1, 2, ... , n - 1} the restriction Ai of the operator Fi to the subspace V(n -1) gives a linear isomorphism from V(n -1) to V(n -1- 2i). Hence we can identify V (n - 1) and V (n - 1 - 2i) as vector spaces via the action of Ai. Set A= An-1· As C commutes with H, all V(>.)'s are invariant with respect to C. Denote by C 1 and H 1 the restrictions of C and H to V (n - 1), respectively. Denote by C 2 and H 2 the restrictions of C and H to V(l - n) respectively. Restricting cpn- 1 = pn- 1 c to V(n - 1) we get

AC1

=

C2A.

(1.13)

Analogously, using F H = (H + 2)F (see Exercise 1.16) we get AH1 = (H2

+ 2(n -

l))A.

(1.14)

As Ker(E) = V(n -1) and C = (H + 1) 2 + 4FE, we have C1=(H1+1) 2 .

(1.15)

As Ker(F) = V(l - n) and C = (H - 1) 2 + 4EF, we have

C2 = (H2 -1) 2.

(1.16)

Lectures on sb(.) =Vi for all>. E {-n + 1, -n + 3, ... , n - 1}. Let { V1, ... 'vk} be a basis of Vn-1· For i E {1, ... 'k} denote by withe linear span of {Vi, FVi, ... , pn- 1 vi}. From the above we have

and, by Corollary 1.19, each Wi is a submodule of V. Since V is indecomposable by our assumptions, we get k = 1 and thus dim Vn-1 = 1. In this case Corollary 1.19 and (1.9) imply that V ~ y(n), which completes the

D

~~

Corollary 1.34. Let V be a finite-dimensional g-module. Then

V ~ EBmn y(nl, nEN

where m n =dim Hom g (V(n) ' V) =dim Hom g (V' y(n)) . Proof. From Theorem 1.29 it follows that we can decompose V into a direct sum of simple modules, say V ~ Vi E9 · · · E9 Vk, where all the Vi's are simple. Now dim Hom 9 (V(n), V) = dim Hom 9 (V(n), EEli Vi) = LdimHom9 (V(nl, Vi) (by Schur's lemma)= l{i :

y(n)

~ Vi}I.

This proves the first equality for mn and the second equality is proved similarly. D

Finite-dimensional modules

1.4

15

Tensor products of finite-dimensional modules

Given two g-modules V and W the vector space V ® W can be endowed with the structure of a g-module as follows:

E(v ® w) = E(v) ® w + v ® E(w), F(v ® w) = F(v) ® w + v ® F(w), H(v ® w) = H(v) ® w + v ® H(w).

(1.17)

The module V ® W is called the tensor product of V and W. For n E N and any g-module V we denote by V®n the g-module

V®V®···®V. n factors

Exercise 1.35. Check that the formulae (1.17) do indeed define on V ® W the structure of a g-module. Exercise 1.36. Let V and W be two g-modules. Check that the map v ® w f--+ w ® v induces an isomorphism between V ®Wand W ® V. Exercise 1.37. Let Vi, V2 and W beg-modules. Prove that

(Vi EB V2) ® W

~

(Vi ® W) EB (V2 ® W).

Exercise 1.38. Let V, Wand Ube g-modules. Prove that V ® (W ® U) ~ (V ® W) ® U.

If both V and W are finite-dimensional, the module V ® W is finitedimensional as well. Due to Corollary 1.34, it is natural to ask how V ® W decomposes into a direct sum of simple modules (depending on V and W). Exercises 1.36 and 1.37 mean that to answer this question it is sufficient to consider the case when both V and 1¥ are simple modules. This is what we will do in this section. Our main result is the following: Theorem 1.39. Let m, n EN be such that y(n) ® y(m) ~ y(n-m+l) EB y(n-m+3) EB ..

m::; n. Then

·EB y(n+m-3) EB y(n+m-1). (1.18)

16

Lectures on .st2( 2 and that (1.18) is true for all m = 1, ... , k-1. Let us compute y(n) ® y(k-l) ® vC 2) in two different ways. On the one hand we have y(n)@ y(k-1)@ y(2)

(by Exercise 1.37) (by inductive assumption) .

(1.20) ~

C>
0 we have (vi, Vn-1-i) (by (1.25)) (by (1.9))

(by induction)

The claim follows.

(~)

(F(vi_i), - 1-E(vn-i)) an-i

1 -(vi-1, F(E(vn-i))) an-i 1

-(vi-1,an-iVn-i) an-i (vi-li Vn-i) c.

D

21

Finite-dimensional modules

For a direct sum of simple modules, the description of bilinear forms analogous to Propositions 1.45 and 1.46 will be more complicated. In particular, as an obvious observation one could point out that it is possible to independently rescale the restrictions of the bilinear form to pairwise orthogonal direct summands.

1.6

Bilinear forms on tensor products

Let V and W be two vector spaces and (·, · )i and (·, · )2 be bilinear forms on V and W respectively. Then the assignment

(v ® w, v' ® w') = (v, v')i · (w, w')2

(1.26)

extends to a bilinear form on the tensor product V ® W. Exercise 1.47. Check that the form(-,·) is symmetric provided that both (-, ·)i and (-, ·)2 are symmetric; that the form (-, ·) is non-degenerate provided that both (·, · )i and ( ·, ·)2 are non-degenerate; and that the form (-, ·) is Hermitian provided that both(·, ·h and(·, ·)2 are Hermitian. Proposition 1.48. Assume that V and W are unitarizable modules (resp. -modules) for the forms (-, ·h and (-, ·h respectively. Then V ® W is unitarizable (resp. a -module) for(·,·).

Proof. We prove the statement for unitarizable modules. For o-modules the proof is similar. Due to Exercise 1.4 7 it is sufficient to check (1.23) for XE {E,F,H}. For v,v' EV and w,w' E W we have

+ v ® X(w), v' ® w') (X(v) ® w, v' ® w') + (v ® X(w), v' ® w') (X(v), v')i · (w, w'h + (v, v')i(X(w), w')2 (v,X*(v'))i · (w,w')2 + (v,v')i(w,X*(w')h (v ® w, X*(v') ® w') + (v ® w, v' ® X*(w')) (v ® w, X*(v') ® w' + v' ® X*(w'))

(X(v ® w), v' ® w') C\;7 ) (X(v) ® w (by linearity) (by (1.26)) (by (1.23)) (by (1.26)) (by linearity) (by (1.17)) The claim follows.

(v ® w, X*(v' ® w')). D

We know that the tensor product of two simple finite-dimensional gmodules is not simple in general (see Theorem 1.39). Hence the bilinear

22

Lectures on sl2( 0. From (1.9) we have pn- 1(vo) = Vn-1· The claim follows. D Let m = 2, n ?: 2 and assume that V~n) is given by (1.9) and V~2 ) is given by (1.19). As all coefficients in (1.17) are positive, we get that Fn(vo@ ei) = CVn-1@ e2, where c > 0. As

(vo@ el, Vn-1@ e2) = (vo, Vn-1)i(e1, e2)2 = 1 > 0

23

Finite-dimensional modules

(here we used that both (-, ·)i and (-, ·h are standard), from Lemma 1.51 we obtain that the restriction of (-, ·) to the direct summand V~n+l) of v~n) ® V~2 ) is standard. For the element w = v 1 ®e 1 -(n- l)v 0 ®e 2 -j. 0 we have E(w) = 0, sow generates the direct summand V~n-l) of V~n) ® v~l. A direct computation shows that pn- 2 (w)

= Vn-l

®el - Vn-2 ® e2.

Another direct computation then shows that (v1 ®el -

(n -

l)vo ® e2,Vn-l ®el -Vn-2 ® e2)

= -n < 0.

Hence from Lemma 1.51 we obtain that the restriction of(-,·) to the direct summand V~n+i) of V~n) ® V~2 ) is non-standard. This completes the proof of the theorem in the case m = 2. Fork E N let us denote Vf,;,+l and V~k,-) the module V~k) endowed with a standard and non-standard (up to a positive real scalar) form, respectively. Then we have just proved that y(n,+) f7\ y(2,+) co,; yCn+l,+) "'y(n-1,-) (1.27) IR'.

\Yffi'.

-IR'.

'CDJR'..

Note that we obviously have V (n,+) IR'.

f7\

y(l,+)

co,;

y(n,+)

y(n,+)

-IR'.'

\Yffi'.

IR'.

f7\

y(l,-)

co,;

y(n,-)

(1.28)

-IR'..

\Yffi'.

In this notation the statement of our theorem can be written as follows: V IR'(n,+) .

f7\ \Y

y(m,+) IR'.

co,;

-

y(n+m-1,+) IR'.

LD \l)

y(n+m-3,-) IR'.

LD \.D

y(n+m-5,+) IR'.

LD

\J.J····

The induction step now follows using (1.27) and (1.28) and, rewriting in this new notation, the calculations in (1.21) and (1.22). We leave the details to the reader. D

1. 7

Addenda and comments

1.7.1 Alternative expositions for the material presented in Sections 1.1-1.4 can be found in a large number of books and articles, see for example [37, 46, 49, 57, 106]. Many of the results are true or have analogs in much more general contexts (which also can be found in the books listed above). In particular, simple finite-dimensional modules are classified (see Theorem 1.22) and Weyl's Theorem (Theorem 1.29) is true for all simple finite-dimensional complex Lie algebras. For all such algebras there is also an analog of Theorem 1.39, however its formulation is more complicated, as higher multiplicities appear on the right hand side.

24

Lectures on sl2( 1 consider some x 1 x 2 · · · Xk as above. A pair (i,j), 1 :::; i < j :::; k, will be called an inversion provided that one of the following holds: Xi

=hand

Xj

= f;

Xi=

e and x 1 = f;

Xi

= e and x 1 = h.

We proceed by induction on the number of inversions in x 1 x 2 · · · Xk· If there are no inversions, the monomial x 1 x 2 · · · Xk is standard and we have nothing to prove. Otherwise, we can fix some inversion (i, i + 1). We have

(2.1)

=

X1 ... Xi-IXi+IXiXi+2 ... Xk

+ X1 ... Xi-I [xi, Xi+1lxi+2 ... Xk·

As [xi, xi+1] E {±h, ±2e, ±2f}, the second summand is a linear combination of monomials of degree k - 1 and hence is dealt with by induction on k. The first summand, in turn, has one inversion less than x 1 x 2 · · · Xk and hence is dealt with by induction on the number of inversions. Thus these two inductions complete the proof. D

39

The universal enveloping algebra

Consider the vector space V = C[a, b, c]. Define, using the induction on the degree of a monomial, the following linear operators F, H and E on V:

(2.5) (2.6) i,j i i

= 0,

= O,j -1- 0, -1- 0.

(2.7)

where i,j, k E No. Exercise 2.15. Check that the formulae (2.5)-(2. 7) do give well-defined linear operators on C[a, b, c]. Exercise 2.16. Check that the formulae (2.5)-(2.7) can be rewritten as follows:

F(aibj ck) =

ai+ 1 ~ ck; j+l k

H(aibjck)

= {

E(aibjck)

=

b c ' F(H(ai-lbjck))

i = 0,

+ [H, F]ai-lbjck,

{~;~(bj-lck)) + [E, F(E(ai-l~ck))

i

-I- O; i,j

H](bj-lck),

+ [E, F](ai-l~ck),

= 0,

= O,j -I- 0, i -1- 0. i

Lemma 2.17. The formulae (2.5)-(2.7) define on V the structure of a g-module.

Proof. We have to check the three relations from (1.2). Let us start with the relation [H, F] = -2F. For i,j, k E No we have H(F(aibjck)) (~) H(ai+ 1 ~ck) (by (2.6)) = F(H(aibjck)) - 2ai+ 1 ~ck (by (2.5)) = F(H(aibjck)) - 2F(ai~ck) and the relation [H, F] = -2F is proved. The relation [E, F] =His proved using the following computation: for i, j, k E N0 we have

E(F(aibjck)) (~) E(ai+ 1 bjck) (by (2.7))

= F(E(aibjck)) + H(aibjck).

Lectures on

40

s[2 ( IC)-modules

Finally, let us prove the relation [H, E] = 2E, which we first write in the form EH - HE= -2E. For any j, k E No we have E(H(bick)) (~) E(bi+ 1 ck)

(by (2.7))

=

H(E(bick)) - 2E(bick)

and the relation [H, E] = 2E is proved on monomials of the form bi ck. The really tricky thing is to prove this relation on monomials aibj ck, where i EN and j, k E N0 . We do this by induction on i. The case i = 0 is already established, so we prove the induction step. We rewrite [H,E] = 2E asHE-EH-2E = 0. ApplyingHE-EH-2E to aibick, where i EN and j, k E N0 , and using Exercise 2.16 we obtain (HE - EH - 2E)(aibick) = (HFE

+ H[E,F]

- EFH - E[H,F] - 2FE - 2[E,F])(ai-lbjck).

(2.8) By induction we have - 2F E = F [E, H]. Using the definition of the commutator and the relation [H, F] = -2F, which we proved above, we also have H[E,F]

=

HEF- HFE,

E[H,F] = EHF- EFH,

-2[E, F]

=

[E, [H, F]].

This reduces the equality (2.8) to

As we have already proved that [E, F] = H, we can add the following zero term: 0 = [H,H] = -[H,H] = [H, [F,E]] to the equality (2.9) and obtain (HE - EH - 2E)(aibjck)

= ([F, [E, HJ]+ [E, [H, F]] + [H, [F, E]])(ai- 1 b5ck). The right-hand side of the latter is equal to zero because of the Jacobi identity for .C(V)(-). This completes the proof. D

The universal enveloping algebra

41

Now we are ready to prove the PBW Theorem 2.13.

Proof. To prove that standard monomials form a basis in U(fJ), we have to check that they generate U(fJ) and that they are linearly independent. The fact that they generate U(fJ) was proved in Lemma 2.14. To prove that standard monomials are linearly independent, consider the U(fJ)-module V from Lemma 2.17. Note that for all i,j,k E No for the constant polynomial 1 E V we have FiHj Ek(l)

= ait?ck.

Now it is left to observe that the elements aibjck EV are linearly independent. Hence the linear operators Fi Hj Ek are also linearly independent. Since these linear operators are exactly the images of standard monomials under the homomorphism defining the U(fJ)-module structure on V, we conclude that standard monomials are linearly independent as well. This completes the proof. D Exercise 2.18. Let x 1 , x 2 and x 3 be the elements e, f and h written in some order. Show that the standard monomials xtx~x~, i,j,k E No, also form a basis of U (fJ). Corollary 2.19. The canonical embedding

F:

of fJ into U(fJ)(-) is injective.

Proof. This follows from the fact that the elements e, f and h form a basis of fJ and the fact that the elements c( e) = e, c( f) = f and c(h) = h are linearly independent in U(fJ) by Theorem 2.13. D After Corollary 2.19 it is natural to identify fJ with c(fJ). Remark 2.20. There exists an alternative (and somewhat easier) argument for Corollary 2.19. The elements e, f and h, which form a basis of fJ, act linearly independently on the natural module (since it is given by the identity map). From Proposition 2. 7 we have that this action coincides with the action of s(e) = e, c(f) = f and s(h) = h. Hence the latter elements must be linearly independent in U(fJ) and thus the map E must be injective.

2.3

Filtration on U(g) and the associated graded algebra

As usual, for a monomial x 1 x 2 · · · Xk E U(fJ) (where Xi E {f, h, e} for all i) the number k is called the degree of the monomial. The degree of the

Lectures on .s[2(C)-modules

42

monomial u is usually denoted by deg(u). For i E N0 denote by U(g)Cil the linear span of all monomials of degree at most i (we also set U (g) (-l l = 0). This gives us the following filtration on U(g): U(g)

=

LJ U(g)Cil. iENo

Note that U(g)C 0 l = . E C denote

V;.={vEV: H(v)=>-·v}. The number >. is called a weight and the space V;. is called the corresponding weight space. The module V is called a weight module provided that

v

=EBY;..

(3.1)

.AEIC

For a weight g-module V we define the support supp V of Vas follows: supp(V)

= {>.

E C : V;. -=/:- O}.

Example 3.1. From (1.9) we have that n-1 vcn)

=

EB v~~l-2i· i=O

Hence

y(n)

is a weight module and supp y(n) = {1 - n, 3 - n, ... , n - 3, n - 1}. 59

Note that supp y(n) is invariant with respect to the action of the Weyl group S 2 of fJ (see Section 2.5). Exercise 3.2. Let V be a finite-dimensional g-module. Prove that V is a weight module and that supp V is invariant with respect to the action of S 2 (as described in Section 2.5). Exercise 3.3. Let V be a g-module. Identify b* = C as in Section 2.5. Show that for every x Eb, A Eb* and v EVA we have x(v) = A.(x) v. Exercise 3.4. Let V and W be two weight g-modules and cp: V ___, W be a homomorphism. Show that cp(V;_) C WA for any A. E C. Denote by 2U the full subcategory of the category g-mod, consisting of all finitely generated weight modules. To understand the structure of 2U we start with some elementary observations. Lemma 3.5. Let V be a weight g-module. Then for every A.EC we have

Proof.

This is analogous to Lemma 1.15: For v EVA we have

H(E(v))

(~) E(H(v)) + 2E(v) v~>- A.E(v) + 2E(v) =(A.+ 2)E(v).

The second inclusion is proved similarly.

D

Exercise 3.6. Let V be a weight module, A. E C and i E Z. Show that U(g)2i VA C VA+2i· Consider the additive subgroup 2Z of C and the corresponding set C/2Z of cosets. For a weight g-module V and~ E C/2Z set

v~

=EB vA. AE~

Denote by W~ the full subcategory of W, consisting of all modules V such that supp V c ~Corollary 3. 7. (i) Let V be a weight g-module. Then for every ~ E C/2Z the subspace v~ is a submodule of and we have

v

(3.2)

61

Weight modules

(ii} We have

Proof. Let A E ( Then A± 2 E ~ by definition. Thus Lemma 3.5 implies that V~ is invariant with respect to the action of both E and F. As H = EF - FE, V~ is also invariant with respect to the action of H. This means that V~ is a submodule of V. The decomposition (3.2) follows now from the decomposition (3.1). This proves the statement (i). The statement (ii) follows from the statement (i) and Exercise 3.4. D Proposition 3.8.

(i} (ii} (iii} (iv}

Every submodule of a weight module is a weight module. Every quotient of a weight module is a weight module. Any direct sum of weight modules is a weight module. Any finite tensor product of weight modules is a weight module.

Proof. Let V be a weight module and W C V be a submodule. Since V is a weight module, for any w E W C V we can write w = w1 + w2 + · · ·+Wk, where Wi E V are weight vectors for all i = 1, ... , k. Without loss of generality we may assume that all Wi =/= 0, that Wi has weight Ai and that the Ai 's are pairwise different. For every i E {1, ... , k} consider the element hi= (h

Ai)(h - A2) ... (h - Ai-1)(h - Ai+i) ... (h - Ak) E U(g).

Then we have i=/=j; i

=

j.

Hence

W 3 hi(w)

= hi(wi) = IJ(Ai - As)wi =/= 0. sopi

This means that wi E W and thus that every vector from W is a sum of weight vectors from W. This proves the statement (i). Let V be a weight module and W c V be a submodule. Then the image of any [J-eigenbasis of V in V /W is, obviously, a generating system consisting of [)-eigenvectors. This proves the statement (ii). The statement (iii) follows from the observation that H (v) = Av and H(w) =Aw implies H(v EB w) = A(v EB w).

Lectures on

62

.s[2 (C)-modules

Finally, the statement (iv) follows from the observation that H (v) = AV and H (w) = µw implies

H(v ® w) (as H(v) =Av and H(w) = µw)

ci;;7l H(v) ® w + v ® H(w) AV® w + v ® µw

= (A+µ)v®w. D

This completes the proof. Exercise 3.9. Let V and W be weight modules. Show that (V ® W) A

=

EB

Vµ ® Wv.

µ+v=A

Exercise 3.10. Let V and W be weight modules. Show that supp V EB W =supp VU supp W;

supp V ® W =supp V +supp W.

Exercise 3.11. Let a:: M '----t N---» K be a short exact sequence of weight modules. Prove that for every A E C the sequence a: induces a short exact sequence MA '----t NA ---» KA. Proposition 3.12. A module generated by weight vectors is a weight module.

Proof. It is enough to prove the statement for a module V, which is generated by one weight vector v, say of weight A. Then, by the PBW Theorem, the module V is generated, as a vector space, by the elements of the form Phjek(v), i,j,k E N0 . We have h(fihjek(v))

= thjek(h(v)) + [h,fihjek](v)

(by Lemma 2.29) = Afihjek(v) + 2(k - i)fihjek(v)

=(A+ 2(k - i))fihjek(v). Hence all vectors

3.2

Ji hJ ek (v)

are weight vectors. The claim follows.

D

Verma modules

We already know many abstract properties of weight modules. However, we do not know any other example of weight modules apart from the modules y(n), n EN, and their direct sums (possibly infinite). In particular, we have

Weight modules

63

no idea whether the categories ®~, considered in the previous section, are nontrivial. So, it is a good time now to construct new examples of weight modules. So far the only simple modules we know are the modules y(n), n EN, given by (1.9). Let us try to think how we can extend the construction given by (1.9). A good suggestion is made by the proof of Lemma 1.18. Indeed, the induction described in this proof works for all i E N, not only for i E {1, 2, ... , n-1 }. This motivates to extend the picture (1.9) to simply consider vectors Vi such that i E N0 (that is not only i E {1, 2, ... , n - 1}) and define on them the action of fl similar to (1.9). As we will see below, this works. It even admits a straightforward generalization, that the original vector v 0 may have any weight. With this in mind, fix>. E C. Consider the vector space M(>.) with the formal basis {vi : i E N0 }. For i EN set ai = i(>. - i + 1). Consider the linear operators E, F and Hon M(>.) defined as follows:

i

# O;

i

=

(3.3)

0.

This can be depicted as follows:

(3.4) Proposition 3.13. {i) The formulae (3.3) define on M(>.) the structure of a weight g-module. {ii) suppM(>.) = {>. - 2i : i E N0 }. {iii) The Casimir element c acts on M(>.) as the scalar(>.+ 1) 2 .

Proof. First we observe that, by definition, the vector vi, i E No, is an eigenvector for H with eigenvalue >. - 2i. Since E increases the eigenvalue by 2 and F decreases the eigenvalue by 2, the relations [H, E] = 2E and [H, F] = -2F are obviously satisfied. Hence we need to check only the relation [E, F] = H. It is enoi"i.gh to check this relation on the elements Vi·

Lectures on sb(C)-modules

64

If we set a 0 = 0, then for every i

E

N0 we have

(EF - FE)( vi) (~) (ai+l - ai)vi

((i + 1)(>. - (i + 1) + 1) - i(A. - i + l))vi

(by definition of ak)

(>. - 2i)vi

H(vi)·

(by (3.3))

Hence the formulae (3.3) define on M(>.) the structure of a g-module. This module is a weight module as {vi} is an eigenbasis for H. The claim (i) follows. The claim (ii) then follows immediately from the definitions. For the claim (iii) we note that E(v 0 ) = 0 and hence c(v0 ) = (>.+ 1) 2 v 0 by Proposition 2.35. For i EN we then have

c(vi) (~) c(Fi(vo)) (by Exercise 2.1) (by above) (by (3.3))

Fi(c(v 0 )) (>.+ 1) 2 Fi(v0 )

(>.

+ 1) 2 v;.

This completes the proof.

D

If Vis a weight g-module andµ E supp Vis such thatµ+ 2 ~supp V, then the weight µ is called a highest weight and any non-zero v E Vµ is called a highest weight vector. A module, generated by a highest weight vector, is called a highest weight module. For example, the weight >. is the unique highest weight of the module M(A.) and the vector v 0 is a highest weight vector of weight >.. In particular, M(A.) is a highest weight module. From the construction we have that the module M(A.) is uniquely determined by its highest weight. The module M(A.) is called the Verma module with highest weight >.. By Proposition 3.13(iii) the Casimir element c acts on M(>.) as the scalar T = (>. + 1) 2 . Define the homomorphism XA : Z(g)

= C[c]

-+

.) as the scalar g(T). The homomorphism X>.. is called the central character of M(>.). More generally, if every element u E Z(g) acts on some module Mas a scalar XM(u), then XM : Z(g) -+ .)

65

Weight modules

as a U(g)-module, the module M(>.) must then be a quotient of the free U(g)-module U(g). This gives us the following alternative description of

M(>.): Proposition 3.14. Let I denote the left ideal of U(g), generated bye and

h - >.. Then M(>.)

~

U(g)/ I.

Proof. As the U(g)-module U(g) is free of rank one, the assignment 1 t--+ v 0 extends to a homomorphism r.p : U(g) __, M(>.). As M(>.) is generated by v 0 , this homomorphism is surjective. Let K denote the kernel of r.p. From (3.3) we have E( v 0 ) = 0 and (H - >.)( v0 ) = 0. Hence e, h- >. E K. This means I c Kand the homomorphism r.p factors through U(g)/ I. Let fj5: U(g)/ I_...,. M(>.) denote the induced epimorphism. We claim that U (g) /I is spanned by the images of {Ji : i E No}. By the PBW Theorem it is enough to show that the image in U(g)/ I of any standard monomial from U(g) can be written as a linear combination of (the images of) some Ji's. For i,j E No and k EN, every standard monomial fhJek belongs to I. Hence such monomials are zero in U(g)/I. For i E No and j E N we have

Jihj = Ji(h - ).. + >.)j =

~

C)

;..J-s Ji (h - >.) 8 = )..J f

+ U,

where u E J. Therefore U(g)/I is generated by the images of {Ji : i E N0 }. At the same time, the images of these generators under fj5 are exactly the vi's, which are linearly independent. Hence {Ji : i E N0 } is a basis of U (g) /I and fj5 is an isomorphism. This completes the proof. D Corollary 3.15 (Universal property of Verma modules).

(i) Let V be a g-module and v EV be such that E(v) = 0 and H(v) = >.v. There then exists a unique homomorphism r.p E Hom 9 (M(>.), V) such that r.p(vo) = v. (ii) Let V be a g-module, generated by a highest weight vector of weight >.. Then V is a quotient of M(>.). Proof. Consider U(g) as a free left U(g)-module of rank one. Then we have a unique homomorphism '!/; E Hom 9 (U(g), V) such that 1/;(1) = v. The equalities E(v) = 0 and H(v) = >.v imply that e and h - >. belong to the kernel of'!/; and hence '!/; factors through the module U(g)/ I from

66

Lectures on sC2(C)-modules

Proposition 3.14. The statement (i) now follows from Proposition 3.14. The statement (ii) follows directly from the statement (i). D Now we can describe the structure of Verma modules. Theorem 3.16 (Structure of Verma modules).

(i) The module M(>.) is simple if, and only if, >. tj_ No. (ii) For n E N0 the module M(n) is indecomposable. Furthermore, the module M(-n - 2) is a (unique) simple submodule of M(n) and we have M(n)/M(-n - 2) ~ y(n+l). Proof. Let >. tj_ No, V C M(>.) be a non-zero submodule and v E V, v #- 0. Then for some k E No we have v = 2=7=o aivi and ak #- 0. As >. tj_ N0 , in (3.3) we have ai #- 0 for all i E N. Hence from (3.3) we obtain that Ek (v) = akEk (Vk) is a non-zero multiple of v 0 . Therefore v 0 E V and thus V = M(>.). This means that the module M(>.) is simple for >. tj_ N0 . Let now n E No. In (3.3) we have an+l = 0, which means that the vector Vn+l of M(n) satisfies E(vn+d = 0 (by (3.3)). We also have H(vn+d = (-n - 2)vn+l by (3.3). Hence, by the universal property of M(-n - 2) (Corollary 3.15), we have a non-zero homomorphism from M(-n - 2) to M(n). In particular, this implies that M(n) is not simple, proving (i). From the previous paragraph we have that the module M(-n - 2) is simple. The submodule M(-n - 2) of M(n) has the basis {vn+1,Vn+2, ... } and hence the quotient M(n)/M(-n-2) has the basis {v 0 , v1 , ... , vn}· The action of g in this basis is given by (1.9). This means that M(n)/M(-n-2) ~ y(n+l). Now let V be any non-zero submodule of M(n) and v E V, v #- 0. As the action of Fon M(n) is injective by (3.3), we have that Fi(v) #- 0 for all i. On the other hand, the vector Fn+l(v) is obviously a linear combination of {Vn+l, Vn+2, ... } and hence belongs to the submodule M(-n - 2). This implies that every non-zero submodule of M(n) intersects M(-n - 2). In particular, the submodule M(-n - 2) is a unique simple submodule and the module M(n) is indecomposable. This completes the proof. D

By Theorem 3.16 for n E No the non-simple module M(n) is uniserial and has the following tower of subquotients (say in the radical filtration):

M(n):

y(n+l)

I

M(-n- 2).

Weight modules

67

Exercise 3.17. For>..,µ E ..), M(µ))

=

.), { .. = -n - 2, µ = n, n

E

No;

otherwise;

where !fn : M(-n - 2) ___, M(n) is some fixed non-zero homomorphism. Deduce that every non-zero homomorphism between Verma modules is injective.

Corollary 3.18 (Classification of simple highest weight modules). For every>.. E ..) with highest weight >... Moreover, we have

L >.. - {M(>..), ( ) -

y(n+l),

>.. tf_ No;

(3.5)

>.. = n E No.

Proof. Let V be a simple module with highest weight >... The universal property of Verma modules gives us an epimorphism M(>..) ---» V. From Theorem 3.16 we have that M(>..) has a unique simple quotient, which is given by (3.5). D Corollary 3.19. For n E N 0 we have a non-split short exact sequence 0 ___, M(-n - 2) ___, M(n) ___,

Proof.

y(n+l) ___,

This follows immediately from Theorem 3.16.

0.

D

Exercise 3.20. For >.. E ..) be the formal vector space with the basis {wi : i E N0 }. For i EN set bi= -i(>.. + i - 1) and define operators E, F and Hon M(>..) via:

E(wi) = Wi+l; H(wi) = (>.. + 2i)wi; F(wi)

(3.6)

= {biwi-1, 0,

i

=

0.

Check that this defines on M(>..) the structure of a weight g-module with support {>.. + 2i : i E No}.

Exercise 3.21. Show that the Casimir element c acts on M(>..) as the scalar (>.. - 1) 2 .

llectures' on sb(iC)-modules

68

If V is a weight g-m0clul(j) and µ E supp Vis smd:lt: tnait µ- 2 tf. supp V, then the weight µ is called a lowest weight and any non-zero v E Vµ is camed' a lowest weight vector. A module generated by a lowest weight vector is called a lowest weight module.

Exercise 3.22. Give an alternative description of the module M(.\), analogous to Proposition 3.14 .. Formulate and prove for M(.\) a lowest weight analog of the universal property. Exercise 3.23. Show that M(.\) has a unique simple quotient L(.\) and thaii

-

L(.\) =

{M(.\), y(n+1),

-.\ tf.No;

-A= n E No.

Exercise 3.24. Show that modules {L(.\) : A E C} classify all simple lowest weight modules.

3.3

Dense modules

In the previous section we have constructed many examples of simple weight g-modules. However, they all had either a highest or a lowest weight. A natural question is: Is it possible to construct a weight module without both highest and lowest weights? Later on in this section we will show that this is possible. A motivating example for our constructions is the following observation: For some fixed A E C consider the vector space V = M (A) E9 M (A + 2). Let the action of E, F and Hon V be given as for the usual direct sum of modules M(.\) and M(.\+2) with one exception: instead of F((O,w0 )) = 0 we set F((O,wo)) = (va,O). Exercise 3.25. Check that the above defines on V the structure of an indecomposable g-module. Show further that Supp V =A+ 2Z. There is, of course, a dual version of the above construction. Consider the vector space V' = V. Let the action of E, F and H on V' be given as for the usual direct sum of modules M (,\) and M (A +2}with one exception: instead of E((v 0 ,0)) = 0 we set E((v 0 ,0)) = (0,w 0 ). Exercise 3.26. Check that the above defines on V' the structure of an indecomposable g-module, that V' '¥- V and that Supp V' = A+ 2Z.

Weight modules

69

From Corollary 3. 7(i) we have that both modules V and V' constructed above have maximal possible supports for indecomposable modules. Modules with this property are called dense modules. The aim of this section is to construct many more examples of dense, especially of simple dense modules. To make things explicit, we call a weight g-module V dense provided that Supp V = >. + 2.Z for some >. E C. Fix now~ E C/2.Z and T E C Consider the vector space V(~, T) with the basis {Vµ : µ E 0- Consider the linear operators E, F and H on V (C T) defined as follows:

F(vµ) = Vµ-2; H(vµ) = µvµ; E(vµ) = ~(T - (µ Setting aµ= ~(T - (µ depicted as follows:

+ 1) 2 )

(3.7)

+ 1) 2)vµ+2·

forµ E ~and fixing some>. E ~'this can be

Lemma 3.27.

(i) The formulae (3.7) define on the vector space V(~, T) the structure of a dense g-module with support~(ii) The Casimir element c acts on the module V(~, T) as the scalar T.

Proof. Just as in the proof of Proposition 3.13 we only have to check the relation [E, F] = H, when applied to the element v;_. Formulae (3.7) reduce this to the following obvious identity: a;__ 2-a;_ = >.. That Supp V(~, T) = ~ now follows from the definition. This and all definitions imply the statement (i). The statement (ii) follows then by a direct calculation. D Exercise 3.28. Let ~ E C/2.Z and T E C Show that every g-module V satisfying the following conditions is isomorphic to the module V(~, T):

(a) (b) (c) (d)

Supp V = ~; c acts on V as the scalar Ti dim V;_ = 1 for some>. E ~; Facts bijectively on V.

For T E C set g7 (>.) = T - (>.+ 1) 2 E C[>.]. Note that g7 (>.) is a quadratic polynomial, so it has at most two different complex roots. Now we are ready to describe the structure of the modules V(~, T).

70

Lectures on sl2(C)-modules

Theorem 3.29 (Structure of

V(~,

T)).

Let~

E C/2Z and TE .). (iii) If~ contains exactly one root of the polynomial gr(>.), say µ, then the module V (~, T) contains a unique simple submodule M (µ) and the quotient V(~, T)/M(µ) ~ M(µ + 2) is also simple. (iv) If~ contains two different roots of the polynomial gr(>.), say µl and µ2, then T = n 2, µ 1 = n - 1 and µ2 = -n - 1 for some n E N. Moreover, V(~, T) is a uniserial module of length three; it contains a unique simple submodule M(-n - 1), a unique non-simple proper submodule M (n- l) (and hence the subquotient M (n- l) / M ( -n- l) ~ y(n)) and the quotient V(~, T)/M(n - 1) ~ M(n + 1) is simple.

Proof. Let µ E ~ and cp E End 9 (V (~, T)). Since cp commutes with the action of h, and V(~, T)µ has basis vµ by definition, we have cp(vµ) =avµ for some a E C. Using this, and the fact that cp commutes with the action off, for every i > 0 we have

Similarly we have Fi(cp(vµ+2i))

=

cp(Fi(vµ+2i))

= cp(vµ)

=avµ= aFi(vµ+2i)·

As pi : V(~, T)µ+ 2i ---+ V(~, T)µ is bijective by our construction of V(~, T), we derive cp(vA) =av,\ for all>. E ~· The statement (i) follows. Assume that ~ does not contain any root of the polynomial gr (>.). Let v E V(~, T), v # 0, and M denote the minimal submodule of V(~, T), containing v. Then we have v = :Z::::µE~ aµvµ, where only finitely many aµ's are non-zero. Let

where the µi's are different pairwise. Then v then we have vµ, EM. Otherwise set u

#

0 implies k 2 l. If k

= (h - µ2)(h - µ2) · · · (h - µk)

E

U(g).

Then u(v) EM and u(v)

= 0'.µ 1 (µ1 - µ2)(µ1 - µ2) · · · (µ1 - µk)vµ, # 0.

=

1,

Weight modules

71

This again implies that vµ, E M. Applying F inductively and using (3.7) we get Vµ 1 _ 2i E M for all i E N0 . As ~ does not contain any root of the polynomial gr(>..), applying E inductively and using (3.7) we get vµ, + 2i EM for all i E N 0 . This yields M = V(~, T), which proves that V(~, T) is simple. Assume that ~ contains exactly one root of the polynomial gr (>..), say µ. Then we have E(v1;,} = 0 and H(vµ) = µvµ by (3.7). Hence, by the universal property of Verma modules (Corollary 3.15), we have a non-zero homomorphism from M(µ) to V(~, T) (which sends the generator v 0 of M(µ) to the element vµ of V(~, T)). Note thatµ tf. N0 for otherwise-µ- 2 would be a second root of gr(>..) in f Hence M(µ) is simple by Theorem 3.16(i) and thus M(µ) is a simple submodule of V(~, T). The quotient V(~, T)/M(µ) has a basis consisting of the images 1Yµ+2i of Vµ+2i, i EN. We have F(vµ+ 2 ) = 0 and H(vµ+ 2 ) = (µ + 2)1!µ+ 2. Sinceµ was the only root of gr(>..) in~' we have Ei(vµ+ 2 ) # 0 for all i E N and hence the elements Ei(vµ+ 2 ), i E N0 , form a basis of V(~, T)/ M(µ). Using Exercise 3.22 we get V(C T)/M(µ) ~ M(µ + 2), which is simple by Exercise 3.23. This proves the statement (iii). Finally, assume that ~ contains two different roots of the polynomial gr(>..), say µi and µ2. Then we may assume µ2 = µi - 2n for some n EN. This gives us

which yields µ 1 = n - 1 E N0 , T = n 2 and µ 2 = -n - l. As in the previous paragraph, we get the inclusion M(n - 1) '---* V(~, T) and the quotient V(~, T)/M(n - 1) ~ M(n + 1), the latter being simple by Exercise 3.23. That M(-n - 1) '---* M(n - 1) follows from Theorem 3.16. This proves the statement (iv), and also completes the proof of the statement (ii) and thus of the whole theorem. 0 By Theorem 3.29, the module V(~, T) is always uniserial. If ~ contains exactly one root of the polynomial gr(>..), sayµ, then V(~, T) has the following tower of simple subquotients in the radical filtration: V(~, T)

:

M(µ

+ 2)

I M(µ). For every n E No the module V(n - 1 + 2Z, n 2) has the following tower of simple subquotients in the radical filtration:

72

Lectures on s{z(IC)-modules

V(n -1+2Z,n2 ):

M(n+ 1)

I y(n)

I M(-n-1). Corollary 3.30.

(i) If the coset ~ contains exactly one root of the polynomial g7 (>.), say µ, then we have a non-split short exact sequence

0--+ M(µ) --+

V(~, T)

--+ M(µ + 2) --+ 0.

(ii) For every n E N we have a non-split short exact sequence

0--+ M(n - 1)--+ V(n - 1+2Z, n 2 )--+ M(n + 1) --+ 0. Proof.

This follows directly from Theorem 3.29.

D

Exercise 3.31. Let n EN. Then we have M(-n -1) C V(n - 1+2Z, n 2 ) by Theorem 3.29. Show that the quotient V(n-1 +2Z, n 2 )/M(-n-1) has a lowest weight vector of weight -n+l but is not isomorphic to M(-n+l).

3.4

Classification of simple weight modules

In the previous sections we constructed many examples of weight modules. Most of these examples were, in fact, simple weight modules. Now we are ready to give a complete classification of such modules. Theorem 3.32 (Classification of simple weight s( 2 -modules). Each simple weight g-module is isomorphic to one of the following (pairwise non-isomorphic) modules:

(i) y(n) for some n E N. (ii) M(>.) for some>. EC\ N0 . (iii) M(->.) for some>. EC\ No. (iv) V(~, T) for some~ E C/2Z and TE C such that T =f. (µ µ E ~-

+ 1) 2

for all

Weight modules

73

To prove this theorem we will need some preparation. Lemma 3.33. Let V be a simple weight g-module and>. E supp V. Then

Vi. is a simple C[c]-module. Proof. Assume that this is not the case and let W' c V.\ be a proper C[c]submodule. Set W = U(g)W'. We daim that Wis a proper submodule of V. Obviously, Wis a non-zero submodule as W ~ W'-/=- 0. The module W is a weight module by Proposition 3.8(i). Let us show that W.\ = W'. The inclusion W' c W.\ is obvious. From Exercise 3.6 we have W.\ = U(g) 0 W'. By Proposition 2.30(iii), the algebra U(g) 0 is generated by c and h. The space W' is invariant with respect to h, as V is a weight module and W' C V.\. The space W' is invariant with respect to c by our assumption. Hence W.\ = W'-/=- V.\, which means that W-/=- V. Therefore Wis a proper submodule of V and hence V is not simple; a contradiction. This completes the proof. 0 Lemma 3.34. Every simple C[c]-module is one-dimensional.

Proof. Let V be a simple C[c]-module. If V is finite-dimensional, then the linear operator c on V has an eigenvector, which generates a onedimensional C[c]-submodule of V. As Vis simple, it must therefore coincide with this submodule. Let us now show that every infinite-dimensional C[c]-module is not simple. Assume that V is a simple infinite-dimensional C[c]-module. As the kernel of c is always a submodule, it must then either be V or 0. In the first case any subspace of V is a submodule and hence V is not simple. This means that c is injective. As the image of c is always a submodule, it must then either be V or 0. In the second case, any subspace of V is a submodule and hence Vis not simple. This means that c is surjective and, in particular, bijective. Let v E V, v -/=- 0. Consider B = { ci (v) : c E Z} (this is well-defined as c acts bijectively on V by the previous paragraph). Assume that the elements in B are linearly dependent. Then, applying some power of c, if necessary, we have

for some k E N and a 0 , ... , ak E C, a 0 ,ak -/=- 0. It follows that the linear span of {v,c(v), ... ,ck-l(v)} is a finite-dimensional submodule of V; a contradiction.

74

Lectures on sb(rc)-modules

As a result, the elements in B are linearly independent. Their linear span is obviously invariant with respect to c and hence must coincide with V. Hence B is in fact a basis of V. But the linear span W of {v,c(v),c2 (v), ... } is then different from V and obviously invariant with respect to c, that is forms a proper submodule of V. This contradicts our assumption that V is simple and completes the proof. D Now we are ready to prove Theorem 3.32.

Proof. Let V be a simple g-module. In particular, Vis indecomposable and hence supp V C ~ for some ~ E C/27L by Corollary 3.7(i). Moreover, for any >. E supp V we have dim Vi, = 1 by Lemmas 3.33 and 3.34. Consider the actions of E and F on V. If the action of E is not injective, then there must exist v E V, v -=I- 0, such that E(v) = 0. As non-zero elements in different weight spaces are linearly independent, we may assume that vis a weight vector. But then the universal property of Verma modules (Corollary 3.15) gives a nontrivial homomorphism from some Verma module to V. Since Vis simple, this homomorphism must be an epimorphism, and V is thus a simple highest weight module. Thus V is either of the form (i) or of the form (ii) by Corollary 3.18. If the action of F on V is not injective, then we similarly get that V is a simple lowest weight module. In this case V is either of the form (i) or of the form (iii) by Exercise 3.23. Assume now that the action of both E and F on V is injective. Let >. E supp V and v E Vi_, v -=I- 0. Then Ei(v) -=I- 0 and Fi(v) -=I- 0 for all i E PJ and we have supp V = {>. + 2i : i E Z} = ~ by Lemma 3.5. In particular, it follows that both E and F act bijectively on V. For i E 7L set Wi = pi (v), which is well-defined as F acts bijectively on V. Since VA is one-dimensional and the action of c commutes with the action of h, we have c( v) = TV for some T E C. Since the action of c commutes with the action of F, we have

c(wi) = c(Fi(v)) = Fi(c(v)) =Fi(TV)= TFi(v) = TWi for any i E Z. Hence c acts on Vas the scalar T. From Exercise 3.28 we then derive V ~ V(~, T); that is Vis given by (iv). The necessary restrictions on ~ and T follow from Theorem 3.29(ii). This shows that every simple weight g-module is isomorphic to some module from the list (i)-(iv). Now let us prove that the modules in the list (i)-(iv) are pairwise nonisomorphic. Let V and W be two different modules from the list and assume that they are isomorphic. Then, in particular, supp V = supp W.

Weight modules

75

From Example 3.1, Proposition 3.13(ii), Exercise 3.20 and Lemma 3.27(i) we have that supp V = supp W is possible only in the case V = V(~, T) and W = V(~, T') for some T -=I- T 1. But then V '# W by Lemma 3.27(ii); a contradiction. Hence the modules in the list (i)-(iv) are pairwise nonisomorphic, which completes the proof. D Exercise 3.35. Show that every simple highest (or lowest) weight module is uniquely determined (up to isomorphism) by its support. Exercise 3.36. Show that every simple weight module is uniquely determined (up to isomorphism) by its support and the eigenvalue of the Casimir element. Exercise 3.37. Let ~ E C/2Z and T E C. Denote by yss(~,T) the same vector space as V (~, T). Define the linear operators E, F and H on V ss ( ~, T) by (3. 7) with the following exception: we set F (v µ) = 0 provided that E(vµ-2) = 0. Show that this defines on yss(~, T) the structure of a dense g-module with support ~· Show further that the module yss(C T) is semi-simple and has the same simple subquotients (with the same multiplicities) as the module V(~,T). The module yss(~,T) is called the semi-simplification of the module V(~, T). Exercise 3.38. Show that every simple weight g-module is isomorphic to a simple subquotient of some (uniquely determined) module V(~,T) (or yss(~, T)). Derive from this that the module EB~,T yss(~, T) is a multiplicityfree direct sum of all simple weight g-modules (the so-called Gelfand model for weight modules). Exercise 3.39. Show that every simple weight g-module has only scalar endomorphisms.

3.5

Coherent families

Fix TE C and consider the module

V(T) =

EB

V(~, T).

~EC/2Z

The module V(T) is called the coherent family corresponding. to T. The weight module V( T) has the following properties and is uniquely determined by them due to Exercise 3.28:

76

Lectures on st2(C)-modules

(I) dim V(T)_x = 1 for all>. E .) ~ V(>. + 2'/L, T). Proof. First we recall that the action of Fon V(C T) is bijective. Therefore V(~, T) carries the natural structure of a uUl_module. By definition, the module Bz V(~, T) coincides with V(~, T) as the vector space, but the action of U(g) is twisted by 8z. Let>. E ~and v EV(~, T) be some weight vector. Then h(v) = >.v. Since 8z(h) = h + 2z (see (3.12)), we have

Gz(h)(v) = (h + 2z)(v) = (>. + 2z)v. This implies that the support of the module Bz V(~, T) equals~+ 2z and that all non-zero weight spaces of Bz V(~, T) are one-dimensional. The action of Fon Bz V(~, T) is unchanged and hence obviously bijective. The element c acts on Bz V(C T) as the scalar T by Proposition 3.49(v). Hence the statement (i) follows from Exercise 3.28. The statement (ii) follows directly from the statement (i). Assume that M(>.) is given by (3.4). Then the module M(>.) has a basis given by {pi (v 0 ) : i E N0 }. From the definition of Bo we get that the elements {Fi(v 0 ) : i E 'IL} form a basis of Bo M(>.). Note that for i E 'IL the element pi (v0 ) has weight >. - 2i. This implies that the support of Bo M(>.) equals >. + 2'/L and that all weight spaces of Bo M(>.) are onedimensional. By definition, the element F acts bijectively on Bo M(>.). By Exercise 3.51, the natural transformation lM(>.) is injective. The Casimir element c acts on M(>.) as the scalar(>.+ 1) 2 = T by Proposition 3.13(iii). By Proposition 3.49(v), the element c acts on Bo M(>.) as the scalar T. Thus Bo M(>.) ~ V(>. + 2'/L, T) follows from Exercise 3.28. This proves the statement (iii) and completes the proof. D

Lectures on sb (C)-modules

82

Theorem 3.52 allows us to produce coherent families from Verma modules in the following way: Corollary 3.53. Let T,).. E .

EB

+ 1) 2 = T.

Then

BzM(.A).

z EC

:S !R(z)
.(T) = {v EM>. : (c - T)k(v) = 0 for some k EN}. For TE .(T). >.EiC

Weight modules

83

-eT -e -e Denote by ® ' the full subcategory of ® consisting of all M E ® such that M = M(T). Lemma 3.54. Let

~ E

C/2Z.

(i) For every ME ®e and TE C the space M(T) is a submodule of M and we have (3.15) (ii) We have

Proof. That M(T) is a submodule of M follows from Exercise 1.31 since c commutes with the operators E, F and H. This implies the decomposition (3.15) and the claim (i) follows. If Mand N are two g-modules and 'P E Hom 9 (M, N), then the operator 'P intertwines, by definition, the action of the Casimir element c on M and N. This yields 'P(M(T)) c N(T) for any TE C. Hence the claim (ii) follows from the claim (i). This completes the proof. D The aim of the next few sections is to describe the categories ®e,T for all ~ E C/2Z and T E C. Some elementary properties of these categories are given by the following proposition: Proposition 3.55.

Let~ E

C/2Z and TE C.

(i) The category ®e,T is an abelian category. (ii) If T =f. (µ + 1) 2 for allµ E ~' then ®e,T has only one simple object, namely V(~, T). (iii) If T = (µ+1) 2 for exactly oneµ E ~' then ®e,T has two simple objects, namely M(µ) and M(µ + 2). (iv) If T = (µ + 1) 2 = (µ + 2n + 1) 2 for some n E N, thenµ = -n - 1, T = n 2 and ®e,T has three simple objects, namely M(-n - 1), y(n) and M(n + 1). (v) Every object in ®e,T has finite length .. (vi) dimHom 9 (M,N) < oo for all M,N E ®e,T. (vii) The category ®e,T is a Krull-Schmidt category, that is, every object in ®e,T decomposes into a finite direct sum of indecomposable objects, moreover, such decomposition is unique up to isomorphism and permutation of summands.

84

Lectures on sb(IC)-modules

Proof. The statement (i) follows from Proposition 3.8 and definitions. The statements (ii)-(iv) follow from the classification of simple weight modules (Theorem 3.32). Let M E W~' 7 be arbitrary and L E W~' 7 be a simple module. Then all non-zero weight spaces of L are one-dimensional (see Theorem 3.32) and hence the multiplicity [M : L] of Lin M cannot exceed dimM.x for any >. E C such that dim L .x ::/= 0. The statement (v) now follows~ from the fact that M has finite-dimensional weight spaces and the fact that we haveonly finitely many simple objects in W~' 7 by statements (ii)-(iv). -~T

Let A, B, C, D E fill ' are such that there is a short exact sequence A '--+ B --+> C. Then the left exactness of the Hom 9 (-, _) bifunctor gives us the following exact sequences: 0 ___, Hom 9 (D, A)___, Hom 9 (D, B) ___, Hom 9 (D, C), 0 ___, Hom 9 (C, D) ___, Hom 9 (B, D) ___, Hom 9 (A, D). It follows that dimHom 9 (D, B)::::; dimHom 9 (D, A)+ dimHom 9 (D, C), dimHom 9 (B, D)::::; dimHom 9 (A, D) + dimHom 9 (C, D).

(3.16)

By Exercise 3.39, the endomorphism algebra of any simple weight g-module is isomorphic to C and hence is finite-dimensional. Since every object in W~' 7 has finite length by the statement (v), the statement (vi) now follows from (3.16) by induction on the sum of the lengths of Mand N. If M E W~' 7 and cp is an idempotent endomorphism of M, then the morphism idM - cp is also an idempotent endomorphism of M and we have the obvious decomposition M ~ cp(M) EB (idM -cp)(M) into a direct sum of g-modules. Now the statement (vi) follows from the abstract Krull-Schmidt Theorem (see for example Theorem 3.6 in [7]). D Although the categories W~' 7 are abelian and have finitely many simple objects, later on we will see that they do not have projective objects and hence cannot be described as categories of modules over some finitedimensional complex associative algebras. However, a satisfactory description can be provided using the (infinite-dimensional) algebra C[[x]] of formal power series in x with complex coefficients. The description of W~' 7 obviously depends on the number of simple objects in W~' 7 •

Weight modules

3. 7

85

Structure of We,-r in the case of one simple object

First we consider the case when the category 2lJ~'T has only one simple object; that is T -1- (µ+ 1) 2 for allµ E ~- In this case, the description of 2lJ~'T will be especially nice. Fix some ,\ E ~- Consider the category C[[x]]-mod of all finite-dimensional C[[x]]-modules. For the action of any power series on such module to be well-defined, the action of the element x must be given by a nilpotent matrix. From the Jordan decomposition theorem it follows that indecomposable objects in C[[x]]-mod naturally correspond to nilpotent Jordan cells (which represent the action of x). Let VE C[[x]]-mod be such that the action of the power series x E C[[x]] on V is given by the linear operator X. We define on V the structure of a .+2i

=

~Xq -

i(>. + i + 1),

a>.-2i

=

~XP + i(>. - i + 1)

for all i E N. Denote by F V the vector space underlying the diagram (3.20) (that is the direct sum of all vector spaces on the diagram). Define the linear operator H on F V as indicated by the dotted arrows (each such arrow denotes a scalar operator on the corresponding vector space). Define the linear operator E on F V as indicated by the regular arrows (going from the left to the right). Finally, define the linear operator Fon F Vas indicated by the double arrow. Lemma 3.62. The above assignment defines on F V the structure of a -~T g-module. Moreover, F V E W ' .

Proof. To prove the lemma we have to check the relations (1.2). By definition we have that the operator H is diagonalizable on F V, that E increases the H-eigenvalues by 2 and that F decreases the H-eigenvalues by 2. This means that the relations [H, F] = -2F and [H, E] = 2E are satisfied. So, it remains to check the relation [E, F] = H, which we can check separately on eigenspaces of H. For the eigenspaces corresponding to the eigenvalues >. + 2i, i E N, i > 1, the relation [E, F] = H follows from the relation a>.+2i - a>.+ 2 (i+l) = >. + 2(i + 1), which, in turn, follows from the definition of a>.+ 2i, i E N. For the eigenvalue >. + 2, the relation [E, F] = H reduces to the following computation: 1

4' V(a)V(b) -

a>.+2

1

= 4xq -

1

4xq + (>. + 2) = >. + 2.

Similarly, we can check the relation [E, F] = H for all eigenvalues >. - 2i, i E N0 . Thus F V is indeed a g-module.

Weight modules

89

As supp F v c ~ by definition, to check that F v E wi,;,T we have to compute all eigenvalues of the operator con F V. It suffices to restrict c to Vp and Vq. From the definition of F V we have that the restriction of c to Vp equals (,\ + 1) 2 + Xp and that the restriction of c to Vq equals (,\ + 1) 2 + Xq. As both Xp and Xq are nilpotent, we derive that the only eigenvalue of c is(,\+ 1) 2

= T.

Hence FV

E wi,;,T

and the proof is complete.

D

Let V and W be two Qt-modules and cp : V __, W a homomorphism. Repeating lpp on all components Vp in (3.20) and repeating rpq on all components Vq in (3.20) we extend cp to a linear map F(rp): FV __, FW.

Exercise 3.63. Check that the linear map F(rp) defined above is a ghomomorphism and derive from this that F: Qt-mod__, wi,;,T is a functor.

Theorem 3.64. The functors F and G are mutually inverse equivalences Of Categories, in particular, the Categories wi,;,T and Qt-mod are equivalent.

Proof. That GF ~ IDm-mod follows immediately from the construction. By Proposition 3.55, every module in wi,;,T has finite length with simple modules M(,\) and M(,\ + 2) as subquotients. From the construction of these modules we see that in both M(,\) and M(,\ + 2) the restriction of F to any weight space ,\ + 2i, i E Z, i -f. 1, is an isomorphism to the weight space,\+ 2(i - 1). Hence, for any V E Wi;'T, we can identify all V.\+ 2 i, i E N, with V>,+ 2 using the action of F. Similarly we can identify all V>.- 2 i, i E No, with V>,. Since the Casimir element c commutes with F, the above identification also identifies the actions of con all V>.+ 2 i, i E N, with the action of con V>.+ 2 ; and the actions of con all V>.- 2 i, i E N0 , with the action of con V.A. Thus, using c = (µ + 1) 2 + 4FE, we can uniquely determine the action of E when restricted to all V>.+ 2 i, i EN, by the formula E = ~F- 1 (c - (,\ + 2i + 1) 2 ). Similarly, the action of Eon all V>.- 2 i, i EN is also uniquely determined. This shows that for any g-module V there is a unique way to reconstruct the g-module FG V from the Qt-module G V, in particular, that FG V ~ V. This and the construction of G and F yield the existence of an isomorphism FG ~ ID®i;,T, which completes the proof. D

90

3.9

Lectures on

5[2

(C)-modules

Structure of fill~,-r in the case of three simple objects -n-1+2Z n 2

Finally, in this section we are going to describe the category 2!J ' for each (fixed) n E N. By Proposition 3.55(iv), it has three simple modules, namely M(-n - 1), y(n) and M(n + 1). Consider the C[[x]]-category SB with three objects p, q and r, generated by morphisms aESB(p,q),

bESB(q,p),

eESB(q,r),

dESB(r,q),

subject to the relations ba = xlp,

ab= de = x1q,

ed = x1r·

The category SB can be depicted as follows: a

ba = x1P, ab= de= x1q, ed = X1r.

c

~~

SB:

p ...,,.______ q ...,,.______ r ' b

d

The notions of SB-modules, their homomorphisms and the category SB-mod are defined in the usual way (as in the case of the category 2l from Section 3.8). Exercise 3.65. Show that the path algebra of SB is isomorphic to the algebra of all those 3 x 3-matrices

au ai2 ai3) ( a 21 a 22 a 23

with coefficients from

a31 a32 a33

C[[x]], which satisfy the condition For V E

onn-1+2z,n2 ~

a 12 , a 23

E xC[[x]],

a 13

E x 2 C[[x]].

. d'iagram: cons1'der t h e £o11owmg (c-n 2 )

Gv :

>2

(c-n )

2E

v -v

~

2EF 1 - n

____,._v.

-n+l - -

-n-1 -

2F

n+l


.), M) = M,\. By Exercise 3.11, the functor M t--+ M,\ is an exact functor from Qi;;",\ to the category C-mod of all complex vector spaces. Hence the functor Hom 9 (M(>.), _)from Qi;;",\ to C-mod is exact. This means that the module M(,\) is projective. D Exercise 5.18. Let ,\ EC\ {-2, -3, ... }. Show that the module M(>.)® is injective in 0. Corollary 5.19. Let,\ EC\ {-2, -3, ... }. Then the simple module L(>.) has a projective cover in 0.

140

Lectures on .sb(IC)-modules

Proof. The module L(>.) is a quotient of M(>..) by Corollaries 3.18 and 3.19. For >.. E C \ { -2, -3, ... } the module M(>.) is projective by Lemma 5.17. This completes the proof. D Lemma 5.20. For every finite-dimensional g-module V and every ME 0 we have V®M E 0. In particular, the endofunctor V@_ of g-mod restricts

to an exact and self-adjoint endofunctor of 0.

Proof. The module V ® M is weight by Proposition 3.8(iv). By Exercise 5.9, there exists >.. 1 , ... , Ak EC such that k

supp M

c

LJ (>..i - 2No). i=l

By Exercise 3.10, we have k

supp V ®MC LJ(supp V +Ai - 2No). i=l

As V is finite-dimensional, the set supp V is finite. It follows that condition (III) for the module V ® M is satisfied. Since V ® _ is exact, from Theorem 3.81 it follows that for every simple module LE 0 the module V ® L has finite length. From Corollary 5.12 we know that M has finite length. Hence, using the exactness of V ® _ again, we conclude that V ® M has finite length and so is finitely generated. This means that V ® M E 0 and thus V ® _ restricts to an endofunctor of 0. That V ®- is exact follows from the fact that it is self-adjoint, which, in turn, follows from Exercise 3.72. D Corollary 5.21. For every finite-dimensional g-module V the endofunctor V ® _ of 0 sends projective modules to projective modules.

Proof. Let P E 0 be projective. By the self-adjointness of V ® _ we have the natural isomorphism of functors (from 0 to C-mod) as follows: Hom 9 (V ® P, _) = Hom 9 (P, V ® _). Now the functor V@_ is exact by Lemma 5.20, and the functor Hom 9 (P, _) is exact since the module Pis projective. Hence the functor Hom 9 (V ®P, -) is exact as a composition of two exact functors. This means that the module V ® P is projective and the claim follows. D Exercise 5.22. Show that for every finite-dimensional g-module V, the endofunctor V ® _ of 0 sends injective modules to injective modules.

Category 0

141

Corollary 5.23. For every n E {2, 3, 4, ... } the simple module L(-n) has a projective cover in 0.

Proof. First we observe that L(-n) = M(-n) by Theorem 3.16. The module M(O) is projective in 0 by Lemma 5.17. Hence, by Corollary 5.21, the module y(n+l) ® M(O) is projective in 0 as well. At the same time, from the self-adjointness of y(n+l) ® _ we have Hom 9 (V(n+l) ® M(O), L(-n))

= Hom9 (M(O), y(n+l)

® M(-n)).

(5.2)

By Proposition 3.13(ii) we have supp M(-n) = -n - 2N0 . From (1.9) we also have supp y(n+l) = {-n, -n + 2, ... , n - 2, n}. Hence from Exercise 3 .10 we obtain supp y(n+l) ® M(-n) = -2No. In particular, there exists a non-zero v E (V(n+l) ® M(-n))o and this vector satisfies e( v) = 0. By the universal property of Verma modules (Corollary 3.15), we thus get a non-zero homomorphism from the module M(O) to y(n+l) ® M(-n). From (5.2) it thus follows that

Hom 9 (V(n+l) ® M(O), L(-n)) which means that

y(n+l) ®

#- 0,

M(O) is a projective cover of L(-n).

D

Now we are ready to prove Theorem 5.16.

Proof. Since every object in 0 has finite length (Corollary 5.12), we can prove the existence of a projective cover for M E 0 by induction on the length of M. If the module M is simple, the statement follows from Proposition 5.5 and Corollaries 5.19 and 5.23. Now assume that Mis not simple and consider any short exact sequence L '--+ M _...,. N, where L is simple. Then the length of N is strictly smaller than the length of M. Let P _...,. L and Q _...,. N be projective covers, which exist by the inductive assumption. Using the projectivity of Q we can lift the surjection Q _...,. N to a homomorphism Q -+ M such that the following diagram commutes: P--PEBQ------Q

t

~

!

0 _____..,.. L _ ___.,.. M ____.,.. N _____..,.. 0

(here the maps P EB Q-+ P and P EB Q-+ Qare natural projections). The above diagram gives as a surjection from the projective module P EB Q to M. Now the claim of the theorem follows by induction. D

Lectures on sl2(C)-modules

142

Exercise 5.24. Show that for any n EN the module y(n) ® M(-1) is projective in 0 and that it contains, as a direct summand, the projective cover of the module L(-n). Exercise 5.25. Show that for any >.. E C \ { ... , -4, -3, -2, 0, 1, 2, 3, ... } the module M(>..) is both projective, injective and simple in 0. Exercise 5.26. Show that 0 has enough injectives; that is, that every module in 0 has an injective envelope in 0. As a result of Theorem 5.16 and Exercise 5.26, for every>.. E C we have the indecomposable projective cover P(>..) of L(>..) and the indecomposable injective envelope I(>..) of L(>..). By Lemma 5.17 and Exercise 5.18, for >.. E C \ { -2, -3, ... } we have P(>..) = M(>..) and J(>..) = M(>..)®. For >.. E { -2, -3, ... } the module P(>..) is called the big projective module. For such >.. the structure of P(>..) and J(>..) is described in the following statement: Proposition 5.27. For>.. E { -2. -3, ... } we have the following:

(i) P(>..) ~I(>..). (ii} The module P(>..) ha.'i a h.sis {vµ: µE · >.

2-2N0 }U{wµ: µEA.-2N 0 }

such that the action oj g m this basis can be depicted as follows: A-2

>-·2

.\

>-+2

>-+4

(5.3) (here aµ = t(µ + >.. + 2)(>.. - µ) for allµ E ->.. - 2 - 2No, and an absence of some arrow means that the corresponding linear operator acts on this basis vector as zero). (iii} The module P(>..) is uniserial of length 3. Its simple top and simple socle are isomorphic to L(>..) and the intermediate subquotient is isomorphic to L(->.. - 2), which may be depicted as follows:

143

Category 0

P(>..):

L(>..)

I

L(->..- 2)

I L(>..)

Proof. Let >.. = -n for n E N, n =/= 1. By our proof of Theorem 5.16, the module P(-n) is a submodule of the module y(n+l) Q9 M(O). As the module M(O) is a submodule of V(2Z, 12 ), using the exactness of the functor yCn+l) Q9 _ we have that P(-n) is a submodule of y(n+l) Q9V(2Z,1 2 ). --n+2Z (n-1) 2

Observe that P(-n) E W ' and hence from Theorem 3.81 we obtain that P(-n) is a submodule of W(-n + 2Z, (n -1) 2 ). If the module W(-n + 2Z, (n - 1) 2 ) is given by (3.32), then a direct calculation shows that the maximal submodule of W(-n + 2Z, (n - 1) 2 ), which belongs to 0, is given by (5.3). Call this module N. A direct calculation shows that the module N, given by (5.3), is a uniserial module satisfying the assertion of (iii). By the above paragraph, the module P(-n) is a submodule of N with simple top L(-n). Hence we either have P(-n) = N or P(-n) = L(-n). On the other hand, the module N has simple top L(-n) and belongs to 0, hence N is a quotient of the projective module P(-n). This yields P(-n) = N, proving both (ii) and (iii). The module M(-1) is projective in 0 by Lemma 5.17 and is injective in 0 by Exercise 5.18. The functor y(n) Q9 _ sends projective modules to projective (Corollary 5.21) and injective modules to injective (Exercise 5.22). Hence every direct summand of the projective module y(n) Q9 M(-1) is ;:tlso injective. Using Exercise 5.24, we thus get that the module P(-n) is injective. By (iii) it has simple socle L(>..), which yields (i). This completes the proof. D Corollary 5.28. Let>.. E C. The indecomposable projective module P(>..) is injective if, and only if, >.. ~ N 0 . Moreover, for>..~ N 0 we have P(>..) =I(>..).

Proof. This follows directly from Proposition 5.27, Lemma 5.17, Theorem 3.16 and Exercise 5.18. D

Lectures on sb(C)-modules

144

Exercise 5.29. Show that for any A E {-2, -3, ... } the module P(,\) is isomorphic to £(W(A + 2Z, (,\ + 1) 2 )). Exercise 5.30. Let A E {-2, -3, ... }. Show that multiplication with the element c - (A +1) 2 defines a non-zero endomorphism 'P of P(A), which satisfies t.p 2 = 0. Show further that the endomorphism algebra of P(,\) is isomorphic to the algebra C[x]/(x 2 ).

5.3

Blocks via quiver and relation

The summands 0~,T from the decomposition (5.1) are called blocks of 0. From previous sections we have that every block of 0 is an abelian category with enough projective objects, moreover, all objects of 0 have finite length. From Exercise 5.15 we also have that every block o~,T has only finitely many simple objects (up to isomorphism). This suggests that blocks of 0 can be described using finite-dimensional associative algebras. Theorem 5.31 (Description of blocks of 0).

Let~ E

C/2Z, TE C.

(i) If (A+ 1) 2 -I- T for all A E ~' then the block o~,T is zero. (ii) If (A+ 1) 2 = T for a unique A E ~' then the block o~,T is semi-simple and equivalent to the category C-mod of complex vector spaces (or, equivalently, C-modules). (iii) If (A1 + 1) 2 = (A2 + 1) 2 = T for A1, A2 E ~' A1 -=f. A2, then T = n 2 for some n E N and the block o~,T is equivalent to the category of modules over the following C-category, whose path algebra is finitedimensional:

'.'.D:

ab= 0.

The blocks o~,T from Theorem 5.31(iii) are called regular blocks. This terminology stems from the observation that simple objects in such blocks are indexed by regular orbits of the dot-action of the Weyl group on ~*, described in Section 2.5. For similar reasons, the block o-i+ 2z,o is called singular. As the module V(~, T) never belongs to 0 (since for the module condition (III) obviously fails), the claim (i) follows directly from Proposition 3.55(ii).

Proof.

V(~,T)

Category 0

145

for a unique A E ~'then, comparing Proposition 3.55(iii) and Proposition 5.5, we obtain that o~,T contains a unique simple object, namely M(.\). In this case we also have T -1- n 2 for any n E N (otherwise we would be in the situation of claim (iii)). Hence,\ tf. Z \ {-1} and from Lemma 5.17 we get that the simple object M(.\) is also projective. Hence the category QE,T is a semi-simple category with one simple object, whose endomorphism algebra is .), L(µ)(j)) = C, 0, C, { Ext~-gmod(L(p.)(j), V'(.A)) = C, 0,

>.=µ,i=O,j=O; A= p, µ = q, i = l,j = -1; otherwise;

>. = µ, i = 0, j = O; A= p, µ = q, i = l,j = 1; otherwise.

D

164

Lectures on s[2(C)-modules

The following statement says that standard and costandard D-modules form homologically dual families: Corollary 5. 73. For any i E N0 , j E Z and>.,µ E {p, q} we have Extb-gmoa(fl(.A), \J(µ)(j)) =

{C, 0,

A=µ, i = j = O; otherwise.

Proof. If ,\ = p, then fl(.\) is simple and the claim follows from Exercise 5.72. If µ = p, then \J(µ) is simple and the claim follows from Exercise 5.72. It remains to consider the case A = µ = q. In this case fl(.\)= P(q) and \J(µ) = I(q). In particular, Extb-gmoa(P(q),J(q)(j))-/=- 0 implies i = 0. For i = 0 we have the obvious degree zero map from P( q) to I(q), which sends the top of P(q) to the socle of I(q) and is unique up to a scalar. This proves the statement in the case j = 0. If j -/=- 0 then Homo-gmod ( P( q) ,I (q) (j)) = 0 follows from the graded filtration of I (q), given by Figure 5.1. This completes the proof. D Exercise 5. 7 4. Show that for any i E N0 , j E Z and .\, µ E {p, q} we have

Ext~-gmoa(P(.A), L(µ)(j))

=

{C, 0,

~

>. = µ, = j = O; otherwise.

Exercise 5. 75. Show that for any i E N 0 , j E Z and .\, µ E {p, q} we have

Ext~-gmoa(L(.A),I(µ)(j)) =

5. 7

{C, 0,

A=µ,~= j

= O;

otherwise.

Category of bounded linear complexes of projective graded D-modules

Recall that for a (graded) algebra A and a (graded) A-module M, the additive closure of M is a full subcategory add(M) of the category of all (graded) A-modules. This consists of all (graded) modules, isomorphic to finite direct sums of direct summands of M. We will use the standard notation (x•, d.) (or, simply x•) for a complex di-2

----~

xi-1

di-1

di

----~xi----~

xH1

d;+1

----~

Category 0

165

of (graded) D-modules. Denote by £If) the category, whose objects are all bounded linear complexes of graded finite-dimensional projective Dmodules, and morphisms are all possible morphisms (chain maps) between complexes of graded D-modules. Then a finite complex x• of graded projective D-modules belongs to £If) if, and only if, for every i E Z the top of the graded projective module xi is concentrated in degree -i (if xi =I 0). Equivalently, we may require Xi E add(D(i)) for all i E Z. For example, from the previous section we have that minimal projective resolutions of both simple D-modules belong to £If) (here we mean the genuine projective resolutions, which are obtained from the exact sequences mentioned in Proposition 5.65 by deleting the bold elements). These and some other objects of £If) are presented on Figures 5.5-5.9 (only non-zero maps and components are shown). Deg.\Pos.

0

0 1

2 Fig. 5.5

Linear complex C(l)• of projective D-modules.

Deg.\Pos.

0

0

Cq

1

CP

+b Fig. 5.6

Linear complex C(2)• of projective D-modules.

Let x• and y• be two complexes from £If) and i E Z. By the definition of £If), the top of Xi is concentrated in degree -i. At the same time, from the definition of £If) we also have that the graded component Y~i 1 is zero.

166

Lectures on sl2(C)-modules

Deg.\Pos.

-1

0

0

Cq

1

CP ____ ,,_cP

2

Cq

3

Cp

+b +a +b Fig. 5.7

Linear complex C(3) 0 of projective D-modules.

Deg.\Pos.

-1

0

0

Cp

1

cqc - - - .... cq

2

CPL - - - ,,... CP

ta +b Fig. 5.8

+b

Linear complex C(4) 0 of projective D-modules.

In particular,

yi- I) = 0, from x· to y· is

Homo-gmod (Xi,

implying that the only homotopy the zero map. It follows that £s+l" is equal to the corresponding homotopy category. Note that the category of all complexes of D-modules is abelian, while the homotopy category is not. Since for £s+l" the two categories coincide, one might hope that £s+l" should be abelian. This turns out to be the case. Proposition 5. 76. The category

£s+l"

is abelian.

Proof. Since the category of all complexes is abelian, it is enough to show that for any two complexes x• and y• from £s+l" and any homomorphism . + 1)2 - (h + 1)2 x-1x 4 (>. + 1)2~(h+1)2

2h

=4+4 = h. The claim (i) follows. To prove the claim (ii) we compute:

.A(c)

=

>J(h + 1) 2

+ 4fe)

= (h + 1)2 + 4 (>. + 1)2 - (h + 1)2 x-1 x 4

=

(>.+1)2.

Hence .>,(c-(,\+ 1) 2) = 0 and I>, belongs to the kernel of.>, contains the infinite-dimensional space C[h]. This yields that the kernel of-)-Mod

is just the functor, which restricts the action from A to U(I>,). Exercise 6.19. Show that for any a EA there exists g(h) E C[h] such that g(h) =/=- 0 and g(h)a E U(I>,). Exercise 6.20. Consider A as a C(h)-U(I>,)-bimodule by restriction. Show that the multiplication map mult defines an isomorphism of the following C(h)-U(I>,)-bimodules:

mult: C(h)

Q9 U(I>-)--+ A. IC[h]

Exercise 6.21. Consider the endofunctor T of C[h]-Mod given by tensoring with the C[h]-C[h]-bimodule C(h). Show that the embedding C[h] '-+ C(h) defines a natural transformation from the identity functor to T, whose kernel coincides with the functor of taking the maximal C[h]-torsion submodule.

Consider the endofunctor F of U(I>,)-Mod given by the tensoring with the U(I>-)-U(I>,)-bimodule A. Exercise 6.22. Show that F =Go F. Lemma 6.23. The embedding U(I>-) '-+A defines a natural transformation from the identity functor to F, whose kernel coincides with the functor of taking the maximal generalized weight submodule.

Proof. First we remark that C[h]-torsion U(I>,)-modules are exactly generalized weight modules. Let M E U(I>-)-Mod. If we consider FM as a C(h)-module, then, applying Exercise 6.20, we obtain that FL~C(h)Q9L. IC[h]

The claim follows from Exercise 6.21.

D

Description of all simple modules

203

Now we are ready to describe simple nonweight U(I.x)-modules. Theorem 6.24.

(i) The functor F induces a bijection, F, from the set of isomorphism classes of simple nonweight U(I.x)-modules to the set of isomorphism classes of simple A-modules. (ii) The inverse of the bijection from (i) is the map, which sends a simple A-module N to its U(I.x)-socle socu(I;.)(N). Proof. Let L be a simple torsion-free U(I.x)-module. From Lemma 6.23 we obtain that L C FL as a U(I.x)-module. In particular, FL -=/= 0. Now we claim that the A-module FL is simple. Since L = U(I.x) · L, by Exercise 6.20 we have FL= ..)-module U(I>..)/(U(I>..)g(h)a). By Theorem 4.26, the module U(I>..)/(U(I>..)g(h)a) has finite length and hence a simple U('I>..)-submodule, say N. As the action of every p(h) E C[h], p(h) -/= 0, is invertible on A/(Aa), the kernel of p(h) on N is zero and thus N is torsion-free. This completes the proof. D Now let N be a simple A-module and L be a simple U('I>..)-submodule of N, which exists by Lemma 6.26. Using adjunction, we have

0-/= Homu(I>-)(L, G N)

= HomA(F L, N).

The module FL is simple by the above and the module N is simple by our assumption. Hence FL~ N. It follows that the map Fis surjective. Assume FL'~ N for some simple torsion-free U('I>..)-module L'. Then, by Lemma 6.25, both L and L' are isomorphic to the simple U('I>..)-socle of N and hence L ~ L'. Therefore the map F is injective, which completes the proof. D Corollary 6.27. For every ,.\ E IC there is a natural bijection between the

isomorphism classes of simple U(I>..)-modules and the similarity classes of irreducible elements in A. Proof. This statement follows from Theorem 6.24 and Propositions 6.14 and 6.15. D Exercise 6.28. Show that for any simple U('I>..)-module L there exists an irreducible (as element of A) element a E U(I>..) C A such that L ~ U(I>..)/(U(I>..) n A.a).

6.4

Finite-dimensionality of kernels and cokernels

Throughout this section we fix..\ E IC. Let 6£>.. denote the full subcategory of the category of all U('I>..)~modules, which consists of all finite length modules. For a U('I>..)-module M and u E U(I>..) we denote by UM the linear operator, representing the action of u on M. Exercise 6.29. Show that 6£>.. is an abelian Krull-Schmidt category in which simple objects are simple U('I>..)-modules and every object has finite length.

Description of all simple modules

205

Our goal for the present section is to prove the following result:

#- 0. Then both the kernel and the cokernel of the linear operator UM are finite-dimensional.

Theorem 6.30. Let ME~£.>. and u E U(I>.), u

First we prove the claim of Theorem 6.30 for kernels.

Proof.

We will need the following lemmas:

Lemma 6.31. For any M, N E finite-dimensional.

~£.>.

the vector space

Hom~-,c-"

(M, N) is

Proof. Let 0 ---+ V ---+ Y ---+ Z ---+ 0 be a short exact sequence in Then the following two sequences are exact:

~£.>..

HomJ.£-" (Z, N) ---+ HomJ.£-" (Y, N) ---+ HomJ.£-" (V, N), HomJ£A (M, V) ---+ HomJ£A (M, Y) ---+ HomJ£A (M, Z). In particular, the middle term in both sequences is finite-dimensional provided that both the left and the rights terms are. Since every module in ~£>. has finite length, it is enough to prove the assertion for simple U(I.>.)modules. For such modules the assertion follows from Schur's lemma and Exercise 4.10. D Denote by Mu the left U(I.>.)-module U(I>.)/(U(I>.)u). Lemma 6.32. Let M be a U(I>.)-module. Then there is an isomorphism

Proof.

Let

-)(Mu,N)---+ Ker(uM)· On the other hand, if v E Ker( UM), we have a unique homomorphism 'ljJ from the free U(I.>.)-module U(I>.) to M, given by 'l/;(1) = v. As u(v) = 0, the homomorphism 'ljJ factors through Mu. This gives us a map g': Ker(uM)---+ Homu(I>-)(Mu,N).

From the definitions it follows that g and g' are mutually inverse linear maps. This completes the proof. D

206

Lectures on sl2(-. By Lemma 6.32 we have

Ker(uM) ~ Homu(I>.)(Mu,M). Since u -1- 0, by Theorem 4.26 the module Mu has finite length. This means that

Homu(I>.)(Mu, M) = HomJ_eJMu, M), which is finite-dimensional by Lemma 6.31. dimensional.

Hence Ker( UM) is finiteD

To prove the finite-dimensionality of the cokernels we will have to work much harder. Our first reduction is the following: Exercise 6.33. Let u E U(I.>-), u -1- 0. Then the cokernel of UM is finitedimensional for any M E J£.>- if and only if the cokernel of UL is finitedimensional for any simple U(I.>-)-module L.

Our next reduction is given by the following observation: Lemma 6.34. Assume that for any a, (3 E U(I.>-), a, (3 space

-1- 0, the vector

is finite-dimensional. Then for any u E U(I.>-), u -1- 0, the cokernel of UL is finite-dimensional for any simple module L.

Proof. Let L be a simple U(I.>-)-module and v E L be non-zero. Then the map 1 c-+ v extends uniquely to an epimorphism cp from the free module U(I.>-) to L. As the free module U(I.>-) is not simple (the algebra U(I.>-) contains non-invertible elements, for instance h by Theorem 4.15(ii)), the morphism cp is not injective. Let a be any element in the kernel. The map cp factors through the module U(I.>-)/(U(I.>-)a). Let Cj5 be the induced epimorphism. This gives us the following short exact sequence: (6.5) Consider now the module N = U(I.>-)/(U(I.>-)a). The cokernel of UN is isomorphic to the vector space U(I.>-)/(U(I.>-)a + uU(I.>-)), which is finitedimensional by assumption. Factoring out the subspace Ker(cp) from (6.5), we obtain that the cokernel of UL is finite-dimensional as well. This completes the proof. D

Description of all simple modules

207

Lemma 6.34 reduces our problem to the study of U(I>.)-modules of the form U(I>.)/(U(I>.)a), where a E U(I>.), a-=/:- 0. Lemma 6.35. Leta= a0 (h)+a 1 (h)e+· · ·+ak(h)ek, where k

2 0, ai(h)

E

C[h] and a0 (h), ak(h) -=/:- 0. Let Va denote the subspace of U(I>.), spanned by the following monomials: hifj, i E No, j EN, i < deg(ao(h));

(6.6)

hiej, i E No, j E No, i < deg(ak(h)), j 2 k;

(6.7)

hiej, i E No, j E No, j < k.

(6.8)

Then U(I>.) =Va EEl U(I>.)a. Proof. Let us show that, modulo U(I>.)a, every element from U(I>.) can be written as a linear combination of the elements from the formulation of the lemma. By Theorem 4.15(ii), it is enough to prove the claim for the monomials of the form hij1 and hiej, where i,j E N0 . We will prove the claim for the monomials of the form hiej. For the monomials of the form hi J1 the arguments are similar. We proceed by double induction on i and j. If we have j < k or i < deg(ak(h)), the monomial hiej appears in the above list and we have nothing to prove. Assume that j 2 k and i 2 deg(ak(h)). If we have ak(h) = cohdeg(ak(h)) + c1hdeg(ak(h))-l + ... (here Ci E C for all i and c0 -=/:- 0), then the element

hiej _ .!_hi-deg(ak(h))ej-ka Co is equal to hiej modulo U(I>.)a and is a linear combination of monomials of the form hi' ej' such that either i' < i or j' < j. This completes our induction. On the other hand, for any (3 E U(I>.), (3 -=/:- 0, we either have the equality (3 = 9s(h)f8 + 9s-1(h)j8- 1 + ... for some s E N and 9s(h) -=/:- 0, or the equality (3 = 9t(h)et + 9t-1(h)et-l + ... for some t E No and some 9t(h) -=/:- 0. In the first case, the product (3a contains in its decomposition the monomial hdeg(gs)+deg(ao) f8. In the second case, the product (3a contains in its decomposition the monomial hdeg(g,)+deg(ak)ek+t. These monomials are not listed in the formulation. This yields Va n U(I>.)a = 0 and the claim follows. D Lemma 6.36. Let

208

Lectures on s[2(lC)-modules

where k, m?: 0, ai(h), bi(h) E C[h] and ao(h), ak(h), bo(h), bm(h) -1- 0. Set W = U(I>Ja + f3U(I>.). The space U(I>.)/W is finite-dimensional. Proof. As U(I>.)a CW, by Lemma 6.35 the space U(I>.)/W is spanned by the images of the monomials from the lists (6.6), (6.7) and (6.8). For all j EN that is sufficiently large, the polynomials ak(h - 2(j - k)) and bm(h) are coprime and hence there exist Xj(h) and Yj(h) such that

Xj(h)ak(h - 2(j - k))

+ bm(h)yj(h -

2m) = 1

Using g(h)e = eg(h+2) for all g(h) E C[h] (see (2.1)), by a direct calculation we obtain for all i E N0 the following:

hixj(h)ej-ka + f3(h

+ 2m)iyj(h)ej-m =

hiej

+

terms of smaller degree.

By induction we get that there exists n E N such that for all j > n, all monomials from the list (6.7) can be reduced modulo W to monomials with smaller j. Similarly treating the list (6.6), we can assume that n is a common bound for both lists and that n > k + m. Let Y denote the finite-dimensional subspace of U(I>.), spanned by the monomials from the lists (6.6) and (6.7) such that j::; n. Fors E N0 , denote by Vs the linear span of hiej, 0::; i::; s, 0::; j < k. Then Vs is a subspace in U(I>.) of dimension k(s + 1). Furthermore, we have the flag V0 C V1 C V'.l C . . . and can thus define V = Vi. From

LJ

iENo

Lemma 6.35 and the previous paragraph we obtain

U(I>.) = W

+ Y + V.

(6.10)

Let a be the maximal degree among the degrees of the polynomials ai ( h) and b be the maximal degree among the degrees of the polynomials bi (h).

Lemma 6.37. For every l?: 0 we have

f3Vz C Vi+ma+b

+ Y + U(I>.)a.

(6.11)

Proof. Let hiej be some monomial from Vi. Then i ::; l and j ::; k and we have

(3hiej

=

bo(h)hiej

+ b1(h)(h -

2)ieJ+ 1 + · · · + bm(h)(h - 2m)iem+j.

All coefficients from C[h] in the above element have degrees at most b + l. In particular, f3hiej E Vi+ 1 if m + j < k. Assume now that m + j?: k. As m + j < m + k, the space Y contains, by construction, all monomials hsem+j wheres< deg(ak(h)). Hence, there exists some y E Y and g(h) E C[h] of degree at most b + l, such that

bm(h)(h - 2m)iem+j

= y + g(h)em+j-kak(h)ek

Description of all simple modules

209

(as writing y and g(h) with unknown coefficients reduces the above equality to a triangular system of linear equations with non-zero elements on the diagonal). Therefore, modulo Y and U(I>.)a, the element {3hiej is equal to the element

f31 = {3hiej - y - g(h)em+j-ka. has the form q(h) + r(h)e + · · · + p(h)em+j-l where all C[h] have degrees, at most, b + l +a. Now we can proceed

The element {31 coefficients from inductively at most m times. We end up with an element from V, in which all coefficients from C[h] have degrees at most b + l +ma. The claim of Lemma 6.37 follows. D Now let d denote the dimension of the kernel of the linear operator {3 on the module Ma. We have d < oo by the first part of Theorem 6.30, proved above. By (6.11), the rank of f3(Vz + U(I>.)a) in Vz+ma+b + Y + U(I>.)a is at least dim(Vz) - d. Hence, using (6.10) and (6.11), we get

+ Y/((Vz+ma+b + Y) n W)) ::; (l +ma+ b + l)k + dim(Y) - (l + l)k + d =(ma+ b)k + dim(Y) + d. dim(Vz+ma+b

Observe that the bound we get does not depend on l. As U(I>.)a C W, taking the limit we thus obtain dim(V

+ Y/((V + Y) n W))::; (ma+ b)k + dim(X) + d < oo.

From (6.10) it follows that dim(U(I>.)/W) < oo, which completes the proof of Lemma 6.36. D Exercise 6.38. Show that for any non-zero a, {3 E U(I>.) there exists nonzero x, y E U(I>.) such that xa and {3y have the form (6.9). Now we can prove Theorem 6.30 for cokernels.

Proof. Let a, {3 E U(I>.), a, {3 -j. 0. Then for any non-zero x, y E U(I>.) we have the inclusions U(I>.)xa c U(I>.)a and {3yU(I>.) C {3U(I>.). In particular, (6.12) By Exercise 6.38 we can choose x and y such that xa and {3y have the form (6.9). By Lemma 6.36 we get that U(I>.)/(U(I>.)xa + {3yU(I>.)) is finite-dimensional. From (6.12) it follows that U(I>.)/(U(I>.)a + {3U(I>.)) is finite-dimensional. Theorem 6.30 now follows from Lemma 6.34 and Exercise 6.33. D

Lectures on sl2(C)-modules

210

6.5

Finite-dimensionality of extensions

Let J£ denote the full subcategory of the category of all g-modules, which consists of all g-modules of finite length. Exercise 6.39. Show that J£ is an abelian Krull-Schmidt category in which simple objects are simple U(g)-modules and every object has finite length.

Our aim in the present section is to prove the following result: Theorem 6.40. For every M, NE

J£ the vector space

EB Exthcol (M, N) iENo

is finite-dimensional. To prove this theorem we need some preparation. We start with the following standard reduction: Exercise 6.41. Assume that the statement of Theorem 6.40 is true for all simple modules Mand N. Show, by induction on the length of Mand N, that the statement is true for all M, N E J£.

Let >.. E C. Set c;.. = c - (>.. + 1) 2 . As the element c;.. is central in U (g), the left ideal generated by c;.. coincides with the two-sided ideal generated by c;... Therefore, from the definition we have the following free resolution of the left U(g)-module U(I;..): 0-+ U(g)

_ ·c;. ----+

U(g)

proj ----+

U(I;..) -+ 0,

(6.13)

where proj denotes the canonical projection. Let now a E U(g) \I;.. be arbitrary. Then the element a = a+ I;.. is non-zero in U(I;..). From the definition of the U(I;..)-module Ma = U(I;..)/U(I;..)a we have the following free resolution of Ma over U(I;..): (6.14) again where proj denotes the canonical projection. Resolving each copy of U(I;..) in (6.14), using the resolution (6.13), we obtain the following free resolution of Ma over U(g): 0-+ U(g)

'P

----+

U(g) E9 U(g)

..p

----+

proj

U(g) ----+Ma -+ 0,

(6.15)

Description of all simple modules

where for x, y

E

U(g) the maps cp and 'lj; are given by the following: cp(x)

= ( -xa) ' XC>.

Lemma 6.42. Let L be a simple U(g)-module, Then

i

(

211

)

Extu(g) Ma, L =

>.

!

E I['.

and a E U(g) \I;...

Ker(aL),

c;..L = O,i = O;

Ker(aL) EB Coker(aL), Coker(aL),

C>.L

0,

otherwise.

= 0, i = 1; c;..L = 0, i = 2;

(6.16)

Proof. As Homu(g)(U(g)., L) ~ L, applying the functor Homu(g)(-, L) to the resolution part of the sequence (6.15) we obtain the following complex: 0

---+

L

1/; -----+

-

L EB L ~ L

---+

0,

(6.17)

where the maps 'lj; and (j5 are given by

?fa(x) = ( ~:),

(j5

(~) =-ax+ c;..y.

The element c;.. is central and hence defines an endomorphism of the simple module L. Notably, if c;..L # 0, then this endomorphism is invertible and the sequence (6.17) is exact. In this case all extensions vanish. In the case c;..L = 0, the homology of (6.17) is obviously given by (6.16). D Corollary 6.43. The claim of Theorem 6.40 is true in the case M and N = L is a simple module.

Proof.

This follows from Lemma 6.42 and Theorem 6.30.

= Ma D

Corollary 6.44. The claim of Theorem 6.40 is true in the case M is a simple dense weight module and N = L is a simple module.

Proof. Let>. EI['., and assume that M ~ V(~, (>.+1) 2 ) for some~ E C/2Z such that >., ->. - 2 tf. ~ (see Theorem 3.32). Fix µ E ~ and let v E V(~, (>.+ 1)2)µ be non-zero. Consider a= h-µ E U(g). Then (h-µ)v = 0 and hence we obtain a surjection Ma_..., V(~, (>.+ 1)2). Since V(~, (>.+ 1)2) is dense and has one-dimensional weight spaces (see (3.8)), the elements {b · v : b E B 1 } form a basis in V(~, (>. + 1)2). Applying Lemma 6.35 we find that the surjection Ma_..., V(~, (>. + 1) 2 ) is an isomorphism. Now the necessary claim follows from Corollary 6.43. D

212

Lectures on sl2(.) and M(-,\ - 2). Corollary 6.46. The claim of Theorem 6.40 is true in the case where M is a Verma module and N = L is a simple module.

Proof. Using Exercise 6.45 and the additivity of Exth(g) (-, L) we can prove this corollary using the same arguments as in the proof of Corollary 6.44. D Exercise 6.47. Show that the claim of Theorem 6.40 is true in the case when M is the universal lowest weight module M(>.) for some ,\ E .) use AB - BA= A 2 to show by induction on the minimal k such that (A - >.)k(v) = 0 that B(v) E V(>.) and hence that B V(>.) C V(>.). For example, if Av = >.v, then (AB - BA)(v) = A 2 (v) implies (A - >.)B(v) = A 2 (v) E V(>.). As A - ,\acts bijectively on V(µ), µ -=f. >., it follows that B(v) EV(>.). Then use that the trace of any commutator is always zero to deduce that ,\ = 0. (c) Hint: This is a special case of Kleineke-Shirnkov's Theorem (see [54]). 1.63. Hint: Use Exercise 1.54. 1.67. (b) Hint: Use the basis {vi} from (1.9) and write the matrix of the form in this basis with unknown coefficients. Then use the definition of Q to obtain linear relations for these unknown coefficients. The resulting matrix of the form will have a constant that alternates in sign on the second diagonal and all other entries equal to zero. 1.70. Hint: Use Theorem 1.50. 1.71. (a) See Proposition 3.4.2 in [20]. (c) Hint: Use induction on the length of a.

241

242

Lectures on sb(IC)-modules

1.72. Hint: Use Sections 3.4 and 7.1 from [20]. 1.73. Hint: Use Theorem 1.22 and Corollary 1.34.

2.1 Hint: Use Lemma 1.30. 2.2 Hint: Use Lemma 1.30. 2.11. Hint: Use Exercise 2.1 and definitions. 2.16 Hint: Use induction on the degree of the monomial. 2.18 Hint: Observe that we have only finitely many monomials for each degree. Moreover, the number of the "old" and "new" standard monomials of each degree coincide. Now show that for every n, every "old" standard monomial of degree at most n can be written as a linear combination of "new" standard monomials of degree at most n, and vice versa. 2.27 Answer: For example x = e + h. 2.45 Hint: Show that this defines the structure of a U(g)-module and use Proposition 2. 7. 2.46 Hint: Show that the linear span of all standard monomials of positive degree in U(g) is a submodule, which does not have any complement. 2 ). 2.48 Answer: dim U(g)(i) /U(g)(i-l) = 2.49 Hint: Use Lemma 2.22. 2.51 Hint: Consider the actions of u, v and 1 on the one-dimensional U(g)module. 2.54 Hint: Use the PBW Theorem. 2.56 Hint: For example, take as Ii the left ideal generated by ei and use that U(g) is a domain to show that ei tf. Ii+l· 2.57 Hint: Consider the image of U(g) in the algebra of all linear operators on y(n).

e!

2.58 Hint: To show that V ~ EBiEN vC 2 i-l) consider the intersections of V with every U(g)(k).

3.2 Hint: Use Example 3.1 and Weyl's Theorem. 3.4 Hint: Use that cp commutes with the action of H. 3.9 Hint: Use computation from the proof of Proposition 3.8. 3.10 Hint: Use Exercise 3.9. 3.17 Hint: Show that any homomorphism between Verma modules is uniquely determined by the image of the generator. 3.31 Hint: Check that the linear operator E does not act injectively on the

Answers and hints to exercises

243

quotient V(n -1+2Z,n2 )/M(-n -1). 3.38 Hint: Use Theorem 3.32. 3.39 Hint: Consider the action of the endomorphism on some weight generator. 3.40 Hint: Multiply the last and the second formulae from (3.9) with 1- 1 from both sides. 3.41 Hint: Use induction on i. 3.50 Hint: Show that Bo V = 0 for any finite-dimensional g-module V. 3.51 Hint: Use that the adjoint action off on U(g) is locally nilpotent. 3.56 Hint: Use Theorem 2.33. 3.57 Hint: Show that the functor of taking a weight subspace (of a fixed weight) is exact on the category of weight modules. 3. 72 Hint: Compare the dimensions of V and vw. 3.74 Hint: Use Exercise 1.38. 3.78 Hint: Construct an injective homomorphism from V(~, T) to V. 3. 79 Hint: Use Proposition 3. 77(ii). 3.88 Hint: Argue similarly to the proof of Theorem 3.81(i). 3.89 Hint: Use that U(g) is a domain. 3.90 Hint: Use Lemma 3.5. 3.91 Hint: For example U(g)®qh,c] V, where Vis a two-dimensional IC[h, c]module on which c acts as a scalar and h acts as a nontrivial Jordan cell. 3.92 Hint: Take the module, constructed in Exercise 3.91, and show that it has a filtration whose subquotients are weight modules. 3.93 Hint: For example EEliENM(2i). 3.94 Hint: Use Theorem 2.33. 3.95 Hint: Use the universal property of Verma modules. 3.97 Hint: Use the universal property of Verma modules. 3.98 Hint: Use that E acts injectively on M(µ), while it acts locally nilpotent on M(>.). 3.99 Hint: The module V(~, T) is generated by V(~, T).>- by definition. 3.101 Hint: Look at the kernel of Fon M(-n+ 1)®. 3.102 Hint: The assumption is satisfied for any V(C T). 3.105 Hint: The category ®i; contains simple modules of three or four possible kinds: those whose support is ~' those whose support is a "one half" of ~' that is a ray in one of two directions, or, sometimes, those whose support is a segment from~ (finite-dimensional modules). Assume that our module has infinite length. At least one type of simple modules should occur infinitely many times in the composition series. Adding up

244

Lectures on .sC2(C)-modules

their supports one can show that the dimensions of weight spaces are not uniformly bounded. 3.106 Hint: Use that every module VE xis generated by V>- for any>. such that Ei acts injectively on V>- for all i E N. Then use arguments similar to those used in the proof of Theorem 3.58. 3.109 Answer: The category from Exercise 3.108(a) has one indecomposable object; the category from Exercise 3.108(b) has four indecomposable objects; the category from Exercise 3.108(c) has nine indecomposable objects. All these objects are multiplicity-free. 3.110 Hint: Use Theorem 3.81(i) and exactness of y(n) ® _. 3.111 Hint: Use Propositions 3.77 and 3.80, and exactness of y(n) ® -· 3.112 Hint: Realize N(-n - 1) as a submodule of W(-n - 1+2Z, n 2 ). 3.113 Hint: Use Theorem 3.81. 3.115 Hint: Use that the adjoint action of g on U(g) is locally nilpotent. 3.116 Hint: Use Theorem 3.81. 3.118 Hint: Use that W® So: W. 3.119 Hint: First prove this for simple modules, then use Weyl's theorem. 3.120 Hint: Use Exercise 3.28 and the definition of®· 3.121 Hint: Use Exercise 3.28.

4.5 Hint: Argue similarly to the proof of Theorem 4.2. 4.6 Hint: Argue similarly to the proof of Theorem 4.2. 4.10 Hint: Show that every endomorphism of L is algebraic over . E . + 1) 2 annihilates M(>.)® and then use Theorem 4.15. 4.30 Hint: Use Proposition 3.77(iv). 4.31 Hint: Follow the proof of Theorem 4.28. 4.32 Hint: Follow the proof of Theorem 4.28. 4.33 Hint: Follow the proof of Theorem 4.7. 4.34 Hint: Use Theorem 4.28. 4.35 Hint: Use Theorem 4.19. 4.36 Hint: Use Theorem 4.19. 4.37 Hint: Use Theorem 4.19. 4.40 Hint: Use Theorem 4.19. 4.42 Hint: Use Exercise 4.38.

Answers and hints to exercises

245

5.6 Hint: Use Exercise 3.92. 5.7 Hint: First reduce the problem to the case where v is a weight vector. Then use the arguments from the proof of Proposition 5.5. 5.9 Hint: Use the proof of Proposition 5.8. 5.10 Hint: Use the proof of Proposition 5.8. 5.15 Hint: Use Proposition 3.55. 5.18 Hint: Use that the duality ® is a contravariant self-equivalence and hence interchanges projective and injective objects. 5.22 Hint: As in the proof of Corollary 5.21 show that for every injective I E 0 the functor Hom 9 (-, V Q9 I) is exact. 5.24 Hint: Use the same arguments as in the proof of Theorem 5.16. 5.25 Hint: Use that the duality ® is a contravariant self-equivalence and hence swaps projective and injective objects. 5.26 Hint: Use ®· 5.35 (d) Hint: Use the usual adjunction between the functors Hom and @. 5.42 Hint: Construct an isomorphism from D to this matrix algebra mapping the generator a of D to the matrix (

~ ~ ~)

and the generator b of D to

000 the matrix

(~ ~ ~).

000 5.45 Hint: Apply ® to the claim of Proposition 5.43. 5.48 Hint: Apply® to the claim of Proposition 5.47. 5.50 Hint: Argue similarly to the proof of Lemma 5.32. 5.57 Hint: The sum of non-zero homogeneous maps of different degrees is not homogeneous. 5.62 Hint: Use arguments, similar to those used in the proof of Theorem 5.31(iii). 5.68 Hint: Apply ® to the claim of Proposition 5.65. 5.72 Hint: Use Proposition 5.65. 5.96 Hint: Compute the value of both '!9i o '!91 and '!91 EB '!91 on M(i) and use Corollary 5.95. 5.97 (a) Hint: Use Figure 5.2. (b) Hint: Use Exercise 3.110 or Exercise 5.92. 5.104 (a) Hint: Use the functor @ o ®· 5.105 (b) Hint: Use the fact that supp(N) Ci - 2N for all NE Oi. 5.107 Hint: Use that this property is additive and check it on all indecomposable modules. Alternatively, use that every module is a quotient of a projective module and check the property on all projective modules.

Lectures on sl2(1C)-modules

246

5.108 5.109 thing 5.111

Hint: Argue similarly to Sections 5.1-5.3. Hint: Use that all these categories are fully additive and check everyfor indecomposable modules. Answer: position: L(p):

-1

0 L(q): 0 ---+ T(p) (-1) P(p): 0 P(q): 0 J(q): 0 ---+ T(p) (-1)

0 ---+ ---+ ---+ ---+ ---+

1

T(p) ---+ 0 T( q) ---+ T(p) (1) ---+ 0 T( q) ---+ 0 T(q)---+ T(p)(l)---+ 0 T( q) ---+ 0

5 .114 Hint: Prove that the center of 0 0 is isomorphic to the center of D and then use 5.113. 5.118 Hint: Prove this for indecomposable projective modules first. 5.119 Hint: Apply both sides to L(O) and show that the cohomologies of the resulting complexes are concentrated in different positions. 5.120 Hint: Apply both sides to L(O) and show that the cohomologies of the resulting complexes are concentrated in different positions. 5.121 (a) Hint: Use that V® ~ V. 5.122 See, for example, [73, 91]. 5.123 Hint: Use 5.122 and the fact that Z ~ ® o Z o ®· 5.124 Hint: Use that Z commutes with both ID and tJg. 5.125 Hint: Use 5.124 and the facts that Z ~ ® o Zo ® and C ~ ® o Ko®· 5.126 Hint: Use that T commutes with both ID and tJg. 5.131 (c) Hint: This is not obvious only in the case .A E N 0 . In this case use induction on n.

6.6 Hint: Use the fact that C[h] is preserved by c::J. 6.12 Hint: Use that any a EA such that n(a) = 0 is invertible. 6.16 Hint: Use Proposition 6.15 and the observation that the condition La~ Lf3 is symmetric. 6.17 Hint: See Chapter 3 in [61]. 6.19 Hint: Take g(h) to be the product of all denominators in all non-zero coefficients of a. 6.28 Hint: Let L be the simple A-module such that L is the simple U(I>.)socle of L. Take any non-zero v E L c L and consider the epimorphism 1/;: A_,. L, which sends 1 to v. Then pick a E U(I>.) such that a generates the kernel of 1/; in A.

Answers and hints to exercises

247

6.38 Hint: One can even choose x,y E B 1 . 6.45 Hint: Show that M(a) surjects onto each of these Verma modules and hence on their direct sum and then use Lemma 6.35. 6.51 Hint: The left regular U(g)-module, which has the trivial module as a quotient. 6.52 Hint: Use Exercise 6.5 and the fact that N ® _ is a self-adjoint functor preserving the category of weight modules. 6.54 Hint: Argue similarly to the proof of Theorem 6.18. 6.57 Hint: Show that the dimension of any simple A-module over -. and m>-., which depend only on the weight of v. If x and y are admissible bases of y(n), then, up to a global scalar, the transformation from x toy is given in terms of (products of) divisors of k>-.'s and m>-.'s. 7.18 Hint: The identity functor is not adjoint to the direct sum of two identity functors. 7.27 Hint: Use that for a local algebra the dimensions of indecomposable projective and injective modules coincide, and that a functor, which is both left and right adjoint to an exact functor, preserves both the additive category of projective modules and the additive category of injective modules. 7.28 Hint: Use Exercise 7.23. 7.38 Hint: Use arguments similar to those used in the proof of Lemma 7.20. 7.42 (a) Hint: Let L 1 , ... , Lk be a complete list of pairwise non-isomorphic simple B-modules and e 1 , ... , ek be the corresponding complete list of pairwise orthogonal primitive idempotents. Then e is the sum of all those ei 's, such that eiM = 0 for all M E D. (d) Hint: Show that B-mod/D is equivalent to EndB(Be)-mod.

248

7.46 Hint: Use that the itself. 7.47 Hint: Use the same 7.49 Hint: Show that isomorphic. 7.52 Hint: Describe first

Lectures on sl2(C)-modules

identity functor is both left and right adjoint to arguments as in the proof of Theorem 7.37. the corresponding associative algebras are not all exact functors between semi-simple categories.

Bibliography

[l] Andersen, H. and Stroppel, C. (2003). Twisting functors on 0, Representation Theory 7, pp. 681-699. [2] Amitsur, S. (1958). Commutative linear differential operators, Pacific J. Math. 8, pp. 1-10. [3] Arkhipov, S. (2004). Algebraic construction of contragradient quasi-Verma modules in positive characteristic, Adv. Stud. Pure Math. 40, pp. 27-68. [4] Amal, D. and Pinczon, G. (1973). Ideaux a gauche clans les quotients simples de l'algebre enveloppante de sl(2), Bull. Soc. Math. France 101, pp. 381-395. [5] Amal, D. and Pinczon, G. (1974). On algebraically irreducible representations of the Lie algebra sl(2), J. Mathematical Phys. 15, pp. 350-359. [6] Barthel, G., Brasselet, J.-P., Fieseler, K.-H. and Kaup, L. (2007). HodgeRiemann relations for polytopes: a geometric approach, Singularity theory, (World Sci. Pub!., Hackensack, NJ), pp. 379-410. [7] Bass, H. (1968). Algebraic K-theory, (W. A. Benjamin, Inc.) [8] Bavula, V. (1990). Classification of simple sl(2)-modules and the finitedimensionality of the module of extensions of simple sl(2)-modules, Ukrain. Mat. Zh. 42, 9, pp. 1174-1180. [9] Bavula, V. (1991). Finite-dimensionality of Extn and Torn of simple modules over a class of algebras, Funkt. Anal. i Prilozhen. 25, 3, pp. 80-82. [10] Bavula, V. (1992). Generalized Wey! algebras and their representations, Algebra i Analiz 4, 1, pp. 75-97. [11] Bavula, V. and Bekkert, V. (2000). Indecomposable representations of generalized Wey! algebras, Comm. Algebra 28, 11, pp. 5067-5100. [12] Beilinson, A. and Bernstein, J. (1981). Localisation de g-modules, C. R. Acad. Sci. Paris Ser. I Math. 292, 1, pp. 15-18. [13] Beilinson, A., Ginzburg, V. and Soergel, W. (1996). Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9, 2, pp. 473-527. [14] Benkart, G. and Roby, T. (1998). Down-up algebras, J. Algebra 209, 1, pp. 305-344. [15] Bergman, G. (1978). The diamond lemma for ring theory, Adv. Math. 29, 2, pp. 178-218.

249

250

Lectures on sl2(.), 194 6, 179

Bi,

257

Hx, 188 E,

34

'19l, 176 {e, f, h}, 2 {h}, 44 c, 45 UM,

204

Index

g-homomorphism, 4 g-submodule, 4

antihomomorphism of Lie algebras, 25 antisymmetric operation, 2 Artinian algebra, 57 associated graded algebra, 43

action, 24 locally finite, 135 weak, 181 additive closure, 164 algebra Z-graded, 153 Artinian, 57 coinvariant, 193, 232 Euclidean, 198 filtered, 42 generalized Weyl, 55 graded, 43, 153 associated, 43 positively, 153 Koszul, 162 standard, 162 Noetherian, 50 positively graded, 153 quasi-hereditary, 151 similar to U(sb), 55 skew Laurent polynomials, 197 symmetric, 231 alternating bilinear form, 22 annihilator, 115 anti-involution, 17 antiautomorphism principal, 56 antihomomorphism, 25

basis admissible, 223 canonical, 234 dual, 234 equitable, 31 equivalent, 223 Kazhdan-Lusztig, 188 dual, 189 natural, 2 standard, 2 structure constant of, 223 big projective module, 142 bilinear form non-standard, 22 standard, 22 block, 144 integral, 171 on the wall, 176 principal, 150 regular, 144 singular, 144 Brorn~'s conjecture, 235 canonical basis, 234 dual, 234 259

260

Lectures on sb(IC)-modules

canonical embedding, 34, 77 Cartan subalgebra, 44 Casimir operator, 11 categorification, 181, 219, 235 de-, 219 lk-, 219 geometric, 236 homomorphism of, 220 naive, 220 isomorphism of, 220 naive, 220, 223 proper, 237 strong, 236 weak, 226 category R-, 86 £:-graded, 153 0-, 135 parabolic, 184 thick, 179 center of, 192 graded, 153 positively, 153 highest weight, 151 Krull-Schmidt, 84 of projective functors, 172 positively graded, 153 subSerre, 238 with enough injectives, 142 with enough projectives, 139 Cayley table, 2 center, 46 center of category, 192 central character, 64 centralizer, 125 centralizer of Cartan, 45 character, 108 characteristic tilting module, 152 characters, 107 co-Zuckerman's functor, 184 coherent family, 75 coinvariant algebra, 193, 232 commutant, 1 completion functor, 183 coshuffiing, 182

costandard module, 151 decategorification, 219 lk-, 219 decomposable module, 10 degree of monomial, 42 dense module, 69 Diamond lemma, 53 dimension global, 160 direct sum, 10 domain, 44 dual canonical basis, 234 duality, 102 Koszul self-, 170 element centralizer of, 125 irreducible, 198 similar, 200 elementary projective functor, 172 enveloping algebra, 33, 34 epimorphism, 4 equitable basis, 31 Euclidean algebra, 198 exterior power, 29 factor module, 5 filtered algebra, 42 filtration costandard, 151 standard, 151 functor co-Zuckerman's, 184 completion, 183 coshuffiing, 182 Mathieu's twisting, 80 projective, 92, 171 elementary, 172 Serre, 186 shuffling, 182 twisting, 183 Zuckerman's, 184 Gelfand model, 75 generalized weight module, 104

261

Index

generalized weight space, 104 generalized Weyl algebra, 55 global dimension, 160 gradable module, 156 graded algebra, 43 graded lift, 156 Grothendieck group, 181 group Grothendieck, 181 Harish-Chandra homomorphism, 49 Hecke algebra, 188 highest weight, 64 highest weight category, 151 highest weight module, 64 highest weight vector, 64 homologically dual families, 164 homomorphism, 4, 24, 220 Harish-Chandra, 49 identity, 4 naive, 220 zero, 4 homomorphism of Lie algebras, 24 ideal primitive, 115 identity homomorphism, 4 indecomposable module, 10 integral block, 171 integral part, 171 inverse lexicographic order, 45 inversion, 38 involution anti-, 17 skew-linear, 17 isomorphism, 4 Jacobi identity, 2 Karlsson-Minton identity, 26 Kazhdan-Lusztig basis, 188 dual, 189 Khovanov homology, 235 Koszul algebra, 162 standard, 162 Koszul dual, 170

Koszul self-duality, 170 Krull-Schmidt category, 84 left module, 25 lemma Diamond, 53 lexicographic degree, 45 lexicographic order, 45 Lie algebra, 1 antihomomorphism, 25 homomorphism, 24 module, 24 representation, 24 underlying, 24 lift graded, 156 standard, 157 localization, 76 lowest weight, 68 lowest weight module, 68 lowest weight vector, 68 Mathieu's twisting functor, 80 module ~-filtered, 151 --, 18