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United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

IRREDUCIBLE NON-ZERO LEVEL MODULESWITH FINITE-DIMENSIONAL WEIGHT SPACES

FOR AFFINE LIE ALGEBRAS

V. Futorny1

Department of Mathematics, Kiev University,64, Vladimirskaya St., Kiev 252033, Ukraine

andInternational Centre for Theoretical Physics, Trieste, Italy

and

A. TsylkeDepartment of Mathematics, Kiev University,64, Vladimirskaya St., Kiev 252033, Ukraine.

ABSTRACT

We study irreducible weight Affine Lie algebra modules with finite-dimensional

weight spaces on which the central element acts non-trivially. In particular, we show

that any such module is a quotient of a generalized Verma module. Moreover, the

classification of such irreducible modules is reduced to the classification of irreducible

torsion-free modules for algebras of type An and Cn.

MIRAMARE - TRIESTE

August 1997

Regular Associate of the ICTP. Funded in part by the CRDF Grant.E-mail: [email protected]

1

1. INTRODUCTION

Let L be an Affine Lie algebra, H be its Cartan subalgebra and c G H be the

central element of L. Also let A be the root system, A r e (respectively A i m ) be the

set of real (respectively imaginary) roots, π be a basis of A and Q be a free abelian

group generated by π.

If V is an irreducible L-module then c acts scalarly on V and the corresponding

scalar is called the level of V. For an L-module, V, set Vλ := {v G V| hv =

λ(h)v, for all h G H}, where λ G H*. If Vλ is non-zero then λ is a weight of V and

Vλ is the corresponding weight space. We denote by P(V) the set of all weights

of V. An L-module is called weight if V = V λ , λ G P(V). Clearly, if V is an

indecomposable module then P(V) C λ + Q for some λ G H*.

A weight L-module V is said to be dense if P(V) = λ + Q for some λ G P(V) and

to be torsion-free if for all real roots the action of the corresponding root elements

£ on V is injective.

Similarly the notion of weight, dense and torsion-free modules can be defined for

any reductive finite-dimensional Lie algebra.

Clearly, any indecomposable torsion-free module is dense. Irreducible dense mod-

ules with all 1-dimensional weight spaces were described in [?] for algebra A(1)1 and

in [?] in general.

We study irreducible modules of a non-zero level with finite-dimensional weight

spaces. It was shown in [?] that a module with bounded dimensions of weight

spaces has a zero level implying, in particular, that any non-zero level irreducible

module with finite-dimensional weight spaces is not torsion-free. We generalize this

implication and show that in fact every such module is non-dense.

In the case of A(1)1 , a complete classification of irreducible non-dense modules

with a finite-dimensional weight subspace was obtained in [?]. The case of algebra

A(1)n was treated in [?]. In the present paper we reduce the classification problem

to the description of irreducible torsion-free modules with finite-dimensional weight

spaces for simple finite-dimensional Lie algebras. Due to the result of Fernando

[?] the latter modules exist only for algebras of type An and Cn. Their complete

classification is known for algebra A1, where the result is standard, and for algebra

A2 [?].

Our proof is based on the reduction of the initial problem to some combinatorial

problem for the root system A. A similar approach was used in [?] to study the

support of ireeducible weight modules for reductive finite-dimensional Lie algebras.

Note that irreducible zero-level modules with finite-dimensional weight spaces

were studied in [?]. In particular, all such modules which are no longer irreducible

for the derived subalgebra of L were classified.

The structure of the paper is the following. In section 3 we show that any ir-

reducible non-zero level module with finite-dimensional weight spaces is non-dense.

The main result of section 4 and the paper is Theorem ?? which states that an

irreducible non-dense module with finite-dimensional weight spaces is a quotient of

the generalized Verma module. Section 5 contains the proof of an important combi-

natorial Proposition ?? which implies Theorem ??. This result is proved for every

Affine series separately. All other Affine Lie algebras were treated by making use of

a computer. These cases are left out.

2. PRELIMINARIES

Let U(L) be the universal enveloping algebra of L. It has a natural Q-gradation:

For α E A denote by Lα the root subspace corresponding to α.

For any y C π we will denote by add(y) the additive closure in A.

Let A + = A+(vr) (respectively A_ = A_(vr)) be a set of all positive (resp. neg-

ative) roots in A with respect to vr, Ar^ = A r e n A±, Alj11 = A i m n A±, δ is the

indivisible positive imaginary root. Hence, A i m = {kδ| k E Z \ {0}}.

Consider a subset S C π and let L(S) be a subalgebra generated by C±tp, <p E S.

Also set N+S = E Lα, α e A+ \ add(S). A subalgebra PS = (L(S) + H) 0 N+S is a

parabolic subalgebra associated with π and S. It has a reductive finite-dimensional

Levi factor L(S) +H. UN is an irreducible weight PS-module with a trivial action

of NS then we construct a L-module

MS(N) = U(L ) NU(PS)

which is called a generalized Verma module associated with π, S and N. It has a

unique irreducible quotient which we denote by LS(N).

itive if N+An element v of an L-module V is called PS-semiprimitive if N+Sv = 0.

Lemma 2.1. IfV is an irreducible L-module, v EV\ is a P S-semiprimitive element

then any v' E Vλ is P S-semiprimitive.

Proof. Since V is irreducible then clearly Vλ is an irreducible module over U(L)0.

Thus v' = uv for some u E U(L)0 and Nfv' = N+Suv = 0 completing the proof. •

For S C π let QS be a free abelian group generated by S.

Proposition 2.2. Let V be an irreducible weight L-module with a PS-semiprimitive

element of weight λ. Denote N = J2ueQs Vλ+ν. Then N is an irreducible PS-module

and V is isomorphic to LS(N). Moreover, either V is a highest weight module or

there exist a basis vr' of A, a subset S' C vr' and an irreducible weight torsion-free

C(S') + H-module N' such that V ~ LS>(N').

Proof. It easily follows from Lemma ?? that any element v E N is PS-semiprimitive

which implies that N is an irreducible PS-module and V is a quotient of MS(N).

Hence V is isomorphic to LS(N). Moreover, N is an irreducible weight module

with finite-dimensional weight spaces for the reductive finite-dimensional Levi factor

L(S)+HofPS. Hence the rest of the Proposition follows from [?], Theorem 4.18. •

Consider the Heisenberg subalgebra G = Cc © L Cks C .fcez\{o}

Clearly, G is a Z-graded algebra. If V is an irreducible Z-graded G- module then

the central element c acts scalarly on V and a corresponding scalar is called the level

of V.

Let G+ = ^Ck&1 G- = ^Ck&. For a e C* = C \ {0}, let Cva be the 1-k>0 k<0

dimensional Gε © Cc-module for which Gεva = 0, cva = ava, ε E {+, —}. Consider

a G-module

Mε(a)=U(G) (g) CvaU(GE®Cc)

associated with a and ε.

Theorem 2.3. (i) Let V be an irreducible Z-graded G-module of level a E C*

such that dimVi < oo for at least one i E Z. Then V ~ Mε(a) for some

e E {+, —} up to the shift of gradation.

(ii) Every finitely-generated Z-graded G-module V of a non- zero level such that

dim Vi < oo for at least one i E Z is completely reducible.

Proof. Since for every k > 0 there exists a non-degenerated pairing between Lkδ

and C-us (cf.[?], Theorem 2.2), one can choose a basis {e\&) e\&)..., e^} in Lkδ,

sk = dimLkδ such that [ e ^ e i ^ ] = kδijc for all i and j . Hence the proof of the

Theorem immediately follows from the proofs of Proposition 4.3, (i) and Proposition

4.5 in [?]. •

3. NON-DENSITY OF IRREDUCIBLE NON-ZERO LEVEL MODULES WITH

finite-dimensional weight spaces

From now on, unless otherwise stated, all L-modules are assumed to be non-zero

level weight modules with finite-dimensional weight spaces.

Let V be an L-module, λ E H* and V± = {v E V| L±nδv = 0 for all n > 1}.

A non-zero element v E Vλ is called maximal (respectively minimal) if v E V+

(respectively v E V~) and Vλ+kδ H V+ = 0 for all k > 0 (respectively V\+fc<5 V F~ = 0

for all k < 0).

Proposition 3.1. Let V be an L-module of a non-zero level, λ E H* and 0 <

dim VΛ < oo. Then the G-module V = ^2keZ Vλ+kδ contains a maximal or minimal

element.

Proof. Consider a G-module V" in V generated by Vλ. Then the module V" is

completely reducible and all its irreducible submodules are of type Mε(λ(c)),ε E

{+, -} up to the shift of gradation by Theorem ??. Suppose that V+ C\V ^ 0.

Since (V'/V")x = 0 we immediately conclude that V+ n Vλ+kδ C V" for all k > 0

and there exists only a finite number of integer k, k > 0, for which V+ Pi Vλ+kδ 7 0.

This shows the existence of a maximal element in V. If V+ n V = 0 then clearly

V~ f i y ' / 0 and the same arguments as above show the existence of a minimal

element in V' completing the proof. •

It follows from Proposition ?? that every L-module of a non-zero level with finite-

dimensional weight spaces has either maximal or minimal element. Without loss

of generality we will always assume that all considered modules have a maximal

element.

Lemma 3.2. Let V be an L-module and v be a non-zero maximal element in V.

Then for any α E A there exists m 0 E Z such that for all m > m0, Lα+mδv = 0 if

a + mδ E A.

Proof. Obviously, one can assume that α E A r e . Let v E Vλ. For each ip E A r e fix

a non-zero element X^ E Cv. Suppose that Xα+mδv ^ 0 for all m E Z such that

a + mδ E A. Choose k E {1, 2, 3} for which α + ikδ E A for all i E Z. Denote

wi = Xα+ikδv = 0,i 6 Z. Since [Likδ,Lα] = Lα+ikδ, the elements wi belong to a

G-submodule N generated by W0. It follows from Theorem ?? that module N is

completely reducible with irreducible submodules of type M ε(λ(c)), ε E {+, —} up

to shift of gradation. Since dim Vλ < oo, there is only a finite number of components

of type M+(λ(c)) in N, and thus there exists an index i0 E Z + such that a G+-

submodule U(G+)wi0 is U(G+)-free. On the other hand,

[• 2fc(5) £a+ioks] = [£k5, [£-k5, £a+ioks]] = £-a+(io+2)k5

and hence xwi0 is proportional to the element yy'wi0 for all x E L2kδ and all y,y' E

Lkδ. The obtained contradiction shows that Xα+m0δv = 0 for some m0 E Z, which

implies that Xα+mδv = 0 for all M > m 0, where α + mδ E A. •

Proposition 3.3. Let V be an irreducible non-zero level L-module with finite-

dimensional weight subspace, v E Vλ\ {0}, λ E H* and v is a maximal element.

Then

(i) A subspace V = ^2keZ V^^ contains a maximal element for any \i E P(V);

(ii) λ + nS^ P(V) for alln>0.

Proof. Let \i E P(V). Since V is irreducible, there exists u E U(L) such that

0 7 uv E Vfj,. Then a G-submodule U(G)uv is completely irreducible by Theorem ??

and every irreducible component L C U(G)uv is isomorphic to Mε(λ(c)), ε E {+, —}

up to the shift of gradation. Applying Lemma ??, we conclude that Lnδuv = 0 for

sufficiently large n. Thus V+ Pi V ^ 0 and there exists at least one irreducible

component L of type M+(λ(c)) such that L n V^ = 0. Now the same arguments as

in the proof of Proposition ?? show the existence of a maximal element in V. Thus

(i) is proved. If λ + nδ E P(V) for some n > 0 then applying (i) to the weight λ + nδ,

we conclude that V+ n (^2keZ V\-\-(k+n)s) = 0. But this contradicts our assumption

of the maximality of v which completes the proof of (ii). •

The main result of this section is the following statement which follows immedi-

ately from Proposition ??, (ii).

Proposition 3.4. Any irreducible weight L-module of a non-zero level with finite-

dimensional weight subspaces is non-dense.

4. EXISTENCE OF A TVSEMIPRIMITIVE ELEMENT IN IRREDUCIBLE NON-ZERO

LEVEL m o d u l e w i t h f i n i t e - d i m e n s i o n a l w e i g h t s p a c e s

The mail result of the paper is the following

Theorem 4.1. Let V be an irreducible weight L-module of a non-zero level with

finite-dimensional weight spaces. Then V is a quotient of a generalized Verma mod-

ule. Moreover, either V is a highest weight module or there exist a basis π of A, a

subset S C π and an irreducible weight torsion-free L(S) + H-module N such that

V ~LS(N).

Remark 4.2. It follows from [?] that the classification of irreducible torsion-free

modules with finite-dimensional weight spaces for a reductive finite-dimensional Lie

algebra can be reduced to the classification of irreducible torsion-free modules for

algebras of type An and Cn. Since L(S) + H is a finite-dimensional reductive Lie

algebra, Theorem ?? therefore reduces the classification of irreducible non-zero level

modules with finite-dimensional weight spaces for Affine Lie algebras to the classifi-

cation of irreducible torsion-free modules for algebras of type An and Cn.

It follows from Proposition ?? that in order to prove Theorem ?? it is enough to

show the existence of a PS-semiprimitive element in V with S = π \ {α} for some

basis π of A and some «GTT. Hence Theorem ?? is a corollary of the following

Proposition 4.3. Let V be an L-module. Then there exist a basis Π, λ E P(V)

and α E π such that Vλ contains a non-trivial PS-semiprimitive element, where

S = π\{α}.

Lemma 4.4. Let V be an L-module, if E A r e

; λ E P(V). Then there exist only

finitely many integers m E Z + for which V\+mLp contains a non-zero element v,

satisfying the condition C^v = 0.

Proof. Consider a subalgebra C(<p), generated by C±tp. Then C(ip) ~ sl(2). If

m E Z and 0 ^ v E V\+rmp, such that C^v = 0, then v generates a £(t/?)-submodule

with a highest weight A([X¥,,X_¥,]) + 2m. Suppose that there exist infinitely many

m E Z + satisfying Lemma ??. But then there exist infinitely many non-isomorphic

£(<p)-modules with highest weight that have a non-trivial intersection with Vλ. We

conclude that dimVλ = oo which contradicts the conditions. Lemma ?? is proved.

•

Let V be an L-module, /i E P(V), v E V^, v = 0, A C A. A pair (v, A) is called

admissible if Lβv = 0 for all β E A and A™ C A.

Lemma 4.5. Let V be an L-module, v E Vλ, λ E P(V), A C A and suppose

that (v,A) is admissible. Then for any γ1,γ2 E Are, such that γ1 + γ2 E A and

(3 + nγ1 + /C72 ^ (AU {0}) \ A for all β E A, n,k > 0, at least one of the pairs

(V, A U {γ1}) and (V, A U {γ2}) is admissible for some v' E Vy, A' E P(V).

Proof. Fix non-zero elements Xγ i Clt, i = 1,2 and consider elements vn =

Xγn1v,n > 1. It follows that for any β E A, n, k > 0 holds XβXγk2Xγn1v = 0. If

there exists n such that vn = 0, wra_i 7 0 then (vn-i,A U {γ1}) is admissible. Sup-

pose now that vn = 0 for all n > 1. Consider elements vn,k = Xγk2vn, n > 1,

1 < k < n. If vn,k = 0, vn>k-i 7 0 for some n, k then (vn>k-i, AU {γ2}) is admissible,

implying the lemma. Assume finally that vn,k = 0 for all 1 < k < n, n > 1. Then

0 7 vn,n E Vλ+n(γ1+γ2) and Lγ1+γ2vn,n = 0 for all n > 1 which contradicts Lemma ??

and hence completes the proof. •

If a pair (v',A U {γ1}) or (v',A U {γ2}) is admissible where v', A, γ1, γ2 satisfy

the conditions of Lemma ?? then we will say that it is obtained from (v, A) by an

admissible transformation.

Proposition ?? follows from Lemma ?? and Proposition ?? from the next section.

5. COMBINATORICS OF THE ROOT SYSTEM.

Define a set fl = {A G 2 A | |A+ \ A| < oo and A D A™1}. An element A e fl

will be called final if there exists a basis vr' and β G ir' such that A D A + \ add(S)

where S = ir'\ {β}.

Let W be the Weyl group of A. If F 1 , F 2 G fl and there exists w E W such that

F1 C add(wF2) then we will write F1 jZ F 2 . If F1, F2 G Q, F1 jZ F2 and F2 |Z F1

then the elements F 1 , F 2 are called W-equivalent and we write F1 ~ F 2 in this case.

We will denote by F the equivalence class of F.

Consider an oriented graph Γ = (Γ0,Γ1 U Γ2) with vertices Γ0 corresponding to

the equivalent classes fl/ ~ and arrows Γ1 U Γ2 C Γ0 x Γ0 defined by the following

rules:

(i) if F1 H F2, F1,F2Efl then (F2, A) G Γ1 ;

(ii) if F 2 = add(F1), Fl,F2ett then {F1,F2) G Γ 1 ;

(iii) if F G Q, γ1,γ2 G A r e such that γ1 + γ2 G F and β + nγ1 + kj2 & (AU{0})\F

for all β G F, n, k > 0 then (F, F^, (F, F2) G Γ2, where F1 = Dγ1(F) := F U {γ1},

F2 = D*(F):=FU{l2},1=(ll,l2);

(iv) if (F1,F2),(F2,F3) G Γ1 t h e ^ A , ^ ) G Γ1 ;

(v) if (F^F^ G F 2 and (F2, F3) G Γ1 then (F^Fs) G Γ2.

Consider the graph universal covering: Π : Γ —> Γ.

Definition 5.1. A vertex F G F o is called primitive if there exists a finite subgraph

T = (T0,Ti U T2) C F satisfying the following conditions:

(i) F G n(f0),(ii) every zero out-valency (sink) vertex of the graph T is a final element,

(iii) if (F, Fi) G T2 then there exist a sequence γ = (γ1, γ2), γi G A r e

; i = 1, 2 and

F 2 G f0 sucht h a t (F,F2) e f2, F1 C Dγi(F), F 2 C ^ ( F ) ; {i, j } = {1,2}.

Proposition 5.2. A is primitive for any A E fl.

Note that Proposition ?? immediately implies Proposition ??. Indeed, let V be

an irreducible L-module. Then Proposition ?? implies the existence of a maximal

(with respect to a proper choice of a basis of A) element in V. Applying Lemma ??

we conclude that there exists an admissible pair (v,A) with v G V and A C A,

|A + \ A| < 00. It follows from Proposition ?? and Lemma ?? that there exists a

sequence of pairs P1,... ,Pl such that P1 = (v, A), Pl = (V, A') where A' is a final

element and Pi+1 is obtained from Pi by an admissible transformation, i = 1,...,l.

Now Proposition ?? clearly follows.

To prove Proposition ?? we construct for each Affine Lie algebra a certain subset

M C Q satisfying the following conditions:

(A1) for any A EQ there exists F E M such that F C A,

(A2) for any non-final element F E M there exist F1,F2,.. E M and

k such that

•D2

j(1)(F), i= and

(A3) for any A E M, |{F E M| F D A}| < oo.

One can easily check that conditions (A1)-(A3) ensure that any vertex A E F o is

primitive.

The sets M are given below for all Affine series. All other Affine Lie algebras

were treated by making use of a computer. Due to a very complicated structure of

M these cases are left out.

5.1. Case A(1)n.

Let L be of type A(1)n , π = {α0,..., αn} and δ = α0 + α1 + . . . + αn.

For convenience we will use the set Z n + 1 = Z/(n+ 1)Z to index the elements of

a basis π = {αi| i E Zn +1} of A. An order < on Z n +1 is defined by a natural order

on the set of non-negative minimal representatives.

For i,j E Z n + 1 and l G Z set a\ ) = αi + αi+1 + . . . + α j + lδ if i < j andl)

* = -a.i+1

E Z set a g =

+ . . . + αn + α0 + . . . + ΑJ 18 if i > j . Clearly, E Zn + 1 ,

JOexhaust all roots in A (cf. [?], Proposition 6.3). Note also that (l+1)δ = a0 = a^ i

for a l l l E Z,i E Zn+1.

Let i1, i2, . . ., ik ^ Z n + 1 . We write i1 -< i2 ^ . . . ^ ik ^ i1 if there exists 0 < m < k

such that im < im+1 < ... < ik < i1 < • • • i %m-\-

Define a partial order on A as follows. For α1,α2 E A set α1 > α2 if there exist

H, i-2,ji,J2 ^ Z n + 1 and l1, l2 E Z such that αm = αi(mlm,j)m, m = 1, 2 and either l1 > l2

or l1 = l2 and i1 -< i2 ^ j 2 ^ j1 -< i1.

T h e W e y l g r o u p W is g e n e r a t e d b y t h e r e f l e c t i o n s si, i E Z n + 1 . F o r i,j, k E Z n + 1

a n d l E Z w e h a v e t h a t si(αi,k(l)) = αi(+1l),k if i ^ k,i

i^Jii^J-U Si(aV) = ajjlj.! and Si(o;g) = JJ

Denote

k + 1; si(αj,i(l)) = aj}-i if

in all other cases.

p ai,-,jkn\ — ir,jr E Zn+1,r =1,...,k.

10

For convenience we will identify ik+m (resp. jk+m) w i t h im (resp. j m ) for all integer

m.

Set M{1) = {Pil';;£k(l) \n<i2<---<ik, 3i<]2<---<3k< Ji, 3m < 3m+i •<

im ^ j m , jm+1 ^ im -< + 1 ^ jm+1, m=1,...,k, k = 2 , . . . , n + 1} U {Pii(l)| i =

0,.. . , n} and consider

M={JM(l)i=0

It follows immediately from the construction of M that the conditions (A1) and

(A3) are satisfied.

Note that F E M is final if and only if F D F/(0) for some 0 < i < n. Note

also that the minimal element of M(l — 1) with respect to the usual sets inclusion

is the maximal element of M(l). In particular, F^'" '^ 7 "" (I) (resp. P00,,11,...,n,...,n (0) i s the

minimal (resp. maximal) element of M(l).

Suppose that F = Pij11,...,i,...,jkk(l) E M(l) is not a final element. Consider two sets of

intervals h = {[im, im+1]| m = 0,.. ., k} and I2 = {[jm,jm+1]| m = 0,. . . , k} and

define the length of an interval [t, s] E I1 U I2 as s — t if s > t and n + 1 + s — t

otherwise.

Let [jp, jp+1] G I2 be an interval of the maximal length in / [ U / j .

If ip = j p = ip+1 — 1 = jp+1 — 1 then l > 0, F = ^'^''.'.'^(O and we can view

F as an element of M(l — 1). Otherwise set γ1 = a\ + l j - + l - 1 and consider F1 =

If α( l) 4. F then choose f such that α( ,-, 4. F and α( -,,, E F and) ,-, 4. F a n d a- -,,,

set γ2 = Q! •/. Fix r such that j r = j ' + 1 and i r _i -< j p + 1 ^ ir ^ i r_i (such

r clearly exists). Since [jp,jp+1] has the maximal length and smF = F for all

m 4. {i1,...,ik,j1,... ,jk} t h e n F2 = s i p + 1 _ i • sjp+1-2 • ... • sir_1+i(F U {γ2}) =

F U {^î^.J = F U itriiCO e Mil).I f αj(lp)+1,jp+1 G F t h e n l > ° a n d w e s e t γ2 = a£j j- p + 1 -2- A g a i n F 2 = s J p + 1_! •

sjp+1-2 ••••• s i r _ 1 + i ( F U { γ 2 } ) = F 1 G M ( l ) .

We see that in all cases above Fi = Dγi (F), i = 1, 2 where γ = (γ1, γ2), and hence

(A2) follows.

The case when [ip, ip+1] E I1 has the maximal length in I1 U I2 is similar.

5.2. Cases C^1^, Dn(2) and A ^ _ 1 ) 5 n > 3.

Let π = {α1,.. . ,αn}. First we describe the root system in each case (cf. [?],

Proposition 6.3).

Let L be of type C ^ and δ = α1 + 2(α2 + . . . + avi-i) + ΑN.

11

0L\ Q.2 OL3 OLn—\ 0Ln

We will use t h e following n o t a t i o n for t h e r o o t s in A :

j = | i | + l , . . . , n - 1 , n + l , . . . , 2 n - | i | - 1, l e Z

α ( l - a ( 0 + α ( l 7 - 2 2n - 2

+1 , i = - n + 2,. . . , n - 2,

= 2lδ + α,

where

_

1 0, otherwise,

% • =

I αk+1 + . . . + αm, i < j , k = m a x ( 1 , i ) , m = min(n — 1,j),

I 0, o therwise ,

Oi2n-j+i + ... + αn, 2n> j > r

, otherwise.

Now let L = Dn(2) and δ = α1 + . . . + αn.

Oi\ OL2 OL'i OLn-\ an

Then the roots of L are exhausted by the following elements:

a g = a~ + a°id + a+ + 2/5, i = -n + 1,..., n - 1, j = \i\ + 1,..., 2n - \i\ - 1, / e Z

where

Q!l + . . . + Q!_j, — n j > n + 1a+ j =

0, otherwise.

12

I (2)Suppose that L is of type A2/n_^ and δ = αn + 2(an-\ + . . . + α1).

o o—> <—c< Q

The the roots of L are exhausted by the following elements:

a?] = a~ + a", + αj + 15, i = -n + 2,... ,n - 2,

=|i 2n-\i\-l, l eZ

13 = n otherwise,

αi,n(l) = ai,n-l + αi,n(l)+1, i = ~n + 2, . . . , n - 2,

leZ

where

f a\ + . . . + Q!_j, — n j > n + 1c i •' = •{

0, otherwise.

Set in all the above cases a§n = (2l + 1)5, a ^ ^ = (2l + 2)δ, i = 0,...,n.

Define a partial order on A as follows: α > β, α, β E A if and only if there exist

h,ji,h,i2,J2, h^Z such that α = af^, (3 = αi(2l2,j)2, and either h > l2 or h = l2 and

i1 <i2< j2 <h-

The Weyl group generators act by the following rules:

(l

- n

and identically in all other cases.

Set

13

S e t

{Pij11,...,i,...,jkk(l)| - n < - j i < n < . . . < i k < n , l < j i < . . . < j k < 2 n - i k < 2 n ,

|im| < jm, m = 1,..., k, k = 1,..., n + 1}

andoo

M=\jM(l)i=0

It follows from the construction that M satisfies (A1) and (A3). Note that F G M

is final if and only if F D Pii+1(0) for some 0 < i < n - 1. The sets P^-n+'CA1)

and -Po'i' n-i(0 a r e respectively the minimal and the maximal elements of M(l)

with respect to the sets inclusion. Also note that the minimal element of M(l — 1)

equals the maximal element of M(l).

Denote [s] = |s| if |s| < n and [s] = 2n — |s| otherwise.

Let F = Pij11,...,i,...,jkk (l) e M(l) be a non-final element. Set i0 = 1 - j1 and jk+1 =

2n — ik — 1. The next statement can easily be checked by direct computations.

L e m m a 5.3. For Pij11,...,i,...,jkk(l) E M(l) either j m = im + 1 for all m = 1 , . . . , k and

l > 0 or there exist p and r such that one of the following conditions holds:

( 1 ) [jr] < [jp+1 - 1] < [jr+1 - 1] < [jp], jr+1 - 1 < n < j p + 1 - 1, 1 < r < p < k .

(2) [ir+1] < [jp+1 - 1] < [ir + 1] < [jp], ir + K 0, n < j p + 1 - 1. 0 < r < p < k,

(3) [jr] < [ip + 1] < [jr+1 - 1] < [ip+1], jr+1 ~ 1 < n, 0 < ip + 1, 1 < r + i > n , « p - l < 0 ,

1 <r <p<k.

Now for each of the cases in Lemma ?? we provide F1 and F2 for which the

condition (A2) holds.

Suppose that j m = im + 1 for all m = 1,..., k and l > 0. If ip + 1 ip+1 for some

1 < p < k then (A2) holds for

_ (0 _ C-1)

F1 = F U {γ 1} G M(l) and F2 = w(F U {γ 2}) = F1, where w = sJp+1_2 • . . . • s j p + 1 .

If i1 = 0 then

_ (0 _ C-1)γ1 = tt-ii+l,^, γ2 = «-i1+l,2n-ii)

F1 = FU {γ1} G M(l) and F2 = w(FU {γ2}) = F1, where w = s^-i •... • s1 •... • si1.

\iik±n-\ then 7 l = a ^ ^ ^ . a , 72 = a ^ x ^ ^ - a , ^ = F U {71} G Mil)

and F2 = w(F U {γ2}) = F1, where w = sik+3 • ... • sn • ... • sik+2. Finally, if i1 = 0,

ik = n ~ 1 and ip + 1 = ip+1 for all 1 < p < k then F is the maximal element of M(l)

14

and hence the minimal element of M(l — 1). Thus the induction on l (the maximal

element of M(0) is final) completes the proof of (A2) in this case.

Assume that F satisfies Lemma ??, ??. In this case we can choose

_ (0 _ (0γ1 = aip+l,jp+i-V 72 = air+lt\jp+1-l]i

F1 = F U {γ1} G M(l) and F2 = w(F U {γ2}) = F U {a v + i J r + 1 - i } G M(l), where

w = S[jp+1-i] • . . . • S[Jr+1-i], for which (A2) holds.

If F satisfies Lemma ??, ?? then set

_ (0 _ (0γ1 = aip+l,jp+i-V 72 = ajp+1-2n-l,jr+l-l>

F1 = F U {γ1} G M(l) and F 2 = w(F U {γ2}) = F U {a v + i J r + 1 - i } G M(l), where

w = s[ir + 1] • • • • s S\jp+1-1] + 1.

If F satisfies Lemma ??, ?? then

_ (0 _ (0γ1 = îp+ijp+i-D 72 = αir+1,ip.

F1 = F U {γ1} G M(l) and F 2 = w(F U {γ2}) = F U {a v + i J r + 1 _i} G M(l), where

If F satisfies Lemma ??, ?? set

_ (0 _ (0γ1 = aip+i,jp+1-v 72 = a_i p _i ) [ i r + 1 _i],


Assume now that F satisfies Lemma ??, ??. Denote

T1 = {a4 m | ir < k < 0, m = j s + 1 - 1 where is < k < is+1,r < s <p},

T2

( k ) = {a\m\ m = 2n-jp + 1,...,n}, k = ir + 1 , . . . , 0,

T2 = T 2

( i r + 1 ) U . . . U T 2

( 0 ) .

Consider any sequence of all possible pairs 7 ^ = (7J , 7 2 )), i = 1, . . ., q such

that 7 j = αk,m G Ti, γ 2 E T^ . Then it is easy to check that for any γ(i) holds

M 3 F U {a^+i^^- i } = «>i(F U {7?}) E D^D^ • • • D*w (F) for some proper

r<s<p)Wl eW&ndD2

i(q)---D2

i(1)(F) = FUT2,w2(FUT2) D FUPÛP^1 eM

for some proper w2 G W.

5.3. Case D^, n>5.

Let £ be of type Dn_XJ vr = {a\,..., an} and ^ = a\ + o;2 + 2(a3 + . . . + ara_2) +

Q!ra_i + αn. We will use the following notation for the roots in A:

15

0L2 0Lr

j =|i

where

,n-l,n+l,...,2n-\i\-l, l eZ

+ ... + αm,k = m a x ( 2 , i ) , m = min(n — 2,j),

1 α* 1

= α 2 + α3

α0,

(an-i,

= α Oi2n+l-j

0,

α1,

i0,

+ ..

+ ...

if i<0

otherwise,

if i = 1,

. + a_i, ifi < - 2 ,

otherwise,

if j = n —

+ an_i, if j > n +

otherwise,

1,

2,

# =α n , if j > n

0, otherwise.

b e t Q J . J 2ri-i = (.' + J-J") i = i-, • • • ,n — l, I E ZJ.

A partial order on A is defined as follows: α > β, α, β E A if and only if there

exist i1,j1,l1,i2,j2,l £ Z such that α = a^ 1 ^, (3 = αi(2l2,j)2 and either l1 > l2 or

l1 = l2, i1 < i2 < j 2 < j1, {i1, i2} 7 { — 1,1} and {j1, j2} 7 {n + 1, n — 1}.

Consider a group W = W x U of the root system A, where U is generated by

two automorphisms u1, u2 of the Dynkin diagram, which permute the simple roots

α1, α2 and an-i, αn respectively. The action of the generators of W is defined by

the following rules:

1 < i < n,0 < |k| < i - 1,

2<i<n-l,

16

0 < |i| < ^ - 2,

ag) = a^-, j = 2 , . . . , n - 1, n + 1,.. . , 2n - 2,

) S « = l ) + 1,. . . , - 1 , 1 , . . . , n - 1,

and trivially in all other cases.

Let

and

Set

= {Pij11,...,i,...,jkk(l)| -n<ii<i2<...<ik<n-l,l<j1<...< 3k.x < jk

jm+1 ~ im < 2n, 1 < m < k - 1, j 1 < n - 1,

ik > — 1, |im| < jm, m = 1 , . . . , k, k = 1 , . . . , n}

and

i=0

It follows from the construction that M satisfies (A1) and (A3). Note that

F E M is final if and only if F D /^+1(0) for some l < i < n - 2 or FD

P-i(0) or F D Pn-2(0). The minimal and the maximal elements of M(l) are

P-nlCn+t2-nin:!-Xi1(l) ^d P^C^^W respectively. The minimal element

of M(l — 1) equals the maximal element of M(l).

Denote [s] = |s| if |s| < n and [s] = 2n — |s| otherwise.

Let F = Pij11,...,i,...,jkk(l) G M(l) be a non-final element. Set i0 = 1 — j1 and jk+1 =

2n — ik — 1 . The next statement can easily be checked by direct computations.

Lemma 5.4. For Pij11,...,i,...,jkk(l) e M(l) either j m = im + 1 or {im,jm} = {-1,2} or

{im,jm} = {n — 2,n + 1} for all m = 1 , . . . , k and l > 0 or there exist p and r such

that one of the following conditions holds:

(1) [ j r ] < [jp+1 - 1 ] < [ j r + 1 - 1 ] < [jp], jr+1 - l < n < j p + 1 - l , l < r < p < k .

( 2 ) [ir+1] < [ j p + i - 1 ] < [ir+1] < [jp], i r + 1 < - I , n + 1 < j P + i - l , 0 < r < p < k .

( 3 ) [jr] < [ip + 1] < [jr+1 - 1] < [ip+1], jr+1 - 1< n,1< ip + 1, 1 < r + i > n + 1,

zp_i < - I , 1 < r < p < k.

Now for each of the cases in Lemma ?? we provide F1 and F 2 for which the

condition (A2) holds.

Suppose that j m = im + 1 or {im,jm} = {-1,2} or {im,jm} = {n - 2, n + 1} for

all m = 1,..., k and l > 0. If ip + 1 =£ ip+1, ip =£ — 1, j p + 1 =£ n + 1 for some 1 < p < k

then (A2) holds for_ (0 _ C-1)

7 S ^ ^F 1 = F U { γ 1 } G M ( l ) a n d F 2 = w ( F U { γ 2 } ) = F 1 , w h e r e w = s j p + 1 - 2 ••••• s j p + 1 .

Ifii^-1 t h e n

_ (0 _ C-1)γ1 = tt-ii+l,^, γ2 = «-i1+l,2n-ii)

F1 = FU{7i} G M(l) a n d F 2 = w(FU{72}) = F1, where w = s^-i-.. .•s2-« rs2-.. .•

s^. If jk ± n+ 1 then γ 1 = ««+i)2n-,fc-2, γ2 = c^h2n_lk_2, F1 = FU {γ1} G M(l)

and F 2 = w(F U {γ2}) = F 1 , where w = sik+3 •... • sra_i • U 2 • sra_i •... • sik+2. Finally,

if k = n, i1 = — 1, jk = k + 1 and j m = im + 1 = m for all 1 < m < k then F is

the maximal element of M(l) and hence the minimal element of M(l — 1). Thus the

induction on l (the maximal element of M(0) is final) completes the proof of (A2)

in this case.

Assume that F satisfies Lemma ??, ??. In this case we can choose

_ (0 _ (0T a v 72 = air+1,[jp+1-i]>

F1 = F U {γ1} G M(l) and F 2 = w(F U {γ2}) = F U {a v + i J r + 1 - i } G M(l), where

w = S[Jp+1-i] • • • • s Sbv+i-i], for which (A2) holds.


_ (0 _ (0γ1 =α îp+ljp+i-l' γ2 = ajp+1-2n-l,jr+1-H

= F U {γ1} G M(l) and F 2 = w(F U {γ2}) = F U {a v + i J r + 1 _i} G M(l), where


_ (0 _ (07 ^ ^ a


If F satisfies Lemma ??, ?? set

_ (0 _ (07i - %+ijp+i-i' 72 - a_ip_i)[ i r+1_i],

1 = F U {γ1} G M(l) and F 2 = w(F U {γ2}) = F U {a v + i J r + 1 _i} G M(l), where

Assume now that F satisfies Lemma ??, ??. Denote

4 ir < k < - 1 , m = j s + 1 - 1 where is < k < is+1,r < s <p},

, m | m = 2n-jp + l,...,n-l}, k = ir + 1,..., - 1 ,

18

Ti /TJ(V + 1) II | | rji{—l)

2 = J-2 U . . . U J 2

Consider any sequence of all possible pairs 7W = (7} , 72 )), i = 1, . . ., q such

that 7^ = αk,l(l E T l ' γ E TP • T h e n i t i s e a s y t o c h e c k t h a t f o r a n y 7 W h o l d s

M 9 F U {^Vi^^.J = ^ i ( F U {7«}) C D^D^ • • • £>2(1) ( F ) f o r s o m e p r Oper

r < s 4.

Let L be of type Bn_x with the diagram

Oin-l

α2α3

and δ = 2(α1 + . . . + are_2) + a n_i + αn.

We will use the following notation for the roots in A:

-XA A d>4 \ d>4 A \ QA , i = -n + 2,. . . , n - 2,

. . , 2 n - | z | - l , Z e Z

where

O-i =

αm,k

f

0,

A; = max(0,i), m = min(n —

otherwise,

a.n-i, ifj = n-l,

/3™"1 = α a2n+i_j + . . . + an_i, if j > n + 2,

0, otherwise,

3

αn, if i > n

0, otherwise.

Suppose that L is of type A2r^_3 with the diagram

aOin-l

OLr

19

and δ = α1 + 2(α2 + . . . + an_2) + o-n-i + αn.

The following elements exhaust the roots in A:

of] = a~ + a°id + (3]~l + $ + 15, % = -n + 2,..., - 1 , 1 , . . . , n - 2,

j = | i | + l , . . . , n — l , n + l , . . . , 2 n — | i | — 1, l G Z

« S = « « J + « g , j = 2,...,2n-2,

a $ l _ 1 = 2(l + l ) 8 - a 1 , l e Z

where

a^- = αk+1 + . . . + αm, k = max(1,i), m = min(n - 2, j ) ,

α1 + . . . + a_i, -n<i<0

0, otherwise,

an-i, i f j = n - l ,

hi-i + . . . + an-i, if j > n + 2,

0, otherwise,

j0, otherwise.

Set in all the above cases a_\ 2n_i = (l + 1)δ, i = 1,... ,n — 1 , / G Z .

A partial order on A is defined as follows: α > β, α, β E A if and only if there

exist i1,j1,l1,i2,j2,l £ Z such that α = α(i1l1,j )1,β = αi(2l2,j)2 and either l1 > l2 or

h = k, i 1 k <J2 <ji and {j1,j2} ^ {n+l,n- 1}.

Consider a group W = W x U where U is generated by the automorphism u of

the Dynkin diagram. The action of the generators of W is defined as follows:

I (I) \ W 1 • • ; o • ; /si(α±(l)i,k) = a±(i-i),k> K i <n,i < k < zn-i,K^n,

M±(i-n-i)> Ki<n,\k\<i-1,

Sr2> || w - 2,/ ( l ) (l+1)

•5n(,Q!2_ra)ra_lJ — an-2,n+li

«(a t>_i) = a £+i> « = -w + 2,. . . , n - 2,

and trivially in all other cases.

Let

20

and

Set

M(l)= {Pij11,...,i,...,jkk(l)| -n<ii<...<ik<n-2,l<j1<...<jk.

jm+1 ~ im < 2n, 1 < m < k - 1, j 1 < n - 1,

ik > 0, |im| < j m , m=1,...,k, k = 1 , . . . , n}

andoo

M=\jM(l).i=0

It follows immediately that M satisfies (A1) and (A3). Note that F e M is final

if and only if F D Pi

i+1(0) for some 0<i<n-2orFD P^{Q). The minimal and

the maximal elements of M(l) are P ^ - n S - n S l S T ^ O and P^-tt-^respectively.

The rest of the proof is simply a combination of the proofs in 5.2 and 5.3. We

leave the details out.

5.5. Case A(2

Let π = {α1, α2} where α1 and α2 are short and long roots respectively. Denote

(l) = α1 + (i - 1)α2 + lδ, i = 1,2,

n{V) _ (l-l) (I) n{V) _ (1) (1+1) (I) _ n{V) _ , . , „Pi — O-2 \ C 2 ) P2 — 1 ' 1 ' 0 — Po — ' I 6 i i

where δ = 2α1 + α2. Then A ^ = {a^l I = 1,2,...}, A^ = {af]\ i = 1,2, l =

0,1,...} is the set of positive short roots, A\_ = {/?} | i = 1, 2, l = 0,1,...} is the

set of positive long roots and A + = A™ + A _ s Al

+.

We write αi( 1 a\2 and /^ < p\^' if l2 > h or l1 = l2 and i2 > i1.

The Weyl group generators s1, s2 act as follows:

Let

Pf] = { a > af]\ a E A s } U {0 > p f \ /3 E A 1 } U A™, i , j = 0,1,2, 1 = 0 , 1 , . . . .

S e t

00

M =i=0

21

Note that F G M i s final if and only if F D P0(0), 1 or F I) P1(0) , • The maximal and1,0 .

the minimal elements of M(l) with respect to the sets inclusion are PoQ and P2(,l2respectively and P0(,L0) = P 2 2 .

The tables below provide a required pair γ = (γ1,γ2) and the sets F1 and F 2 for

any non-final F G M(l) and any integer l > 0.

In the case l > 0:

i

1.

2.

3.

4.

5.

6.

F

(l+1)P0,0

P ( 01,2

P2,1

P ( 01,1

p(0P0,1p(0

γ 1

ft1"a?

γ 2

1 ( l )

«?>P?P?og>

F1 = FU{7i}

p«1 , 2p(0

1,1p(0

1,1p(0

0,1p(0

P ( 0P0,0

w2

1

s1

s 2

1

s2

s1

F2 = w2(F U

{γ2})

P2,1p(0

1,1p(0P1,1p(0

1,0p(0

P ( 0P0,0

In the case l = 0:

1.

2.

3.

4.

F

(1)P0,0

(0)P1,2

(0)2,1(0)

P1,1

γ 1

af0)af>a?>

γ 2

1(0)

Pf"

F1 = FU{7l}

) ( 0 )1,2

) ( 0 )1,1

) ( 0 )1,1

P0

(°),1 - final

w2

1

s1

s2

1

F 2 = w2(F U

{γ2})) ( 0 )

2,1) ( 0 )P1,1) ( 0 )

1,1

P1(0),0 - final

Hence, the proof of (A2) can be completed by the induction on l. Statements

(A1) and (A3) follow directly from the construction of M.

6. ACKNOWLEDGEMENT

First author would like to thank the Erwin Schroedinger Institute for Mathemati-

cal Physics, the International Centre for Theoretical Physics in Trieste and Carleton

University for the hospitality and support during his visits in 1996-1997.

R E F E R E N C E S

[1] D.Britten, V.Futorny, F.Lemire, Simple A2-modules with a finite dimensional weight space,Commun. in Algebra, 23(2) (1995), 467-510.

[2] D.Britten, F.Lemire, On level 0 Affine Lie modules, Canad. Math. Bull., 37(3) (1994), 310-314.[3] D.Britten, F.Lemire, F.Zorzitto, Pointed torsion-free modules for Affine Lie Algebras, Com-

mun. in Algebra, 18 (1990), 3307-3321.[4] A.Cylke, V.Futorny, S.Ovsienko, On the support of irreducible non-dense modules for finite-

dimensional Lie algebras, University of Bielefeld, Preprint, 1996.

[5] V.Futorny, Irreducible non-dense A(1)1-modules, Pacific J. of Math. 172 (1996), 83-99.

[6] V.Futorny, Irreducible level zero A(1)n -modules with finite-dimensional weight spaces, PreprintESI 325 (1996).

22

[7] S.Fernando, Lie algebra modules with finite dimensional weight spaces I, Trans. Amer. Math.Soc. 322 (1990), 757-781.

[8] V.Kac, Infinite dimensional Lie algebras, Cambridge University Press, third edition, 1990.[9] S.E.Rao, Classification of Loop modules with finite-dimensional weight spaces, Math. Ann.

(to appear).[10] S.Spirin, Z2-graded modules with 1-dimensional components over Lie algebra A(1)1 , Functional

Analysis and its appl., 21 (1985), 84-85.

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