Indecomposable representations of the Diamond Lie algebra

JOURNAL OF MATHEMATICAL PHYSICS 51, 033515 共2010兲

Indecomposable representations of the Diamond Lie algebra Paolo Casati,1,a兲 Stefania Minniti,2,b兲 and Valentina Salari2,c兲 1

Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via degli Arcimboldi 8, I-20125 Milano, Italy 2 Politecnico di Milano, Piazza Leonardo da Vinci 32, 20126 Milano, Italy 共Received 6 May 2009; accepted 14 January 2010; published online 18 March 2010兲

We study classes of indecomposable representations of the Diamond Lie algebra: a four-dimensional solvable Lie algebra, which is the central extension of the Poincaré Lie algebra in two dimensions. These representations are obtained first 共inspired by Douglas and Premat 关Commun. Algebra 35, 1433 共2007兲兴兲 by embedding the Diamond Lie algebra in the simple Lie algebras A2 and C2, and then by regarding it as a subalgebra of a truncated current Lie algebra. © 2010 American Institute of Physics. 关doi:10.1063/1.3316063兴

I. INTRODUCTION

The Diamond Lie algebra D 共Ref. 8兲 is a four-dimensional Lie algebra with bases J, P1, P2, T, and nonzero commutation relations 关J, P1兴 = P2,

关J, P2兴 = − P1,

关P1, P2兴 = T,

共1.1兲

which can be realized as subalgebra of the simple Lie algebra sp共4 , R兲 through the map

␪J + ␣ P1 + ␤ P2 + ␥T 哫

冢

␤ 2␥ 0 0 −␪ ␤ 0 ␪ 0 −␣ 0 0 0 0 0 ␣

冣

,

共1.2兲

and which is also known in literature under the name Nappi–Witten Lie algebra or simply as the central extension of the Poincaré Lie algebra in two dimensions. This algebra could play an important role in conformal field theory,13,16,17 mainly for two reasons: it is the central extension of the Poincaré Lie algebra in two dimensions and it is a solvable quadratic Lie algebra 共i.e., D admits a bilinear symmetric adinvariant form兲. These properties permit the use of D to construct a Wess–Zumino–Witten model, which describes a homogeneous four-dimensional Lorentz-signature space time.13 However, despite its importance and interest, very little is known about its finite dimensional indecomposable representations. The realization given by the map 共1.2兲 is an example. 共D is a solvable Lie algebra, and therefore all its finite dimensional irreducible representations are one-dimensional and quite uninteresting.兲 In this paper, we shall examine some classes of such representations. In a recent article, Douglas and Premat4 examined the finite dimensional indecomposable representations of the Euclidean Lie algebra e共2兲关i.e., the complexification of the Lie algebra of the group of Euclidean motions in the plane E共2兲 ⯝ SO共2兲 › R2兴. They further developed the investigations made some a兲

Electronic addresses: [email protected] and [email protected]. Electronic mail: [email protected]. Electronic mail: [email protected].

b兲 c兲

0022-2488/2010/51共3兲/033515/20/$30.00

51, 033515-1

© 2010 American Institute of Physics

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Casati, Minniti, and Salari

years before by Repka and de Guise14 on the same subject. The complexification of Diamond Lie algebra can be regarded as a central extension of the complexification of the Euclidean Lie algebra e共2兲. In their work, Douglas and Premat developed a wide class of indecomposable representations of the Lie algebra e共2兲 by embedding it in the simple Lie algebra sl共3 , C兲共i.e., in the Lie algebra of traceless 3 ⫻ 3 complex matrices兲. We shall show how such strategy also works in our case by giving rise to a class of indecomposable representations. Moreover, we shall also prove how a technique developed to obtain representation of the so-called truncated current Lie algebras3 provides a tool to obtain a second class of indecomposable representations of the Diamond Lie algebra. Finally, we shall use Kirillov’s embedding 共1.2兲 of the Diamond Lie algebra D in sp共4 , R兲 to construct a third inequivalent class of indecomposable representations. This paper is organized as follows. In Sec. I we summarize some important facts about the Diamond Lie algebra. In Sec. II we show how the Diamond Lie algebra can be embedded in two nonequivalent ways in the Lie algebra sl共3 , C兲: first by generalizing the embedding of e共2兲 considered by Douglas and Premat; second by proving that the 共complexification兲 of Diamond Lie algebra is a subalgebra of a Borel subalgebra of sl共3 , C兲. Further, we show its embedding in the truncated current Lie algebra sl共2 , C兲共2兲 and a more suitable form of its embending 共1.2兲 in the Lie algebra sp共4 , R兲. In Sec. III we shall derive some general properties of the indecomposable finite dimensional representations of the Diamond Lie algebra. In Sec. IV we shall construct a first class of indecomposable representations corresponding to the two embeddings in sl共3 , C兲. We show how in this case, the irreducible finite dimensional representations of sl共3 , C兲 remain indecomposable once restricted to the Diamond Lie algebra. In Sec. V we shall use some of the tools of the theory of the current truncated Lie algebras in order to construct some new representations which will decompose into the direct sum of two indecomposable representations. With a few exceptions these are inequivalent with respect to those obtained in the previous sections. Finally, Sec. VI is devoted to constructing a new inequivalent class of indecomposable representations by using the embedding 共1.2兲 of the Lie algebra D in sp共4 , R兲. II. DIAMOND LIE ALGEBRA

In this section we summarize the most important properties of the Diamond Lie algebra and describe its embeddings in the Lie algebras sl共3 , C兲, sp共4 , C兲, and in a particular truncated current Lie algebra, which will be used in the construction of its indecomposable representations. Maybe the most interesting property of the Diamond Lie algebra D is the presence of a bilinear symmetric adinvariant nondegenerate form on it, which, being D solvable, cannot be its Killing form. It can easily be shown that the bilinear form ⍀ on D, whose matricial expression with respect to the bases P1, P2, J, and T in 共1.1兲, is given by13

冢冣 1 0 0 0

⍀=

0 1 0 0

0 0 0 1

,

0 0 1 0

is a bilinear symmetric adinvariant nondegenerate form on D. This is actually not an accident but a direct consequence of the fact that D is the double extension of an Abelian Lie algebra.11 Let us first consider the two-dimensional Poincaré Lie algebra: P = so共1 , 1兲 › R2; its second cohomology group H2共P , R兲 has dimension 1 and, therefore, the Lie algebra P admits a nontrivial central extension P 丣 RT, which coincides with the Diamond Lie algebra D. Two other constructions of D are important for our purposes in this paper 共or more precisely in the first case of its complexification兲: one as a central extension of the complexification of the Euclidean Lie algebra in two dimension e共2兲 and the other as a semidirect product between the three dimensional Heisenberg Lie algebra H and the Abelian Lie algebra R.

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Indecomposable modules of the Diamond Lie algebra

The compexification 共for which we shall keep the same symbol D兲 of the Diamond Lie algebra: D 丢 RC displays the following 共complex兲 bases: P+ = P1 − iP2,

P− = P1 + iP2,

T,

共2.1兲

J,

where i is the imaginary unit, whose nonzero commutators are 关J, P+兴 = iP+,

关J, P−兴 = − iP−,

关P+, P−兴 = 2iT.

共2.2兲 4

Now the complexification of the Euclidean Lie algebra in two dimension e共2兲 is the threedimensional Lie algebra with bases P+, P−, J, and commutation relations 关P+, P−兴 = 0,

关J, P⫾兴 = ⫾ P⫾ .

This algebra, as in the case of the Poincarè Lie algebra, has the second group of cohomology group H2共e共2兲 , C兲 of dimension 1 and, therefore, admits a nontrivial central extension, which coincides with the complexification of the algebra D described above. This observation, as we shall see, allows us to extend 共or maybe better to construct a modified extension兲 of the embedding of e共2兲 in the simple Lie algebra sl共3 , C兲 considered by Douglas and Premat for the Lie algebra D. The second realization, beyond its intrinsic interest, suggests a possible application of the Diamond Lie algebra in harmonic analysis and allows us to embed the algebra D in sl共3 , C兲 in a way which is inequivalent to the previous one. Let us describe it in more detail. The threedimensional Heisenberg Lie algebra H is the nilpotent Lie algebra with bases P1, P2, T, and nonzero commutator 关P1, P2兴 = T. This algebra admits a noninner derivation ␦ given by the relations

␦ P 1 = P 2,

␦ P 2 = − P 1,

␦T = 0.

Using the derivation ␦, we can construct the semidirect product: H’␦R, which coincides with the Lie algebra D. We have already observed that a way to construct representations of the Diamond Lie algebra D is to embed it in the Lie algebra sl共3 , C兲. We shall do it in two inequivalent ways: first generalizing the embedding considered by Douglas and Premat for the Lie algebra e共2兲 and then by extending the canonical three-dimensional representation of the Heisenberg Lie algebra H. To describe such embeddings in detail, let us first briefly recall the definition of the simple complex Lie algebra sl共3 , C兲. The Lie algebra sl共3 , C兲 of traceless 3 ⫻ 3 matrices is the simple complex Lie algebra characterized by the Cartan matrix,7 A = 共aij兲1艋i,j艋2 =

冉

2

−1

−1

2

冊

.

Let H1, H2, E1, E2, E12, F1, F2, and F12 be the Chevalley basis of sl共3 , C兲 defined by

冢

a

冣

c e aH1 + bH2 + cE1 + dE2 + eE12 + fF1 + gF2 + hF12 = f b − a d . −b h g Following Douglas and Premat, we want to construct indecomposable finite dimensional representations of D by restricting those of sl共3 , C兲 to the two inequivalent embeddings of D in sl共3 , C兲 given in the next lemma which can be easily proven by direct computation. Lemma 2.1: The maps ⌽ : D → sl共3 , C兲 and ⌿ : D → sl共3 , C兲 defined on the basis (1.1) of D by 1 ⌽共P1兲 = 共E1 + F12兲, 2

i ⌽共P2兲 = 共E1 − F12兲, 2

i ⌽共J兲 = 共2H1 + H2兲, 3

i ⌽共T兲 = F2 , 2

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1 ⌿共P1兲 = 共E1 + E2兲, 2

i ⌿共P2兲 = 共E1 − E2兲, 2

i ⌿共J兲 = 共H1 − H2兲, 3

i ⌿共T兲 = − E12 2

共2.3兲

are inequivalent Lie algebra embeddings. Remark 2.2: 共1兲

It is easily shown that the maps ⌽ and ⌿ regarded as representations of D are one the contragredient of the other,

␰⌽共X兲␰−1 = − ⌿共X兲T,

␰−1⌿共X兲␰ = − ⌽共X兲T,

∀ X 苸 D,

共2.4兲

where ␰ is the matrix

冢

0 −1

␰= 1 0 共2兲

共3兲共4兲

0

0

0

0

−1

冣

and XT denotes the transpose of X. Two representations of D obtained composing two equivalent embeddings of D in sl共3 , C兲 with the same representation ␲ of sl共3 , C兲 are equivalent. In particular, the two representations of D : ␲⌽ = ␲ ⴰ ⌽ and ␲⌫ = ␲ ⴰ ⌫ obtained composing a given representation ␲ of sl共3 , C兲, respectively, with the two equivalent embeddings ⌽ and ⌫ = ␰⌽␰−1 given in 共2.4兲 and 共2.3兲 are related by the equivalence: ␲⌽ = ␲共␰兲␲⌽␲共␰−1兲, where we have denoted with the same letter ␲ the corresponding representation of the group SL共3 , C兲. We shall see that as a consequence of the previous remark, although the embeddings ⌽ and ⌿ are inequivalent, the corresponding classes of representations of D overlap each other. On the bases P+, P−, J, and T of the complexification of D, the maps ␾ and ⌿ are given by ⌽共P+兲 = E1,

⌽共P2兲 = F12,

⌿共P+兲 = E1,

⌿共P−兲 = E2,

i ⌽共J兲 = 共2H1 + H2兲, 3 i ⌿共J兲 = 共H1 − H2兲, 3

i ⌽共T兲 = F2 , 2 i ⌿共T兲 = − F12 . 2

共2.5兲

A further inequivalent class of representations may be obtained by embedding our algebra in a truncated current Lie algebra. The truncated current Lie algebras are quotients of loop algebras recently considered in literature 共sometimes under a different name兲, which may be defined explicitly as follows. Definition 2.3: (Reference 18) Let g be a Lie algebra over a field K of characteristic zero; for any positive integer number n 苸 N, the Lie algebra g共n兲 = g 丢 K共n兲

共2.6兲

is called a truncated current Lie algebra 关where K共n兲 = K关t兴 / 共t兲n+1, K关t兴 is the ring of polynomials in the variable t and 共t兲n+1 is its principal ideal generated by tn+1兴. We can identify g共n兲 with the vector space of all polynomials in t with coefficients in g of a degree less or equal to n endowed with the Lie bracket,

冋

n

n

j=0

i=0

册

n

k

兺 X jt j, 兺 Y iti = 兺兺关X j,Y k−j兴tk . k=0 j=0

In this paper, we shall use the embedding of the Diamond Lie algebra D in the truncated Lie algebra sl共2 , C兲共2兲 given by the map ⌰ : D → sl共2 , C兲共2兲共which can easily be seen to be an isomorphism兲,

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⌰共J兲 = 21 共E − F兲丢 1,

⌰共P1兲 = 共E − F兲丢 t,

⌰共P2兲 = H 丢 t,

⌰共T兲 = − 2共E + F兲丢 t2 , 共2.7兲

where H, E, and F are the canonical bases of sl共2 , C兲,

H=

冉冊 1

0

0 −1

,

E=

冉冊 0 1 0 0

,

F=

冉冊 0 0 1 0

.

Using the map sl共2 , C兲共2兲 → sl共2 , C兲共2兲 given by X 丢 tk 哫 ␹X␹−1 丢 tkk = 0 , 1 , 2, where X 苸 sl共2 , C兲 and

␹=

冉冊

1 i 1 , 2 1 i

the embedding ⌰ may be transformed into the equivalent one, ⌶共J兲 = 21 H 丢 1,

⌶共P+兲 = 2E 丢 t,

⌶共P−兲 = 2F 丢 t,

⌶共T兲 = − 2iH 丢 t2 .

共2.8兲

Now the importance of such embeddings is due to the fact that it is easy to construct representations of the truncated current Lie algebras g共n兲 provided one knows representations of the Lie algebra g.3 Theorem 2.4: If ⌸ : g → Aut共Km兲 for some m is a true representation of g, then the map ˆ :g共n兲哫 Aut共Km共n+1兲兲 ⌸ given by

ˆ ⌸

冉兺冊

X it i =

i=0

冢

⌸共X0兲

0

0

0

0

0

⌸共X1兲 ]

⌸共X0兲

0

0

0

0

0

0

]

]

]

]

. . . . . . ⌸共X0兲

⌸共Xn−1兲 ⌸共Xn兲

0

⌸共Xn−1兲 . . . . . . ⌸共X1兲 ⌸共X0兲

冣

is a true representation of g共n兲. The map 共1.2兲 embeds D in sp共4 , R兲 and therefore its complexification in sp共4 , C兲兲. Since we shall see that this latter embedding gives rise to a 共new兲 class of finite dimensional indecomposable representations of the Diamond Lie algebra D, for the convenience of the reader, we will finish this section by briefly describing the simple Lie algebra sp共4 , C兲. The Lie algebra sp共4 , C兲 is the Lie algebra of the subgroup SP共4 , C兲 of GL共4 , C兲, which preserves the canonical symplectic form of C4. This Lie algebra is a simple complex Lie algebra of type C2 共recall that C2 coincides with B2兲 whose Cartan matrix is7 A = 共aij兲1艋i,j艋2 =

冉

2

−1

−2

2

冊

.

Actually, we are not going to use directly the embedding 共1.2兲 but rather an equivalent one. For our purposes, it is better to consider the “realization” of the simple Lie algebra C2 共see, for example, Ref. 7 or even more explicitly Ref. 6兲 given in Appendix 1 of Ref. 5. In this formulation, the Chevalley basis of sp共4 , C兲 is defined by

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aH1 + bH2 + cE1 + dE2 + eE12 + fE112 + gF1 + hF2 + iF12 + jF112 =

冢

a

c

e

−f

g

b−a

d

−e

i

h

a−b

0

−j

−i

0

−a

冣

,

and the extension to the complexification of the Diamond Lie algebra of the Kirillov’s embedding 共1.2兲 ␩ : D → sp共4 , C兲 simply becomes

␩共P+兲 = E1,

␩共P−兲 = F12,

␩共J兲 = i共H1 + H2兲,

␩共T兲 = iF2 ,

共2.9兲

where the elements P+, P−, and T are mapped in root vectors of C2. For example, it is easy to check that the map 共2.9兲 defines an indecomposable true four-dimensional representation of D. III. GENERAL INDECOMPOSABLE FINITE DIMENSIONAL MODULES

We discuss in this section some general properties of the indecomposable finite dimensional D-modules, and we introduce the notations and the terminologies used in this paper. Theorem 3.1: (i) (ii)

In any finite dimensional complex representations of D, the element T acts as a nilpotent operator. If V is a nontrivial complex finite dimensional module of D, then T has at least two Jordan blocks. Proof:

共i兲

共ii兲

This claim follows from the Engels theorem7 and the fact that T is at the center of a nilpotent ideal 共the Heisenberg Lie algebra兲 of D but it can be also checked directly. Since the representation space V is a finite dimensional complex space, T has at least an eigenvector v with a corresponding eigenvalue ␭. It is enough to prove that ␭ = 0. On the subspace W of V spanned by the vectors Pi1 ¯ Pikvk 苸 N i1 ¯ ik = 1 , 2, T is the identity multiplied by ␭ because T commutes both with P1 and P2. Therefore, it holds ␭ dim共W兲 = Tr兩W共T兲 = Tr兩W共关P1 , P2兴兲 = 0, which implies that ␭ = 0. Let us suppose that T has only one Jordan block. There exists a basis 兵v1 , v2 , , . . . , vn其 of V such that vi = TVi−1 and Tvn = 0. Now since P1, P2, and J commute with T, the vector vn is an eigenvector for them. But then 关J , P1兴 = P2 and 关J , P2兴 = −P1 imply that P1vn = P2vn = 0. Therefore, we have TPivn−1 = PiTvn−1 = Pivn = 0 i = 1 , 2. Our hypothesis states that Pivn−1 = ␮ivn i = 1 , 2 ␮i 苸 C and then since 关P1 , P2兴 = T that Tvn−1 = 关P1 , P2兴vn−1 = ␮2 P1vn − ␮2 P2vn = 0, which is a contradiction. 䊐

If we further suppose 共as it happens in all representations considered in this paper兲 that J acts diagonally then it holds Proposition 3.2: Let V be a finite dimensional D-module on which J acts diagonally, then P+, P−, P− P+, and P+ P+ are nilpotent operators. Proof: In our hypothesis V decomposes as V = 丣 V k, k苸Z

Vk = 兵v 苸 V兩Jv = ␭kv␭k 苸 C其

with Vk = 兵0其 for all but a finite number of k. From the commutations relations 关J , P⫾兴 = ⫾ iP⫾, it follows that P+ and P− act, respectively, as “rising” and “lowering” operators,14 P+共Vk兲傺 Vk+1,

P+共Vk兲傺 Vk−1 ,

and therefore in any finite dimensional representation are nilpotent operators. Let us now show that in this case P+ P− 共and P− P+兲 are also nilpotent operators. Using the comutator 关P+ , P−兴 = 2iT, we have by induction

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n

共P− P+兲 = 兺 ank P−k P+k Tn−k ,

共3.1兲

n

k=1

where the coefficients ank can be computed recursively by n n an+1 k = 2ikak + ak−1 , n = 0 for m ⬎ n. Indeed, for n = 1 we have that P− P+ is already written in where we define an0 = 0, am the desired form, while if this formula is true for n, then for n + 1, we have

n

共P− P+兲

n+1

= 共P− P+兲 P− P+ = 兺 n

n

ank P−k P+k Tn−k P− P+

k=1

=兺

n

ank P−k 关P+k , P−兴P+Tn−k

k=1

+ 兺 ank P−k+1 P+k+1Tn−k k=1

n

n

n+1

k=1

k=1

k=1

k k n+1−k = 兺 2kiank P−k P+k Tn+1−k + 兺 ank P−k+1 P+k+1Tn−k = 兺 an+1 . k P − P +T

Since P−, P+, and T are nilpotent operators, there exists a minimal positive integer n0 such that P−n0 = P+n0 = Tn0 = 0. Then, for n 艌 2n0 − 1 from 共3.1兲 follows that 共P− P+兲n = 0, and therefore that P− P+ is nilpotent. Since P+ P− = P− P+ + T, and P− P+ and T are commuting nilpotent operators, P+ P− is also nilpotent. 䊐 Henceforth, in this paper, modules that decompose in the direct sum of eigenspaces of J will be called weight D-modules and an eigenvector v of J corresponding to the eigenvalue ␭ will be called a weight vector of weight ␭. The subspace of V spanned by the weight vectors corresponding to the same weight ␭ is called the weight space of ␭ and is denoted by V␭. In a weight D-module V a set v0 , . . . , vn 苸 V is called a set of generators if each element is a n U共D兲vi兴. The set will be called a weight vector and V is generated by v0 , . . . , vn 关i.e., V = 艛i=1 minimal set of generators if fewer than n + 1 weight vectors will not generate V. A nonzero element v 苸 V is called terminal if P+v = P−v = 0. In Ref. 4 Douglas and Premat observed that a e共2兲 representation may be completely characterized in terms of generators and relations among them, i.e., equations of the type n

m

P+m +j P+n +j␣ijv j = 0. 兺兺 j=0 i=0 i

i

Unfortunately, our situation is more complicated because in our case the operators P+ and P− do not commute, which forces us to consider relations where some generators appear more than one time. Despite these complications, the graphical representations of some modules of the algebra e共2兲 introduced by Repka and de Guise in Ref. 14 and eventually developed in Ref. 4 may be usefully generalized to the modules of the Lie algebra D on which J acts diagonally. The idea is to describe a finite dimensional D-module with a graph, whose vertices correspond to the eigenvectors of J and vertices at the same height form a basis of a weight space. Finally, arrows pointing diagonally upward represent the actions of P+ and arrows pointing diagonally downward represent the action of P−. For example, the graphical representation of the eight-dimensional D-module studied in the next section is15 as follows 共see Fig. 1兲.

IV. sl„3 , C…-MODULES AS D-MODULES

By means of the embedding of D in sl共3 , C兲 constructed in Lemma 2.1, every finite dimensional sl共3 , C兲-module becomes a finite dimensional complex D-module. A detailed description of these latter modules is therefore useful for our purposes.

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> 7

•

P+ R

•

•

R•

•

3

P− ~ w•

R•

FIG. 1. The graph of V共⌳1 + ⌳2兲.

A. Finite dimensional representations of sl„3 , C…

If V is a finite dimensional module of a simple complex Lie algebra g, then V is a weight module, i.e., under the action of the Cartan subalgebra h of g it decomposes as V = 丣 V␮ , ␮苸h*

where h* is the complex dual of h and V␮ = 兵x 苸 h* 兩 H · x = ␮共H兲x其. If x 苸 V␮, we say that x has weight ␮ and following Douglas and Premat,4 we write wt共x兲 = ␮. The universal enveloping algebra U共sl共3 , C兲兲 is a sl共3 , C兲 module when sl共3 , C兲 acts on it by left multiplication and any irreducible finite dimensional representations of sl共3 , C兲 can be constructed as quotients of such a module. Defining for i = 1 , 2 ␣i, ⌳i 苸 h* by ␣i共H j兲 = aij⌳i共H j兲 = ␦ij 关i.e., ␣1 and ␣2 are the simple roots of sl共2 , C兲兴 and for each pair n , m 苸 N set ⌳ = n⌳1 + m⌳2, then i兲共i = 1 , 2兲 is an the quotient of U共sl共3 , C兲兲 by the left ideal J⌳ generated by Ei, Hi − ⌳共Hi兲, F1+⌳共H i irreducible finite dimensional sl共3 , C兲-module which we shall denote by V共⌳兲. If ⌳ = n⌳1 + m⌳2 ⫽ ⌳⬘ = n⬘⌳1 + m⬘⌳2 关i.e., if 共n , m兲 ⫽ 共n⬘ , m⬘兲兴, then V共⌳兲 and V共⌳⬘兲 are inequivalent sl共3 , C兲-modules. Finally, any finite dimensional module may be realized in such a way.9,7 If we denote by u⌳ the coset 1 + J⌳, then it can be shown that the module V共⌳兲 is generated by the r u⌳其, where p , q , r 苸 N, as well4 by the monomials 兵Fi1 ¯ Fiku⌳其 with k 苸 N, monomials 兵F1pFq1F12 p q r F1F2u⌳ has the weight ⌳ − 共p + q兲␣1 − 共p i1 ¯ ik = 1 , 2. It is easy to show that the monomial F12 9,7 + r兲␣2, while the monomial Fi1 ¯ Fiku⌳ has the weight ⌳ − 兺kj=1␣i j i j = 1 , 2.4 In the setting of these latter generators, Littelman proved in Ref. 10 that if ⌳ = n⌳1 + m⌳2, then defining F共a兲 i = 共1 / a!兲Fai , i = 1 , 2, the sets 共b兲共c兲 B121 = 兵F共a兲 1 F2 F1 兩a,b,c 苸 N

and

a 艋 n + b − 2c,

c 艋 b 艋 c + m,

c 艋 n其

共b兲共c兲 B212 = 兵F共a兲 2 F1 F2 兩a,b,c 苸 N

and

a 艋 m + b − 2c,

c 艋 b 艋 c + n,

c 艋 m其

and

are both the bases for the sl共3 , C兲-module V共⌳兲. However, these bases are not well tailored for the action of D on V共⌳兲, forcing us to construct from Littelmann’s basis the basis given in the following. Theorem 4.1: A basis for the representation space V共⌳兲 with ⌳ = n⌳1 + m⌳2 is given by the monomials r u ⌳, F1pFq2F12

0 艋 p 艋 n,

0 艋 q 艋 m,

0 艋 p + q + r 艋 n + m.

Proof: First we observe that the number of monomials 共4.1兲 is exactly the dimension of V共⌳兲, namely: 21 共n + 1兲共m + 1兲共n + m + 2兲. Therefore, it suffices to show that any element of the Littelman basis B121 is a linear combination of such monomials. To do that, we need to prove the following. Lemma 4.2: In U共sl共3 , C兲兲 yields

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Indecomposable modules of the Diamond Lie algebra n

Fb2Fn1

= 兺 k! k=0

冉冊冉冊 n k

b n−k b−k k F F F . k 1 2 12

共4.1兲

Proof: We proceed by induction on n. n = 1 coincides with the relation 关Fb2 , F1兴 = bFb−1 2 F12, which follows easily by induction. Let us now assume 共4.2兲 for n and let us check it for n + 1,

冉冊冉冊冉冊

n

Fb2Fn+1 1 = 兺 k! k=0

n k

冉冊冉冊兺冉冊冉冊兺冉冊冉冊兺冉冊冉冊 n

n

b n−k b−k n k F 关F2 ,F1兴F12 = k! ⫻ k 1 k k=0 n

−

b−k−1 k+1 F12 k兲Fn−k 1 F2

⫻

冉冊

冉冊冉冊冉冊冉冊 n

n b n−k+1 b−k k F1 F2 F12 + 兺 k! k k k=0

b n−k b−k k n F1 F2 F12F1 = 兺 k! k k k=0

n

b n−k+1 b−k k n F F2 F12 + 兺 k! k 1 k k=0 n

b 共b k

n b n−k+1 b−k k F1 F2 F12 + 兺共k + 1兲! k k k=0

n = k! k k=0

n+1

b b b−k−1 k+1 Fn−k F12 = k! 1 F2 k+1 k k=0

n+1 k

k Fn−k+1 Fb−k 1 2 F12 ,

b 兲 . 䊐 where we have used the identity k!共 bk 兲共b − k兲 = 共k + 1兲!共 k+1 The weights of the module V共⌳兲 are of the form ⌳ − i␣1 − j␣2 with 0 艋 i 艋 n + m, 0 艋 j 艋 n + m and for a fixed pair 共i , j兲 it is easy to determine the elements of the basis B121 which span the corresponding weight space. The element Fa1Fb2Fc3u⌳ has the weight ⌳ − 共a + c兲␣1 − b␣2 and taking into account the constraints on a, b, and c, we have

V共⌳兲⌳−i␣1−j␣2

=

冦

j h span兵Fi−h 1 F2F1u⌳其h=0,. . .,i ,

0 艋 i 艋 n,

0 艋 j 艋 m,

i艋j

j h span兵Fi−h 1 F2F1u⌳其h=0,. . .,j , F2j F1j−m+hu⌳其h=0,. . .,n+m−j , span兵Fi−j+m−h 1 F2j F1j−m+hu⌳其h=0,. . .,m+n−i , span兵Fi−j+m−h 1 j h span兵Fi−h 1 F2F1u⌳其h=0,. . .,n , j h span兵Fi−h 1 F2F1u⌳其h=0,. . .,n+j−i , F2j F1j−m+hu⌳其h=0,. . .,m , span兵Fi−j+m−h 1 F2j F1j−m+hu⌳其h=0,. . .,m+i−j , span兵Fi−j+m−h 1

0 艋 i 艋 n,

0 艋 j 艋 m,

i⬎j

n ⬍ i 艋 n + m, m ⬍ j 艋 n + m, i 艋 j n ⬍ i 艋 n + m, m ⬍ j 艋 n + m, i ⬎ j n ⬍ i 艋 n + m, 0 艋 j 艋 m,

i艋j

n ⬍ i 艋 n + m, 0 艋 j 艋 m,

i⬎j

0 艋 i 艋 n,

m ⬍ j 艋 m + n, i ⬎ j

0 艋 i 艋 n,

m ⬍ j 艋 n + m, i 艋 j,

冧

where the fifth and seventh cases are not empty only if m ⬎ n and n ⬎ m, respectively. Now in the case when 0 艋 i 艋 n, 0 艋 j 艋 m, since we have i − k 艋 i 艋 n, j − k 艋 j 艋 m, and i + j − k 艋 i + j 艋 n + m, we may apply directly for any h = 0 , . . . , i 共respectively, h = 0 , . . . j兲 formula 共4.1兲, h

j h Fi−h 1 F2F1 = 兺 k! k=0

冉冊冉冊 h k

j i−k j−k k F F F k 1 2 12

to obtain the corresponding elements of the basis B121 as a linear combination of elements 共4.1兲. Let us consider the case when n ⬍ i 艋 n + m, m ⬍ j 艋 n + m, and i 艋 j. In this case the corresponding element of the basis B121 can be written using formula 共4.1兲 and the fact that elements of the type l u⌳ with j ⬎ m are zero as Fi1F2j F12

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Casati, Minniti, and Salari h+j−m

Fi−j+m−h F2j F1j−m+h 1

=

兺 k! k=j−m

冉冊冉冊 h k

j i−k j−k k F F F , k 1 2 12

h = 0, . . . ,n + m − j.

j h These latter n + m − j + 1 equations together with those arising from the fact that Fi−h 1 F2F1u⌳ = 0 if h⬎n n+h

0=

Fi−n−h F2j Fn+h 1 1 u⌳

= 兺 k! k=1

冉冊冉冊 n+h k

j i−k j−k k F F F u ⌳, k 1 2 12

h = 0, . . . ,i − n

make a triangular system of m + i − j + 1 equations which allows us to write the elements of j−k k Fi−j+m−h F2j F1j−m+hu⌳ 共h = 0 , . . . , n + m − j兲 and the elements Fi−k 1 1 F2 F12u⌳ 共k = i + j − n − m + 1 , . . . , n i−k j−k k − i兲 as a linear combination of the elements F1 F2 F12u⌳ with k = i , . . . , i + j − n − m, i.e., of elements of the basis 共4.1兲. The remaining cases can be treated in a similar way and are left to the reader. 䊐 B. The D-modules V„⌳…

We may now regard the sl共3 , C兲-module as a D module using the embeddings ⌽ and ⌿ of Sec. II. In principle, one should obtain for each of these embeddings a corresponding class of representation of D. However, formula 共2.4兲 and Remark 2.2 show that if ␲ is a representation of sl共3 , C兲, then the representation of D given by the composition of ␲ with the embedding ⌿: ␲ ˜ ⴰ ⌽, where ␲ ˜ is obtained from ␲ by setting ␲ ˜ 共X兲 ⴰ ⌿ is equivalent to the representation ␲ = ␲共␪共X兲兲, ∀X 苸 sl共3 , C兲. ␪ is the Cartan involution of sl共3 , C兲. Therefore, since the set of all irreducible finite dimensional representations of sl共3 , C兲 is preserved by the transformation ␲ ˜ defined above, the two classes of D-modules coincide. This fact allows us to consider only →␲ the D-modules obtained by means of the embedding ⌽ by simply letting the element P+, P−, T, and J of D act on V共⌳兲, respectively, as E1, F12, 共i / 3兲共2H1 + H2兲, and 共i / 2兲F2. We shall see that for any fixed ⌳ = n⌳1 + m⌳2 the corresponding module V共⌳兲 viewed as D-module by means of the embedding ⌽ 共2.3兲 is indecomposable and that if ⌳ ⫽ ⌳⬘ then V共⌳兲 and V共⌳⬘兲 are inequivalent as D-modules. Lemma 4.3: The D-module V共⌳兲 is generated by the element Fn1u⌳. F共k兲 Proof: In Ref. 4 Douglas and Premat showed that the set G = 兵F共n+k兲 1 1 u⌳ 兩 0 艋 k 艋 m其 is a set of generators of V共⌳兲 as e共2兲-module. This means, in our setting, that V共⌳兲 is generated by acting with E1 and F2, i.e., by the operators P+ and −2iT on the set G. Therefore, it suffices to show that Fn1u␭ generates G. In order to do that, let us first prove by induction over k the formula n+k−1 k−1 k F2 u⌳ . Fn+k 1 F2u⌳ = − 共n + k兲F12F1

共4.2兲

Indeed, if k = 1, we have n+1 n+1 n Fn+1 1 F2u⌳ = 关F1 ,F2兴u⌳ + F2F1 u⌳ = − 共n + 1兲F12F1u⌳ ,

then if it is true for k we have for k + 1 using repeatedly the induction hypothesis n+k+1 n+k k k Fk+1 ,F2兴Fk2u⌳ + F2Fn+k+1 Fk2u⌳ = − 共n + k + 1兲F12Fn+k Fn+k+1 1 1 F 2u ⌳ + F 2F 1F 1 F 2u ⌳ 1 2 u⌳ = 关F1 共4.2兲

n+k−1 k−1 k = − 共n + k + 1兲F12Fn+k F2 u⌳ = ¯ 1 F2u⌳ − 共n + k兲F12F2F1F1

共4.2兲

k

k k+1 = − 共n + k + 1兲F12Fn+k 兿共n + k − i兲F12k F2Fn+1 1 u⌳ 1 F2u⌳ + 共− 1兲 i=0

k but proving the claim. Now formula 共4.2兲 show that Fn+k 1 F2u⌳ is generated by applying n+k−1 k−1 F2 u⌳ and therefore by recursion that the set G can be generated by the F12 to the element F1 element Fn1u⌳. 䊐 Theorem 4.4: The D-module V共⌳兲 is indecomposable.

Fn+1 1 u⌳ = 0,

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Proof: We have shown that V共⌳兲 = U共D兲Fn1u⌳, i.e., that V共⌳兲 is a cyclic D-module and therefore indecomposable. 䊐 Theorem 4.4 may be also proven in a different way using the equivalent embedding 共2.4兲. Module V共⌳兲 as a quotient of the Verma sl共3 , C兲-module M共⌳兲 is generated by acting on the element u⌳ with F1, F2, and F12, i.e., by the operators P+, P−, and −2iT. This implies that V共⌳兲 = U共D兲u⌳. An explicit computation of such action may be found in Refs. 12. Theorem 4.4 may also be derived by the results of Douglas and Premat4 because their Theorem 1 may be reformulated in the present context by saying that V共⌳兲 is an indecomposable module of the ideal of D spanned by the element P+ and T and therefore of the whole Lie algebra D. It is also possible 共and useful to obtain its graphical representations兲 to determine the terminal elements of the D-module V共⌳兲. n Lemma 4.5: The element Fm+n 2 F1u⌳ is the unique terminal element of V共⌳兲. Proof: Since 关P+ , P−兴 = 2iT, any terminal element is annihilated by T as well. In other words, the set of terminal elements is a subset of the elements annihilated simultaneously by E1 and F2, which in Ref. 4 Douglas and Premat found to be F共k兲 T = 兵F共m+k兲 1 u⌳兩0 艋 k 艋 n其. 2 Therefore, we may look for our terminal elements only among the elements of the set T. We shall n see that the unique element of T is also annihilated by F12 is Fm+n 2 F1u⌳. This is done first by showing by induction the formula m+k−1 k−1 k F 1 u ⌳, Fm+k 2 F1u⌳ = 共m + k兲F12F2

which implies that have indeed

0 艋 k 艋 n,

m+k+1 k+1 k F1 u⌳ ⫽ 0 F12Fm+k 2 F1u⌳ = 关1 / 共m + k + 1兲兴F2

共4.3兲

for k = 0 , . . . , n − 1. For k = 1, we

m+1 m+1 m Fm+1 2 F1u⌳ = 关F2 ,F1兴u⌳ + F1F2 u⌳ = 共m + 1兲F12F2 u⌳

because Fm+1 2 u⌳ = 0. Now assuming that formula 共4.3兲 is true for n, we have for n + 1 k

Fm+k+1 Fk+1 2 1 u⌳

= 关共m + k + +k−

k 1兲F12Fm+k 2 F1

k F1Fm+1 i兲F12 2 兴u⌳

+

k F1F2Fm+k 2 F1兴u⌳

= 关共m + k +

= ¯ = 关共m + k +

k 1兲F12Fm+k 2 F1

+ 兿共m i=0

k 1兲F12Fm+k 2 F 1u ⌳兴

Fm+1 2 u␭ = 0.

n It remains to check that F12Fm+n because 2 F1u␭ = 0, but this follows from the fact that there is no element in V共⌳兲 of the weight ⌳ − 共n + 1兲␣1 − 共m + 1兲␣2. 䊐

V. sl„2 , C…„2…-MODULES AS D-MODULES

In Sec. II we have seen that the Diamond Lie algebra D may also be embedded in the truncated current Lie algebra sl共2 , C兲共2兲. Therefore, one may hope to obtain new indecomposable modules of D by studying the irreducible finite dimensional representations of sl共2 , C兲共2兲. A wide class of this latter representations may be constructed using Theorem 2.4. The starting point is the irreducible finite dimensional representations of the simple Lie algebra sl共2 , C兲. These representations are classified by the positive integers. More precisely, for any n 苸 N, there exists a sl共2 , C兲 module Vn equipped with a basis 兵vk其k=0,. . .,n such that7 Hvk = 共n − 2k兲vk, 0 艋 k 艋 n, Evk = 共n − k + 1兲vk−1, 1 艋 k 艋 n, Ev0 = 0 Fvk−1 = kvk, 0 艋 k 艋 n − 1 Fvn = 0, where H, E, and F are the elements of sl共2 , C兲 defined in Sec. II. Theorem 2.4 and the embedding 共2.8兲共where we use, respectively, 21 P+, 21 P−, and 41 T instead of P+, P−, and T兲 allow us to construct that for any n 苸 N a 3n + three-dimensional D-module Wn j=0,1,2 is endowed with a basis 兵vkj其k=0,. . .,n such that the action of the 共new兲 elements J, P+, P−, and T is given by Jvkj = i/2共n − 2k兲vkj,

k = 0, . . . ,n,

j = 0,1,2,

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Casati, Minniti, and Salari j+1 P+vkj = 共n − k + 1兲vk−1 ,

j+1 P−vkj = 共k + 1兲vk+1 ,

P+v0j = 0,

P−vnj = 0,

Tv0k = 2i共n − 2k兲v2k ,

P+v2k = 0,

P−v2k = 0

Tvkj = 0

k = 1, . . . ,n,

j = 0,1,

k = 1, . . . ,n − 1,

j = 0,1,

k = 1, . . . ,n,

j = 1,2.

共5.1兲

Proposition 5.1: Let Vn be the D-module constructed above and let us denote by 关x兴 the integer part of x, then 共1兲共2兲共3兲

if n = 0, then Vn decomposes in the direct sum of three trivial one-dimensional modules. If n = 2j, j 苸 N, j 艌 1, then Vn decomposes into the direct sum Vn = U1n 丣 U2n of two modules, respectively, of dimension 3共n / 2兲 + 2 and 3共n / 2兲 + 1, given by U1n 1 2 0 2 0 0 2 2 1 1 = span兵v0 , v2k , v2k−1 , v0 , v2k , 其k=1,. . .,关n/2兴 and Un = span兵v2k−1 , v0 , v2k , v2k−1 , 其k=1,. . .,关n/2兴. If n = 2j + 1 j 苸 Nj 艌 0, then Vn decomposes into the direct sum Vn = W1n 丣 W2n of two modules 0 1 2 , v2k+1 , v2k , 其k=0,. . .,关n/2兴 and of equal dimension 3关n / 2兴, respectively, given by W1n = span兵v2k 2 0 1 2 Wm = span兵v2k+1 , v2k , v2k+1 , 其k=0,. . .,关n/2兴.

1 Proof: We have only to check that the action of P+ and P− leaves invariant the subspaces Um 2 j and Um. This follows almost immediately from the fact that that P+vk is either zero or a multiple j+1 j+1 of vk−1 and, similarly, that P−vkj is either zero or a multiple of vk+1 . 䊐 These latter modules cannot be further decomposed. 1 2 1 2 , Um , Wm , and Wm are indecomposable D-modules. Theorem 5.2: The modules Um Proof: Let M be any of such modules, and let us denote by M j the subset of M spanned by the elements of the “canonical basis” of the form vkj so that M = 丣 2j=0M j. Suppose now that M is not indecomposable. Then there exist two submodules P and Q such that M = P 丣 Q. Now by inspection of the action of P+ and P− 共5.1兲 on the modules defined in Proposition 5.1, we find there exists 2 1 a unique nonzero element m in M such that either P+m = 0 共i.e., M is a module of type Um or Wm 2 2 0 2 and m is a nonzero multiple of v0兲 or P+m = 0 共i.e., M is a module of type Um or Wm and m is a nonzero multiple of v01兲. In the first case, we may decompose v00 as v00 = p00 + q00, where p00 苸 P0 = P 艚 M 0 and q00 苸 Q0 = Q 艚 M 0. Now P 艚 Q = 兵0其 and P+v00 = 0 imply that P+ p00 = −P+q00; and since P and Q are two submodules such that P 艚 Q = 兵0其 and P+v00 = 0, this, in turn, implies that P+ p00 = P+q00 = 0. But since by inspection Ker共P+兩M 0兲 is one dimensional, we have that either p00 = v00 and q00 = 0 or p00 = 0 and q00 = v00. We may suppose without loosing generality that p00 = v00; in this case P− p00 = v11 苸 P. Now if v11 spans the whole space M 1 then from Eq. 共5.1兲 it follows that v00 spans M 0 which then coincides with P0. From Eq. 共5.1兲, it follows that P = M, Q = 兵0其, i.e., that M is indecomposable. Otherwise, there exists an element v02 in M 0 such that P−v02 is a nonzero multiple of v11. If we decompose v02 as v02 = p02 + q02 with p02 苸 P and q02 苸 Q, we have that P−q02 = P−v02 − P− p02 = ␣v11 − P− p02 苸 P, which again implies that P−q02 = 0, i.e., q02 苸 Ker共P+兩M 0兲傺 P0, which forces q20 = 0. Therefore, v02 belongs to P. Iterating the procedure we may show step by step that any element of M 1 belongs to the submodule P, which implies that M 0 coincides with P0. Finally, using Eq. 共5.1兲 we show that P = M or, in other words, that M is indecomposable. If vice versa, we are in the second case decomposing again v01 as v01 = p01 + q01 with p01 苸 P0 and q01 苸 Q0. We have that P+2v02 = 0 implies P+2 p01 = −P+2q01, which again recalling that in the present case dim共Ker共P+兩M 0兲兲 = 1 implies p02 = v02 and q02 = 0 or p02 = 0 and q02 = v02. Thus, the same argument of the previous case also works in this second case, showing that M is indecomposable. 䊐 Formula 共5.1兲 tell us also which is a minimal set of generators for each indecomposable D-module described in this section. Proposition 5.3: The indecomposable D-modules U1n, U2n, W1n, and W2n n = 1 , 2 , . . . admit the 0 0 0 其k=1,. . .,n/2, 兵v2k−1 其k=1,. . .,n/2, 兵v2k , 其k=1,. . .,关n/2兴, and minimal set of generators given, respectively, by 兵v2k 0 兵v2k+1其k=1,. . .,关n/2兴. Proof: That the sets written above are sets of generators is clear. We have to show that for any module M of the type above a set of generators has at least dim共M 0兲 elements. Suppose that M admits a set of generators S j = 兵w0 , . . . , w j其, any element w j can be decomposed as wi = w0i + w1i

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+ w2i , where whj 苸 M h, h = 0 , 1 , 2 and i = 0 , . . . , j. Now, since any element generated by this set of generators must be in the span of S j and of the elements P+wi = P+w0j + P+w1j , P−2wi = P−2w0i ,

P−wi = P−w0j + P−w1j ,

P− P+wi = P− P+w0i ,

P+2wi = P+2w0j ,

P+ P−wi = P+ P−w0i ,

i = 0, . . . j,

any element of M 0 generated by the action of D on S j must be in the span of 兵w0i 其, i = 0 , . . . , j. But 䊐 then S j generates M 0 only if j 艌 dim共M 0兲. Theorem 5.4: The D-modules U1n, U2n, W1n, and W2n with n 艌 3 are inequivalent indecomposable modules, which are also inequivalent with respect to the modules V共⌳兲 studied in the previous section. Proof: That such modules are indecomposable was already shown. Now the modules Win i = 1 , 2 and the modules Uin i = 1 , 2 are inequivalent because the dimensions of the first ones are 0 mod 3, while those of the latter ones are 1 mod 3 or 2 mod 3. Further, since two modules Wni1 1 and Wni2 have different dimensions if i1 ⫽ i2 or n1 ⫽ n2 such modules are all inequivalent. Finally, 2 since two modules Uni1 and Wni2 may have the same dimension only if n1 = n2, we only have to 1 2 check that Win1 is inequivalent to Win2 for any n. However, this follows from the fact that by Proposition 5.1 these modules have a different minimal number of generators 共namely 关n / 2兴 and 关n / 2兴 − 1兲. A similar argument shows that these modules and the D-modules V共⌳兲 are inequivalent. These latter modules can be generated just by one element while for the modules treated in this section for n 艌 3 it happens only for the module W23 which has dimension 4, while no module V共⌳兲 has such dimension. 䊐 Remark 5.5: • The modules W12 and W22 are equivalent, respectively, to V共⌳1兲 and V共⌳2兲. • The modul W23 is untrue because on it T acts as a zero operator; therefore, the representation of D found by Kirillov does not belong to this class of module.

VI. sp„4 , C…-MODULES AS D-MODULES

The embedding 共2.9兲 and the finite dimensional sp共4 , C兲-modules may be regarded as finite dimensional D-modules. This last section is devoted to showing that these new D-modules are indecomposable modules inequivalent to those already constructed in the previous sections. Basically, the theory developed by Douglas and Premat in Ref. 4 also works in this case. To implement such a theory in our setting, we need first to point out some properties of the irreducible finite dimensional representations of sp共4 , C兲. A. Finite dimensional representations of sp„4 , C…

The construction of the irreducible finite dimensional modules of sl共3 , C兲 presented in the Sec. IV B may be applied also to the Lie algebra sp共4 , C兲. Let ⌳i 苸 h* i = 1 , 2 again be defined by ⌳i共H j兲 = ␦ij. Then, the irreducible finite dimensional modules of sp共4 , C兲 are in one to one correspondence with the “highest weight” ⌳ = n⌳1 + m⌳2n , m 苸 N. For any fixed pair n , m 苸 N the corresponding module, denoted by W共⌳兲, may be constructed as the quotient of the universal i兲共i enveloping algebra U共sp共4 , C兲兲 and by the left ideal J⌳ generated by Ei, Hi − ⌳共Hi兲, F1+⌳共H i = 1 , 2兲. If we denote by u␭ the element of W共⌳兲 corresponding to the coset 1 + J⌳, then the c d F112u⌳其 with a , b , c , d 苸 N elements of W共⌳兲 are linear combinations of the mononials 兵Fa1Fb2F12 a b c d and the monomial F1F2F12F112u⌳ has weight ⌳ − 共a + c + 2d兲␣1 − 共b + c + d兲␣2, where ␣i 苸 h* i = 1 , 2 are the simple roots of sp共4 , C兲共␣i共H j兲 = aij兲. Moreover, in Ref. 1 Bliem showed that a basis for W共⌳兲 is given by the set

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B1234 =

冦

Fa11Fa22Fa13Fa24u␭

冤

0 艋 a4 艋 m,2a4 艋 a3 艋 n + 2a4 , 1 a3 艋 a2 艋 m − a3 − 2a4,0 艋 a1 艋 n + 2a2 − 2a3 + 2a4 . 2

冥冧

Once again, this basis is not very well suited for our purposes. Theorem 6.1: The sp共4 , C兲-module W共⌳兲 is spanned by the (maybe not linearly independent) elements 0 艋 p 艋 n,

r s F1pF112 u ⌳, Fq2F12

0 艋 q 艋 m,

0 艋 p 艋 n0 艋 s 艋 n + mq,r 苸 N,

共6.1兲

moreover, the two sets k u⌳兩0 艋 k 艋 m其 G = 兵gk = Fn1F112

and

n k T = 兵tk = Fm 2 F12F112u⌳兩0 艋 k 艋 m其

are two sets of linearly independent elements of W共⌳兲. c d F112u⌳其 with Proof: We have already noticed that W共⌳兲 is spanned by elements 兵Fa1Fb2F12 a , b , c , d 苸 N. Since F1 commutes with F112 and H1u⌳ = nu⌳, 共H1 + H2兲u␭ = 共n + m兲u⌳ from the theory of the representations of sl共2 , C兲7 it follows that W共⌳兲 is spanned by the elements r s s F1pF112 u⌳ with q , r 苸 N, 0 艋 p 艋 n and 0 艋 s 艋 n + m. Further, the weight of F1pF112 u⌳ is 共n Fq2F12 − 2p − 2s兲⌳1 + 共m + p兲⌳2 and this a weight of W共⌳兲 only if p + s 艋 n + m 共see Refs. 1 and 2兲. Let us now prove the claim concerning first the set T and then the set G. Since the elements tkk = 0 , 1 , . . . , m belong to different weight spaces, they are linearly independent if they are different from zero. In order to prove that this is the case, let us consider the action on W共⌳兲 of the three embedding of sl共2 , C兲 given by g112 = span兵H1 + H2 , E112 , F112其, g12 = span兵H1 + 2H2 , E12 , F12其, and g2 = span兵H2 , E2 , F2其, respectively. We have already observed that u⌳ is the highest weight vector k u⌳ with of an irreducible n + m + 1 dimensional g112-module, and therefore that the elements F112 i−1 i 0 艋 k 艋 m are different from zero. Now from the relations 关E12 , F112兴 = iF1F112 and 关E12 , Fi1兴 = −2iFi−1 1 E2, which can be easily proven by induction we may show that j−1

k−j j k F112 u⌳ = 兿共k − i兲F1j F112 u ⌳, E12

0 艋 k 艋 m,

0 艋 j 艋 Min兵n,k其.

共6.2兲

i=0

k−1 k k Indeed, for j = 1 we have E12F112 u⌳ = 关E12 , F112 兴u⌳ = kF1F112 u⌳ now suppose 共6.2兲 true for j, then we have for j + 1 j−1

j+1 k F112u⌳ E12

= E12兿共k −

j−1

k−j i兲F1j F112 u⌳

i=0

= 兿共k −

j−1

k−j i兲关E12,F1j 兴F112 u⌳

i=0

k−j + 兿共k − i兲F1j E12F112 u⌳ i=0

j−1

j−1

j−1

i=0

i=0

i=0

k−j k−j k−j−1 = − 2j 兿共k − i兲F1j−1E2F112 u⌳ + 兿共k − i兲F1j 关E12,F112 兴u⌳ = 兿共k − i兲kF1j+1F112 u⌳ . k F112 u⌳

Therefore, is a weight vector in a g12-module whose highest vector is, respectively, either k−n Fk1u⌳ if k ⬍ n or Fn1F112 u⌳ if k 艌 n. The corresponding weight 共with respect to H1 + 2H2兲 is, respecn k F112u⌳ ⫽ 0 in both cases. tively, either n + 2m or n + 2m − 2k + 2n and therefore F12 n−1 n−2 n Similarly, using the relations 关E2 , F12兴 = nF1F12 + n共n − 1兲F12 F112 共which can still be proven by induction兲, we can prove by induction, min兵j,n−j其 n k F112u⌳ = E2j F12

兺 i=0

冉冊兿 j i

j+i−1 n−j−i k+i 共n − h兲F1j−iF12 F112u⌳,

0 艋 k 艋 m,

0 艋 j 艋 n.

h=0

Indeed, for j = 1, n−1 k n−2 k+1 k n k n 兴F112 u⌳ = nF1F12 F112u⌳ + n共n − 1兲F12 F112 , F112u⌳ = 关E2,F12 E2F12

then using the induction hypothesis, we have for j + 1,

共6.3兲

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冉兺冉冊兿 j

n k E2j+1F12 F112u⌳

= E2

j i

i=0

j+i−1

共n −

n−j−i k+i h兲F1j−iF12 F112

h=0 j

n−j−i k+i 兴F112u⌳ = 兺 − h兲F1j−i关E2,F12 j

+兺 i=0

冉冊兿 j i

j+i−1 h=0

i=0

冊

u⌳ = 兺

冉冊兿 j i

j

i=0

冉冊兿 j i

j+i−1

共n

h=0

j+i−1 n−j−i−1 k+i 共n − h兲共n − i兲F1j−i+1F12 F112u⌳

h=0 j+1

n−j−i−2 k+i+1 共n − h兲共n − i兲共n − i − 1兲F1j−iF12 F112 u⌳ = 兺 i=0

冉冊兿 j i

j+i

共n

h=0

n−j−i−1 k+i F112u⌳ . − h兲F1j−iF12

Obviously, if n − j ⬎ j, then only the first n − j + 1 elements of the summation survive. Now from n k F112u⌳ is a weight vector of weight m − n, in g2-module whose highest 共6.3兲, it follows that F12 k m k u⌳ of weight m + n. Also, Fm weight vector is the element Fn1F112 2 F12F112u⌳ is different from zero. Again to prove that the elements gk 0 艋 k 艋 m are linearly independent, it is enough to show that they are different from zero since they have different weights. This leads us to compute the k action of E2k 1 on F112u⌳, but since we are going to use it later on, let us actually show by induction the most general formula 共6.4兲

k Fn1F112 u⌳ = n!共n + 2k兲!Fk2u⌳ . En+2k 1

Indeed, for k = 0 using the formula 关E1 , Fn1兴 = nFn−1H1 − n共n − 1兲Fn−1, we have easily by induction 共see also Ref. 3兲 En1Fn1u⌳ = 共n!兲2u⌳. Now supposed 共6.4兲 is true for k using 关Eh1 , F112兴 = −hF12Eh−1 1 + h共h − 1兲F2Eh−2 1 , we have for k + 1 k+1 k k En+2k+2 Fn1F112 u⌳ Fn1F112 u⌳ = E21关En+2k ,F112兴Fn1F112 u⌳ + E21F112En+2k 1 1 1 k k = − 共n + 2k兲E21F12En+2k−1 Fn1F112 u⌳ + 共n + 2k兲共n + 2k − 1兲E21F2En+2k−2 Fn1F112 u⌳ 1 1 k k + E21F112En+2k Fn1F112 u⌳ = − 共n + 2k兲关E21,F12兴En+2k−1 Fn1F112 u⌳ − 共n 1 1 k + 2k兲F12En+2k+1 Fn1F112 u⌳ + 共n + 2k兲共n + 2k − 1兲n!共n + 2k兲!Fk+1 1 2 u⌳ + n!共n k + 2k兲!Fk2E21F112u⌳ = 4共n + 2k兲F2En+2k kFn1F112 u⌳ + 2n!共n + 2k兲!Fk+1 1 2 u⌳ + n!共n k+1 + 2k兲共n + 2k − 1兲n!共n + 2k兲!Fk+1 2 u⌳ = n!共n + 2k兲!4共n + 2k兲F2 u⌳ + 共共n + 2k兲共n k+1 + 2k − 1兲 + 2兲n!共n + 2k兲!Fk+1 2 u⌳ = n!共n + 2共k + 1兲兲!F2 u⌳ . k u⌳ = 共2k兲!Fk2u⌳. Now, since E1 commutes with F2, we have that In particular, we have E2kF112 k F112u⌳ belongs with weight n − 2k to a g1-module, whose highest weight vector is Fk2u⌳ of weight k n + 2k. Therefore, the elements Fn1F112 u⌳0 艋 k 艋 m are all different from zero. 䊐

B. The D-modules W„⌳… n k Lemma 6.2: The set T = 兵tk = Fm 2 F12F112u⌳ 兩 0 艋 k 艋 m其 is a set of terminal elements of W共⌳兲. Proof: Using the previous theorem, we have only to check that E1tk = F12tk = 0 for 0 艋 k 艋 m. h−1 h−1 h h 兴 = −2hF2F12 and 关E1 , F112 兴 = −2hF12F112 , we have Using the relations 关E1 , F12 n−1 k m n+1 k−1 E1tk = − 2nFm+1 2 F12 F112u␭ − 2kF2 F12 F112 u␭ = 0,

k = 0, . . . ,m

m n+1 m n+1 because Fm+1 2 u␭ = 0 by construction and F2 F12 u⌳ = 0 because the weight of F2 F12 u⌳ does not 1,2 belong to the sp共4 , C兲-weight of W共⌳兲. 关This latter claim may also be proven directly by using the theory of the sl共2 , C兲-modules.兴 For the same reason, we have n+1 k F12tk = Fm 2 F12 F112u␭ = 0,

k = 0, . . . ,m.

k u⌳ 兩 0 艋 k 艋 m其 is a set of generators of W共⌳兲. Lemma 6.3: The set G = 兵gk = Fn1F112

䊐

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Proof: We have already seen that the elements of T are linearly independent. Now from the relations j−1 i−1 j i j j j i 2 u⌳ = i共n − i2 + i兲Fi−1 E1Fi1F112 1 F112u⌳ − jF1F12F112u⌳ = i共n − i + i + 2j兲F1 F112u⌳ + F12F1F112u⌳ ,

共6.5兲

j j F12 · Fi1F112 u⌳ = F12Fi1F112 u⌳ ,

j u⌳, 0 艋 i 艋 n, 0 艋 j 艋 m belong to the which hold for any i and j, it follows by recursion that Fi1F112 j u⌳ = 0 for D-module generated by the set G. The same relations together with the fact that Fn1F112 m + 1 艋 j 艋 m + n 共because the corresponding weights are not weights of the sp共4 , C兲-weight of j u⌳, 0 艋 i 艋 n, 0 艋 j 艋 m are 共maybe zero兲 W共⌳兲 see Refs. 2 and 1兲 implies that the elements Fi1F112 elements of a D-module generated by the set G. Finally, the relations

∀ l,h,i, j 苸 N

j h i j u⌳ = ihFl2F12 F1F112u⌳, P−l · Th · Fi1F112

show that the D-module generated by the set G coincides with W共⌳兲. 䊐 Lemma 6.4: For all 0 艋 k 艋 m, P−m−k P+2k+ngk = ck0t0 for some nonzero constant ck0. Therefore, the D-module generated by gk contains t0. Proof: Using Eq. 共6.4兲, we have k n Fn1F112 u⌳ = Tm−k P−n n!共n + 2k兲!Fk2u⌳ = n!共n + 2k兲!Fm Tm−k P−n P+n+2kgk = Tm−k P−n En+2k 2 F12u⌳ . 1

䊐 Lemma 6.5: Let 兵w0 , . . . , wh其 be a set of generators of W共⌳兲. Then, for each 0 艋 k 艋 m there exists an 0 艋 l 艋 h such that wt共gk兲 = wt共wl兲. Proof: Following Douglas and Premat4 suppose that for some 0 艋 k 艋 m wt共gk兲 ⫽ wt共wl兲 for all 0 艋 l 艋 h. Hence, gk can be written as gk =

Pl共T, P−, P+兲P+wt共w 兲−wt共g 兲共wl兲 + Pl共T, P−, P+兲P−wt共w 兲−wt共g 兲共wl兲, 兺兺 wt共w 兲⬍wt共g 兲 wt共w 兲⬎wt共g 兲 k

l

k

k

k

l

k

k

where Pl共T , P− , P+兲 are polynomials in the operators T, P−, and P+ of the type 兺pikTh P−k P+k pik 苸 C h , k 苸 N. If we substitute in this equation the expression of the elements wl 0 艋 l 艋 h in terms of the generators G, we may write using formulas of the type 共3.1兲 gk as gk = Q0共T, P−, P+兲T共gk兲 + Q⫾共T, P−, P+兲P− P+共gk兲 + 兺 Qi共T, P−, P+兲P+2共k−i兲共gi兲 i⬎k

+ 兺 Qi共T, P−, P+兲P−2共i−k兲共gi兲, i⬍k

where Qi共T , P− , P+兲 are still polynomials of the type 兺qikTh P−k P+k . Then, the sp共4 , C兲 weight on the left hand side does not equal the sp共4 , C兲 weight of any of the summands on the right hand side. This is a contradiction; therefore, for each 0 艋 k 艋 m, there exists an l such that wt共wl兲 = wt共gk兲.䊐 Corollary 6.6: G is a minimal set of generators. Corollary 6.7: The D-modules W共⌳兲 are pairwise nonisomorphic. Proof: This proof goes exactly as in the case of Ref. 4. Let n⬘, m⬘, n, m in N and ⌳ = n⌳1 + m⌳2, ⌰ = n⬘⌳1 + m⬘⌳2. Since the cardinality of a minimal set of generators is an invariant, W共⌳兲 ⯝ W共⌰兲 as D-modules implies that m = m⬘. But then W共⌳兲 can only have the same dimen䊐 sion of W共⌰兲 if n = n⬘. Lemma 6.8: Let 兵w0 , . . . , wm其 be a minimal set of generators of W共⌳兲. Then for each 0 艋 k 艋 m, there exists a ik, 0 艋 ik 艋 m such that wik = 共ck + Pk共T, P−, P+兲兲gk + 兺 Ql共T, P−, P+兲P+2共k−i兲共gi兲 + 兺 Ql共T, P−, P+兲P−2共i−k兲共gi兲, i⬎k

i⬍k

where ck is a complex number different from 0, Pk共T , P− , P+兲 is a polynomial in T, P−, and P+ of

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the type 兺h,k艌0,k+h艌1 phkTh P−k P+k (i.e., always “divisible” either for T or for P− P+) and finally the polynomials Ql共T , P− , P+兲 are polynomials of the type 兺qijTi P−j P+j . Proof: Lemma 6.5 claims that for any 0 艋 k 艋 m there exists at least a wl such that wt共wl兲 = wt共gk兲 and therefore such that wl = Cl共T, P−, P+兲共gk兲 + 兺 Ali共T, P−, P+兲P+2共k−i兲共gi兲 + 兺 Bli共T, P−, P+兲P−2共i−k兲共gi兲 i⬍k

i⬎k

兺dikTh P−k P+k ,

for some polynomials Ck, Ali, Bli of the type dik 苸 C, h , k 苸 N. If all the operators Cl共T , P− , P+兲 are nilpotent then gk is generated by elements of weight different from wt共gk兲 and elements of the type C1⬘共T, P−, P+兲T共gk兲 + C2⬘共T, P−, P+兲P− P+共gk兲 + 兺 Ali⬘ 共T, P−, P+兲P+2共k−i兲共gi兲 + 兺 Bli⬘ 共T, P−, P+兲P−2共i−k兲 i⬎k

i⬍k

⫻共gi兲 for some polynomials Cl⬘, C2⬘, Ali⬘ and Bli⬘ . But then reasoning as in the proof of Lemma 6.5, we have that gk can be written as gk = C1⬙共T, P−, P+兲T共gk兲 + C2⬙共T, P−, P+兲P− P+共gk兲 + 兺 Ali⬙ 共T, P−, P+兲P+2共k−i兲共gi兲 i⬎k

+ 兺 Bli⬙ 共T, P−, P+兲P−2共i−k兲共gi兲 i⬍k

for some polynomials Cl⬙, C2⬙, Ali⬙ , and Bli⬙ . But once again the sp共4 , C兲-weight of the left side cannot coincide with that of the right side, giving rise to a contradiction. Hence, there exists some 䊐 0 艋 ik 艋 m with invertible Cik. Lemma 6.9: Let W be a D-module with generators 兵v0 , . . . , vm其 such that wt共vk兲 = wt共vk−1兲 − 2. Define wk = vk + Ck共T, P−, P+兲vk + 兺 Aki共T, P−, P+兲P−2共k−i兲共vi兲 + 兺 Bki共T, P−, P+兲P+2共i−k兲共vi兲 i⬍k

共6.6兲

i⬎k

with 0 艋 k 艋 m and Ck共P+ , P− , T兲 nilpotent polynomial operator in T, P−, and P+. Then 兵w0 , . . . , wm其 generates W. Proof: We proceed by induction over m. If m = 0, then 共6.6兲 becomes w0 = v0 + C0共T , P− , P+兲v0. If C0 = 0, there is nothing to prove. Otherwise, since C0共T , P− , P+兲 is a nilpotent r operator there exists an integer r such that Cr+1 0 共T , P− , P+兲v0 = 0 and C0共T , P− , P+兲v0 ⫽ 0, then from r+1 r r r r C0w0 = C0v0 + C0 v0 = C0v0, it follows that C0v0 is generated by w0. Now using the relations r−i r−i+1 Cr−i v0 i = 0 , . . . r we see by recursion that w0 generates v0 and hence the whole 0 w0 = C0 v0 + C0 module V. For 0 艋 k 艋 m − 1, define wk = wk − Bkm共T, P−, P+兲P+2共m−k兲共wm − Ck共T, P−, P+兲共vm兲兲. Substituting Eq. 共6.6兲 in this latter one, we get wk = vk + Ck共T, P−, P+兲共vk兲 + 兺 Aki共T, P−, P+兲P−2共k−i兲共vi兲 + 兺 Bki共T, P−, P+兲P+2共i−k兲共vi兲 i⬍k

− Bkm共T, P−, P+兲兺 P+2共m−k兲Ami共T, P−, P+兲P−2共m−i兲共vi兲 = i⬍m

where

i⬎k

兺 D iv i ,

i⬍m

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冦

I + Ck共T, P−, P+兲 − Bkm P+2共m−k兲Amk P−2共m−k兲 if i = k if i ⬍ k

Di = Aki − Bkm P+2共m−k兲Ami P−2共k−i兲 Bki P+2共i−k兲

−

if i ⬎ k.

Bkm P+2共m−k兲Ami P−2共m−i兲

冧

Reasoning as in Proposition 3.2, it follows that Nk = Ck共T , P− , P+兲 − Bkm P+2共m−k兲Amk P−2共m−k兲 is a nilpotent operator. Now since Dk = I + Nk, where I is the identity and nk is nilpotent, we may apply the induction hypothesis on the elements 兵w0 , . . . , wm−1其 and claim that the D-module generated by 兵w0 , . . . , wm−1其 is the same D-module generated by the elements 兵v0 , . . . , vm−1其. Hence, the elements 兵v0 , . . . , vm−1其 are in the module generated by 兵w0 , . . . , wm其. Finally, since wm = 共vm + Cmvm兲 + 兺 Aik共P+, P−,T兲P−2共m−i兲共vi兲 i⬍k

with Cm nilpotent operator, the same argument used in the case m = 0, it shows that vm is also in the module generated by 兵w0 , . . . , wm其. Therefore the set 兵w0 , . . . , wm其 generates the same D-module 䊐 generated by v0 , . . . , vm, i.e., W. Theorem 6.10: The module W共␭兲 is D-indecomposable. Proof: Suppose that W共␭兲 can be decomposed as the direct sum W = U 丣 V. Let 兵w0 , . . . , ws其 and 兵ws+1 , . . . , wm其 generate U and V, respectively. Then 兵w0 , . . . , wm其 generates the whole W共␭兲. Using Lemma 6.8, for any 0 艋 k 艋 m, there exists a ik such that wik = Ck共T, P−, P+兲共gk兲 + 兺 Ai共T, P−, P+兲P−2共k−i兲共gi兲 + 兺 Bi共T, P−, P+兲P+2共i−k兲共gi兲 i⬍k

i⬎k

with Ck = C0k + Ck共T , P− , P+兲, where C0k is a complex number different from 0 and Ck共T , P− , P+兲 is a nilpotent operator. Using similar computations of those used in Lemma 6.4, we may show that wik generates the element t0. From P⫾共dhkTh P−k P+k 兲 = 共共 ⫾ 1兲dhk2iTh+1kP−k−1 P+k−1 + dhkTh P−k P+k 兲P⫾, it follows that if D共T , P− , P+兲 is a polynomial sum of monomials of the type dhkTh P−k P+k , then i eiTl P−j−l P+i−l for some P⫾hD = D⬘ P⫾h with D⬘ polynomial of the same type. Moreover, P+i P−j = 兺l=0 coefficients el 苸 C. From these facts and Lemma 6.4, it follows that Tm−k P−n P+n+2k共wik兲 = ck0C0k t0 + 兺 Ai⬘共T, P−, P+兲Tm−k P−n P+n+2k P−2共k−i兲共gi兲 i⬍k

+ 兺 Bi⬘共T, P−, P+兲Tm−k P−n P+n+2i共gi兲 = ck0C0k t0 i⬎k

+兺

冉兺

2共k−i兲

Ai⬘共T, P−, P+兲Tm−k P−n

i⬍k

l=0

冊

elTl P−2共k−i兲−l P+n+2k−l 共gi兲 + 兺 Bi⬘共T, P−, P+兲 i⬎k

⫻共ci0t0兲 = ck0C0k t0 + 兺 Ai⬘共T, P−, P+兲e2共k−i兲T2共k−i兲Tm−k P−n P+n+2i共gi兲 i⬍k

+ 兺 Bi⬘共T, P−, P+兲共ci0t0兲 = ck0C0k t0 + 兺 Ai⬘共T, P−, P+兲e2共k−i兲共ci0t0兲 i⬎k

i⬍k

+ 兺 Bi⬘共T, P−, P+兲共cit0兲 = ck0C0k t0 ⫽ 0. i⬎k

Suppose now that there exist two indices k , l k ⫽ l such that wik 苸 U and wil 苸 V. In this case, the element t0 is generated both by wik and by wil, and hence t0 苸 U 艚 V. This is in contradiction with U 艚 V = 兵0其, hence wk 苸 U for all k’s or wk 苸 V for all k’s. Then by Lemma 6.9, either W共␭兲 = U and 䊐 V = 兵0其 or W共␭兲 = v and U = 兵0其.

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FIG. 2. 共Color online兲 The graph if W共2⌳1兲.

We conclude our investigation of these latter classes of D-modules by showing that they are inequivalent up to the case V共⌳1兲 with those found in the previous sections. Theorem 6.11: The D-modules W共⌳兲 for ⌳ ⫽ ⌳2 are inequivalent (up to the trivial case) to the D-modules V共⌳兲 of Sec. IV and to the D-modules U1n, U2n, W1n, and W2n of Sec. V. Proof: From Lemma 4.3, it follows that a D-module W共n⌳1 + m⌳2兲 may be equivalent to a D-module V共n⬘⌳1 + m⬘⌳2兲 only if m = 0. In this case, however, using the basis of Refs. 4 and 1, it can be shown that the maximal nonzero power of T in W共n⌳1兲 it is Tn while in V共n⬘⌳1 + m⬘⌳2兲 is Tn⬘+m⬘. This fact forces n = n⬘ + m⬘ but then the equation dim共V共⌳兲兲 = dim共W共⌳兲兲 becomes 共n⬘ + 1兲共n − n⬘ + 1兲 = 3共n + 1兲共n + 3兲 which for any n ⬎ 0 has no positive solutions in n⬘. 1 may be equivalent to a module W共n⌳1 + m⌳2兲 only if From Proposition 5.3, a D-module U2l 1 m = l. However, dim共W共n⌳1 + m⌳2兲兲 = 6 共n + 1兲共m + 1兲共n + m + 2兲共n + 2m + 3兲艌 21 m3 + m2 + m + 1 ⬎ 3m 1 1 兲 for m ⬎ 1; hence, U2l could be equivalent to a a module W共n⌳1 + m⌳2兲 only if + 4 艌 dim共U2m m = 0 , 1. Similar arguments show that for any n , m 苸 C the D-module W共n⌳1 + m⌳2兲 is also in1 1 W2l+1 , and U22l for l ⬎ 1. A direct check shows that in the equivalent to all the modules W2l+1 䊐 remaining cases the only equivalence is W共⌳2兲 ⯝ U12 共See Fig. 2兲. ACKNOWLEDGMENTS

The authors would like to thank the anonymous referee who has substantially contributed to improve the first version of this paper. 1

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