OSCILLATOR REPRESENTATIONS OF VIRASORO ALGEBRA

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OSCILLATOR REPRESENTATIONS OF VIRASORO ALGEBRAPunita Batra aa School of Mathematics , Tata Institute of Fundamental Research , Homi Bhabha Road,Mumbai, Colaba, 400 005, IndiaPublished online: 01 Sep 2006.

To cite this article: Punita Batra (2002) OSCILLATOR REPRESENTATIONS OF VIRASORO ALGEBRA, Communications in Algebra,30:4, 1903-1919, DOI: 10.1081/AGB-120013223

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OSCILLATOR REPRESENTATIONS OF

VIRASORO ALGEBRA

Punita Batra

School of Mathematics, Tata Institute of FundamentalResearch, Homi Bhabha Road, Colaba,

Mumbai 400 005, IndiaE-mail: [email protected]

ABSTRACT

This paper describes Oscillator representations of Virasoroalgebra over L, which is a part of an extension of a simplylaced lattice by a hyperbolic lattice of rank 2n. These resultsare generalization of the results of Fabbri and Okoh.

0. INTRODUCTION

The purpose of this paper is to generalize oscillator representations ofVirasoro algebra over Lm, where Lm ¼ L is described below. Let _QQ be alattice of type Am;Dm and Em with root length normalized to two. Let

Q ¼ _QQZd1 Zd2 � � � Zdm;

G ¼ QZm1 Zm2 � � � Zmm;

L ¼ Zd1 Zd2 � � � Zdm Zm1 Zm2 � � � Zmm;

1903

Copyright # 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUNICATIONS IN ALGEBRA, 30(4), 1903–1919 (2002)

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where

ðdi j _QQÞ ¼ 0; ðmj j _QQÞ ¼ 0; ðdi j djÞ ¼ 0; ðmi j mjÞ ¼ 0;

ðdi j mjÞ ¼ dij for all pairs i and j. In [1], this case is handled when L is ahyperbolic lattice of the type L ¼ ZdZm, where ðQ j dÞ ¼ ð _QQ j mÞ ¼ ðm jmÞ ¼ 0 and ðd j mÞ ¼ 1.

In Sec. 1, we give the definition of generalized Heisenberg algebraHðL; nÞ associated to a geometric lattice L, which is a free Z-module of finiterank n-together with a non-trivial symmetric Z-bilinear form. We introducethe notation HðL; 1Þ ¼ AðLÞ.

In Sec. 2, we introduce a canonical representation VLðlÞ; l 2 C�Z Lof AðLÞ. A representation of AðLÞ is irreducible if and only if L is a non-degenerate lattice.

In Sec. 3, we describe Virasoro algebra and Virasoro operator Lk. Wedefine a representation of Virasoro algebra on VLðlÞ.

In Sec. 4, we prove our main results. Proposition 4.1 gives simplerexpressions for the oscillator operators, we give the definition of VLðlÞ‘ andin Theorem 4.2, we prove that VLðlÞ‘ is a proper Vir- submodule of VLðlÞand we give a filtration of Vir-submodules. In Theorem 4.4, we prove thatVir-module VLð0Þ‘=VLð0Þ‘�1 ¼ VLð0Þ‘ is completely reducible and using theresult of Proposition 4.3, we conclude that VLðlÞ‘ ¼ VLðlÞ‘=VLðlÞ‘�1 is alsocompletely reducible.

1. GENERALIZED HEISENBERG ALGEBRA

Let ðL; ðjÞÞ be a geometric lattice, i.e., a free Z-module L of finite ranktogether with a non-trivial symmetric Z-bilinear form ðjÞ : L� L! Z. LetL ¼ C�Z L, the complexificaton of L. Extend ðjÞ to a symmetric bilinearfrom on L also denoted by ðjÞ.

For r 2 Zn * Cn, let LðrÞ be an isomorphic copy of L while CnðrÞ is anisomorphic copy of Cn. The isomorphism is given by x! xðrÞ. If x 2Cn; zxðrÞ will denote the element xðrÞ to distinguish it from the elements ofLðrÞ. For r 2 Zn; g; g0 2 L; s; s0 2 Cn and a 2 C, we have

zsðrÞ þ zs0 ðrÞ ¼ zsþs0 ðrÞ; ð1Þ

azsðrÞ ¼ zasðrÞ; ð2Þ

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gðrÞ þ g0ðrÞ ¼ ðgþ g0ÞðrÞ; ð3Þ

agðrÞ ¼ ðagÞðrÞ: ð4Þ

Now let Cn ¼ r2Zn CnðrÞ;Dn ¼ r2Zn CzrðrÞ where CzrðrÞ is the one-dimensional complex vector space with basis zrðrÞ. Let Zn ¼ Cn=Dn. Con-sider the C-space

HðL; nÞ ¼ r2ZnLðrÞ

� Zn: ð5Þ

Introduce a bracket operation on HðL; nÞ as follows

½gðr1Þ; Zðr2Þ� ¼ ðg j ZÞzr1ðr1 þ r2Þ ð6Þ

Zn central: ð7Þ

By (6) and (7), HðL; nÞ is a two step nilpotent algebra and hence the mul-tiplication satisfies the Jacobi identity. From (1), (2), (6) and (7), we deducethat HðL; nÞ is a Lie algebra. We call it the generalized Heisenberg algebraassociated to L and n.

We use the notation AðLÞ for HðL; 1Þ.

HðL; 1Þ ¼ AðLÞ: ð8Þ

2. A CANONICAL REPRESENTATION OF AðLÞ

By Proposition 1.2 (a) of ½1�;Z1 is one-dimensional. Let c denote thefixed generator of Z1. Then

AðLÞ ¼ r2ZLðkÞ

� Cc;

and in AðLÞ Eqs. (6) and (7) are as follows

½aðk1Þ; bðk2Þ� ¼ k1dk1þk2;0ða j bÞc

c central

VIRASORO ALGEBRA 1905

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where d denotes Kronecker delta. Here Lð0Þ is an abelian subalgebra ofAðLÞ. It has a complement ðn2Znf0g LðnÞÞ Cc satisfying AðLÞ ¼ðn2Znf0gLðnÞ CcÞ � Lð0Þ, where � denotes the direct product of Liealgebras. We shall construct a canonical representation of AðLÞ by firstdefining a representation of the subalgebra ðn2Znf0g LðnÞÞ Cc. Let

AðLÞ� ¼a

n>0

Lð�nÞ ð9Þ

with corresponding symmetric algebra SðAðLÞ�Þ. Let faigmi¼1 is an ortho-normal basis of L ¼ C�Z L. Then SðAðLÞ�Þ may be considered as thepolynomial ring in the indeterminates faið�nÞ : 1 � i � m; n > 0g.

Let a; b 2 L. Let m; n be positive integers. We denote by @aðnÞ theunique derivation of SðAðLÞ�Þ satisfying

@aðnÞðbð�mÞÞ ¼ ndn;mða j bÞ ð10Þ

where dn;m is Kronecker delta. Let lað�nÞ be the map on SðAðLÞ�Þ defined byf 7! að�nÞf, multiplication by að�nÞ. Then we get the following repre-sentation on SðAðLÞ�Þ of the Lie algebra ðn2Znf0g LðnÞÞ Cc.

cf ¼ f; ð11Þ

að�nÞf ¼ lað�nÞf; ð12Þ

aðnÞf ¼ @aðnÞf: ð13Þ

Let M be any nondegenerate lattice containing L. Let M be thecomplexification of M. Fix l 2M and let Cel be the one-dimensionalC-space. Consider the C-space

VLðlÞ ¼ Cel �C SðAðLÞ�Þ :

We make VLðlÞ an AðLÞ-module by defining

cðel � f Þ ¼ el � f; ð14Þ

aðnÞðel � f Þ ¼ el � aðnÞf; n 6¼ 0 ð15Þ

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að0Þðel � f Þ ¼ ða j lÞel � f ð16Þ

where aðnÞf; n 6¼ 0 is given by (12) and (13). According to Proposition 1.5 of[1], VLðlÞ affords a representation of AðLÞ which is irreducible if and only ifL is a nondegenerate lattice. The module VLðlÞ is called a canonical repre-sentation of AðLÞ.

3. VIRASORO ALGEBRA AND ITS OSCILLATOR

OPERATORS

The Virasoro algebra Vir is an infinite7dimensional Lie algebra withgenerators fdk: k 2 Zg and bracket relations

½dk; d‘� ¼ ðk� ‘Þdkþ‘ þ1

12dkþ‘;oðk3 � kÞC

where C is a central symbol.Let L be an arbitrary non-degenerate geometric lattice of rank m with

complexification L ¼ C�Z L. Let faigmi¼1 be an orthonormal basis for Lover C. Our aim is to define a representation of Vir on VLðlÞ. For u; v 2 Z,we define a normal ordering : : of aiðuÞaiðvÞ, as in [1], by

: aiðuÞaiðvÞ :¼ aiðuÞaiðvÞ if u � v ð18Þ

: aiðuÞaiðvÞ :¼ aiðvÞaiðuÞ if u > v: ð19Þ

Now for k 2 Z, consider

Lk ¼1

2

X

j2Z

Xm

i¼1

: aið�jÞaið jþ kÞ : ð20Þ

Due to the normal ordering each Lk is an operator of VLðlÞ using (14), (15)and (16). The operator Lk is called a oscillator operator or Virasoro operator.According to Proposition 1.8 of [1], the map dk ! Lk; C ! mI, where m isthe rank of L and I is the identity operator, gives a representation of Vir onVLðlÞ.


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4. OSCILLATOR REPRESENTATIONS OF VIR OVER

Km ¼ K

Let _QQ be a lattice of type Am;Dm or Em with root lengths normalizedto two. Let

Q ¼ _QQZd1 Zd2 � � �Zdm; ð21Þ

G ¼ QZm1 Zm2 � � � Zmm; ð22Þ

L ¼ Lm ¼ Zd1 Zd2 � � � Zdm Zm1 Zm2 � � �Zmm; ð23Þ

where ð _QQ j diÞ ¼ 0; ð _QQ j mjÞ ¼ 0; ðdi j djÞ ¼ 0; ðmi j mjÞ ¼ 0, ðdi j mjÞ ¼ dijðdijis Kronecker delta) for all pairs ði; jÞ. Recalling (9), let

S ¼ SðAðLÞ�Þ: ð24Þ

An orthonormal basis for C�Z L is

a1 ¼1

2d1 þ m1;

a2 ¼i

2d1 � im1;

a3 ¼1

2d2 þ m2;

a4 ¼ �i

2d2 þ im2;

..

.

a2m�1 ¼1

2dm þ mm;

a2m ¼i

2dm � imm; where i2 ¼ �1:

We use the notation Hk; k 2 Z for the corresponding oscillatoroperators. So (20) becomes

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Hk ¼1

2

X

j2Z: a1ð�jÞa1ð jþ kÞ : þ

X

j2Z: a2ð�jÞa2ð jþ kÞ :

(

þ � � � � � � � � � þX

j2Z: a2m�1ð�jÞa2m�1ð jþ kÞ :

þX

j2Z: a2mð�jÞa2mð jþ kÞ :

): ð25Þ

Proposition 4.1. For every n 2 Z, we have that

ðiÞ Hn ¼1

2

X

j2Z: m1ð�jÞd1ðjþ nÞ : þ

X

j2Z: d1ð�jÞm1ðjþ nÞ :

(

þX

j2Z: m2ð�jÞd2ðjþ nÞ : þ

X

j2Z: d2ð�jÞm2ðjþ nÞ :

þ� � � þX

j2Z: mmð�jÞdmð jþ nÞ : þ

X

j2Z: dmð�jÞmmð jþ nÞ :

):

ðiiÞ Hn ¼ H�n Hþn ; where

H�n ¼22m1

n

2

� d1

n

2

� þ 2

2m2

n

2

� d2

n

2

� þ � � � 2

2mm

n

2

� dm

n

2

�

þX

j>�n2

m1ð�jÞd1ðjþ nÞ þ � � � þX

j>�n2

mmð�jÞdmðjþ nÞ;

Hþn ¼22d1

n

2

� m1

n

2

� þ � � � þ 2

2dm

n

2

� mm

n

2

�

þX

j>�n2

d1ð�jÞm1ð jþ nÞ þ � � � þX

j>�n2

dmð�jÞmmð jþ nÞ

where

E ¼ 1 if n is even,0 if n is odd.

�


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Proof. For any j 2 Z,

: a1ð�jÞa1ð jþ nÞ : þ : a2ð�jÞa2ð jþ nÞ :þ � � � þ : a2m�1ð�jÞa2m�1ð jþ nÞ : þ : a2mð�jÞa2mð jþ nÞ :¼ a1ð�jÞa1ð jþ nÞ þ a2ð�jÞa2ð jþ nÞ þ � � � þ a2m�1ð�jÞa2m�1ð jþ nÞþ a2mð�jÞa2mð jþ nÞ if � j � jþ n;

or

a1ð jþ nÞa1ð�jÞ þ a2ð jþ nÞa2ð�jÞ þ � � � þ a2m�1ð jþ nÞa2m�1ð�jÞþ a2mð jþ nÞa2mð�jÞ if � j > jþ n:

Thus for (i), it is sufficient to show that

a1ð�jÞa1ð jþ nÞ þ a2ð�jÞa2ð jþ nÞ þ � � � þ a2m�1ð�jÞa2m�1ð jþ nÞþ a2mð�jÞa2mð jþ nÞ ¼ m1ð�jÞd1ð jþ nÞ þ d1ð�jÞm1ð jþ nÞþ � � � þ mmð�jÞdmð jþ nÞ þ dmð�jÞmmð jþ nÞ

and

a1ð jþ nÞa1ð�jÞ þ a2ð jþ nÞa2ð�jÞ þ � � � þ a2m�1ð jþ nÞa2m�1ð�jÞþ a2mð jþ nÞa2mð�jÞ ¼ d1ð jþ nÞm1ð�jÞ þ m1ð jþ nÞd1ð�jÞ þ � � � þ dmð jþ nÞmmð�jÞ þ mmð jþ nÞdmð�jÞ:

We show only the first, since the second is similar. We have

a1ð�jÞa1ð jþ nÞ þ a2ð�jÞa2ð jþ nÞ þ � � � þ a2m�1ð�jÞa2m�1ð jþ nÞ

þ a2mð�jÞa2mð jþ nÞ ¼ 1

2d1ð�jÞ þ m1ð�jÞ

� �1

2d1ð jþ nÞ

�

þ m1ð jþ nÞoþ i

2d1ð�jÞ � im1ð�jÞ

� �i

2d1ð jþ nÞ � im1ð jþ nÞ

� �

þ � � � þ 1

2dmð�jÞ þ mmð�jÞ

� �1

2dmð jþ nÞ þ mmð jþ nÞ

� �

þ i

2dmð�jÞ � immð�jÞ

� �i

2dmð jþ nÞ � immð jþ nÞ

� �

¼ 1

4d1ð�jÞd1ð jþ nÞ þ 1

2d1ð�jÞm1ð jþ nÞ þ m1ð�jÞd1ð jþ nÞ

þ m1ð�jÞm1ð jþ nÞ � 1

4d1ð�jÞd1ð jþ nÞ þ 1

2m1ð�jÞd1ð jþ nÞ

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þ 1

2d1ð�jÞm1ð jþ nÞ � m1ð�jÞm1ð jþ nÞ þ � � � þ 1

4dmð�jÞdmð jþ nÞ

þ 1

2mmð�jÞdmð jþ nÞ þ 1

2dmð�jÞmmð jþ nÞ þ mmð�jÞmmð jþ nÞ

� 1

4dmð�jÞdmð jþ nÞ þ 1

2mmð�jÞdmð jþ nÞ

þ 1

2dmð�jÞmmð jþ nÞ � mmð�jÞmmð jþ nÞ

¼ d1ð�jÞm1ð jþ nÞ þ m1ð�jÞd1ð jþ nÞ þ � � � þ mmð�jÞdmð jþ nÞþ dmð�jÞmmð jþ nÞ:

This proves (i).(ii) To prove (ii), we use (i) and the definition of normal ordering.

Hn ¼1

2

X

�j� jþnm1ð�jÞd1ð jþ nÞ þ 1

2

X

�j� jþnd1ð�jÞm1ð jþ nÞ

þ 1

2

X

�j�jþnm2ð�jÞd2ð jþ nÞ þ 1

2

X

�j�jþnd2ð�jÞm2ð jþ nÞ

þ � � � þ 1

2

X

�j� jþnmmð�jÞdmð jþ nÞ þ 1

2

X

�j� jþndmð�jÞmmð jþ nÞ

þ 1

2

X

�j>jþnd1ð jþ nÞm1ð�jÞ þ

1

2

X

�j>jþnm1ð jþ nÞd1ð�jÞ

þ 1

2

X

�j>jþnd2ð jþ nÞm2ð�jÞ þ

1

2

X

�j>jþnm2ð jþ nÞd2ð�jÞ

þ � � � þ 1

2

X

�j>jþndmð jþ nÞmmð�jÞ þ

1

2

X

�j>jþnmmð jþ nÞdmð�jÞ:

As �j � jþ n) j � � n2. We split the terms having �j � jþ n as j >

� n2 and j ¼ � n

2. Also replacing j by �j� n in the terms containing�j > jþ n, we get

Hn ¼1

2

X

j>�n2

m1ð�jÞd1ð jþ nÞ þ 1

2

X

j>�n2

d1ð�jÞm1ð jþ nÞ

� � � þ 1

2

X

j>�n2

mmð�jÞdmð jþ nÞ þ 1

2

X

j>�n2



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þ E2m1

n

2

� d1

n

2

� þ E

2d1

n

2

� m1

n

2

� þ � � � þ E

2mm

n

2

� dm

n

2

�

þ E2dm

n

2

� mm

n

2

� þ 1

2

X

j>�n2

d1ð�jÞm1ð jþ nÞ

þ 1

2

X

j>�n2

m1ð�jÞd1ð jþ nÞ þ � � � þ 1

2

X

j>�n2


þ 1

2

X

j>�n2

mmð�jÞdmð jþ nÞ:

Regrouping, we get

Hn ¼E2m1

n

2

� d1

n

2

� þ E

2m2

n

2

� d2

n

2

� þ � � � þ E

2mm

n

2

� dm

n

2

�

þX

j>�n2

m1ð�jÞd1ð jþ nÞ þ � � � þX

j>�n2

mmð�jÞdmð jþ nÞ

þ 22d1

n

2

� m1

n

2

� þ � � � þ E

2dm

n

2

� mm

n

2

�

þX

j>�n2

d1ð�jÞm1ð jþ nÞ þ � � � þX

j>�n2

dmð�jÞmmð jþ nÞ:

If we replace �j by i and jþ n by j, then we get the following alter-native way of expressing H+n .

H�n ¼E2m1

n

2

� d1

n

2

� þ � � � þ E

2mm

n

2

� dm

n

2

�

þX

i<jiþj¼n

m1ðiÞd1ð jÞ þ � � � þX

i<jiþj¼n

mmðiÞdmð jÞ ð26Þ

Hþn ¼E2d1

n

2

� m1

n

2

� þ � � � þ E

2dm

n

2

� mm

n

2

�

þX

i<jiþj¼n

d1ðiÞm1ð jÞ þ � � � þX

i<jiþj¼n

dmðiÞmmð jÞ: ð27Þ

Since C�Z L ¼ Cd1 Cd2 � � � Cdm Cm1 Cm2 � � � þCmm;we have

C�Z LðnÞ ¼ Cd1ðnÞ Cd2ðnÞ � � � CdmðnÞ Cm1ðnÞ Cm2ðnÞ � � � CmmðnÞ:

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The algebra S in (24) contains the following C-subspaces

M1 ¼ Sa

n>0

Cm1ð�nÞ !

; D1 ¼ Sa

n>0

Cd1ð�nÞ !

;

M2 ¼ Sa

n>0

Cm2ð�nÞ !

; D2 ¼ Sa

n>0

Cd2ð�nÞ !

;

..

.

..

.

Mm ¼ Sa

n>0

Cmmð�nÞ !

; Dm ¼ Sa

n>0

Cdmð�nÞ !

:

So S ¼M1D1M2D2 � � � � � �MmDm and hence for l 2 C�Z L, we have thefollowing canonical representation of AðLÞ.

VLðlÞ ¼ Cel �C M1D1M2D2 � � �MmDm:

VLðlÞ is a Vir-module via Hn according to proposition 1.8 of [1]. We shallshow that it has a filtration of Vir-submodules. We haveM1 ¼ S

‘n>0 Cm1ð�nÞ

� �, which has the following C-basis

fm1ð�nÞ; n 2 Zsþ; s � 1; n1 � � � � � nsg [ f1g;

where Zþ is the set of natural numbers, n ¼ ðn1; n2; . . . nsÞ and m1ð�nÞ ¼m1ð�n1Þm1ð�n2Þ � � � m1ðnsÞ. We say that m1ð�nÞ has m1-length s. By replacingm1 by m2; m3; . . . mm; d1; d2; . . . dm; we get m2; m3; . . . ; mm; d1; d2; . . . dm-length.The length of the zero polynomial is taken to be �1 < the length of everynonzero polynomial.

Let

ðM1Þj ¼ 0 if j < 0

ðM1Þ0 ¼ C:

For j > 0; ðM1Þj ¼ C-span of all monomials in M1 of m1-length j. Set

ðM1Þ�j ¼a

k� j

ðM1Þk:


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Then

M1 ¼a

j�0

ðM1Þj:

We use ðD1Þj to denote the analogous C-space with m1, replaced by d1. ThenD1 ¼

‘j�0ðD1Þj. Similarly, we use ðM2Þj; . . . ; ðMmÞj to denote the analogous

C-spaces with m1 replaced by m2; . . . ; mm. With ‘ an arbitrary integer, let

S‘ ¼a1

j1¼0

a1

j2¼0

� � �a1

jm¼0

ðM1Þ�j1þ‘ðD1Þj1ðM2Þ�j2þ‘ðD2Þj2

� � � � ðMmÞ�jmþ‘ðDmÞjm :

For l 2 Zd1 Zd2 � � � Zdm, we let

VLðlÞ‘ ¼ Cel �C S‘: ð28Þ

We note that

ðMiÞ�jiþ‘ � ðMiÞ�jiþ‘þ1 8i ¼ 1; . . .m ð29Þ

S‘ 6¼ S; and S‘ � S‘þ1: ð30Þ

Theorem 4.2. For any integer ‘;VLðlÞ‘ is a proper Vir-submodule of VLðlÞand VLðlÞ‘ � VLð‘Þ‘þ1.

Proof. Since S‘ 6¼ S, we have thatVLðlÞ‘ 6¼ VLðlÞ. The inclusion followsfrom Eq. (30). We now show that VLðlÞ‘ is closed under the action of Hþnand H�n . Using Eqs. (26) and (27), we need to check closure under m1ðiÞd1ð jÞi < j; m2ðiÞd2ð jÞ i < j; . . . mmðiÞdmð jÞ i < j; d1ðiÞm1ð jÞ i < j; d2ðiÞm2ð jÞi <j; . . . dm ðiÞmmð jÞ i < j; m1ðn2Þd1ðn2Þ; . . . mmðn2Þdmðn2Þ; d1ðn2Þm1ðn2Þ; . . . dmðn2Þmmðn2Þ:

Let f ¼ el � x; x 2 S‘.

Here x ¼ m1ð�n1Þm1ð�n2Þ � � � m1ð�nsÞd1ð�k1Þ � � � d1ð�kj1Þ� � � � mmð�‘1Þ � � � mmð�‘rÞdmð�p1Þ � � � dmð�pjmÞ

where

s � j1 þ ‘; . . . ; r � jm þ ‘:

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Let, z ¼ m1ðiÞd1ð jÞ; i < j. We shall be using Eqs. (10) to (16) in theproof below. If j > 0; d1ð jÞðel � xÞ

¼ jel �Xs

t¼1

dj;ntm1ð�n1Þ � � � m1ð�ntÞ � � � m1ð�nsÞd1ð�k1Þ � � � d1ð�kj1Þ

� � � � dmð�pjmÞ;

here overbar denotes omission. Call

xt ¼ dj; ntm1ð�n1Þ � � � m1ð�ntÞ � � � m1ð�nsÞd1ð�k1Þ � � � d1ð�kj1Þ� � � � dmð�pjmÞ:

Now each summand here is either zero or its m1-length is one less than thatof x. If i < 0, then m1-length of m1ðiÞxt will be restored to that of x. If i > 0,then the effect of m1ðiÞ on each summand xt is to break it into summandsthat are zero or have d1-length one less than d1-length of xt. So zx is in

ðM1Þ�ð j1�1Þþ‘ðD1Þð j1�1ÞðM2Þ� j2þ‘ðD2Þj2 � � � ðMmÞ� jmþ‘ðDmÞjm

i.e., zx is in S‘. So m1ðiÞd1ð jÞf 2 VLðlÞ‘.Suppose j < 0 then d1ð jÞx has d1-length one more than that of x. Since

i < j, we have i < 0. So m1ðiÞd1ð jÞx will have m1-length one more than that ofx. So m1ðiÞd1ð jÞx is in

ðM1Þ�ð j1þ1Þþ‘ðD1Þð j1þ1ÞðM2Þ� j2þ‘ðD2Þj2 � � � ðMmÞ� jmþ‘ðDmÞjm :

So zx is in S‘ and m1ðiÞd1ð jÞf 2 VLðlÞ‘.Suppose j ¼ 0, then

m1ðiÞd1ð0Þel � x ¼ m1ðiÞðd1 j lÞðel � xÞ¼ m1 ðiÞ 0 since l 2 Zd1 �Zd2 � � � Zdm ¼ 0:

So m1ðiÞd1ð0Þel � x ¼ 0. Rest of the cases are handled in a similar fashion.Our next goal is to show that VLðlÞ‘ ¼ VLðlÞ‘=VLðlÞ‘�1 is a com-

pletely reducible representation of the Virasoro algebra. Denote the quotientS‘=S‘�1 by S‘ and we have that

S‘ ¼a1

j1¼0

� � �a1

jm¼0

ðM1Þj1þ‘ðD1Þj1ðM2Þj2þ‘ðD1Þj2 � � � ðMmÞjmþ‘ðDmÞjm :


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Proposition 4.3. The Vir-modules VLðlÞ‘ and VLð0Þ‘ are isomorphic.

Proof. Let f : VLð0Þ‘ ! VLðlÞ‘ be the map given by f0 ! fl.

Let

f0 ¼Xr

k¼0

ckðe0 � xkÞ 2 VLð0Þ‘; ck 2 C

and

fl ¼Xr

k¼0

ckðel � xkÞ 2 VLðlÞ‘; ck 2 C:

The map f is clearly one-one and onto. It is easy to see that it is a Vir-homomorphism. Hence VLðlÞ‘ andVLð0Þ‘ are isomorphic.

We now define a positive definite Hermitian form hji on VLðlÞ. Forai; bi 2 fm1; d1; m2; d2; . . . mm; dmg, let

ða1ð�n1Þ � � � asð�nsÞ j b1ð�m1Þ � � � brð�mrÞÞ

¼ dr;sX

s2pðrÞ

Yr

k¼1

nkdnk;msðkÞ ðak j bsðkÞÞ ð31Þ

where dx;y; x; y 2 Z, denotes the Kronecker delta and pðrÞ denotes thesymmetric group on r symbols. Let i : S ! S be the unique antilinear mapsatisfying

iðm1ð�rÞd1ð�sÞ � � � mmð�qÞdmð�kÞÞ ¼ m1ð�sÞd1ð�rÞmmð�kÞdmð�qÞ

ið1Þ ¼ 1, where r 2 Zrpþ ; s 2 Zsa

þ ; � � � q 2 Zqkþ ; k 2 Zkt

þ , rp; sa; qk; kt � 1. Themap i is an involution. Now we define a Hermitian form on VLðlÞ usingEq. (31). Let x; x0 2 S; z ¼ el � x; z0 ¼ el � x0. Then hz j z0i ¼ ðx j iðx0ÞÞ:

Proposition 4.4. a) The set

P ¼fz ¼ el � m1ð�rÞd1ð�sÞ � � � mmð�qÞdmð�kÞ :r 2 Z

rpþ ; s 2 Zsa

þ ; q 2 Zqkþ ;k 2 Zkt

þ ;

r1 � r2 � � � � � rrp ; s1 � s2 � � � � � ssa

q1 � q2 � � � � � qqk ; k1 � k2 � � � � � kktg

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[fel � 1g is an orthogonal basis of VLðlÞ with respect to hji. b) The form hjiis positive definite on VLðlÞ and

k z k2¼ cYrp

i¼1

riYsa

j¼1

sj � � �Yqk

‘¼1

q‘Ykt

r¼1

kr;

where c is some positive constant. We call k z k the norm of z.

Proof. a) It is easy to see that the set P is an orthogonal basis of VLðlÞ withrespect to hji using Eq. (31).

b) Let

z ¼ el � m1ð�r1Þ � � � m1ð�rrpÞd1ð�s1Þ � � � d1ð�ssaÞ � � � mmð�q1Þ� � � � mmð�qqkÞdmð�k1Þdmð�k2Þ � � � dmð�kktÞ:

Then

k z k2¼ hz j zi ¼ ðm1ð�r1Þ � � � m1ð�rrpÞd1ð�s1Þ � � � d1ð�ssaÞ� � � � mmð�q1Þ � � � mmð�qqkÞdmð�k1Þ � � � dmð�kktÞj m1ð�s1Þ � � � m1ð�ssaÞd1ð�r1Þ � � � d1ð�rrpÞ � � � mmð�k1Þ� � � � mmð�kktÞdmð�q1Þ � � � dmð�qqkÞÞ¼

X

sEPðrrpþssaþqqkþkkt Þr1ðm1 j d1Þr2ðm1 j d1Þ

� � � � rrpðm1 j d1Þs1ðd1 j m1Þ � � � ssaðd1 j m1Þ � � �� q1ðmm j dmÞqqkðmm j dmÞk1ðdm j mmÞ� � � � kktðdm j mmÞ; ð32Þ using ð31Þ

¼ cYrp

i¼1

riYsa

j¼1

sj � � �Yqk

‘¼1

q‘Ykt

r¼1

kr;

where c is some positive constant depends upon the number of permutationsin Pðrrp þ ssa þ qqk þ kktÞ which contribute a nonzero term in (32). Thedegree of z in Proposition 4.4 is defined as

Xrp

i¼1

ri þXsa

j¼1

sj � � � þXqk

‘¼1

q‘ þXkt

r¼1

kr:


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Let VLð0Þ‘ð jÞ denote the suspace of VLð0Þ‘ spanned by elements ofdegree j. This is finite-dimensional vector space. It is easy to check that thisfinite dimensional space is the eigenspace of the eigenvalue j of the oscillatoroperator H0 in Proposition 4.1.

Theorem 4.5. Let ‘ be any integer. Then the Vir-module VLð0Þ‘ is completelyreducible.

Proof. Let VLð0Þ‘ð jÞ ¼ Bj and B ¼ VLð0Þ‘. Now B can be written as adirect sum of finite-dimensional subspaces of elements of degree j. So

B ¼ j2Z

Bj: ð33Þ

Let U be any subrepresentation of B. Using Lemma 1.1 of [4],

U ¼ j2ZðU \ BjÞ:

Using [4], it follows that the eigenspace of H0 appearing in (33) aremutually orthogonal with respect to the Hermitian form hji on B. We denoteby Uj-the subspace U \ Bj. Then we can define a subspace U? by

[? ¼ j2Z

U?j

where U?j is the finite dimentional orthogonal complement of Uj in Bj.Clearly, we have

B ¼ UU?;

since

U? ¼ fv 2 B j hu j vi ¼ 0g

It is now clear that U? is an invariant subspace for Vir. Since hU j U?i ¼ 0and LjU * U, we have

hLjU j U?i ¼ 0

) hU j LjU?i ¼ 0

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So U? is also invariant under Lj; j 2 Z. So VLð0Þ‘ is completely reducible asa Vir-module.

Corollary 4.6. For any integer ‘, the Vir-module VLðlÞ‘ is completelyreducible.

Proof. Proof follows by Proposition 4.3 and Theorem 4.5.

ACKNOWLEDGMENT

I thank Dr. S. Eswara Rao for his help and useful suggestions infinding the results of this paper.

REFERENCES

1. Fabbri, M.A.; Okoh, F. Representations of Virasoro-Heisenberg Al-gebra and Virasoro-Toroidal Algebras. Canad. J. Math. 1999, 51 (3),5237545.

2. Fabbri, M.A.; Moody, R.V. Irreducible Representations of Virasoro-Toroidal Lie Algebras. Comm. Math. Phys. 1994, 159, 1713.

3. Eswara Rao, S.; Moody, R.V. Vertex Representations for n-ToroidalLie Algebras and a Generalization of the Virasoro Algebra. Comm.Math. Phys. 1994, 159, 2397264.

4. Kac, V.G.; Raina, A.K. Bombay Lectures on Highest Weight Re-presentation of Infinite-Dimensional Lie-Algebras; Singapore WorldScientific, 1987.

Received October 2000Revised April 2001


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OSCILLATOR REPRESENTATIONS OF VIRASORO ALGEBRA