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a

rXiv:1602.05602v1

[math.QA]17F

eb2016

2-permutations of lattice vertex operator algebras:

Higher rank

Chongying Dong

Department of Mathematics, University of California, Santa Cruz, CA 95064

USA

Feng Xuand Nina Yu

Department of Mathematics, University of California, Riverside, CA 92521

USA

February 19, 2016

Abstract

The fusion rules of the 2-permutation orbifold of an arbitrary lattice vertex operator

algebra are determined by using the theory of quantum dimension.

1 Introduction

This paper is a continuation of our investigation on 2-permutation of lattice vertex operator

algebras [DXY]. In particular, the quantum dimensions of irreducible modules and the fusion

rules are determined. If the rank of the lattice is one, these results have been obtained previously

in [DXY].

Let V be a vertex operator algebra and n a fixed positive integer and consider the ten-sor product vertex operator algebra Vn [FHL]. Then the symmetric groupSn acts naturallyonVn as automorphisms. The permutation orbifold theory has been studied extensively inphysics [KS, FKS,BHS, Ba]. Conformal nets approach to permutation orbifolds have been

given in [KLX]. Twisted sectors of permutation orbifolds of tensor products of an arbitrary

vertex operator algebra have been constructed in [BDM]. TheC2-cofiniteness of permutationorbifolds and general cyclic orbifolds have been studied in [A3,A4,M]. But the representa-

tion theory such as rationality, classification of irreducible modules, and fusion rules for the

Supported by NSF grant DMS-1404741 and China NSF grant 11371261Partially supported by China NSF grant 11471064

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fixed point vertex operator algebra(Vn)G for anynand any subgroupGofSnhave not beeninvestigated much.

As a starting point, we studied representations of 2-permutation orbifold model of rank one

lattice vertex operator algebras in [DXY]. In this paper, we complete the study of 2-permutation

orbifold model of lattice vertex operator algebras VL for any positive definite even lattice L.Similar to rank one case, the permutation orbifold model (VLVL)Z2 can be realized as asimple current extension of the rational vertex operator algebraV2L V+2L. It follows from[Y,HKL] that(VLVL)Z2 is rational. According to [DRX], every irreducible(VLVL)Z2-module occurs in an irreducibleg-twistedVL VL-module. So the classification of irreducible(VL VL)Z2-modules is known. But this classification result does not suggest how to computethe fusion rules among the irreducible modules. The main idea is to use the general theory of

simple current extension of a rational vertex operator algebra and representations ofVLandV+L

to study the representations of(VL VL)Z2 . We decompose each irreducibleVL VL-moduleinto a direct sum of irreducible(VL VL)Z2-modules by using the fusion rules for both vertexoperator algebrasV

2L

andV+

2L[DL1,A1,ADL]. This decomposition is crucial in computing

the fusion rules. We emphasize that the theory of quantum dimensions introduced and studied

in [DJX,DRX] plays an essential role in computing the fusion rules. It is not clear to us how

to achieve this without using the quantum dimensions. The fusion rules in conformal nets for

any 2-permutation models were computed by using theS-matrix [KLX].

We should mention that the constructions ofg-twisted modules for lattice vertex operatoralgebra VLwhere gis automorphism of finite order induced from an isometry ofL were already

given in [FLM1,FLM2,L,DL2]. In the caseg is of order2, the irreducible modules ofVgL

have been classified recently in [BE]. An equivalence of two constructions of permutation-

twisted modules for lattice vertex operator algebras in [FLM1, L] and [BDM] was given in

[BHL].

The paper is organized as follows: 2 and 3 are preliminaries on the vertex operator alge-

bras theory. In these sections we give some basic notions that appear in this paper and recall

the constructions of the lattice type vertex operator algebras VL andV+

L and their (twisted)

modules. In 4 we study (VL VL)Z2 , the 2-cyclic permutation orbifold models for rankdlattice vertex operator algebras. In particular, we decompose each irreducibleVL VL-moduleinto a direct sum of irreducible (VL VL)Z2-modules. The quantum dimensions of all irre-ducible modules of(VL VL)Z2 are obtained explicitly in 5. Finally, we apply results fromthe previous sections to determine all fusion products in 6.

2 Preliminaries

Let (V,Y, 1, ) be a vertex operator algebra [Bo, FLM2] and g an automorphism of vertexoperator algebraVof orderT. Denote the decomposition ofVinto eigenspaces ofg as:

V =T1r=0 Vr, Vr =

vV|gv = e2ir/Tv .Here are the definitions of weak, admissible, ordinaryg-twistedV-modules [DLM3].

Definition 2.1. Aweakg-twistedV-moduleMis a vector space with a linear map

YM :V(End

M) {z}2

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vYM(v, z) =nQ

vnzn1 (vnEndM)

which satisfies the following: for all0rT 1,uVr ,vV,wM,(1)ulw= 0ifl is sufficiently large,(2)YM(u, z) =

n r

T+Z unz

n1,

(3)YM(1, z) =I dM,

(4) (Twisted Jacobi identity)

z10

z1 z2z0

YM(u, z1) YM(v, z2) z10

z2 z1z0

YM(v, z2) YM(u, z1)

z12 z1 z0z2 r/T

z1 z0z2 YM(Y (u, z0) v, z2) ,where(z) =

nZ z

n.

Definition 2.2. Anadmissible g-twistedV-module M=n 1TZ+M(n)is a

1TZ+-graded weak

g-twisted module such that umM(n) M(wtu m 1 +n) for homogeneousu V andm, n 1

TZ.

Definition 2.3. A (ordinary) g-twistedV-module is a weak g-twisted V-module M whichcarries a C-grading induced by the spectrum ofL(0), where L(0) is the component opera-tor ofY(, z) = nZ L(n)zn2. That is, we haveM = CM, whereM ={wM|L(0)w = w}. Moreover,dim M is finite and for fixed, Mn

T+= 0for all small enough

integersn.A vectorwM is called a weight vector of weight , and write= wtw.Remark 2.4. Ifg = I dVwe have the notions of weak, ordinary and admissible V-modules.

Note that the cyclic groupg generated byg acts on any admissibleg-twistedV-moduleM such that g|M(n) = e2in forn 1TZ andgYM(v, z)g1 = YM(gv,z) for all v V. Inparticular,Mr =nZM( rT+ n)is an admissible Vg-module forr = 0,...,T1.Moreover,ifMis irreducible then eachMr is irreducible admissibleVg-module [DY,MT,DRX].

Definition 2.5. A vertex operator algebra V is called g-rational if the admissible g-twisted

module category is semisimple.V is calledrationalifV is1-rational.Definition 2.6. A vertex operator algebra V is said to beC2-cofinite ifV /C2(V) is finite di-mensional, whereC2(V) =v2u|v, uV.Remark 2.7. If vertex operator algebra V is rational orC2-cofinite, thenVhas only finitelymany irreducible admissible modules up to isomorphism and each irreducible admissible mod-

ule is ordinary [DLM3,Li].

Now we consider the tensor product vertex algebras and the tensor product modules for ten-

sor product vertex operator algebras. The tensor product of vertex operator algebras (V1, Y1, 1, 1)and

(V

2

, Y

2

, 1,

2

) is constructed on the tensor product vector space

V = V

1

V2 where

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the vertex operator Y (, z) is defined byY (v1 v2, z) = Y (v1, z)Y (v2, z) forv i Vi ,i = 1, 2, the vacuum vector is 1 = 11 and the Virasoro element is = 1 2. Then(V,Y, 1, ) is a vertex operator algebra [FHL,LL]. LetWi be an admissibleVi-module fori= 1, 2. We may construct the tensor product admissible module W1 W2 for the tensor prod-uct vertex operator algebraV1

V2 byY (v1

v2, z) =Y (v1, z)

Y (v2, z). ThenW1

W2

is an admissibleV1 V2 -module. We have the following result about tensor product modules[DMZ,FHL]:

Theorem 2.8. LetV1, V2 be rational vertex operator algebras, then V1 V2 is rational andany irreducible V1 V2 -module is a tensor productW1 W2 for some irreducibleVi-moduleWi andi= 1, 2.

LetM=

n 1TZ+

M(n)be an admissible g-twisted V-module, the contragredient module

M is defined as follows:M =

n 1TZ+M(n),

whereM(n) = HomC(M(n),C).The vertex operatorYM(v, z)is defined forvV viaYM(v, z)f, u= f, YM(ezL(1)(z2)L(0)v, z1)u

where f, w= f(w)is the natural paring MM C. Then Mis an admissible g1-twistedV-module [X]. A V-module Mis said to be self dualifMand Mare isomorphic V-modules.

We now recall the notion of intertwining operators and fusion rules [ FHL]:

Definition 2.9. Let (V, Y) be a vertex operator algebra and let (W1, Y1), (W2, Y2) and

(W3, Y3)beV-modules. An intertwining operator of type

W1

W2 W3 is a linear map

I(, z) : W2 Hom(W3, W1){z}

uI(u, z) =nQ

unzn1

satisfying:

(1) for anyuW2 andvW3,unv= 0forn sufficiently large;(2)I(L1v, z) = ( ddz )I(v, z);(3) (Jacobi Identity) for anyuV, vW2

z10

z1 z2z0

Y1(u, z1)I(v, z2) z10

z2+z1z0

I(v, z2)Y

3(u, z1)

=z12

z1 z0

z2

I(Y2 (u, z0)v, z2).

We denote the space of all intertwining operators of type

W1

W2 W3

by IV

W1

W2 W3

.

Let NW1

W2, W3 =NV

W1

W2 W3

= dim IV

W1

W2 W3

. These integers NW

1

W2, W3are usually

called thefusion rules.

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Definition 2.10. Let V be a vertex operator algebra, andW1, W2 be two V-modules. A

module(W, I), whereI IV

WW1 W2

, is called a fusion productofW1 andW2 if for

anyV-moduleM andY IV M

W1 W2 , there is a unique V-module homomorphism

f :W M,such that Y=f I.As usual, we denote(W, I)byW1 V W2.

It is well known that ifVis rational, then the fusion product exists. We shall often considerthe fusion product

W1 V W2 =

W

NWW1, W2W

whereWruns over the set of equivalence classes of irreducibleV-modules.

The fusion rules satisfy the following symmetry [FHL].

Proposition 2.11. LetWi (i= 1, 2, 3)beV-modules. Then

dim IV

W3

W1W2

= dim IV

W3

W2W1

, dim IV

W3

W1W2

= dim IV

(W2)

W1( W3)

.

Definition 2.12. LetVbe a simple vertex operator algebra. A simpleV-moduleM is calleda simple currentif for any irreducible V-moduleW, W Mexists and is also a simple V-module.

LetD be a finite abelian group and assume that we have a set of irreducible simple currentV0 -modules

{V

|

D

}indexed byD. The following definition is from [Y].

Definition 2.13. An extension VD =DV ofV0 is called a D-graded simple currentextensionifVD carries a structure of a simple vertex operator algebra such that Y(u

, z) u V+ ((z))for anyu V andu V .

3 Vertex Operator algebraVLandV+

L

We first review the construction of the vertex operator algebra VL associated with a positivedefinite even latticeL[Bo,FLM2].

We are working in the setting of [DL1, FLM2]. Let L be a positive definite even latticewith bilinear form, and L its dual lattice in h = C ZL. Let{0= 0, 1, 2, } be acomplete set of coset representatives ofL in L.Then the lattice vertex operator algebraVL isrational andVi+L are the irreducibleVL-modules [Bo,FLM2,D1, DLM2].

Now assume M is positive definite even lattice such that, 2Z for , M.In this case, VM+ = S(h t1C [t1])C [ +M] for any M where S() is thesymmetric algebra andC [ +M] =

MCe+is the subspace of the group algebraC [M

]corresponds to+M. ThenVMhas a canonical automorphism of order 2 induced from1isometry ofM. In fact, we can define a linear map fromV+M toV+M for any Msuch that Y+M(u, z)

1 =Y+M(u,z)for any uVMwhere Y+Mdefines a VM-module

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structure onV+M [AD]. Clearly, if2M, is an endormorphism fromV+M toV+M. Forsuch we denote eigenspace of with eigenvalue 1inV+M byV+M.

We now turn our attention to the construction of-twisted VM-modules [FLM1, FLM2,L,DL2]. Note thatM/2M is an abelian group of order 2d whered is the rank ofM. ThenM/2Mhas exactly2d inequivalent irreducible modulesTwhere is irreducible character of

M/2M. It was proved in [FLM2] thatVTM = S

ht 12C [t1] T is an irreducible-twisted module. According to Remark2.4, acts onV

TM . Again we denote eigenspace of

with eigenvalue1by VT,M . Moreover,VM is-rational and

VTM|

gives a complete list

of inequivalent irreducible-twistedVM-modules [D2].

We have the following classification of the irreducibleV+L -modules [AD,DN]:

Theorem 3.1. Let Mbe a positive definite even lattice and {i} be a set of coset representativesofM inM. Then any irreducibleV+M-module is isomorphic to one of the following:

Vi+M(2i /M) , Vi+M(2iM) , VT,

M .Furthermore,Vi+M

=Vj+Mif and only ifi jM.From now on, we fix a rankdlattice L= Z1+ +Zdwith positive definite symmetric

non-degenerate bilinear form, . LetM = 2L. Later we will see that the 2-permutationorbifold model we study is closely related to the rational vertex operator algebras V2L andV+

2L. We now consider the fusion rules for the vertex operator algebras V2L andV

+2L

.

First we notice that the dual lattice of

2L can be written by

2L

=

2|L

.

Thus fusion rules for irreducibleV2L-modules are given by the following [DL1]:

Proposition 3.2. NV

2L

V 2+2LV

2+2L

V 2+2L =+

2 +

2L, 2

+

2L for , andL.

The fusion rules for V+L for any L was obtain in [ADL]. For this purpose, we need toidentify the contragredient modules of the irreducible V+

2L-modules first (see Proposition 3.7

of [ADL]).

Proposition 3.3. Every irreducibleV+2L

-module is self dual.

Remark 3.4. For anyL, we define a characterso that

2i

= (1)

i,i2

+,i

for1

i

d. Then = if and only if

2L. Thus

{

|

L/2L

}gives all

different characters.

Recall the number ,= e,i for , (2L)[ADL]. For any character of2L/22L,

cwas defined in [ADL] and we note that here for anyL we have:

c

2

= (1),

2

=

2

.

For anyL, L, the character 2

is defined in [ADL] by

2

2 = (1),

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a set of coset representatives of2LinL.Then we have decomposition

VLL=S

V12

+L+ V2

2 +L.

It is clear thatL+=L= 2L.Also,(2) =(2) =2 where is the1-isometryonLdefined before. Thus

(VLL)Z2 =

S

V 2

+

2L V+2

+

2L.

For short, we set

U=S

V 2

+

2L V+2

+

2L

and

V=V2L

V+

2L.

It follows from [DL1] and3.5that eachV 2

+

2L V+2

+

2Lis a simple currentV-module. In

particular,U is a simple current extension ofV. By [D1,DLM1,A2,DJL],V is rational. Wealso know thatV isC2-cofinite [ABD,Ya]. From [Y] or [HKL] we haveProposition 4.1. The vertex operator algebraUis rational.

From the classification of irreducible modules ofV[D1, DN,AD], every irreducible modulehas positive weight except the vertex operator algebra itself. A result from [DRX] gives:

Proposition 4.2. Every irreducible

U-module occurs in an irreducible i-twisted VL

VLmod-

ule fori = 0, 1.

As far as representation theory concerns, it remains to compute the fusion rules for U.Butit is not so easy to achieve this goal with the irreducible modules given abstractly in [DRX]. On

the other handUis a simple current extension ofV2L V+2L and we know the fusion rulesfor bothV2L andV

+2L

,it is natural to use these results to determine the fusion rules forU.Inthe rest of this section, we will realize each irreducibleU-module as a direct sum of irreducibleV-modules.

Recall from [D1] that all irreducible VL-modules are given by VL+, L. LetT =

{0 = 0, 1, 2,

}be a complete set of representatives ofL in L. Assume that

|T | =

|L/L|=l. Then there are exactlyl inequivalent irreducibleVL-modules.For any, L,V+L V+L is an irreducibleVL VL-module. If + L= + L, then

V+LV+L andV+LV+L are isomorphic irreducible(VL VL)Z2-modules [DM,DY].The number of such isomorphism classes of irreducible (VL VL)Z2-modules is l2l2 .

When = , V+L V+L + V+L V+L split into two different representations of(VL VL)Z2 by [DY]. The number of such isomorphism classes of irreducible(VL VL)Z2-modules is2l.

It is shown in [BDM] that there is one-to-one correspondence between the category of-twistedVLVL-modules and the category ofVL-modules. Thus the number of isomorphismclasses of irreducible

-twisted

VLVL-module is also

l. From [DY], each twisted module

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hence and define the same character. Thus

U W =

S,(

2)=1

V+2

+

2L VT+ ,+

2L +

S,(

2)=1

V+2

+

2L VT+,

2L

is an irreducibleU-module. We denote this module by (0).We now prove that ( +0) = (0) for anyL. It is clear that for any2L,

V++2

+

2L VT++ ,+

2L =V+

2 +

2L VT+ ,+

2L

for S.So ( +0) = (0).Ifdoes not lie in2L,we can assume SasSis a cosetrepresentatives of2LinL.The result follows immediately from the definition ofU W.

Similarly, we can prove that for any, L,whenW=V 2

+

2L VT,

2L ,

U W = S,(

2)=1

V+2

+

2L VT+ ,

2L +

S,(

2)=1

V+2

+

2L VT+,+

2L

is an irreducibleU-module which we denote by(1).The number of inequivalent irreducibleU-modules of the form( ), T, = 0, 1is2l.Now we have in total l

2l2

+ 2l + 2l = l2+7l

2 irreducibleU-modules. Thus they are the

inequivalent irreducibleU-modules.Remark 4.5. By Proposition 3.7 in [ADL], for any irreducibleV2L-module Vi+

2L, we haveVi+2L = Vi+2L, wherei is any coset representative of 2L in 2L. By Propo-

sition 3.3, it is clear that for any , L with + L = + L, and = 0, 1, we have( )

= ( ),( ) = ( ), and( ) = ( ).

5 The quantum dimensions

Quantum dimensions have been systematically studied in [DXY,DRX]. It is proved that for a

rational,C2-cofinite, self-dual vertex operator algebra of CFT type, quantum dimensions of itsirreducible modules have nice properties that are helpful in determining fusion products. The

2-permuation orbifold model(VL VL)Z2 we study here satisfies all the conditions and hencewe can use quantum dimensions to determine some fusion rules. First we recall some notions

and properties about quantum dimensions.

Definition 5.1. Letg be an automorphism of the vertex operator algebra V with orderT. LetM=n 1

TZ+M+nbe ag-twistedV-module, the formal character ofMis defined as

chqM= trMqL(0)c/24 =qc/24

n 1

TZ+

(dim M+n) qn,

where

is the conformal weight ofM

.

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We denote the holomorphic function chqM byZM(). Here and below, is in the upperhalf plane H andq= e2i.

Definition 5.2. Let V be a vertex operator algebra and M a g-twisted V-module such thatZV()andZM()exists. The quantum dimension ofMoverVis defined as

qdimVM= limy0

ZM(iy)

ZV(iy),

wherey is real and positive.

AssumeV is a rational,C2-cofinite vertex operator algebra of CFT type withV= V . LetM0= V, M1, , Ml be all inequivalent irreducibleV-modules. Moreover, we assume theconformal weightsi ofM

i are positive for alli > 0.Then we have the following propertiesof quantum dimensions [DJX]:

Proposition 5.3. qdimVMi

1,i= 0, , l.Proposition 5.4. For anyi, j = 0, , l,

qdimV

Mi Mj

=qdimVMi qdimVMj .

Proposition 5.5. AV-moduleM is a simple current if and only ifqdimVM= 1.

Remark 5.6. By Proposition4.1and [A4] we see that the vertex operator algebra (VL VL)Z2satisfies all the assumptions required in [DJX] and thus we can apply these properties.

We obtain quantum dimensions of all irreducible (VL

VL)Z2-modules as follows:

Proposition 5.7. For, Lwith +L=+L, = 0, 1, we have

qdimU( ) = 1 (5.1)qdimU( ) = 2 (5.2)

qdimU

( ) =

|L/L| (5.3)

Proof. Using the definition of quantum dimension we see that for any irreducibleU-moduleM,

qdimUM=qdimVM

qdimVU .

For anyL, S, by fusion rules of irreducibleV2L- andV+2L-modules in Proposi-tion 3.2 and Proposition 3.5,we see that V2+

2+

2LV+2

+

2Lis a simple current V-module.

By Proposition5.4 and Proposition5.5 , V2+ 2

+

2LV+2

+

2Lis of quantum dimension

1 as irreducibleV-module. Thus qdimV

( ) = 2d and qdimVU = 2d. Thus we obtain

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For S,, L with+L= +L, we have2

+2

2L. Thus by fusion

rules of irreducibleV+2L

-modules in Proposition3.5, quantum dimension ofV+2

+

2L is2

as irreducibleV+2L

-module. By Proposition5.4,

qdimV(V++2 +2L V+2 +2L) =qdim2L V++2 +2L qdimV+2L V++2 +2L = 2.Therefore we getqdimV( ) = 2 2d = 2d+1 and hence we prove (5.2).To prove (5.3), first we recall from [DJX] that glob (V) =

M(qdim M)

2whereM runs

over all irreducible modules ofV. By Proposition5.4, glob (VL VL) = (glob (VL))2. Nowwe have

glob (VL VL) =

T(qdim V+L)

2

2=|T |2 =|L/L|2 =l2

asqdim V+L = 1 for any irreducibleVL-moduleV+L. SetqdimU( ) = x , by quantumdimensions of irreducibleU-modules( )and( )we obtain above, we have

glob

VG

= l2 l

2 22 + 2l 12 + 2l x2.

It is proved in [DRX] that glob

VG

=|G|2 glob (V) .Therefore we get22 l2l2

+ 2l 12 +2l x2 = 22 l2. Solving the equation givesx= land thusqdimU( ) = |L/L|.

6 Fusion Rules

In this section, we use the quantum dimensions obtained in the previous section and the fusion

rules of irreducibleV2L- andV+

2L-modules in [DL1, ADL] to determine the fusion products

of the 2-permutation orbifold model.

Theorem 6.1. LetL be as before. Let, , ,L, , 1 = 0, 1.(a) (i)

( ) ( 1) = ( + +1), (6.1)

(ii) if+L=+L, then( ) ( ) = ( + +) , (6.2)

(iii) if + L= + L ,+ L=+ L, + + L= + + L, and + + L= + + L,then

( ) ( ) = ( +0) + ( +1) + (+0) + (+1), (6.3)

(iv) if + L= + L,+ L=+ L, + + L= + + L, and + + L= + + L,then

( )

( ) = ( + +) +

(+0) +

( +1),

(6.4)

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(v) if + L= + L,+ L=+ L, + + L= + + Land + + L= + + L,then

( ) ( ) = ( + +) + (+ +) . (6.5)

(b) If +L=+L,

( ) ( ) = ( + +0) + ( ++1), = 0, 1, (6.6)( ) ( 1) = (2 + +1). (6.7)

(c) LetG ={ T |2L},(i) if +

2 L,

( ) ( 1) = G

+2

+ 1 + L,=+2 +,G

( + ) , (6.8)

(ii) if +2 L,

( ) ( 1) = L,=+

( + ) . (6.9)

Proof. Consider fusion product ofV+LV+L and V+LV+L as irreducible VLVL-modules. By fusion rules in Proposition 3.2 and Theorem 2.10 in [ADL], as irreducible VL

VL-

modules, we have the following fusion product:

(V+L V+L) (V+L V+L) =V++L V++L. (6.10)Proof of (6.1):If + L= + Land+ L= + L, then + + L= + + L. Moreover,

V+L V+L = (0) + (1), V+L V+L = (0) + (1), and V++L V++L =( +0) + ( +1)asU-modules.

First we consider fusion ruleNU

(+ )( 0) ( 0)

, {0, 1}. TakeV =VL VL andU =U

in Proposition 2.9 in [ADL], then (6.10) implies

1 =NVLVL

V++LV++LV+LV+L V+LV+L

NU

(+ 0)+ (+ 1)

( 0) ( 0)

.

SoNU

(+ )( 0)( 0)

= 0or 1.

Now takeV =UandU=Vin Proposition 2.9 in [ADL], then

NU

(+ )( 0) ( 0)

NV

(+ )

V2+2LV+2L V2+2LV+2L

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LetPbe the projection ofV++L V++L to ( + )for = 0, 1.ThenPIis an nonzerointertwining operator in IU

( + )

( ) ( )

.SoNU

(+ )

( ) ( )

1 for = 0, 1. Since( )

and( )are isomorphicU

-modules,NU (+ )( ) ( ) 1 for= 0, 1.

By Proposition5.4and quantum dimensions in Proposition5.7we see that

qdimU(( ) ( )) = 4.

So we obtain (6.3).

Proof of (6.4): Now we have V+LV+L= ( ) , V+LV+L= ( ), V++LV++L=( + +), andV++L V++L= (+0) + (+1)asU-modules. TakeV =VL VLandU=U in Proposition 2.9 in [ADL], then (6.10) implies

1 =NVLVL V++LV++LV+LV+L V+LV+L NU(+ +)( ) ( ) .SoNU

(+ +)( ) ( )

1.

Using the condition + + L= + + L and the proof of (6.3) gives NU

(+ )

( ) ( )

1

for = 0, 1.Applying the formulaqdimU(( ) ( )) = 4again to obtain (6.4).

Proof of (6.5): Now we haveV+LV+L= ( ), V+LV+L= ( ), V++LV++L= ( + +) and V++LV++L= (+ +) asU-modules. From theproof of (6.4) we see that

1 =NVLVLV++LV++L

V+LV+L V+LV+L NU(+ +)( ) ( )

and

1 =NVLVL

V++LV++LV+LV+L V+LV+L

NU

(+ +)( ) ( )

.

Notice that as irreducibleU-modules,( )=( ). SoNU

(+ +)( ) ( )

1. The result

follows immediately by the fact thatqdimU( + +) =qdimU( + +) = 2.

Proof of (6.6): First by Proposition5.4and quantum dimensions in Proposition5.7

qdimU( ) ( ) =qdimU( ) qdimU( ) = 2|L/L|.By fusion rules in Proposition3.5, for any , S, NV+

2L

WV+

2 +

2L

VT+,2L

= 0only ifW = V

T,2L

for some L. SoNU

(1 )

( ) ( 0)

= NU

(2 )

( ) ( 0)

= 0 for any

1, 2, Lwith1 + L=+ L and= 0, 1. Using the classification of irreducible modulesin Proposition4.4we see that ( )

( ) = (11) + (22) for some 1, 2 L and

1, 2 {

0, 1}

. So we only need to determine1, 1,1and2.

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Proposition2.11,Remark4.5, and fusion product (6.7), we also haveNU

(+2 + 1)(+2 ) ( 1)

= 1

for all G. Clearly, for 1, 2 G with1= 2, +2 +1 +L= +2 +2+L. So

+

2 +1

1 and

+

2 +2

1 are not isomorphicU-modules. Also notice that forany G, ( + 2 )=( ). Assume|G|=n, then the number of isomorphism classes of

+2

+ 1

such thatNU

(+2 + 1)( ) ( 1)

= 1is equal ton.

By Proposition 2.11, Remark4.5, and fusion product (6.6), we see that NU

(+ )( ) ( 1)

= 1

for any T such that+ L= + +L. That is,NU

(+ )( ) ( 1)

= 1for all Tsatisfying thatcannot be written of the from +

2 + withG. The number of such T

isln. Also note that( )=( )as irreducibleU-modules for any, T . Thus thenumber of isomorphism classes of( +

) such that N

U(+ )( ) ( 1) = 1 is ln2 . Bycounting quantum dimensions, we see that( ) ( 1) =

G

+2

+ 1

+

T,=+2 +,G( + ) .

Proof of (6.9): By Proposition2.11,Remark4.5, and fusion product (6.6), we see that

NU

(+ )( ) ( 1)

= 1for any T such that+L= + +L.

Since +2 L, we see that every T satisfy such condition. Thus the number of such

is equal tol. Notice that( )=( )as irreducibleU-modules for any, T. Thus thenumber of isomorphism classes of( + )such thatNU

(+ )( ) ( 1)

= 1is l

2. Now the

quantum dimension of

T,+L=++L( + )is land the proof of (6.9) is complete.

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