New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.21(2015) 657–698.

An S

-symmetry of the Jacobi identity for intertwining operator algebras

Ling Chen

Abstract. We prove anS3-symmetry of the Jacobi identity for inter- twining operator algebras. Since this Jacobi identity involves the braid- ing and fusing isomorphisms satisfying the genus-zero Moore–Seiberg equations, our proof uses not only the basic properties of intertwining operators, but also the properties of braiding and fusing isomorphisms and the genus-zero Moore–Seiberg equations. Our proof depends heav- ily on the theory of multivalued analytic functions of several variables, especially the theory of analytic extensions.

Contents

1. Introduction 657

2. Preliminaries 658

3. S3-symmetry of the Jacobi identity 670

References 697

1. Introduction

Intertwining operator algebras were introduced and studied by Huang in [H1, H2]. In [C], the author studied intertwining operator algebras in a setting more general than [H2]. In particular, the duality properties, Ja- cobi identity, Moore–Seiberg equations, locality and some other properties of intertwining operator algebras were studied. For the background on in- tertwining operator algebras, we refer the reader to [H1, H2, C].

For vertex operator algebras, the Jacobi identity has an S₃-symmetry which corresponds to the obviousS3-symmetry of the Jacobi identity for Lie algebras [FHL]. For abelian intertwining operator algebras (see [DL2, DL1]), Guo [G] proved that the Jacobi identity for these algebras also has an S₃- symmetry. In this paper, we prove an S3-symmetry of the Jacobi identity for intertwining operator algebras introduced by Huang [H2] and studied by the author [C]. See Theorem 3.1 for the statement of thisS3-symmetry.

Received April 23, 2015.

2010Mathematics Subject Classification. 17B69, 81T40.

Key words and phrases. Intertwining operator algebras, Moore–Seiberg equations, Ja- cobi identity,S3-symmetry.

The author is supported by NSFC grant 11401559.

ISSN 1076-9803/2015

657

LING CHEN

TheS₃-symmetry in this general case is much more complicated but is also much more interesting and much deeper. Note that the Jacobi identity for general intertwining operator algebras in [H2] and [C] involves the braiding and fusing isomorphisms satisfying the genus-zero Moore–Seiberg equations.

The proof of theS3-symmetry in the present paper uses not only the prop- erties of the intertwining operators (for example, the skew-symmetry) but also the properties of braiding and fusing isomorphisms and the genus-zero Moore–Seiberg equations. In particular, our proof depends heavily on the theory of multivalued analytic functions of several variables, especially the theory of analytic extensions.

This paper is organized as follows. In Section 2, we review some pre- liminaries concerning the theory of intertwining operator algebras which we need to formulate and prove the main result of this paper. In Section 3, we prove an S3-symmetry of the Jacobi identity for intertwining operator algebras.

Acknowledgments. The author is very grateful to Professor Yi-Zhi Huang for his support, encouragement, many discussions on the paper and help with the exposition of the paper.

2. Preliminaries

In this section, we first recall some notations and facts in formal calculus and complex analysis (see [FLM, FHL, H2] for more details), then we re- view some definitions and properties in the theory of intertwining operator algebras in [H2, C]. These are necessary preliminaries for formulating and proving the main result of this paper.

In this paper, as in [FHL, H2, C], x, x₀, . . . are independent commuting formal variables. And for a vector space W and a formal variablex, as in [FHL, H2, C], we shall useW[x],W[x, x⁻¹],W[[x]],W[[x, x⁻¹]],W((x)) and W{x} to denote the spaces of all polynomials inx, all Laurent polynomials in x, all formal power series in x, all formal Laurent series in x, all formal Laurent series inx with finitely many negative powers and all formal series with arbitrary powers ofxinC, respectively. For series with more than one formal variables, we shall use similar notations. We shall use Res_xf(x) to denote the coefficient ofx⁻¹ inf(x) for anyf(x)∈W{x}. As in [FHL, H2, C],z, z₀, . . . ,are complex numbers,notformal variables.

Let

(2.1) δ(x) =X

n∈Z

xⁿ.

It has the following important property: For anyf(x)∈C[x, x⁻¹],

(2.2) f(x)δ(x) =f(1)δ(x).

Following [FHL, H2, C], we use the convention that negative powers of a binomial are to be expanded in nonnegative powers of the second summand

3-SYMMETRY OF THE JACOBI IDENTITY 659

so that, for example, (2.3)

x⁻¹₀ δ

x1−x2

x₀

=X

n∈Z

(x1−x2)ⁿ

xⁿ⁺¹₀ = X

m∈N, n∈Z

(−1)^m n

m

x⁻ⁿ⁻¹₀ x^n−m₁ x^m₂ . The following identities are often very useful:

(2.4) x⁻¹₁ δ

x₂+x₀ x1

=x⁻¹₂ δ

x₁−x₀ x2

,

(2.5) x⁻¹₀ δ

x1−x2

x0

−x⁻¹₀ δ

x2−x1

−x₀

=x⁻¹₂ δ

x1−x0

x2

.

As in [FHL, H2, C], C[x₁, x₂]_S is the ring of rational functions obtained by inverting the products of (zero or more) elements of the setS of nonzero homogenous linear polynomials in x₁ and x₂. Also, ι₁₂ is the operation of expanding an element of C[x1, x2]S, that is, a polynomial in x1 and x2

divided by a product of homogenous linear polynomials in x1 and x2, as a formal series containing at most finitely many negative powers ofx₂ (using binomial expansions for negative powers of linear polynomials involving both x₁ and x₂); similarly for ι₂₁, and so on. We need the following fact from [FHL].

Proposition 2.1. Consider a rational function of the form (2.6) f(x₀, x₁, x₂) = g(x₀, x₁, x₂)

x^r₀x^s₁x^t₂ , where g is a polynomial and r, s, t∈Z. Then (2.7) x⁻¹₁ δ

x2+x0

x₁

ι20(f|_x1=x0+x2) =x⁻¹₂ δ

x1−x0

x₂

ι10(f|_x2=x1−x0) and

(2.8) x⁻¹₀ δ

x1−x2

x₀

ι12(f|_x0=x1−x2)−x⁻¹₀ δ

x2−x1

−x₀

ι21(f|_x0=x1−x2)

=x⁻¹₂ δ

x₁−x₀ x2

ι₁₀(f|_x2=x1−x0).

As in [FHL, H2, C], the graded dual of a Z-graded, or more generally, C-graded, vector spaceW =`

nW_(n) is denoted by

(2.9) W⁰ =a

n

W_(n)^∗ .

For any z ∈ C, we use logz to denote the value log|z|+iargz with 0 ≤argz < 2π of logarithm of z. For two multivalued functions f1 and f2

on a region, f₁ and f₂ are equal if on any simply connected open subset of the region, any single-valued branch of f1 is equal to a single-valued branch of f2, and vice versa.

LING CHEN

Now we recall some basic notions and results in the theory of intertwining operator algebras. For the details of the definitions and properties of vertex operator algebras, their modules and intertwining operators, the reader is referred to [FHL, FLM, H2]. And for more details of the properties of intertwining operator algebras, the reader is referred to [H2, C].

Let (V, Y,1, ω) be a vertex operator algebra, and letW₁, W₂, W₃ be mod- ules ofV. The space of all intertwining operators of type _W^W3

1 W2

is denoted by ¯V_W^W3

1W2 instead of V_W^W3

1W2, for as in [C], the latter shall be used to denote a subspace of ¯V_W^W3

1W2 in the definition of intertwining operator algebra. The dimension of this vector space is denoted by ¯N_W^W3

1W2. It is the so-called fusion rule of the same type. Let Y be an intertwining operator of type _W^W3

1W2

. Given anyr ∈Z, as in [HL, H2, C], we define

(2.10) Ω_r(Y) :W₂⊗W₁ →W₃{x}

by

(2.11) Ωr(Y)(w₍₂₎, x)w₍₁₎ =e^xL(−1)Y(w₍₁₎, e^(2r+1)πix)w₍₂₎

forw₍₁₎∈W1,w₍₂₎ ∈W2. We have the following result proved in [HL]:

Proposition 2.2. For any Y ∈ V¯_W^W3

1W2, r ∈ Z, we have Ωr(Y) ∈ V¯_W^W3

2W1. Moreover,

(2.12) Ω−r−1(Ωr(Y)) = Ω_r(Ω−r−1(Y)) =Y.

In particular, the correspondence Y 7→Ω_r(Y) defines a linear isomorphism from V¯_W^W3

1W2 toV¯_W^W3

2W1, and we have

(2.13) N¯_W^W3

1W2 = ¯N_W^W3

2W1.

Now we recall the first definition of intertwining operator algebras in [H2]:

Definition 2.3 (Intertwining operator algebra). An intertwining operator algebra of central charge c∈C consists of the following data:

(1) a vector space

(2.14) W = a

a∈A

W^a

graded by a finite set A containing a special element e (graded by color);

(2) a vertex operator algebra structure of central charge c on W^e, and aW^e-module structure onW^a for each a∈ A;

(3) a subspace V_a^a3

1a2 of the space of all intertwining operators of type

W^a3 W^a1W^a2

for each triple a₁, a₂, a₃ ∈ A, with its dimension denoted byN_a^a3

1a2.

These data satisfy the following axioms for any a1, a2, a3, a4, a5, a6 ∈ A, w_(ai)∈W^ai,i= 1,2,3, andw⁰_(a

4)∈(W^a4)⁰:

3-SYMMETRY OF THE JACOBI IDENTITY 661

(1) The W^e-module structure on W^e is the adjoint module structure.

For any a ∈ A, the space V_ea^a is the one-dimensional vector space spanned by the vertex operator for the W^e-module W^a. For any a₁, a₂ ∈ Asuch thata₁6=a₂,V_ea^a2₁ = 0.

(2) Weight condition: For any a ∈ A and the corresponding module W^a =`

n∈CW_(n)^a graded by the action of L(0), there exists h_a∈R such thatW_(n)^a = 0 for n6∈ha+Z.

(3) Convergence properties: For any m ∈ Z+, a_i, b_j ∈ A, w_(ai) ∈W^ai, Y_i ∈ V_a^bi

ibi+1, i = 1, . . . , m, j = 1, . . . , m+ 1, w⁰_(b

1) ∈ (W^b1)⁰ and w_(bm+1)∈W^bm+1, the series

(2.15) hw⁰_(b

1),Y₁(w_(a1), x1)· · · Y_m(w_(am), xm)w_(bm+1)i_Wb1|_xn

i=e^nlogzi, i=1,...,m, n∈R

is absolutely convergent when|z₁|>· · ·>|z_m|>0 and its sum can be analytically extended to a multivalued analytic function on the region given byz_i 6= 0,i= 1, . . . , m,z_i 6=z_j,i6=j, such that for any set of possible singular points with eitherz_i = 0, z_i =∞ orz_i =z_j fori 6=j, this multivalued analytic function can be expanded near the singularity as a series having the same form as the expansion near the singular points of a solution of a system of differential equations with regular singular points. For anyY₁∈ V_a^a5

1a2 andY₂ ∈ V_a^a4

5a3, the series

(2.16) hw⁰_(a

4),Y₂(Y₁(w_(a1), x₀)w_(a2), x₂)w_(a3)i_W^a4|_xn

0=e^{nlog(z1−z2)}, xⁿ₂=e^nlogz2, n∈R

is absolutely convergent when|z₂|>|z₁−z₂|>0.

(4) Associativity: For anyY₁ ∈ V_a^a₁⁴_a5 and Y₂ ∈ V_a^a₂⁵_a3, there exist Y_3,i^a ∈ V_a^a

1a2 and Y_4,i^a ∈ V_aa^a4

3 fori= 1, . . . ,N_a^a

1a2N_aa^a4

3 and a∈ A, such that the (multivalued) analytic function

(2.17) hw_(a⁰

4),Y₁(w_(a1), x1)Y₂(w_(a2), x2)w_(a3)i_W^a4|_x1=z1,x2=z2

defined on the region|z₁|>|z₂|>0 and the (multivalued) analytic function

(2.18) X

a∈A N_a^a

1a2N_aa^a4

3

X

i=1

hw_(a⁰

4),Y_4,i^a (Y_3,i^a (w_(a1), x0)w_(a2), x2)w_(a3)i_W^a4

x0=z1−z₂,x2=z2

defined on the region|z₂|>|z₁−z₂|>0 are equal on the intersection

|z₁|>|z₂|>|z₁−z₂|>0.

(5) Skew-symmetry: The restriction of Ω−1 to V_a^a₁³_a2 is an isomorphism fromV_a^a3

1a2 toV_a^a3

2a1.

Remark 2.4. The skew-symmetry isomorphisms Ω−1(a1, a2;a3) for alla1, a2, a3 ∈ A

LING CHEN

give an isomorphism

(2.19) Ω−1 : a

a1,a2,a3∈A

V_a^a3

1a2 → a

a1,a2,a3∈A

V_a^a3

1a2,

which, as in [H2, C], is still called theskew-symmetry isomorphism. In this paper, as in [H2, C], we shall omit subscript −1 in Ω₋₁ for simplicity and denote it by Ω.

We denote the intertwining operator algebra just defined by (W,A,{V_a^a₁³_a2},1, ω)

or simply byW.

Next, as in [H2, C], we give the two linear maps corresponding to the multiplication and iterates of intertwining operators, respectively. Let

a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5⊗ V_a^a5

2a3

−P→(Hom(W ⊗W ⊗W, W)){x₁, x₂} (2.20)

Z 7→P(Z)

be the linear map defined using products of intertwining operators as follows:

For

(2.21) Z ∈ a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3,

the element P(Z) to be defined is a linear map from W ⊗ W ⊗W to W{x₁, x2}. We denote the image of w1 ⊗w2 ⊗w3 under this map by (P(Z))(w₁, w₂, w₃;x₁, x₂) for any w₁, w₂, w₃ ∈ W. Then we define P by linearity and by

(2.22) (P(Y1⊗ Y₂))(w_(a6), w_(a7), w_(a8);x1, x2)

=

(Y₁(w_(a6), x1)Y₂(w_(a7), x2)w_(a8), a6 =a1, a7 =a2, a8 =a3,

0, otherwise,

for a₁, . . . , a₈ ∈ A, Y₁ ∈ V_a^a4

1a5, Y₂ ∈ V_a^a5

2a3, and w_(a6) ∈ W^a6, w_(a7) ∈W^a7, w_(a8)∈W^a8. So we have an isomorphism

(2.23) P˜ :

a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3

KerP −→P





a

a1,a2,a3,a4,a5∈A

V_a^a₁⁴_a5 ⊗ V_a^a₂⁵_a3





3-SYMMETRY OF THE JACOBI IDENTITY 663

which makes the following diagram commute:

(2.24) a

a1,a2,a3,a4,a5∈A

V_a^a₁⁴_a5 ⊗ V_a^a₂⁵_a3

πP

P //P





a

a1,a2,a3,a4,a5∈A

V_a^a₁⁴_a5 ⊗ V_a^a₂⁵_a3





a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3

KerP

P˜ 55

,

whereπ_P is the corresponding canonical projective map. As in [C], we also denote πP(Z) by [Z]_P orZ+ KerP forZ ∈`

a1,a2,a3,a4,a5∈AV_a^a4

1a5 ⊗ V_a^a5

2a3

when there is no ambiguity. The second linear map is a

a1,a2,a3,a4,a5∈A

V_a^a₁⁵_a2⊗ V_a^a₅⁴_a3 −→^I (Hom(W ⊗W ⊗W, W)){x₀, x2} (2.25)

Z 7→I(Z)

defined using iterates of intertwining operators as follows: For

(2.26) Z ∈ a

a1,a2,a3,a4,a5∈A

V_a^a5

1a2 ⊗ V_a^a4

5a3,

the element I(Z) to be defined is a linear map from W ⊗ W ⊗ W to W{x₀, x₂}. We denote the image of w₁ ⊗w₂ ⊗w₃ under this map by (I(Z))(w1, w2, w3;x0, x2) for any w1, w2, w3 ∈ W. Then we define I by linearity and by

(2.27) (I(Y₁⊗ Y₂))(w_(a6), w_(a7), w_(a8);x₀, x₂)

=

(Y₂(Y₁(w_(a6), x0)w_(a7), x2)w_(a8), a6 =a1, a7 =a2, a8 =a3,

0, otherwise,

for a₁, . . . , a₈ ∈ A, Y₁ ∈ V_a^a5

1a2, Y₂ ∈ V_a^a4

5a3, and w_(a6) ∈ W^a6, w_(a7) ∈W^a7, w_(a8)∈W^a8. Therefore we have an isomorphism

(2.28)

˜I:

a

a1,a2,a3,a4,a5∈A

V_a^a5

1a2 ⊗ V_a^a4

5a3

KerI −→I





a

a1,a2,a3,a4,a5∈A

V_a^a₁⁵_a2 ⊗ V_a^a₅⁴_a3





LING CHEN

which makes the following diagram commute:

(2.29) a

a1,a2,a3,a4,a5∈A

V_a^a₁⁵_a2 ⊗ V_a^a₅⁴_a3

πI

I //I





a

a1,a2,a3,a4,a5∈A

V_a^a₁⁵_a2 ⊗ V_a^a₅⁴_a3





a

a1,a2,a3,a4,a5∈A

V_a^a5

1a2 ⊗ V_a^a4

5a3

KerI

˜I 55

,

where π_I is the corresponding canonical projective map. As in [C], we also denote πI(Z) by [Z]_I or Z + Ker I for Z ∈ `

a1,a2,a3,a4,a5∈AV_a^a5

1a2 ⊗ V_a^a4

5a3

when there is no ambiguity.

The two linear mapsPand Iare called themultiplication of intertwining operators and theiterates of intertwining operators, respectively.

Moreover, in [H2, C], Huang and the author obtained isomorphisms from the associativity of intertwining operator algebras and from the skew-symme- try isomorphism Ω. The fusing isomorphism which we obtained from the associativity of intertwining operator algebras is a map

(2.30) F :

a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3

KerP −→

a

a1,a2,a3,a4,a5∈A

V_a^a5

1a2 ⊗ V_a^a4

5a3

KerI determined by linearity and by

(2.31) F(Y₁⊗ Y₂+ KerP) =X

a∈A N_a^a

1a2N_aa^a4₃

X

i=1

Y_3,i^a ⊗ Y_4,i^a + KerI fora₁,· · · , a₅∈ A,Y₁∈ V_a^a4

1a5 and Y₂∈ V_a^a5

2a3, where

(2.32) {Y_3,i^a ∈ V_a^a_1a2,Y_4,i^a ∈ V_aa^a4₃ |i= 1,· · ·,N_a^a_1a2N_aa^a4₃, a∈ A}

is a set of intertwining operators satisfying that for anyw1, w2, w3∈W and w⁰ ∈W⁰,

(2.33) X

a∈A N_a^a

1a2N_aa^a4₃

X

i=1

hw⁰,(I(Y_3,i^a ⊗ Y_4,i^a ))

(w1, w2, w3;x0, x2)i_W

xⁿ₀=e^{nlog(z1−z2)},xⁿ₂=e^nlogz2

is equal to

(2.34) hw⁰,(P(Y1⊗ Y₂))(w1, w2, w3;x1, x2)i_W|_xn

1=e^nlogz1,xⁿ₂=e^nlogz2

on the region

3-SYMMETRY OF THE JACOBI IDENTITY 665

S1 ={(z₁, z2)∈C² |Rez1 >Rez2>Re(z1−z2)>0,

Imz1>Imz2>Im(z1−z2)>0}.

It was also proved that πP



 a

a5∈A

V_a^a4

1a5⊗ V_a^a5

2a3





F(a₁,a2,a3,a4)

−−−−−−−−−→πI



 a

a5∈A

V_a^a5

1a2 ⊗ V_a^a4

5a3

 (2.35) 

Y₁⊗ Y₂+ KerP7−→ F(Y₁⊗ Y₂+ KerP) is an isomorphism for any a₁,· · ·, a₄ ∈ A, where Y₁ ∈ V_a^a4

1a5, Y₂ ∈ V_a^a5

2a3. These isomorphisms are also calledfusing isomorphisms. The isomorphisms we obtained from Ω and its inverse are linear isomorphic maps:

(2.36) Ω˜⁽¹⁾,(Ωg⁻¹)⁽¹⁾ : a

a1,a2,a3,a4,a5∈A

V_a^a₁⁵_a2⊗ V_a^a₅⁴_a3

KerI −→

a

a1,a2,a3,a4,a5∈A

V_a^a₂⁵_a1⊗ V_a^a₅⁴_a3

KerI defined by linearity and by

Ω˜⁽¹⁾(Y₁⊗ Y₂+ KerI) = Ω(Y1)⊗ Y₂+ KerI, (2.37)

(Ωg⁻¹)⁽¹⁾(Y₁⊗ Y₂+ KerI) = Ω⁻¹(Y₁)⊗ Y₂+ KerI (2.38)

fora1, . . . , a5∈ A,Y₁ ∈ V_a^a5

1a2,Y₂∈ V_a^a4

5a3; (2.39) Ω˜⁽²⁾,(Ωg⁻¹)⁽²⁾ :

a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5⊗ V_a^a5

2a3

KerP −→

a

a1,a2,a3,a4,a5∈A

V_a^a5

2a3⊗ V_a^a4

5a1

KerI defined by linearity and by

Ω˜⁽²⁾(Y₁⊗ Y₂+ KerP) =Y₂⊗Ω(Y₁) + KerI, (2.40)

(Ωg⁻¹)⁽²⁾(Y₁⊗ Y₂+ KerP) =Y₂⊗Ω⁻¹(Y₁) + KerI (2.41)

fora1, . . . , a5∈ A,Y₁ ∈ V_a^a4

1a5,Y₂∈ V_a^a5

2a3; (2.42) Ω˜⁽³⁾,(Ωg⁻¹)⁽³⁾ :

a

a1,a2,a3,a4,a5∈A

V_a^a5

1a2⊗ V_a^a4

5a3

KerI −→

a

a1,a2,a3,a4,a5∈A

V_a^a4

3a5⊗ V_a^a5

1a2

KerP defined by linearity and by

Ω˜⁽³⁾(Y₁⊗ Y₂+ KerI) = Ω(Y2)⊗ Y₁+ KerP, (2.43)

(Ωg⁻¹)⁽³⁾(Y₁⊗ Y₂+ KerI) = Ω⁻¹(Y₂)⊗ Y₁+ KerP (2.44)

LING CHEN

fora₁, . . . , a₅∈ A,Y₁ ∈ V_a^a5

1a2,Y₂∈ V_a^a4

5a3; (2.45) Ω˜⁽⁴⁾,(Ωg⁻¹)⁽⁴⁾ :

a

a1,a2,a3,a4,a5∈A

V_a^a₁⁴_a5⊗ V_a^a₂⁵_a3

KerP −→

a

a1,a2,a3,a4,a5∈A

V_a^a₁⁴_a5⊗ V_a^a₃⁵_a2

KerP defined by linearity and by

Ω˜⁽⁴⁾(Y₁⊗ Y₂+ KerP) =Y₁⊗Ω(Y₂) + KerP, (2.46)

(Ωg⁻¹)⁽⁴⁾(Y₁⊗ Y₂+ KerP) =Y₁⊗Ω⁻¹(Y₂) + KerP (2.47)

for a₁, . . . , a₅ ∈ A, Y₁ ∈ V_a^a4

1a5, Y₂ ∈ V_a^a5

2a3. And these isomorphisms have relations:

(2.48) ( ˜Ω⁽²⁾)⁻¹ = (Ωg⁻¹)⁽³⁾, ((Ωg⁻¹)⁽²⁾)⁻¹= ˜Ω⁽³⁾, (2.49) ( ˜Ω⁽¹⁾)⁻¹ = (Ωg⁻¹)⁽¹⁾, ( ˜Ω⁽⁴⁾)⁻¹ = (Ωg⁻¹)⁽⁴⁾.

The above isomorphisms are not independent, we proved the following relations in [C]:

Theorem 2.5. The above isomorphisms satisfy the following genus-zero Moore–Seiberg equations:

F ◦Ω˜⁽³⁾◦ F = ˜Ω⁽¹⁾◦ F ◦Ω˜⁽⁴⁾, (2.50)

F ◦(Ωg⁻¹)⁽³⁾◦ F = (Ωg⁻¹)⁽¹⁾◦ F ◦(Ωg⁻¹)⁽⁴⁾. (2.51)

Using the fusing isomorphism and the isomorphism ˜Ω⁽¹⁾, we deduced a braiding isomorphism

(2.52) B=F⁻¹◦Ω˜⁽¹⁾◦ F : a

a1,a2,a3,a4,a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3

KerP −→

a

a1,a2,a3,a4,a5∈A

V_a^a4

2a5 ⊗ V_a^a5

1a3

KerP .

Moreover, we get an isomorphism πP



 a

a5∈A

V_a^a4

1a5 ⊗ V_a^a5

2a3





B(a₁,a2,a3,a4)

−−−−−−−−−→πP



 a

a5∈A

V_a^a4

2a5 ⊗ V_a^a5

1a3

 (2.53) 

Y₁⊗ Y₂+ KerP7−→ B(Y₁⊗ Y₂+ KerP) for any a1,· · · , a4 ∈ A, where Y₁ ∈ V_a^a4

1a5, Y₂ ∈ V_a^a5

2a3. We also call these isomorphismsbraiding isomorphisms.

Before formulating the Jacobi identity for intertwining operator algebras, we need to recall the specifics of one more property, which is about certain special multivalued analytic functions, and were discussed in [H2, C].

3-SYMMETRY OF THE JACOBI IDENTITY 667

First we consider some simply connected regions in C². Cutting the re- gions |z₁| > |z₂| > 0 and |z₂| > |z₁| > 0 along the intersections of these regions with

{(z₁, z₂)∈C²|z₁ ∈[0,+∞)} ∪ {(z₁, z₂)∈C² |z₂∈[0,+∞)}, we obtain two simply connected regions, which, as in [H2, C], are denoted by R1 and R2, respectively. Also, let R3 be the simply connected region obtained by cutting the region|z₂|>|z₁−z₂|>0 along the intersection of this region with

{(z₁, z2)∈C² |z2∈[0,+∞)} ∪ {(z₁, z2)∈C² |z1−z2 ∈[0,+∞)}, and let R₄ be the simply connected region obtained by cutting the region

|z₁|>|z₁−z2|>0 along the intersection of this region with

{(z₁, z₂)∈C² |z₁∈[0,+∞)} ∪ {(z₁, z₂)∈C² |z₂−z₁ ∈[0,+∞)}.

Then we consider some special multivalued analytic functions on (2.54) M² ={(z₁, z2)∈C²|z1, z2 6= 0, z₁ 6=z2}.

For a₁, a₂, a₃, a₄ ∈ A, as in [H2, C], we let G^a1,a2,a3,a4 be the set of multi- valued analytic functions onM² with a choice of a single-valued branch on the regionR₁ satisfying the following property: Any branch of

f(z₁, z₂)∈G^a1,a2,a3,a4

on the regions |z₁| > |z₂| > 0, |z₂| > |z₁| > 0 and |z₂| > |z₁ −z₂| > 0, respectively, can be expanded as

X

a∈A

z₁^{ha4−ha1−ha}z^h₂^{a−ha2−ha3}F_a(z₁, z₂), (2.55)

X

a∈A

z₂^{ha4−ha2−ha}z^h₁^{a−ha1−ha3}G_a(z₁, z₂) (2.56)

and

X

a∈A

z^h₂^{a4−ha−ha3}(z₁−z₂)^{ha−ha1−ha2}H_a(z₁, z₂), (2.57)

respectively, where for a∈ A,

F_a(z₁, z₂)∈C[[z₂/z₁]][z₁, z₁⁻¹, z₂, z₂⁻¹], (2.58)

G_a(z₁, z₂)∈C[[z₁/z₂]][z₁, z₁⁻¹, z₂, z₂⁻¹] (2.59)

and

(2.60) H_a(z₁, z₂)∈C[[(z₁−z₂)/z₂]][z₂, z₂⁻¹, z₁−z₂,(z₁−z₂)⁻¹].

The chosen single-valued branch onR1 of an element ofG^a1,a2,a3,a4 is called thepreferred branch onR₁. As in [C], we use the nonempty simply connected regions

LING CHEN

S₁ ={(z₁, z₂)∈C² |Rez₁ >Rez₂>Re(z₁−z₂)>0,

Imz₁ >Imz₂ >Im(z₁−z₂)>0}

and

S₂ ={(z₁, z₂)∈C² |Rez₂ >Rez₁>Re(z₂−z₁)>0,

Imz₂ >Imz₁ >Im(z₂−z₁)>0}

to determine other special branches of an element of G^a1,a2,a3,a4 on R₂,R₃ and R4 related to the preferred branch on R1. Firstly, the restriction of the preferred branch onR₁of an element ofG^a1,a2,a3,a4to the regionS₁ ⊂R₁∩R₃ gives a single-valued branch of the element on R3, which is then called the preferred branch on R₃. Secondly, the restriction of the preferred branch on R1 to the region S1 ⊂R1∩R4 also gives a single-valued branch of the element on R4, which is then called the preferred branch on R4. Moreover, the restriction of the preferred branch on R₄ to the region S₂ ⊂ R₄∩R₂ then gives a single-valued branch of the element on R2 and we call it the preferred branch onR₂. It was verified in [H2, C] thatG^a1,a2,a3,a4 is a vector space.

For any element ofG^a1,a2,a3,a4, the preferred branches of this function on R1,R2 andR3 give formal series in

(2.61) a

a∈A

x^h₁^{a4−ha1−ha}x^h₂^{a−ha2−ha3}C[[x2/x1]][x1, x⁻¹₁ , x2, x⁻¹₂ ],

(2.62) a

a∈A

x^h₂^{a4−ha2−ha}x^h₁^{a−ha1−ha3}C[[x₁/x₂]][x₁, x⁻¹₁ , x₂, x⁻¹₂ ] and

(2.63) a

a∈A

x^h₂^{a4−ha−ha3}x^h₀^{a−ha1−ha2}C[[x₀/x₂]][x₀, x⁻¹₀ , x₂, x⁻¹₂ ], respectively, which induce linear maps

G^a1,a2,a3,a4 −^ι−¹²→ a

a∈A

x^h₁^{a4−ha1−ha}x^h₂^{a−ha2−ha3}C[[x2/x1]][x1, x⁻¹₁ , x2, x⁻¹₂ ], (2.64)

G^a1,a2,a3,a4 −^ι−²¹→ a

a∈A

x^h₂^{a4−ha2−ha}x^h₁^{a−ha1−ha3}C[[x1/x2]][x1, x⁻¹₁ , x2, x⁻¹₂ ], (2.65)

G^a1,a2,a3,a4 −^ι−²⁰→ a

a∈A

x^h₂^{a4−ha−ha3}x^h₀^{a−ha1−ha2}C[[x0/x2]][x0, x⁻¹₀ , x2, x⁻¹₂ ], (2.66)

generalizingι12,ι21andι20discussed at the beginning of this section. These maps are injective because analytic extensions are unique.