Bimodule and twisted representation of vertex operator algebras

0.18 MB·English

by  Qifen Jiang

Bimodule and twisted representation of vertex
operator algebras
Qifen Jiang1
Department of Mathematics, Shanghai Jiaotong University
Shanghai 200240, China
Xiangyu Jiao2
Department of Mathematics, East China Normal University
5 Shanghai 200241, China
1
0
2
Abstract In this paper, for a vertex operator algebra V with an automorphism g of
n
a order T, an admissible V-module M and a fixed nonnegative rational number n ∈ 1Z ,
J T +
we construct an Ag,n(V)-bimodule Ag,n(M) and study its some properties, discuss the
9
connections between bimodule A (M) and intertwining operators. Especially, bimod-
g,n

] ule A (M) is a natural quotient of A (M) and there is a linear isomorphism
T g,n−1 g,n
T
R between the space IMk of intertwining operators and the space of homomorphisms
MMj
. Hom (A (M) ⊗ Mj(s),Mk(t)) for s,t ≤ n,Mj,Mk are g-twisted V mod-
h Ag,n(V) g,n Ag,n(V)
t ules, if V is g-rational.
a
m Keywords Bimodule, g-twisted module, vertex operator algebra, intertwining opera-
[ tor.

1 MSC (2010): 17B69.
v Remark: It has been accepted for publication in SCIENCE CHINA Mathematics.
9
3
0
2 1 Introduction
0
.
1
It is known that twisted representations are the main ingredients in orbifold conformal
0
5 field theory (refer [DHVW1-DHVW2, LJ1-LJ2, FLM, DM, DLM1, DLM3, DLM5, DLM6
1 and MT]etc). They play a fundamental role in the construction of the moonshine vertex
:
v operator algebra V♮ [FLM] and other orbifold vertex operator algebras [DGM]. Many of
i
X them have been studied in the literature (refer [ABD, DJ2, DLM3, DJM4, HY, MT]). In
r [DLM3, Z], associative algebra A (V) was constructed for a given vertex operator algebra
a g
V withanautomorphismg offiniteorderT. Thisishelpfultoinvestigate abstractorbifold
models. A (V), which was constructed in [DLM5] for nonnegative numbers n ∈ 1Z , is
g,n T +
a generalization of the algebra A (V), where A (V) = A (V) (where A (V) coincides
g g g,0 id,0
with A(V) in[Z], andA (V) isexactly A (V) in [DLM4]). It was shown that A (V)
id,n n g,n−1
T
is a natural quotient of A (V).
g,n
On the other hand, an A (V)-bimodule A (M) for any V-module M was also intro-
n n
duced in [FZ, DR] to deal with the intertwining operators and fusion rules. Moreover
1Supported by NSFC grants 11101269and 11431010.
E-mail:[email protected]
2Supported by NSFC grant 11401213
E-mail:[email protected]
1
an important result in [FZ, L2, DR] is that there is an isomorphism between the space
of intertwining operators among irreducible V-modules and a certain space constructed
from bimodules for A (V). By generalizing the bimodule theory in [FZ], a sequence
id,n
of bimodules of A (V)-A (V) in [DJ1, DJ2] were studied, which were denoted by
g,n g,m
A (V).
g,n,m
Motivated by [DR, DJ1, DJ2], for any g, n as above, in this paper we construct
an A (V)-bimodule A (M) for any admissible V-module M such that A (M)
g,n g,n g,n−1
T
is a natural quotient of A (M). The structure of those bimodules are discussed. We
g,n
also consider the connections between bimodule A (M) and intertwining operators.
g,n
Moreover, it is established that if V is g-rational then there is a linear isomorphism
between the space IMk of intertwining operators and the space of homomorphism
MMj
Hom (A (M)⊗ Mj(s),Mk(t))
Ag,n(V) g,n Ag,n(V)
for s,t ≤ n,Mj,Mk are g-twisted V modules.
The paper is organized as follows. We recall the construction of associative algebras
A (V) and some results from [DLM3]-[DLM5] in Section 2. In Section 3,we give the con-
g,n
struction of A (M) and study its structure. In section 4, we discuss the relationbetween
g,n
A (M) and intertwining operators. As in [FZ, L2, DR], we obtain the isomorphism from
g,n
the space IMk of intertwining operators to Hom (A (M)⊗ Mj(s),Mk(t))
MMj Ag,n(V) g,n Ag,n(V)
if V is g-rational.
2 The associative algebras A (V )
g,n
Let V = (V,Y,1,ω) be a vertex operator algebra ([B], [FLM], [LL]) with an automor-
phism g of finite order T. Then V has the decomposition of eigenspace with respect to
the action g:
V = Vr
r∈Z/TZ
M
where Vr = {v ∈ V|gv = e−2πTirv}. Firstly, we recall definitions of modules of different
types from [DLM3, FLM, FFR, D].
Definition 2.1. A weak g-twisted V-module M is a vector space equipped with a linear
map
YM(·,z) :V → (EndM)[[zT1,z−T1]]
v 7→ Y (v,z) = v z−n−1 (v ∈ EndM)
M n n
n∈Q
X
2
which satisfies the following conditions for all 0 ≤ r ≤ T −1,u ∈ Vr, v ∈ V, w ∈ M.
Y (u,z) = u z−n−1;
M n
n∈r+Z
XT
u w = 0 for n ≫ 0;
n
Y (1,z) = id ;
M M
z −z z −z
z−1δ( 1 2)Y (u,z )Y (v,z )−z−1δ( 2 1)Y (v,z )Y (u,z )
0 z M 1 M 2 0 −z M 2 M 1
0 0
z −z z −z
= z2−1δ( 1z 0)( 1z 0)−TrYM(Y(u,z0)v,z2).
2 2
Definition 2.2. An (ordinary) g-twisted V-module is a weak g-twisted V-module M which
carries a C-grading induced by the spectrum of L(0) where L(0) is a component operator
of
Y (ω,z) = L(n)z−n−2.
M
n∈Z
X
That is
M = ⊕ M ,
λ∈C λ
where M = {w ∈ M|L(0)w = λw}. Moreover one requires that M is finite dimensional
λ λ
and for fixed λ, Mn+λ = 0 for all small enough integers n.
T
Definition 2.3. An admissible g-twisted V-module is a weak g-twisted V-module M which
carries a 1Z -grading
T +
M = ⊕ M(n)
n∈T1Z+
that satisfies the following
v M(n) ⊂ M(n+wtv −m−1)
m
for homogeneous v ∈ V.
Remark 2.4. Assume M is an irreducible addimisible g-twisted V-module, one knows
that M = ⊕n∈T1Z+Mn+h for some h ∈ C such that Mh 6= 0, where h is the conformal
weight of M. We always assume M(n) = M .
n+h
Next we recall some results of the associative algebra A (V) from [DLM3-DLM5].
g,n
Also see [Z].
Fixn = l+ i ∈ 1Z withl anonnegativeinteger and0 ≤ i ≤ T−1. For0 ≤ r ≤ T−1,
T T +
define
1, if i ≥ r,
δ (r) = .
i
( 0, if i < r.
Let O (V) be the linear span of all u◦ v and L(−1)u+L(0)u where for homogeneous
g,n g,n
u ∈ Vr and v ∈ V,
(1+z)wtu+l−1+δi(r)+Tr
u◦ v = Res Y(u,z)v .
g,n z z2l+δi(r)+δi(T−r)+1
3
We also define a second product ∗ on V for u ∈ Vr and v ∈ V as follows:
g,n
l m+l (1+z)wtu+l
u∗ v = (−1)m Res Y(u,z) v
g,n l z zl+m+1
m=0 (cid:18) (cid:19)
X
l ∞
m+l wtu+l
= (−1)m u v.
j−m−l−1
l j
m=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19)
XX
if r = 0 and u∗ v = 0 if r > 0.
g,n
Define A (V) to be the quotient V/O (V). Then A (V) = A (V) and A (V) =
g,n g,n g,0 g id,n
A (V) have already been defined and studied in [DLM3] and [DLM4] respectively.
n
The following theorem summarizes the main results of [DLM5].
Theorem 2.5. Let V be a vertex operator algebra and g an automorphism of V of finite
order T. Let M = ⊕m∈T1Z+M(m) be an admissible g-twisted V-module. Let n ∈ T1Z+.
Then
(1) A (V) is an associative algebra whose product is induced by ∗ .
g,n g,n
(2) Theidentitymap onV inducesanalgebraepimorphismfromA (V) to A (V).
g,n g,n−1/T
(3) Let W be a weak g-twisted V module and set
Ω (W) = {w ∈ W|u w = 0,u ∈ V,k > n}.
n wtu−1+k
Then Ω (W) is an A (V)-module such that v+O (V) acts as o(v) = v for homo-
n g,n g,n wtv−1
geneous v.
(4) Each M(m) for m ≤ n is an A (V)-submodule of Ω (W). Furthermore, M is
g,n n
irreducible if and only if each M(m) is an irreducible A (V)-module.
g,n
(5) The linear map ϕ from V to V defined by ϕ(u) = eL(1)(−1)L(0)u for u ∈ V induces
an anti-isomorphism from Ag,n(V) to Ag−1,n(V).
3 Bimodules A (M) of A (V )
g,n g,n
Motivated by the ideas of A(V)-bimodule A(M) and A (V)-bimodule A (M) from
n n
[Z] and [DR] for any nonnegative integer n respectively, we will construct and study
A (V)-bimodule A (M) for n ∈ 1Z in this section.
g,n g,n T +
Let V be same as vertex operator algebra in section 2 and M be an admissible V-
module, O (M) be the linear span of u ◦ w where for homogeneous u ∈ Vr and
g,n g,n
w ∈ M,
(1+z)wtu+l−1+δi(r)+Tr
u◦ w = Res Y (u,z)w .
g,n z M z2l+δi(r)+δi(T−r)+1
We also define a left bilinear product ∗ for u ∈ Vr and w ∈ M :
g,n
l m+l (1+z)wtu+l
u∗ w = (−1)m Res Y (u,z) w (3.1)
g,n l z M zl+m+1
m=0 (cid:18) (cid:19)
X
l ∞
m+l wtu+l
= (−1)m u w,
j−m−l−1
l j
m=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19)
XX
4
if r = 0 and u∗ w = 0 if r > 0 and a right bilinear product
g,n
l m+l (1+z)wtu+m−1
w ∗ u = (−1)l Res Y (u,z) w (3.2)
g,n l z M zl+m+1
m=0 (cid:18) (cid:19)
X
l ∞
m+l wtu+m−1
= (−1)l u w.
j−m−l−1
l j
m=0 j=0 (cid:18) (cid:19)(cid:18) (cid:19)
XX
if r = 0 and w ∗ u = 0 if r > 0.
g,n
Set A (M) = M/O (M). Onecan see that A (M) isthe A (V)-bimoduleA (M)
g,n g,n 1,n n n
studied in [DR]. Moreover, A (M) is the A(V)-bimodule A(M) studied in [FZ].
1,0
Lemma 3.1. (1) Assume that u ∈ Vr is homogeneous, w ∈ M and m ≥ k ≥ 0. Then
(1+z)wtu+l−1+δi(r)+Tr+k
Res Y (u,z)w ∈ O (M).
z M z2l+δi(r)+δi(T−r)+1+m g,n
(2) For homogeneous u ∈ V0 and w ∈ M,
u∗ w −w ∗ u−Res Y (u,z)w(1+z)wtu−1 ∈ O (M).
g,n g,n z M g,n
Proof: The proof of (1) is similar to that of Lemma 2.1.2 of [Z].
For (2), it follows directly from the definitions (3.1) and (3.2) and using the following
result in the appendix of [DLM4]:
l m+l (−1)m(1+z)l+1 −(−1)l(1+z)m
= 1.
l zl+m+1
m=0(cid:18) (cid:19)
X
(cid:3)
Thus we can obtain the result in (2).
Lemma 3.2.
(L(−1)+L(0))V ∗ M ⊂ O (M),M ∗ (L(−1)+L(0))V ⊂ O (M)
g,n g,n g,n g,n
Proof: By the definition of ∗ , we only need to prove the case u ∈ V0,w ∈ M. From
g,n
(3.1), we have
l m+l (1+z)wtu+1+l
(L(−1)u)∗ w = (−1)m Res Y (L(−1)u,z)w
g,n l z M zl+m+1
m=0 (cid:18) (cid:19)
X
l m+l d (1+z)wtu+1+l
= (−1)m+1 Res Y (u,z)w ( )
l z M dz zl+m+1
m=0 (cid:18) (cid:19)
X
5
l m+l (wtu+1+l)(1+z)wtu+l
= (−1)m+1 Res Y (u,z)w[
z M
l zl+m+1
m=0 (cid:18) (cid:19)
X
(l +m+1)(1+z)wtu+1+l
− ],
zl+m+2
l m+l (1+z)wtu+l
(L(0)u)∗ w = (−1)m Res Y (L(0)u,z)w
g,n z M
l zl+m+1
m=0 (cid:18) (cid:19)
X
l m+l (1+z)wtu+l
= (−1)m wtuRes Y (u,z)w
l z M zl+m+1
m=0 (cid:18) (cid:19)
X
Thus
(L(−1)+L(0))u∗ w
g,n
l m+l (−l−1)(1+z)wtu+l
= (−1)m Res Y (u,z)w[
l z M zl+m+1
m=0 (cid:18) (cid:19)
X
(l +m+1)(1+z)wtu+1+l
+ ]
zl+m+2
l m+l (mz +l+m+1)(1+z)wtu+l
= (−1)m Res Y (u,z)w .
z M
l zl+m+2
m=0 (cid:18) (cid:19)
X
Using Lemma 2.2 in [DLM4]:
l m+l (mz +l +m+1)(1+z)wtu+l 2l+1 2l+1
(−1)m = (−1)l ,
l zl+m+2 l z2l+2
m=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
we obtain
2l+1 (2l+1)(1+z)wtu+l
(L(−1)+L(0))u∗ w = (−1)l Res Y (u,z)w .
g,n z M
l z2l+2
(cid:18) (cid:19)
Since u ∈ V0,δ (r) = 1 and δ (T − r) = 0, it is clear that the right-hand side lies in
i i
O (M). So
g,n
(L(−1)+L(0))V ∗ M ⊂ O (M)
g,n g,n
For the second containment,we only need to use this result and Lemma 3.1(2). This
(cid:3)
completes the proof.
Lemma 3.3. (1) O (V)∗ M ⊂ O (M),M ∗ O (V) ⊂ O (M),
g,n g,n g,n g,n g,n g,n
(2) V ∗ O (M) ⊂ O (M),O (M)∗ V ⊂ O (M).
g,n g,n g,n g,n g,n g,n
Proof: By Lemma 3.2, it remains to prove that (u◦ v)∗ w,w∗ (u◦ v),u∗
g,n g,n g,n g,n g,n
(v ◦ w),(v◦ w)∗ u ∈ O (M) for u,v ∈ V and w ∈ M.
g,n g,n g,n g,n
6
(1) Since Vr ∗ M and M ∗ Vr equal zero for r 6= 0, we only need to consider the
g,n g,n
case u ∈ Vr and v ∈ VT−r. It is not difficult to prove (u◦ v)∗ w ⊂ O (M) according
g,n g,n g,n
to Lemma 3.4(1) in [DR], we omit it. We now prove w∗ (u◦ v) ∈ O (M).
g,n g,n g,n
Let u ∈ Vr,v ∈ VT−r and w ∈ M, we have
wtu+l−1+δ (r)+ r
(u◦ v) = i T (u v).
g,n j j−2l−1−δi(r)−δi(T−r)
j≥0 (cid:18) (cid:19)
X
Then it follows Lemma 3.1(2) that
w ∗ (u◦ v)
g,n g,n
wtu+l−1+δ (r)+ r
≡ − i T Res Y (u v,z )w×
j z2 M j−2l−1−δi(r)−δi(T−r) 2
j≥0 (cid:18) (cid:19)
X
(1+z )wtu+wtv+2l+δi(r)+δi(T−r)−1−j(modO (M))
2 g,n
(1+z2 +z0)wtu+l−1+δi(r)+Tr
= −Res Res Y (Y(u,z )v,z )w ×
z2 z0 M 0 2 z2l+δi(r)+δi(T−r)+1
0
(1+z2)wtv+l+δi(T−r)−Tr(modOg,n(M))
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
= −Res Res Y (u,z )Y (v,z )w
z1 z2 M 1 M 2 (z1 −z2)2l+δi(r)+δi(T−r)+1
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
+Res Res Y (v,z )Y (u,z )w
z2 z1 M 2 M 1 (−z2 +z1)2l+δi(r)+δi(T−r)+1
−2l−1−δ (r)−δ (T −r)
= (−1)j+1 i i Res Res Y (u,z )Y (v,z )w ×
j z1 z2 M 1 M 2
j≥0 (cid:18) (cid:19)
X
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
z2l+δi(r)+δi(T−r)+1+jz−j
1 2
−2l −1−δ (r)−δ (T −r)
+ (−1)δi(r)+δi(T−r)+1 i i Res Res Y (v,z )Y (u,z )w ×
j z2 z1 M 2 M 1
j≥0 (cid:18) (cid:19)
X
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
.
z−jz2l+δi(r)+δi(T−r)+1+j
1 2
Since u ∈ Vr,v ∈ VT−r and from Lemma 3.1 we know that both
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
Res Res Y (u,z )Y (v,z )w
z1 z2 M 1 M 2 z2l+δi(r)+δi(T−r)+1+jz−j
1 2
and
(1+z1)wtu+l−1+δi(r)+Tr(1+z2)wtv+l+δi(T−r)−Tr
Res Res Y (v,z )Y (u,z )w
z2 z1 M 2 M 1 z−jz2l+δi(r)+δi(T−r)+1+j
1 2
7
lie in O (M).
g,n
(2) Let u ∈ V0,v ∈ Vr and w ∈ M, we have
l
m+l
u∗ (v ◦ w) = (−1)m Res Y (u,z )Res Y (v,z )w
g,n g,n l z1 M 1 z2 M 2
m=0 (cid:18) (cid:19)
X
(1+z1)wtu+l (1+z2)wtv+l−1+δi(r)+Tr
× ·
zl+m+1 z2l+δi(r)+δi(T−r)+1
1 2
l
m+l
= (−1)m Res Res Y (v,z )Y (u,z )w
l z2 z1 M 2 M 1
m=0 (cid:18) (cid:19)
X
(1+z1)wtu+l (1+z2)wtv+l−1+δi(r)+Tr
× ·
zl+m+1 z2l+δi(r)+δi(T−r)+1
1 2
+ l (−1)m m+l Res Res Y (Y(u,z )v,z )w × (1+z2 +z0)wtu+l · (1+z2)wtv+l−1+δi(r)+Tr
m=0 (cid:18) l (cid:19) z2 z0 M 0 2 (z2 +z0)l+m+1 z22l+δi(r)+δi(T−r)+1
X
l
m+l
= (−1)m Res Res Y (v,z )Y (u,z )w
l z2 z1 M 2 M 1
m=0 (cid:18) (cid:19)
X
(1+z1)wtu+l (1+z2)wtv+l−1+δi(r)+Tr
× ·
zl+m+1 z2l+δi(r)+δi(T−r)+1
1 2
l
m+l wtu+l −l −m−1
+ (−1)m Res Y (u v,z )w
l j k z2 M j+k 2
m=0 (cid:18) (cid:19)j,k≥0(cid:18) (cid:19)(cid:18) (cid:19)
X X
(1+z2)wtu+wtv+2l−j−1+δi(r)+Tr
×
z3l+δi(r)+δi(T−r)+2+m+k
2
l
m+l
= (−1)m Res Res Y (v,z )Y (u,z )w
l z2 z1 M 2 M 1
m=0 (cid:18) (cid:19)
X
(1+z1)wtu+l (1+z2)wtv+l−1+δi(r)+Tr
× ·
zl+m+1 z2l+δi(r)+δi(T−r)+1
1 2
l
m+l wtu+l −l −m−1
+ (−1)m Res Y (u v,z )w
l j k z2 M k+j 2
m=0 (cid:18) (cid:19)k,j≥0(cid:18) (cid:19)(cid:18) (cid:19)
X X
(1+z2)wt(uk+jv)+l−1+δi(r)+Tr+l+k+1
×
z2l+δi(r)+δi(T−r)+1+l+m+k+1
2
By the definition of O (M) and Lemma 3.1(1), we know that the two resulting terms
g,n
are in O (M).
g,n
Using Lemma 3.1(2), similarly, we also have (v ◦ w)∗ u ∈ O (M). so the proof
g,n g,n g,n
(cid:3)
of the Lemma is complete.
Now we can establish the main results of this section as follows:
8
Theorem 3.4. Let M be an admissible V-module and n ∈ 1Z . Then
T +
(1) The vector space A (M) is an A (V)-bimodule with the left and right actions of
g,n g,n
A (V) on A (M) induced from (3.1) and (3.2) respectively.
g,n g,n
(2) The identity map on M induces an A (V)-bimodule epimorphism from A (M)
g,n g,n
to A (M) if n ≥ 1.
g,n−1
T
(3) The map
φ : w 7→ eL(1)(−1)L(0)w
induces a linear isomorphism from Ag,n(M) to Ag−1,n(M) such that
φ(u∗g,n w) = φ(w)∗g−1,n φ(u),φ(w∗g,n u) = φ(u)∗g−1,n φ(w)
for u ∈ V and w ∈ M.
(4) If V is g-rational, then both Og,n−s(M)/Og,n−s−1(M) and Ag,n−s(M) are Ag,n(V)-
T T T
bimodules for s = 1,...,nT and
nT
Ag,n(M) = Ag,0(M) Og,n−s(M)/Og,n−s−1(M).
T T
s=1
MM
Proof: (1) By Lemmas 3.2 and 3.3 it suffices to show that the following identities hold
in A (M) for u,v ∈ V and w ∈ M :
g,n
(u∗ w)∗ v = u∗ (w∗ v),1∗ w = w ∗ 1 = w
g,n g,n g,n g,n g,n g,n
(u∗ v)∗ w = u∗ (v ∗ w),w∗ (u∗ v) = (w ∗ u)∗ v.
g,n g,n g,n g,n g,n g,n g,n g,n
As the proofs are similar, we only prove the third identity in detail. From the definition,
9
we may assume u,v ∈ V0. Then we have
l
m +l m +l wtu+l
(u∗ v)∗ w = (−1)m1+m2 1 2
g,n g,n
l l i
m1X,m2=0Xi≥0 (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
(1+z )wtu+wtv+2l+m1−i
2
·Res Y (u v,z )w
z2 M i−l−m1−1 2 zl+m2+1
2
l
m +l m +l
= (−1)m1+m2 1 2
l l
m1X,m2=0 (cid:18) (cid:19)(cid:18) (cid:19)
(1+z +z )wtu+l(1+z )wtv+m1+l
2 0 2
·Res Res Y (Y(u,z )v,z )w
z2 z0 M 0 2 zl+m1+1zl+m2+1
0 2
l
m +l m +l
= (−1)m1+m2 1 2
l l
m1X,m2=0 (cid:18) (cid:19)(cid:18) (cid:19)
(1+z )wtu+l(1+z )wtv+m1+l
1 2
·Res Res Y (u,z )Y (v,z )w
z1 z2 M 1 M 2 (z −z )l+m1+1zl+m2+1
1 2 2
l
m +l m +l
− (−1)m1+m2 1 2
l l
m1X,m2=0 (cid:18) (cid:19)(cid:18) (cid:19)
(1+z )wtu+l(1+z )wtv+m1+l
1 2
·Res Res Y (v,z )Y (u,z )w
z2 z1 M 2 M 1 (−z +z )l+m1+1zl+m2+1
2 1 2
l
m +l m +l −l −m −1
= (−1)m1+m2+i 1 2 1
l l i
m1X,m2=0Xi≥0 (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
(1+z )wtu+l(1+z )wtv+m1+l
1 2
·Res Res Y (u,z )Y (v,z )w
z1 z2 M 1 M 2 zl+m1+1+izl+m2+1−i
1 2
l
m +l m +l −l −m −1
− (−1)m2+l+1+i 1 2 1
l l i
m1X,m2=0Xi≥0 (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
(1+z )wtu+l(1+z )wtv+m1+l
1 2
·Res Res Y (v,z )Y (u,z )w .
z2 z1 M 2 M 1 z−iz2l+m1+m2+2+i
1 2
It follows from Lemma 3.1(1) that
(1+z )wtu+l(1+z )wtv+l+m1
1 2
Res Res Y (v,z )Y (u,z )w ∈ O (M).
z2 z1 M 2 M 1 z−iz2l+m1+m2+2+i g,n
1 2
and if i > l−m
1
(1+z )wtu+l(1+z )wtv+m1+l
1 2
Res Res Y (u,z )Y (v,z )w
z1 z2 M 1 M 2 zl+m1+1+izl+m2+1−i
1 2
10