Vertex operator representations of type Cl⁽¹⁾ and product-sum identities
We construct a class of homogeneous vertex representations of Cl⁽¹⁾, l ≥ 2, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory. Побудовано клас рівномірних вершинних зображень Cl⁽¹⁾, l ≥ 2. Отримано низку тотожностей типу сум i добутків. Ці...
Saved in:
| Published in: | Український математичний журнал |
|---|---|
| Date: | 2014 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2014
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/165508 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities / Li-Meng Xia, Naihong Hu // Український математичний журнал. — 2014. — Т. 66, № 2. — С. 226–243. — Бібліогр.: 14 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860215923214909440 |
|---|---|
| author | Li-Meng, Xia Naihong, Hu |
| author_facet | Li-Meng, Xia Naihong, Hu |
| citation_txt | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities / Li-Meng Xia, Naihong Hu // Український математичний журнал. — 2014. — Т. 66, № 2. — С. 226–243. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| description | We construct a class of homogeneous vertex representations of Cl⁽¹⁾, l ≥ 2, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory.
Побудовано клас рівномірних вершинних зображень Cl⁽¹⁾, l ≥ 2. Отримано низку тотожностей типу сум i добутків. Ці тотожності мають змістовну інтерпретацію теорії чисел.
|
| first_indexed | 2025-12-07T18:16:02Z |
| format | Article |
| fulltext |
UDC 512.5
Li-Meng Xia* (Jiangsu Univ., China),
Naihong Hu** (East China Normal Univ., Shanghai, China)
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l
AND PRODUCT-SUM IDENTITIES
ВЕРШИННI ОПЕРАТОРНI ЗОБРАЖЕННЯ ТИПУ C
(1)
l
ТА ТОТОЖНОСТI ТИПУ СУМ I ДОБУТКIВ
The purposes of this work are to construct a class of homogeneous vertex representations of C(1)
l , l ≥ 2, and to deduce a
series of product-sum identities. These identities have fine interpretation in the number theory.
Побудовано клас рiвномiрних вершинних зображень C
(1)
l , l ≥ 2. Отримано низку тотожностей типу сум i добуткiв.
Цi тотожностi мають змiстовну iнтерпретацiю теорiї чисел.
1. Introduction. It is well known that there is a close relationship between representations of affine
Lie algebras and combinatorics. For example, the Jacobi triple product identity can be obtained as the
Weyl – Kac denominator formula for the affine Lie algebra ŝl2 [7]. The famous Rogers – Ramanujan
identities can be realized from the character formula of certain level three representations [8]. Like
the Jacobi triple product identity, the quintuple product identity is also equivalent to the Weyl – Kac
denominator formula for the affine Lie algebra A(2)
2 . In [6], the following infinite product:
∞∏
n=1
1
(1− q6n−1)(1− q6n−5)
(1.1)
is expressed by a sum of two other infinite products in four different ways.
I. Schur [12] (see also [1]) was probably the first person who studied the partitions described
by (1.1). He showed that the number of partitions of n into parts congruent to ±1(mod 6) is equal
to the number of partitions of n into distinct parts congruent to ±1(mod 3) and is also equal to the
number of partitions of n into parts that differ at least 3 with added condition that difference between
multiples of 3 is at least 6. His first result can be briefly described by
∞∏
n=1
1
(1− q6n−1)(1− q6n−5)
=
∞∏
n=1
1 + qn
1 + q3n
. (1.2)
Motivated by product-sum identity provided by [6], we study a generalized product-sum relations
of some special partitions. Our method uses the vertex representations of affine Lie algebras of type
C
(1)
l . For the related topics, one can refer [4, 5, 10, 11, 13] and references therein.
Theorem 1.1. For any odd l ≥ 3, the following product-sum identity holds:
∞∏
n=1
1 + qn
1 + qln
=
l−1
2∑
s=0
q
(l−2s)2−1
8
∏
n6≡±(s+1),0(mod l+2)
1
(1− q2n)(1− qln)
,
particularly, it covers the first result of [6] when l = 3.
* Supported by the NNSF of China (Grant No. 11001110) and Jiangsu Government Scholarship for Overseas Studies.
** Supported in part by the NNSF of China (Grant No. 11271131), the PCSIRT and the RFDP from the MOE of China,
the National & Shanghai Leading Academic Discipline Projects (Project Number B407).
c© LI-MENG XIA, NAIHONG HU, 2014
226 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 227
Our result in Theorem 1.1 implies the following partition theorem.
Theorem 1.2. Suppose that l = 2r+ 1 ≥ 3 is an odd number, Al(n) is the number of partitions
of n into distinct parts without multiples of l, and Bl,s(n) is the number of partitions of n into
2k1 + . . .+ 2ki + lr1 + . . .+ lrj +
(l − 2s)2 − 1
8
with constraints kp, rp 6≡ ±(s+ 1), 0(mod l + 2). Then for any positive integer n, we have
Al(n) = Bl,0(n) +Bl,1(n) + . . .+Bl,r(n).
Proof. Let 1 +
∑∞
n=1
Ana
n be the power series of
∏∞
n=1
1 + qn
1 + qln
. Because
∞∏
n=1
1 + qn
1 + qln
=
∏
n≥1 is not a multiple of l
(1 + qn) =
∑
n1>n2>...>nk≥1
ni is not a multiple of l, k≥0
qn1+...+nk .
Then An is the number of partitions of n into distinct parts without multiples of l and An = A(n).
A similar argument on Bl,s(n) shows that Theorem 1.1 is equivalent to the relation
Al(n) = Bl,0(n) +Bl,1(n) + . . .+Bl,r(n), for all positive integer n.
Theorem 1.2 is proved.
For example,
A5(15) = 16 B5,0(15) = 3 B5,1(15) = 7 B5,2(15) = 6
1 + 14 1 + 6 + 8 2(2× 3) + 3 2(1× 7) + 1 2(1× 5) + 5(1)
2 + 13 2 + 4 + 9 2(2 + 4) + 3 2(1× 4 + 3) + 1 2(1× 3 + 2) + 5(1)
3 + 12 2 + 6 + 7 2(3 + 3) + 3 2(1 + 3× 2) + 1 2(1 + 2× 2) + 5(1)
4 + 11 3 + 4 + 8 2(1× 3 + 4) + 1 2(5) + 5(1)
6 + 9 1 + 2 + 3 + 9 2(1 + 6) + 1 5(1× 3)
7 + 8 1 + 2 + 4 + 8 2(3 + 4) + 1 5(1 + 2)
1 + 2 + 12 1 + 3 + 4 + 7 2(1× 2) + 5(1× 2)
1 + 3 + 11 2 + 3 + 4 + 6
Table 1.1 lists the values of A5(n), B5,0(n), B5,1(n), B5,2(n) for n ≤ 15.
The above results will be proved by the irreducible decompositions of vertex module V (P ) =
= S(Ĥ−)⊗C[P ] of C(1)
l , where 1⊗ 1 has weight Λ0. If we assume that 1⊗ 1 has weight Λ1, then
our method also gives the following result.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
228 LI-MENG XIA, NAIHONG HU
Table 1.1
n A5(n) B5,0(n) B5,1(n) B5,2(n)
1 1 0 1 0
2 1 0 0 1
3 2 1 1 0
4 2 0 0 2
5 2 0 1 1
6 3 0 1 2
7 4 1 2 1
8 4 0 1 3
9 6 1 3 2
10 7 0 1 6
11 8 2 4 2
12 10 0 2 8
13 12 3 6 3
14 14 0 3 11
15 16 3 7 6
Theorem 1.3. For any even l ≥ 2, the following product-sum identity holds:
∞∏
n=1
(1 + qn−1/2)2
(1 + qn)(1 + qln)
=
∏
n≥1
(1− q
(
l
(
l+2
2
))
(2n−1))(1− q(l+2)(2n−1))∏(
l
2 +1
)
6 |n
(1− q2n)(1− qln)
+
+2
l/2−1∑
s=0
q
(l−2s)2
8
∏
n6≡±(s+1),0(mod l+2)
1
(1− q2n)(1− qln)
,
or equivalently,
∞∏
n=1
(1 + q2n−1)2
(1 + q2n)(1 + q2ln)
=
∏
n≥1
(1− q(l(l+2)(2n−1))(1− q2(l+2)(2n−1))∏
( l
2
+1)6 |n
(1− q4n)(1− q2ln)
+
+2
l/2−1∑
s=0
q
(l−2s)2
4
∏
n6≡±(s+1),0(mod l+2)
1
(1− q4n)(1− q2ln)
.
Throughout the paper, we let C, Z present the set of complex numbers and the set of integers,
respectively.
2. Affine Lie algebra of type C
(1)
l . 2.1. Let Ġ be a finite-dimensional simple Lie algebra of
type Cl, A = C[t±1] the ring of Laurent polynomials in variable t. Then the affine Lie algebra of
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 229
type C(1)
l is the vector space
G̃ = Ġ ⊗A ⊕ Cc⊕ Cd,
with Lie bracket:
[x⊗ tm, y ⊗ tn ] = [x, y ]⊗ tm+n +m(x | y)δm+n,0c,
[ c,G ] = 0,
[ d, x⊗ tm ] = mx⊗ tm,
where x, y ∈ Ġ, m, n ∈ Z and (· | ·) is a nondegenerate invariant normalized symmetric bilinear
form on Ġ.
2.2. Suppose that Ḣ is a Cartan subalgebra of Ġ, and Ḣ∗ the dual space of Ḣ. Then there exists
an inner product (· | ·)|Ḣ∗
R
and an orthogonal normal basis {e1, e2, . . . , el} in Euclidean space Ḣ∗R
such that the simple root system
Π =
{
α1 =
1√
2
(e1 − e2), . . . , αl−1 =
1√
2
(el−1 − el), αl =
√
2 el
}
,
the short root system
∆̇S =
{
± 1√
2
(ei − ej), ±
1√
2
(ei + ej)
∣∣∣∣ 1 ≤ i < j ≤ l
}
,
where
1√
2
(ei− ej) = αi + . . .+αj−1 for 1 ≤ i < j ≤ l, 1√
2
(ei + el) = αi + . . .+αl for 1 ≤ i < l,
1√
2
(ei+ ej) = αi+ . . .+αj−1 + 2αj + . . .+ 2αl−1 +αl for 1 ≤ i < j < l; and the long root system
∆̇L =
{
±
√
2 ei
∣∣ 1 ≤ i ≤ l
}
,
where
√
2 ei = 2αi + . . .+ 2αl−1 + αl for 1 ≤ i < l.
Then the root lattice is
Q =
l⊕
i=1
Zαi,
(αi|αi) = 1, 1 ≤ i ≤ l − 1, and (αl | αl) = 2.
Let γ : Ḣ −→ Ḣ∗ be the linear isomorphism such that
αi(γ
−1(αj)) = (αi | αj), i, j = 1, . . . , l,
and
γ(α∨i ) = 2αi, i = 1, . . . , l − 1, γ(α∨l ) = αl.
Then we have (α∨i |α∨j ) = (γ(α∨i )|γ(α∨j )). As usual, we identify Ḣ with Ḣ∗ via γ, i.e., α∨ =
=
2α
(α | α)
.
For any weight Λ ∈ (Ḣ ⊕ Cc⊕ Cd)∗, let L(Λ) denote the irreducible highest weight G̃-module
with highest weight Λ.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
230 LI-MENG XIA, NAIHONG HU
2.3. Define a 2-cocycle ε0 : Q×Q −→ {±1} by
ε0(a+ b, c) = ε0(a, c) ε0(b, c), ε0(a, b+ c) = ε0(a, b) ε0(a, c), a, b, c ∈ L,
and
ε0(αi, αj) =
−1, i = j + 1,
1, other pairs (i, j).
Let P =
⊕l
i=1−1 Zαi ⊕
1
2
Zαl. Extend ε0 to Q× P with
ε0
(
αi,
1
2
αl
)
= 1.
2.4. For α =
∑l
i=1
kiαi ∈ ∆̇∪{0}, define maps p : ∆̇∪{0} → ∆̇S∪{0} and s : ∆̇∪{0} → Q̇L
by
p
(
l∑
i=1
kiαi
)
=
l−1∑
i=1
ρ(ki)αi, s
(
l∑
i=1
kiαi
)
=
l−1∑
i=1
(
ki − ρ(ki)
)
αi,
where Q̇L = SpanZ ∆̇L and ρ(ki) ∈ {0, 1} such that ρ(ki) ≡ ki (mod 2). It is straightforward to
check the following statements.
Lemma 2.1. (i) p (∆̇L ∪ { 0 }) = 0, and p(−α) = p(α) for any α ∈ ∆̇S .
(ii) Suppose that α, β, α+ β ∈ ∆̇, then we have:
(1) if α ∈ ∆̇L, then (α | β) = −1, p(α+ β) = p(β), s(α+ β) = s(α) + s(β);
(2) if α, β ∈ ∆̇S , α+β ∈ ∆̇L, then (α | β) = 0, p(α) = p(β), s(α+β)−s(α)−s(β) = 2p(α);
(3) if α, β, α+ β ∈ ∆̇S , then (α | β) = −1
2
and |(p(α) | p(β))| = 1
2
; moreover,
(a) (p(α) | p(β)) =
1
2
, then p(α + β) = p(α) − p(β), s(α + β) − s(α) − s(β) = 2p(β), or
p(α+ β) = −p(α) + p(β), s(α+ β)− s(α)− s(β) = 2p(α);
(b) if (p(α) | p(β)) = −1
2
, then p(α+ β) = p(α) + p(β), s(α+ β) = s(α) + s(β).
(iii) For any α ∈ ∆̇, we have:
(1) s(α) ∈
{
±
√
2(ei − el) | 1 ≤ i ≤ l
}
⊂ Q̇L;
(2) p(α) ∈
{
1√
2
(ei − ej)
∣∣∣ 1 ≤ i ≤ j ≤ l
}
⊂ ∆̇S ∪ {0};
(3) s(α) + s(−α) = −2p(α) ∈ Q̇L;
(4) α± p(α) ∈ Q̇L.
2.5. Define a map f : Q×Q→ {±1} by
f(α, β) = (−1)(s(α)|β)+(p(α)|p(β)+p(α+β)).
Set ε = ε0 ◦ f, then ε : Q × Q −→ {±1} is still a 2-cocycle, which has the property (ii) in the
following lemma.
Lemma 2.2. (i) For α, β ∈ ∆̇, we have
ε0(α, β) = (−1)(α|β)+(p(α)|p(β))+(s(α)|β)+(s(β)|α) · ε0(β, α).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 231
(ii) For α, β, α+ β ∈ ∆̇, we have ε(α, β) = −ε(β, α).
2.6. We have the following proposition.
Proposition 2.1. The affine Lie algebra G̃ of type C(1)
l has a system of generators{
α∨i ⊗ tn, eα ⊗ tn | 1 ≤ i ≤ l, n ∈ Z
}
and c, d with relations [
α∨i ⊗ tm, α∨j ⊗ tn
]
= m(α∨i | α∨j )δm+n,0c,[
α∨i ⊗ tm, eα ⊗ tn
]
= α(α∨i ) eα ⊗ tm+n,
[
eα ⊗ tm, e−α ⊗ tn
]
= ε(α,−α)
2
(α | α)
[
γ−1(α)⊗ tm+n +mδm+n,0c
]
,
[
eα ⊗ tm, eβ ⊗ tn
]
= ε(α, β)
(
1 + δ1, (p(α)|p(β))
)
eα+β ⊗ tm+n ∀α, β, α+ β ∈ ∆̇,[
eα ⊗ tm, eβ ⊗ tn
]
= 0 ∀α, β ∈ ∆̇, α+ β 6∈ ∆̇ ∪ { 0 },
where γ is the canonical linear space isomorphism from Ḣ to Ḣ∗.
3. Vertex construction of Lie algebra of type C
(1)
l . 3.1. Let H(m), m ∈ Z, be an isomorphic
copy of Ḣ. Set ḢS := SpanC{αi | 1 ≤ i ≤ l−1 } and HS
(
n− 1
2
)
, n ∈ Z, is an isomorphic copy
of ḢS .
Define a Lie algebra
Ĥ =
⊕
m∈Z
H(m)⊕
⊕
n∈Z
HS
(
n− 1
2
)
⊕ Cc,
with Lie bracket [
H̃, c
]
= 0,
[ a(m), b(n) ] = m (a | b) δm,−nc.
Let
Ĥ− =
⊕
m∈Z−
H(m)⊕
⊕
n∈Z−
HS
(
n+
1
2
)
,
and let S(Ĥ−) be the symmetric algebra generated by Ĥ−. Then S(Ĥ−) is an Ĥ-module with the
action
c · v = v, a(m) · v = a(m)v ∀m < 0,
and
a(m) · b(n) = m (a, b) δm+n,0 ∀m ≥ 0, n < 0,
where a, b ∈ H, m, n ∈ 1
2
Z.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
232 LI-MENG XIA, NAIHONG HU
3.2. We form a group algebra C[P ] with base elements eh, h ∈ P, and the multiplication
eh1eh2 = eh1+h2 ∀h1, h2 ∈ P.
Set
V (P ) := S(Ĥ−)⊗ C[P ]
and extend the action of Ĥ to space V (P ) by
a(m) · (v ⊗ er) = (a(m) · v)⊗ er ∀m ∈ 1
2
Z∗;
and define
a(0) · (v ⊗ er) = (a | r) v ⊗ er,
which makes V (P ) into a Ĥ-module.
3.3. For r ∈ P, α ∈ Q, define C-linear operators as
eα · (v ⊗ er) = v ⊗ eα+r,
zα · (v ⊗ er) = z(α|r) v ⊗ er,
εα · (v ⊗ er) = (−1)(s(α)|r) ε0(α, r) v ⊗ er,
a(z) =
∑
j∈Z
a(j) z−2j ,
E±(α, z) · (v ⊗ er) =
(
exp
(
∓
∞∑
n=1
1
n
z∓2nα(±n)
)
· v
)
⊗ er,
F±(α, z) · (v ⊗ er) =
(
exp
(
∓
∞∑
n=0
2
2n+1
z∓(2n+1)α
(
±2n+1
2
))
· v
)
⊗ er.
Then a(z), E±(α, z), F±(α, z) ∈ (EndV (P ))[[z, z−1]].
As usual, we shall adopt the notation of normal ordering product
: a(i)b(j) :=
a(i)b(j), if i ≤ j,
b(j)a(i), if j < i,
where a, b ∈ L and i, j ∈ 1
2
Z.
3.4. Let Ṽ (P ) be the formal completion of V (P ) = S
(
Ĥ−
)
⊗ C[P ]. We give some vertex
operators on Ṽ (P ):
(1) For α ∈ ∆̇ ∪ {0}, set
Y (α, z) = E−(α, z)E+(α, z)F−(p(α), z)F+(p(α), z),
Zε(α, z) = z(α|α)eαz2αεα,
Xε(α, z) := Y (α, z)⊗ Zε(α, z).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 233
(2) For α, β ∈ ∆̇, define
Xε(α, β, z, w) =: Y (α, z)Y (β,w) : ⊗ Zε(α+ β,w).
3.5. The Laurent series of operators Xε(α, z) is denoted by
Xε(α, z) =
∞∑
k=−∞
Xε
k/2(α) z−k.
Then for all k ∈ Z, Xε
k/2(α) is an operator on V (P ). Note that Xε
n(α) acts as an operator on V (P )
in the following way:
Xε
n(α) · (v ⊗ er) = ε(α, r)Y
n+
1
2 (α|α)+(α|r)
(α) (v)⊗ eα+r ∀v ⊗ er ∈ V (P ).
3.6. For v = a1(−n1)a2(−n2) . . . ap(−np)⊗er ∈ V (L), define the degree action of d on V (P )
by
d · (v ⊗ er) =
(
deg (v)− 1
2
(r | r)
)
v ⊗ er,
where deg (v) = −
∑p
i=1
ni.
The number deg (v)− 1
2
(r | r) is called the degree of v ⊗ er and denoted by deg (v ⊗ er).
3.7. We have the following proposition.
Proposition 3.1. The affine Lie algebra G̃ of type C(1)
l is homomorphic to the Lie algebra J
generated by operators α∨(n), Xε
n(α), c, d (α ∈ ∆̇, n ∈ Z) on V (P ) = S(Ĥ−) ⊗ C[P ], i.e., there
exists a unique Lie algebra homomorphism π from G̃ to the Lie subalgebra J of End(V (P )) such
that
π(γ−1(αi)⊗ tn) =
2
(αi | αi)
αi(n),
π(eα ⊗ tn) = Xε
n(α),
π(c) = id,
π(d) = d,
that is, V (P ) is a G̃-module.
4. Some computations needed.
Lemma 4.1. For any l, if Λs is the basic weight of C(1)
l , then we have
dimq(L(Λs)) = dimq(L(Λl−s)),
dimq(L(Λs)) =
∞∏
n=1
(1− q2(l+2)n)(1− q2(l+2)n−2−2s)(1− q2(l+2)n−2l−2+2s)
1− qn
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
234 LI-MENG XIA, NAIHONG HU
For the definition of dimq, one can refer [7, p. 183] (Proposition 10.10).
Define q-series
κq(l, r) =
∑
n∈Z
qln
2−rn, (4.1)
for 0 < r ≤ l. If r = l, then
κq(l, l) = 2
∞∏
n=1
(1− q4n)2
1− q2n
, (4.2)
by Gauss identity
∑
n∈Z
q2n2−n =
∞∏
n=1
(1− q2n)2
1− qn
.
Suppose that V = S(α(−1), α(−2), . . .)⊗C[Zα] with (α|α) = 2, then V is an irreducible A(1)
1 -
module isomorphic to L(Λ0) (one can see [4] for details). The degree of v = α(−n1) . . . α(−nk)⊗
⊗ enα ∈ V is defined as −n1 − . . . − nk − n2 and weight of v is −(n1 + . . . + nk + n2)δ + nα.
Hence
chV = eΛ0
1∏∞
n=1
(1− e−nδ)
∑
n∈Z
e−n
2δ+nα.
Moreover,
chL(Λ0) = eΛ0
∑
n∈Z
e−(3n2+n)δ+3nα −
∑
n∈Z
e−(3n2+n)δ−(3n+1)α∏∞
n=1
(1− e−nδ)(1− e−nδ+α)(1− e−(n−1)δ−α)
.
If e−δ, e−α are specialized as ql, qr, respectively, then V ∼= L(Λ0) implies the following lemma.
Lemma 4.2. If 0 < r < l, then
κq(l, r) =
∞∏
n=1
(1− q2ln)(1− q4ln−2(l−r))(1− q4l(n−1)+2(l−r))
(1− q2ln−l−r)(1− q2l(n−1)+l+r)
.
Proof. This lemma can easily be proved using the quintuple product identity (see [3]).
5. The module structure. 5.1. Let α0 ∈ H∗ such that {α0, α1, . . . , αl} is the simple root
system of affine Lie algebra G̃ and α0(α∨0 ) = 2, α0(α∨1 ) = −2, α0(d) = 1 and α0(α∨j ) = α0(c) = 0,
2 ≤ j ≤ l. Then δ = α0 + 2α1 + 2α2 + . . . + 2αl−1 + αl is the primitive imaginary root of G̃. Let
Λi ∈ H∗ be such that
Λi(α
∨
j ) = δij , Λi(d) = 0, 0 ≤ j ≤ l.
Lemma 5.1. With respect to the Cartan subalgebra H of G̃, V (P ) has the weight space decom-
position
V (P ) =
∑
λ∈weight(V (P ))
V (P )λ,
and the weight space V (P )λ has a basis v ⊗ er, where r ∈ P, v ∈ S(Ḣ−), and
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 235
λ = Λ0 +
(
deg (v)− 1
2
(r | r)
)
δ + r,
so deg (v) and r are uniquely determined by λ.
5.2. The following describes the possible distribution of the maximal weights of G̃-module
V (Q̇).
Lemma 5.2. For any λ ∈ P (V (Q̇)), we have
λ ≤ Λj −
j
4
δ,
for some j ∈ Z, where 0 ≤ j ≤ l.
Proof. By Lemma 5.1, λ = Λ0 −
(
k +
1
2
(r | r)
)
δ + r, where r =
∑l−1
i=1
kiαi +
kl
2
αl ∈ P and
k ∈ 1
2
N. At first, we have
1
2
(r | r)δ − r =
=
1
4
(
k2
1 + (k2 − k1)2 + . . .+ (kl−1 − kl−2)2 + (2kl − kl−1)2
)
δ −
l−1∑
i=1
kiαi −
kl
2
αl =
=
1
4
[(
k2
1δ − 2k1(2α1 + 2α2 + . . .+ 2αl−1 + αl)
)
+
+
(
(k2 − k1)2δ − 2(k2 − k1)(2α2 + . . .+ 2αl−1 + αl)
)
+ . . .
. . .+
(
(kl−1 − kl−2)2δ − 2(kl−1 − kl−2)(2αl−1 + αl)
)
+ (kl − kl−1)2δ − 2(kl − kl−1)αl
]
.
Suppose that
α = 2αi + . . .+ 2αl−1 + αl < δ.
If n < 0, then n2δ − 2nα > 0. If n > 1, then
(n2 − 1)δ − 2(n− 1)α = (n− 1)((n+ 1)δ − 2α) > 0.
Hence we get
n2δ − 2nα ≥ 0
or
n2δ − 2nα ≥ δ − 2α.
So
1
2
(r | r)δ − r ≥
≥ s
4
δ − 1
2
[
(2αp1 + . . .+ 2αl−1 + αl) + . . .+ (2αps + . . .+ 2αl−1 + αl)
]
≥
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
236 LI-MENG XIA, NAIHONG HU
≥ s
4
δ − 1
2
[
(2α1 + . . .+ 2αl−1 + αl) + . . .+ (2αs + . . .+ 2αl−1 + αl)
]
=
=
1
2
(γs | γs)δ − γs,
for some s, where
γs = α1 + 2α2 + . . .+ (s− 1)αs−1 + s(αs + . . .+ αl−1) +
s
2
αl ∈ P,
and it clear that Λs = Λ0 + γs, (γs | γs) =
s
2
. Then we have
λ = Λ0 −
(
k +
1
2
(r | r)
)
δ + r ≤ Λ0 −
1
2
(r|r)δ + r ≤
≤ Λ0 −
1
2
(γs | γs)δ + γs = Λs −
s
4
δ
for some s, 0 ≤ s ≤ l.
Remark 5.1. By the result above, we know that any highest weight of V (P ) belongs to the set
l⋃
s=0
{
Λs −
s
4
− p
2
δ
∣∣∣∣ p ≥ 0, s = 0, 1, . . . , l
}
.
More precisely, any highest weight vector has the form v ⊗ eγs for some s.
Theorem 5.1. V (P ) has the decomposition
V (P ) =
l⊕
s=0
V (P )[s],
where V (P )[s] is the sum of those irreducible submodules whose highest weights λ ≤ Λs −
s
4
.
6. Highest weight vectors. 6.1. Define operators
S(α, z) = exp
(∑
n>0
α(−n+ 1/2)
n− 1/2
z2n−1
)
exp
(
−
∑
n>0
α(n− 1/2)
n− 1/2
z−2n+1
)
,
with series expansion
S(α, z) =
∑
n∈1
2Z
Sn(α)z−2n.
Lemma 6.1. For i = 1, . . . , l − 1, we have
{Sn(αi), Sm(αi)} = Sn(αi)Sm(αi) + Sm(αi)Sn(αi) = −2δm+n,0
and
Sn(α) = (−1)2nSn(−α), n ∈ 1
2
Z.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 237
6.2. Define βi = αi for i = 1, . . . , l − 1 and
βl = −
l−1∑
i=1
i
l
αi,
also let
yi =
l∑
j=i
2βj , i = 1, . . . , l.
Define
Z [s](z) =
∑
i∈Z
Z
[s]
i/2z
−i =
s∑
j=1
S(yj , z)−
l∑
j=s+1
S(yj , z),
for even s. Particularly, Z [l](z) = −Z [0](z).
Remark 6.1. The operators Z [s] are the same as (or isomorphic to) those defined by Lepowsky
and Wilson in [8, 9], where they are generating operators of vacuum spaces of standard A(1)
1 -modules
of level l. For more details, one can refer to those two papers.
Lemma 6.2. For any n ∈ 1
2
Z, if v ⊗ eγs is a highest weight vector and Z [s]
n v ⊗ eγs is not zero,
then Z [s]
n v ⊗ eγs is also a highest weight vector.
Proof. At first, we give the proof for s = 0. For i < l, we have
Sn(yi) + Sn(yi+1) =
∑
j∈Z
Sj(βi)Sn−j(βi + 2βi+1 + . . .+ 2βl),
and (yj |αi) = 0, j 6= i, i+ 1. Hence
−Xε
0(αi)Z
[0]
n (v ⊗ 1) =
= Xε
0(αi)
∑
r 6=i,i+1
Sn(yr) +
∑
j∈Z
Sj(βi)Sn−j(βi + 2βi+1 + . . .+ 2βl)
v ⊗ 1 =
= Y 1
2
(αi)
∑
r 6=i,i+1
Sn(yr) +
∑
j∈Z
Sj(βi)Sn−j(βi + 2βi+1 + . . .+ 2βl)
v ⊗ eαi =
= Y 1
2
(αi)
∑
r 6=i,i+1
Sn(yr)v ⊗ eαi+
+Y 1
2
(αi)
∑
j∈Z
Sj(αi)Sn−j(βi + 2βi+1 + . . .+ 2βl)v ⊗ eαi =
=
∑
r 6=i,i+1
Sn(yr)X
ε
0(αi)v ⊗ 1+
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
238 LI-MENG XIA, NAIHONG HU
+
∑
k∈Z
Ek(αi)S 1
2
−k(αi)
∑
j∈Z
Sj(αi)Sn−j(βi + 2βi+1 + . . .+ 2βl)v ⊗ eαi =
= −
∑
j∈Z
Sj(αi)Sn−j(βi + 2βi+1 + . . .+ 2βl)X
ε
0(αi)v ⊗ 1 = 0.
Moreover, operators Xε
0(αl) and Xε
1(−(2α1 + . . . + 2αl−1 + αl)) commute with Z [0]
n , so Z [0]
n v ⊗ 1
is still a highest weight vector.
The proof for s = l is the same as above.
For Z [s]
n with 0 < s < l,
Sn(yi)− Sn(yi+1) =
∑
j∈Z+1/2
Sj(βi)Sn−j(βi + 2βi+1 + . . .+ 2βl),
then
Xε
0(αs)Z
[s]
n (v ⊗ eγs) =
= Xε
0(αs)
(∑
r<s
−
∑
r>s+1
)
Sn(yr) +
∑
j∈Z+1/2
Sj(βs)Sn−j(βs + 2βs+1 + . . .+ 2βl)
v ⊗ eγs =
= Y1(αs)
(∑
r<s
−
∑
r>s+1
)
Sn(yr) +
∑
j∈Z
Sj(βs)Sn−j(βs + 2βs+1 + . . .+ 2βl)
v ⊗ eγs+αi =
= Y1(αi)
(∑
r<s
−
∑
r>s+1
)
Sn(yr)v ⊗ eγs+αi+
+Y1(αi)
∑
j∈Z
Sj(αs)Sn−j(βs + 2βs+1 + . . .+ 2βl)v ⊗ eγs+αs =
=
∑
r 6=i,i+1
Sn(yr)X
ε
0(αs)v ⊗ eγs+
+
∑
k∈Z
Ek(αi)S1−k(αi)
∑
j∈Z+1/2
Sj(αs)Sn−j(βs + 2βs+1 + . . .+ 2βl)v ⊗ eγs+αs =
= −
∑
j∈Z
Sj(αs)Sn−j(βs + 2βs+1 + . . .+ 2βl)X
ε
0(αs)v ⊗ eγs = 0.
For other Xε(αi) and Xε
0(αl), X
ε
1
(
− (2α1 + . . .+ 2αl−1 +αl)
)
, the proof is similar to the first case.
Then Z [s]
n v ⊗ eγs is also a highest weight vector.
Lemma 6.2 is proved.
For λ = Λ0 −
∑l
i=0
kiαi, define
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 239
deg λ =
l∑
i=0
ki,
and
V (P )i =
∑
λ : deg λ=i
V (P )λ,
then
V (P ) =
∑
V (P )i.
The q-character chq is a map from V (P ) to Z[q±1]
(
to Z
[
q±1/2
]
if l is even
)
defined by
chqV (P ) =
∑
dimV (P )iq
i.
Define the highest weight vector space of V (P )[s] as Ωs ⊗ eγs . Then we have the following
theorem.
Theorem 6.1. Ωs is generated by operators Z [s]
i , i ∈
1
2
Z−. Moreover,
chqΩs =
∞∏
n=1
(1− ql(l+2)n)(1− ql[(l+2)n−s−1])(1− ql[(l+2)n−l+s−1])
1− qln
.
7. Proof of Theorem 6.1. Let
Ĥ− =
⊕
n∈Z−
HS
(
n+
1
2
)
.
Theorem 6.1 will be proved by the following lemmas.
Lemma 7.1. S(ĤS
−
)⊗ 1 can be generated by operators Z [s]
n , n ∈
1
2
Z, s = 0, . . . , l, on 1⊗ 1.
Proof. At first, by the definition of operators Z [s](z),
Z [1]
n − Z [0]
n = 2S(y1),
Z [2]
n − Z [1]
n = 2S(y2),
. . . . . . . . . . . . . . . . . . . . . . . .
Z [l−1]
n − Z [l−2]
n = 2S(yl−1),
moreover, for 0 < s < l and m ∈ Z, ys
(
m+
1
2
)
can be generated by operators Sn(ys), n ∈
1
2
Z.
So S(H−S )⊗ 1 can be generated by the Z [s]
n ’s.
Lemma 7.2. Suppose that v ∈ S(ĤS−), then v ⊗ eγs is a highest weight vector if and only if
for all positive integers m,
Sm−1/2(αi)v ⊗ 1 = 0, 0 < i < l, i 6= s, Sm(αs)v ⊗ 1 = 0 (when (αs|αs) = 1).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
240 LI-MENG XIA, NAIHONG HU
Proof. As we know that v ⊗ eγs is a highest weight vector if and only if
Xε
0(αi)v ⊗ eγs = Xε
1(−2α1 − . . .− 2αl−1 − αl)v ⊗ eγs = 0, i = 1, . . . , l.
For any v ∈ S(ĤS
−
), it always holds that
Xε
0(αl)v ⊗ eγs = Xε
1(−2α1 − . . .− 2αl−1 − αl)v ⊗ eγs = 0.
Let
E(α, z) = E−(α, z)E+(α, z) =
∑
j∈Z
Ej(α)z−j ,
then for 0 < i < l,
Xε
0(αi)v ⊗ eγs = εαiY1/2(αi)v ⊗ eγs+αi = εαi
∑
j∈Z
Ej(αi)S1/2−j(αi)v ⊗ eγs+αi
for i 6= s and
Xε
0(αi)v ⊗ eγs = εαiY1(αi)v ⊗ eγs+αi = εαi
∑
j∈Z
Ej(αi)S1−j(αi)v ⊗ eγs+αi
for i = s. Thus this lemma holds.
Lemma 7.3. If v ∈ S(ĤS
−
) and for all positive integer m,
Sm−1/2(α1)v ⊗ 1 = 0,
then v belongs to the subspace W1 generated by Z [0]
n/2, Z
[2]
n/2, . . . , Z
[l−1]
n/2 , Z
[l]
n/2 = −Z [0]
n/2, n ∈ Z.
Proof. Notice that
Z
[0]
n/2 = −
∑
r 6=1,2
Sn/2(yr)−
∑
j∈Z
Sj(β1)Sn/2−j(β1 + 2β2 + . . .+ 2βl) =
= −
∑
j∈Z
Sj(α1)Sn/2−j(β1 + 2β2 + . . .+ 2βl) + terms commuting with S(α1),
Z
[1]
n/2 =
∑
j∈Z
Sj+1/2(α1)Sn/2−j−1/2(β1 + 2β2 + . . .+ 2βl) + terms commuting with S(α1),
and
Z
[s]
n/2 =
∑
j∈Z
Sj(α1)Sn/2−j(β1 + 2β2 + . . .+ 2βl) + terms commuting with S(α1),
for s ≥ 2. Since (α1|β1 + 2β2 + . . .+ 2βl) = 0, a homogeneous non-zero vector
v = v ⊗ 1 =
∑
aj1,...,jkZ
[s1]
j1
. . . Z
[sk]
jk
⊗ 1
can be written as ∑
bi1,...,irSi1(α1) . . . Sir(α1)⊗ 1, i1 < . . . < ir ≤ 0,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 241
where bi1,...,ir is a non-zero polynomial commuting with S(α1). Then v ∈ W1 if and only if
i1, . . . , ir ∈ Z for any bi1,...,ir . It is easy to show that if bi1,...,ir ⊗ 1 6= 0, then
S−j1(α1) . . . S−jr(α1)v = a scalar of bj1,...,jr ⊗ 1 6= 0.
Condition Sm−1/2(α1)v ⊗ 1 = 0 implies all i1, . . . , ir ∈ Z, so v ∈W1.
Lemma 7.3 is proved.
A similar argument shows the following two lemmas.
Lemma 7.4. If v ∈ S
(
ĤS
−)
and for all positive integer m,
Sm−1/2(α1)v ⊗ 1 = 0, Sm−1/2(α2)v ⊗ 1 = 0,
then v belongs to the subspace generated by Z [0]
n/2, Z
[3]
n/2, . . . , Z
[l−1]
n/2 , Z
[l]
n/2 = −Z [0]
n/2.
Lemma 7.5. If v ∈ S
(
ĤS
−)
and for all positive integer m and 1 < i < l,
Sm−1/2(αi)v ⊗ 1 = 0,
then v belongs to the subspace generated by Z [0]
n .
Similarly to the proof for s = 0 above, for general s, we have the following lemma.
Lemma 7.6. Suppose that v ∈ S
(
ĤS
−)
and 0 < s < l. If
Sm−1/2(αi)v ⊗ 1 = 0, 0 < i < l, i 6= s, Sm(αs)v ⊗ 1 = 0 (when (αs|αs) = 1),
for all positive integer m, then v belongs to the subspace generated by Z [s]
n .
Lemma 7.7. For any 0 ≤ s ≤ l, the element 1⊗ eγs is a highest weight vector.
Lemma 7.8. For odd l ≥ 3, Ωs has basis{
Z [s]
n1
. . . Z [s]
nk
⊗ 1
∣∣∣ np ∈ 1
2
Z−, np ≤ np+1, np ≤ np+r − 1, nk−σ(s) ≤ −1
}
.
For even l ≥ 2, Ωs has basis{
Z [s]
n1
. . . Z [s]
nk
⊗ 1
∣∣∣ np ∈ 1
2
Z−, np − np+r < −1⇒
⇒
r∑
i=0
np+i ∈ Z, np ≤ np+r − 1, nk−σ(i) ≤ −1
}
,
here r =
l − 1
2
if l odd and r =
l
2
if l even, σ(s) = s for s ≤ r, otherwise, σ(s) = r + 1− s.
Lemma 7.9. For any 0 ≤ s ≤ l,
chqΩs =
∞∏
n=1
(1− ql(l+2)n)(1− ql[(l+2)n−s−1])(1− ql[(l+2)n−l+s−1])
1− qln
.
For Lemmas 7.8 and 7.9, one can refer [8] (Theorem 10.4), [9] (Section 14) and [2] (Section 3).
Lemmas 7.1 – 7.9 prove Theorem 6.1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
242 LI-MENG XIA, NAIHONG HU
8. Product-sum identities. Since
V (P ) =
l∑
s=0
Ωs ⊗ L
(
Λs −
s
4
δ
)
,
we have the specialized character
chqV (P ) =
l∑
s=0
chqΩs chqL
(
Λs −
s
4
δ
)
,
the left-hand side is ∑
n1,...,nl∈Z
q
1
2 (ln2
1−n1+ln2
2−3n2+...+ln2
l−(2l−1)nl)∏∞
n=1
(1− qln)l−1(1− q2ln)
which equals
q−l
2/8[κq1/2(l, 1)κq1/2(l, 3), . . . , κq1/2(l, l − 1)]2∏∞
n=1
(1− qln)l−1(1− q2ln)
= q−l
2/8
∞∏
n=1
(1 + qn−1/2)2
1− qln
for even l, and equals
q−
l2−1
8 [κq1/2(l, 1)κq1/2(l, 3), . . . , κq1/2(l, l − 2)]2κq1/2(l, l)∏∞
n=1
(1− qln)l−1(1− q2ln)
= 2q−
l2−1
8
∞∏
n=1
(1− q2ln−l)
(1− q2n−1)2
for odd l. Where κq is defined by Eqs. (4.1) and (4.2).
The right-hand side is
l∑
s=0
chqΩs chqL
(
Λs −
s
4
δ
)
=
l∑
s=0
q
(l−s)s
2 chqΩs dimq L(Λs).
Then by the computation of Ωs and dimq L(Λs) before, the proof for our main theorems is finished.
1. Andrews G. E. q-series: their development and application in analysis, number theory, combinatorics, physics and
computer algebra // CBMS Reg. Conf. Ser. Math. – Providence, RI: Amer. Math. Soc., 1986. – 66.
2. Bressoud D. M. Analytic and combinational generalizations of the Rogers – Ramanujan identities // Mem. Amer.
Math. Soc. – 1980. – 24, № 227.
3. Carlitz L., Subbarao M. V. A simple proof of the quintuple product identity // Proc. Amer. Math. Soc. – 1978. – 32. –
P. 42 – 44.
4. Frenkel I. B., Lepowsky J., Meurman A. Vertex operator algerbas and the monster. – Boston: Acad. Press, 1989.
5. Gao Y. Vertex operators arising from the homogenous realization for ĝlN // Communs Math. Phys. – 1994. – 159. –
P. 1 – 13.
6. Jing N., Xia L. Representations of affine Lie algebras and product-sum identities // J. Algebra. – 2007. – 314, № 2. –
P. 538 – 552.
7. Kac V. G. Infinite-dimensional Lie algebras. – 3 rd ed. – Cambridge, U.K.: Cambridge Univ. Press, 1990.
8. Lepowsky J., Wilson R. The structure of standard modules. I. Universal algebras and the Rogers – Ramanujan identities
// Invent. Math. – 1984. – 77. – P. 199 – 290.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
VERTEX OPERATOR REPRESENTATIONS OF TYPE C
(1)
l AND PRODUCT-SUM IDENTITIES 243
9. Lepowsky J., Wilson R. The structure of standard modules. II. The case A
(1)
1 , principal gradation // Invent. Math. –
1985. – 79. – P. 417 – 442.
10. Liu D., Hu N. Vertex representations of the toroidal Lie algebra of type G
(1)
2 // J. Pure and Appl. Algebra. – 2005. –
198. – P. 257 – 279.
11. Misra K. C. Realization of level one standard C̃2k+1-modules // Trans. Amer. Math. Soc. – 1990. – 321, № 2. –
P. 483 – 504.
12. Schur I. Zur additiven Zahlentheorie // S. B. Preuss. Akad. Wiss. Phys. Math. Kl. – 1926. – S. 488 – 495.
13. Xia L., Hu N. Irreducible representations for Virasoro-toroidal Lie algebras // J. Pure and Appl. Algebra. – 2004. –
194. – P. 213 – 237.
Received 30.12.11,
after revision — 15.11.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 2
|
| id | nasplib_isofts_kiev_ua-123456789-165508 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T18:16:02Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Li-Meng, Xia Naihong, Hu 2020-02-13T20:22:11Z 2020-02-13T20:22:11Z 2014 Vertex operator representations of type Cl⁽¹⁾ and product-sum identities / Li-Meng Xia, Naihong Hu // Український математичний журнал. — 2014. — Т. 66, № 2. — С. 226–243. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165508 512.5 We construct a class of homogeneous vertex representations of Cl⁽¹⁾, l ≥ 2, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory. Побудовано клас рівномірних вершинних зображень Cl⁽¹⁾, l ≥ 2. Отримано низку тотожностей типу сум i добутків. Ці тотожності мають змістовну інтерпретацію теорії чисел. Supported by the NNSF of China (Grant No. 11001110) and Jiangsu Government Scholarship for Overseas Studies.
 Supported in part by the NNSF of China (Grant No. 11271131), the PCSIRT and the RFDP from the MOE of China,
 the National & Shanghai Leading Academic Discipline Projects (Project Number B407). en Інститут математики НАН України Український математичний журнал Статті Vertex operator representations of type Cl⁽¹⁾ and product-sum identities Вершинні операторні зображення типу Cl⁽¹⁾ та тотожності типу сум i добутків Article published earlier |
| spellingShingle | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities Li-Meng, Xia Naihong, Hu Статті |
| title | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities |
| title_alt | Вершинні операторні зображення типу Cl⁽¹⁾ та тотожності типу сум i добутків |
| title_full | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities |
| title_fullStr | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities |
| title_full_unstemmed | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities |
| title_short | Vertex operator representations of type Cl⁽¹⁾ and product-sum identities |
| title_sort | vertex operator representations of type cl⁽¹⁾ and product-sum identities |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/165508 |
| work_keys_str_mv | AT limengxia vertexoperatorrepresentationsoftypecl1andproductsumidentities AT naihonghu vertexoperatorrepresentationsoftypecl1andproductsumidentities AT limengxia veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív AT naihonghu veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív |