Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities
We construct a class of homogeneous vertex representations of $C_l^{(1)},\; l ≥ 2$, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory.
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2014
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2126 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508059160281088 |
|---|---|
| author | Hu, Naihong Xia, Li-Meng Гу, Наіхонг Хія, Лі-Менг |
| author_facet | Hu, Naihong Xia, Li-Meng Гу, Наіхонг Хія, Лі-Менг |
| author_sort | Hu, Naihong |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:24:43Z |
| description | We construct a class of homogeneous vertex representations of $C_l^{(1)},\; l ≥ 2$, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory. |
| first_indexed | 2026-03-24T02:19:11Z |
| 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 | umjimathkievua-article-2126 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:19:11Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/89/0031fe432955f026ce70947d31a91a89.pdf |
| spelling | umjimathkievua-article-21262019-12-05T10:24:43Z Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities Вершинні операторні зображення типу $C_l^{(1)}$ та тотожності типу сум i добутків Hu, Naihong Xia, Li-Meng Гу, Наіхонг Хія, Лі-Менг We construct a class of homogeneous vertex representations of $C_l^{(1)},\; l ≥ 2$, and deduce a series of product-sum identities. These identities have fine interpretation in the number theory. Побудовано клас рівномірних вершинних зображень $C_l^{(1)},\; l ≥ 2$. Отримано низку тотожностей типу сум i добутків. Ці тотожності мають змістовну інтерпретацію теорії чисел. Institute of Mathematics, NAS of Ukraine 2014-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2126 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 2 (2014); 226–243 Український математичний журнал; Том 66 № 2 (2014); 226–243 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2126/1254 https://umj.imath.kiev.ua/index.php/umj/article/view/2126/1255 Copyright (c) 2014 Hu Naihong; Xia Li-Meng |
| spellingShingle | Hu, Naihong Xia, Li-Meng Гу, Наіхонг Хія, Лі-Менг Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title | Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title_alt | Вершинні операторні зображення типу $C_l^{(1)}$ та тотожності типу сум i добутків |
| title_full | Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title_fullStr | Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title_full_unstemmed | Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title_short | Vertex Operator Representations of Type $C_l^{(1)}$ and Product-Sum Identities |
| title_sort | vertex operator representations of type $c_l^{(1)}$ and product-sum identities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2126 |
| work_keys_str_mv | AT hunaihong vertexoperatorrepresentationsoftypecl1andproductsumidentities AT xialimeng vertexoperatorrepresentationsoftypecl1andproductsumidentities AT gunaíhong vertexoperatorrepresentationsoftypecl1andproductsumidentities AT híâlímeng vertexoperatorrepresentationsoftypecl1andproductsumidentities AT hunaihong veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív AT xialimeng veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív AT gunaíhong veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív AT híâlímeng veršinníoperatornízobražennâtipucl1tatotožnostítipusumidobutkív |