Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks
We study certain representations of the quantum toroidal 𝖌𝔩₁ algebra for 𝑞 = 𝘵. We construct explicit bosonization of the Fock modules 𝓕⁽ⁿ′'ⁿ⁾ᵤ with a nontrivial slope 𝑛′/𝑛. As a vector space, it is naturally identified with the basic level 1 representation of affine 𝖌𝔩ₙ. We also study twisted...
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Інститут математики НАН України
2020
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| Цитувати: | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks. Mikhail Bershtein and Roman Gonin. SIGMA 16 (2020), 077, 55 pages |
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| author_facet | Bershtein, Mikhail Gonin, Roman |
| citation_txt | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks. Mikhail Bershtein and Roman Gonin. SIGMA 16 (2020), 077, 55 pages |
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| description | We study certain representations of the quantum toroidal 𝖌𝔩₁ algebra for 𝑞 = 𝘵. We construct explicit bosonization of the Fock modules 𝓕⁽ⁿ′'ⁿ⁾ᵤ with a nontrivial slope 𝑛′/𝑛. As a vector space, it is naturally identified with the basic level 1 representation of affine 𝖌𝔩ₙ. We also study twisted 𝑊-algebras of s𝔩n acting on these Fock modules. As an application, we prove the relation on 𝑞-deformed conformal blocks, which was conjectured in the study of 𝑞-deformation of isomonodromy/CFT correspondence.
|
| first_indexed | 2026-03-14T16:09:49Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 077, 55 pages
Twisted Representations of Algebra
of q-Difference Operators, Twisted q-W Algebras
and Conformal Blocks
Mikhail BERSHTEIN †1†2†3†4†5 and Roman GONIN †2†3
†1 Landau Institute for Theoretical Physics, Chernogolovka, Russia
E-mail: mbersht@gmail.com
†2 Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia
E-mail: roma-gonin@yandex.ru
†3 National Research University Higher School of Economics, Moscow, Russia
†4 Institute for Information Transmission Problems, Moscow, Russia
†5 Independent University of Moscow, Moscow, Russia
Received November 22, 2019, in final form August 01, 2020; Published online August 16, 2020
https://doi.org/10.3842/SIGMA.2020.077
Abstract. We study certain representations of quantum toroidal gl1 algebra for q = t. We
construct explicit bosonization of the Fock modules F (n′,n)
u with a nontrivial slope n′/n. As
a vector space, it is naturally identified with the basic level 1 representation of affine gln.
We also study twisted W -algebras of sln acting on these Fock modules. As an application,
we prove the relation on q-deformed conformal blocks which was conjectured in the study
of q-deformation of isomonodromy/CFT correspondence.
Key words: quantum algebras; toroidal algebras; W -algebras; conformal blocks; Nekrasov
partition function; Whittaker vector
2020 Mathematics Subject Classification: 17B67; 17B69; 81R10
1 Introduction
Toroidal algebra. Representation theory of quantum toroidal algebras has been actively de-
veloped in recent years. This theory has numerous applications, including geometric representa-
tion theory and AGT relation [43], topological strings [1], integrable systems, knot theory [28],
and combinatorics [13].
In this paper we consider only the quantum toroidal gl1 algebra; we denote it by Uq,t
(
g̈l1
)
.
The algebra depends on two parameters q, t and has PBW generators Ek,l, (k, l) ∈ Z2 and central
generators c′, c [12]. In the main part of the text we consider only the case q = t, where toroidal
algebra becomes the universal enveloping of the Lie algebra with these generators Ek,l, c
′, c and
the relation
[Ek,l, Er,s] =
(
q(sk−lr)/2 − q(lr−sk)/2
)
Ek+r,l+s + δk,−rδl,−s(c
′k + cl).
We denote this Lie algebra by Diffq, since there is a homomorphism from this algebra to the
algebra of q-difference operators generated by D, x with the relation Dx = qxD; namely Ek,l 7→
qkl/2xlDk.
There is another presentation of the algebra Diffq (and more generally Uq,t
(
g̈l1
)
) using the
Chevalley generators E(z) =
∑
k∈Z
E1,kz
−k, F (z) =
∑
k∈Z
E−1,kz
−k, H(z) =
∑
k 6=0
E0,kz
−k, see,
e.g., [47].
mailto:mbersht@gmail.com
mailto:roma-gonin@yandex.ru
https://doi.org/10.3842/SIGMA.2020.077
2 M. Bershtein and R. Gonin
In this paper we deal with the Fock representations of Diffq; to be more precise there is
a family Fu of Fock modules, depending on the parameter u (see Proposition 3.1 for a construc-
tion of Fu). They are just Fock representations of the Heisenberg algebra generated by E0,k.
The images of E(z) and F (z) are vertex operators. A construction of this type is usually called
bosonization.
It was shown in [18, 43] that the image of toroidal algebra Uq,t
(
g̈l1
)
in the endomorphisms of
the tensor product of n Fock modules is the deformed W -algebra for gln. There is the so-called
conformal limit q, t → 1, in which deformed W -algebras go to vertex algebras. These vertex
algebras are tensor products of the Heisenberg algebra and the W -algebras of sln. In the case
q = t, the central charge of the corresponding W -algebra of sln is equal to n − 1. These W -
algebras appear in the study of isomonodromy/CFT correspondence (see [23, 24]). This is one
of the motivations of our paper.
The q-deformation of the isomonodromy/CFT correspondence was proposed in [9, 11, 31].
The main statement is an explicit formula for the q-isomonodromic tau function as an infinite
sum of conformal blocks for certain deformed W -algebras with q = t. In general, these tau
functions are complicated, but there are special cases (corresponding to algebraic solutions)
where these tau functions are very simple [5, 9]. These cases should correspond to special
representations of q-deformed W -algebras. The construction of such representation is one of the
purposes of this paper.
Twisted Fock modules. There is a natural action of SL(2,Z) on Diffq. We will parametrize
σ ∈ SL(2,Z) by
σ =
(
m′ m
n′ n
)
.
Then σ acts as
σ(Ek,l) = Em′k+ml,n′k+nl, σ(c′) = m′c′ + n′c, σ(c) = mc′ + nc.
For any Diffq module M and σ ∈ SL(2,Z), we denote by Mσ the module twisted by the
automorphism σ (see Definition 2.5). The twisted Fock modules depend only on n and n′ (up to
isomorphism). These numbers are the values of the central generators c and c′, correspondingly,
acting on Fσu . Therefore we will also use the notation F (n′,n)
u for Fσu . Twisted Fock modules Fσu
(for generic q, t) were used, for example, in [1] and [29].
In Section 4 we construct explicit bosonization of the twisted Fock modules Fσu for q = t.
Actually, we give three constructions: the first one in terms of n-fermions (see Theorem 4.1),
the second one in terms of n-bosons (see Theorem 4.3) and the third one in terms of one twisted
boson (see Theorem 4.4) (here, for simplicity, we assume that n > 0). In other words, any
twisted Fock module will be identified with the basic module for ĝln; these two bosonizations
correspond to homogeneous [21] and principal [32, 35] constructions.
The construction of the bosonization is nontrivial, because it is given in terms of Chevalley
generators (note that the SL(2,Z) action is not easy to describe in terms of Chevalley gener-
ators). The appearance of affine gln is in agreement with the Gorsky–Neguţ conjecture [29].
More specifically, it was conjectured in [29] that there exists an action (with certain properties)
of Up1/2
(
ĝln
)
on Fσu for p = q/t 6= 1; we expect this to be p-deformation of the ĝln-action
constructed in this paper.
It is instructive to look at the formulas in the simplest examples. For simplicity, we give
here only formulas for E(z). Here we introduce the notation in a sloppy way (for details see
Sections 3 and 4).
Example 1.1. In the standard case n = 1, n′ = 0 we have
E(z) = uq−1/2zψ
(
q−1/2z
)
ψ∗
(
q1/2z
)
=
u
1− q
: exp
(
φ
(
q1/2z
)
− φ
(
q−1/2z
))
:,
Twisted Representations of Algebra of q-Difference Operators 3
where ψ(z), ψ∗(z) are complex conjugate fermions (see Section 3.2), φ(z) =
∑
j 6=0
a[j]z−j/j is
a boson and a[j] are generators of the Heisenberg algebra with relation [a[j], a[j′]] = jδj+j′,0 (see
Section 3.1).
Example 1.2. The first nontrivial case is given by n = 2, n′ = 1. We have three formulas
(corresponding to Theorems 4.1, 4.3 and 4.4):
E(z) = u
1
2 q−
1
4
(
z2ψ(0)
(
q−1/2z
)
ψ∗(1)
(
q1/2z
)
+ zψ(1)
(
q−1/2z
)
ψ∗(0)
(
q1/2z
))
, (1.1)
E(z) = u
1
2 q−
1
4
(
z2 : exp
(
φ1
(
q1/2z
)
− φ0
(
q−1/2z
))
:
+ z : exp
(
φ0
(
q1/2z
)
− φ1
(
q−1/2z
))
:
)
(−1)a0[0], (1.2)
E(z) =
z
1
2u
1
2
2
(
1− q
1
2
)
: exp
∑
k 6=0
q−k/4 − qk/4
k
akz
−k/2
:
− : exp
∑
k 6=0
(−1)k
q−k/4 − qk/4
k
akz
−k/2
:
. (1.3)
Here ψ(0)(z), ψ
∗
(0)(z) and ψ(1)(z), ψ
∗
(1)(z) are anticommuting pairs of complex conjugate fermions
(see Section 4.1), φb(z) =
∑
j 6=0
ab[j]z
−j/j+Q+ab[0] log z are commuting bosons, and ab[j] are gen-
erators of the Heisenberg algebra with the relation [ab[j], ab′ [j
′]] = jδj+j′,0δb,b′ (see Section 4.2).
The generators ak in (1.3) satisfy [ak, ak′ ] = kδk+k′,0.
The relation between (1.1) and (1.2) is a standard boson-fermion correspondence. In the
right-hand side of formula (1.3) we have only one Heisenberg algebra with generators ak, but
since we have both integer and half-integer powers of z, one can think that we have a boson with
a nontrivial monodromy. This is the reason for the term ‘twisted boson’; we will also call this
construction strange bosonization. Note that half-integer powers of z cancel in the right-side
of (1.3).
We present two different proofs of Theorems 4.1, 4.3 and 4.4. The first one is given in
Section 5 and is based on the following idea. For any full rank sublattice Λ ∈ Z2 of index n,
we have a subalgebra DiffΛ
q1/n
⊂ Diffq1/n , which is spanned by Ea,b for (a, b) ∈ Λ and central
elements c, c′. The algebra DiffΛ
q1/n
is isomorphic to Diffq; the isomorphism depends on the
choice of a positively oriented basis v1, v2 in Λ. Denote this isomorphism by φv1,v2 .
If the basis v1, v2 is such that v1 = (N, 0), v2 = (R, d), then the restriction of the Fock
module Fu on φv1,v2(Diffq) is isomorphic to the sum of tensor products of the Fock modules
Fu1/N |φv1,v2 (Diffq)
∼=
⊕
l∈Q(d)
Fuqrl0 ⊗ · · · ⊗ Fuqr(αn+lα) ⊗ · · · ⊗ F
uq
r( d−1
d
+ld−1) (1.4)
where r = gcd(N,R) and Q(d) = {(l0, . . . , ld−1) ∈ Zd |
∑
li = 0}. If we choose basis w1, w2 in
Λ which differs from v1, v2 by σ ∈ SL(2,Z), we get an analogue of decomposition (1.4) with
right-hand side given by a sum of tensor products of the twisted Fock modules. For the basis
w1 = (r, ntw), w2 = (0, n), we write formulas for Chevalley generators of Diffq = DiffΛ
q1/n
using
either initial fermion or initial boson for Fu. Applying this for the lattices with d = 1, we get
Theorems 4.1, 4.3 and 4.4.
The secondproof of these theorems is based on the semi-infinite construction. Let Vu denote
the representation of the algebra Diffq in a vector space with basis xk−α for k ∈ Z, where
Diffq acts as q-difference operators (see Definition 3.8). This representation is called vector
4 M. Bershtein and R. Gonin
(or evaluation) representation; the parameter u is equal to q−α. The Fock module Fu is iso-
morphic to Λ∞/2+0(Vu) ⊂ Λ
∞/2
(Vu). After the twist, we get a semi-infinite construction of
Fσu ⊂
(
Λ
∞/2
Vu
)σ
= Λ
∞/2
(V σ
u ). Note that conjecturally the semi-infinite construction of Fσu can
be generalized for q 6= t (cf. [15]).
Twisted W -algebras. Denote by Diff>0
q the subalgebra of Diffq generated by c and Ea,b,
for a > 0. There is an another set of generators Ek[j] of the completion of the U
(
Diff>0
q
)
, defined
by the formula
∑
j∈Z
Ek[j]z−j = (E(z))k (see Appendix A for the definition of the power of E(z)).
The currents H(z) and Ek(z) for k ∈ Z>0 satisfy relations of the q-deformed W -algebra of gl∞
(see [43]). We denote this algebra by Wq(gl∞).
There is an ideal J>0
µ,d in U
(
Diff>0
q
)
= Wq(gl∞) which acts by zero on any tensor product
Fu1 ⊗ · · · ⊗ Fud , here µ = 1
1−q (u1 · · ·ud)1/n. This ideal is generated by relations c = d and
Ed(z) = µdd! exp(ϕ−(z)) exp(ϕ+(z)),
where
ϕ−(z) =
∑
j>0
q−j/2 − qj/2
j
E0,−jz
j , ϕ+(z) = −
∑
j>0
qj/2 − q−j/2
j
E0,jz
−j .
The quotient of Wq(gl∞)/J>0
µ,d is the q-deformed W -algebra of gld. We denote this algebra
by Wq(gld); it does no depend on µ (up to isomorphism) and acts on any tensor product
Fu1 ⊗ · · · ⊗ Fud (see [19, 43]).
In Section 7 we study a tensor product of the twisted Fock modules Fσu1 ⊗ · · · ⊗ F
σ
ud
. We
prove that the ideal J>0
µ,nd,n′d generated by relations c = nd and
End(z) = zn
′dµnd(nd)! exp(ϕ−(z)) exp(ϕ+(z))
acts by zero for µ = (−1)1/n q−1/2n
q1/2−q−1/2 (u1 · · ·ud)1/nd.We denote the quotient Wq(gl∞)/J>0
µ,nd,n′d
by Wq(glnd, n
′d) and call it the twisted q-deformed W -algebra of glnd.
There exists another description of the above using the q-deformed W -algebra of sln intro-
duced in [17]. Define Tk[j] by the formula
Tk(z) =
∑
Tk[j]z
−j =
µ−k
k!
exp
(
−k
c
ϕ−(z)
)
Ek(z) exp
(
−k
c
ϕ+(z)
)
.
The generators Tk[j] are elements of a localization of the completion of U
(
Diff>0
q
)
. These
generators commute with Hi and satisfy certain quadratic relations. The algebra generated
by Tk[j] is denoted by Wq(sl∞).
There is an ideal in Wq(sl∞) which acts by zero on any tensor product Fu1 ⊗ · · · ⊗Fud . This
ideal contains relations c = d, Td(z) = 1, and Td+k(z) = 0 for k > 0. The quotient is a standard
W -algebraWq(sld) [17] (see also Definition 7.1). We have a relationWq(gld) =Wq(sld)⊗U(Heis),
where Heis is the Heisenberg algebra generated by E0,j .
In the case of a product of the twisted Fock modules Fσu1 ⊗ · · · ⊗ F
σ
ud
the situation is similar.
The corresponding ideal contains the relations Tnd(z) = zn
′d, Tnd+k(z) = 0 for k > 0. We present
the quotient in terms of the generators T1(z), . . . , Tnd(z) and relations (this is Theorem 7.7).
We call the algebra with such generators and relations by twisted W -algebra Wq(slnd, n
′d); see
Definition 7.3.1 The quadratic relations in the algebra Wq(slnd, n
′d) are the same as in the
untwisted case (see equation (7.1)–(7.2)), the only difference lies in the relation Tnd(z) = zn
′d.
1One can find a definition of Wq,p(sl2, 1) in [44, equations (37)–(38)].
Twisted Representations of Algebra of q-Difference Operators 5
The algebra Wq(slnd, n
′d) is graded, with deg Tk[j] = j + n′k
n . Let us rename the generators
by T twk [r] = Tk
[
r− n′k
n
]
, for r ∈ n′k
n +Z. The presentations of the algebraWq(slnd, n
′d) in terms
of generators T twk [r] and the presentations of the algebra Wq(slnd) is terms of generators Tk[r]
are given by the same formulas; the only difference is the region of r. Heuristically, one can
think that Wq(slnd, n
′d) is the same algebra as Wq(slnd) but with currents having nontrivial
monodromy around zero.
In order to explain these results in more details, consider an example of sl2.
Example 1.3. As a warm-up, consider the untwisted case n′ = 0. The algebra Wq(sl2) is
q-deformed Virasoro algebra [45]. It has one generating current T (z) = T1(z) and the relation
reads
∞∑
l=0
f [l]
(
T [r−l]T [s+l]− T [s−l]T [r+l]
)
= −2r
(
q
1
2−q−
1
2
)2
δr+s,0, (1.5)
where f [l] are coefficients of a series
∞∑
l=0
f [l]xl =
√
(1− qx)
(
1− q−1x
)
/(1−x). This algebra has
a standard bosonization [45]
T (z) = −
(
q
1
2 − q−
1
2
)
z
×
[
u : exp
(
η(q1/2z
)
− η
(
q−1/2z
))
: +u−1 : exp
(
η
(
q−1/2z
)
− η
(
q1/2z
))
:
]
, (1.6)
where η(z) =
∑
k 6=0
η[k]z−k/k and η[k] are the generators of the Heisenberg algebra [η[k1], η[k2]] =
1
2k1δk1+k2,0; one can also add η[0] related to the parameter u. In terms of the toroidal algebra
Diffq this formula corresponds to the tensor product of two Fock modules Fu1 ⊗ Fu2 , here
u2 = u1/u2.
Example 1.4. Now, consider the twisted case n′ = 1. The algebra Wq(sl2, 1) is generated
by one current T tw(z) = T tw1 (z) =
∑
r∈Z+1/2
T tw1 [r]z−r. The generators T tw[r] = T tw1 [r] satisfy
relation (1.5). The algebra Wq(sl2, 1) is called twisted q-deformed Virasoro algebra.
As was explained above, the representations of Wq(sl2, 1) come from the twisted Fock modu-
les F (1,2)
u .The bosonization of the twisted Fock module leads to the bosonization of theWq(sl2, 1).
Using formula (1.2) we get a bosonization
T tw(z) =
(
q
1
2 − q−
1
2
)
×
[
z1/2 : exp
(
η
(
q1/2z
)
+ η
(
q−1/2z
))
: +z3/2 : exp
(
−η
(
q1/2z
)
− η
(
q−1/2z
))
:
]
.
Using formula (1.3) we get a strange bosonization
T tw(z) = (−1)
1
2
q
1
2 − q−
1
2
2
(
q
1
4 − q−
1
4
)z 1
2
×
: exp
∑
2-r
q−
r
4 − q
r
4
r
Jrz
− r
2
− : exp
∑
2-r
q
r
4 − q−
r
4
r
Jrz
− r
2
:
.
Here η(z) =
∑
k 6=0
η[k]z−k/k + Q + η[0] log z, and Jr are modes of the odd Heisenberg algebra,
[Jr, Js] = rδr+s,0. These formulas for bosonization are probably new.
6 M. Bershtein and R. Gonin
Example 1.5. One can also use embedding DiffΛ
q1/n
⊂ Diffq1/n in order to construct a bosoniza-
tion of the W -algebras. Namely one can take a representation of Diffq1/n with known bosoniza-
tion and then express the W -algebra related to Diffq = DiffΛ
q1/n
in terms of these bosons.
For example, consider Λ generated by v1 = e1, v2 = 2e2 and the Fock representations Fu1/2
of Diffq1/2 . One can show (for example, using (1.4)) that Wq(gl∞) algebra related to Diffq
∼=
DiffΛ
q1/2
acts on Fu1/2 through the quotient Wq(gl2). Therefore, we get an odd bosonization of
non-twisted q-deformed Virasoro algebra Wq(sl2)
T (z) =
q
1
4 + q−
1
4
2
: exp
∑
2-r
q−
r
4 − q
r
4
r
Jrz
− r
2
: + :exp
∑
2-r
q
r
4 − q−
r
4
r
Jrz
− r
2
:
. (1.7)
Here Jr are the odd modes of the initial boson for Fu. The even modes of the boson disappear
in the formula since it belongs to Heis ⊂ DiffΛ
q1/2
.
It follows from the decomposition (1.4) that formula (1.7) gives bosonization of certain special
representationWq(sl2), to be more specific, a direct sum of Fock modules (defined by (1.6)) with
particular parameters u = ql−1/4 for l ∈ Z.
In the conformal limit q → 1 formula (1.7) goes to the odd bosonization of the Virasoro
algebra Lk = 1
4
∑
1
2
(r+s)=k
:JrJs : + 1
16δk,0, see, e.g., [48].
Whittaker vectors and relations on conformal blocks. As an application, in Section 9
we prove the following identity
z
1
2
∑ i2
n2
∏
i 6=j
1(
q1+ i−j
n ; q, q
)
∞
(
q
1
n z
1
n ; q
1
n , q
1
n
)
∞
=
∑
(l0,...,ln−1)∈Q(n)
Z
(
ql0 , q
1
n
+l1 , . . . , q
n−1
n
+ln−1 ; z
)
. (1.8)
Here the lattice Q(n) is as above, (u; q, q)∞ =
∞∏
i,j=0
(
1 − qi+ju
)
. The function Z(u1, . . . , un; z)
is a Whittaker limit of conformal block. By AGT relation it equals to the Nekrasov partition
function. We recall the definition of Z(u1, . . . , un; z) below.
The relation (1.8) was conjectured in [5] in the framework of q-isomonodromy/CFT corre-
spondence. As we discussed in the first part of the introduction the main statement of this
correspondence is an explicit formula for the q-isomonodromic tau function as an infinite sum
of conformal blocks. The left-hand side of (1.8) is a tau function corresponding to the algebraic
solution of deautonomized discrete flow in Toda system, see [5, equation (3.11)]. The right-hand
side of (1.8) is a specialization of conjectural formula [5, equation (3.6)] for the generic tau
function of these flows. In differential case the isomonodromy/CFT correspondanse is proven
in many cases, see [7, 25, 26, 30], but in the q-difference case the main statements are still con-
jectures. The generic formula for tau function of deautonomized discrete flow in Toda system
is proven only for particular case n = 2 [10, 37]. Here we prove formula for arbitrary n but for
special solution.
Let us recall the definition of Z(u1, . . . , un; z). The Whittaker vector W (z|u1, . . . , uN ) is
a vector in a completion of Fu1 ⊗ · · · ⊗ Fun , which is an eigenvector of Ea,b for Nb > a > 0
with certain eigenvalues depending on z, see Definition 9.2. Such vector exists and unique for
generic values of u1, . . . , un. This property looks to be a part of folklore, we give a proof of
this in Appendix D. The proof is essentially based on the results of [42, 43]. The function Z is
proportional to a Shapovalov pairing of two Whittaker vectors
Z(u1, . . . , un; z) = z
∑
(log ui)
2
2(log q)2
∏
i 6=j
1(
quiu
−1
j ; q, q
)
∞
〈
Wu
(
1|qu−1
n , . . . , qu−1
1
)
,W (z|u1, . . . , un)
〉
.
Twisted Representations of Algebra of q-Difference Operators 7
We give a proof of (1.8) using decomposition (1.4). We consider the Whittaker vector W (z|1)
for the algebra Diffq1/n . Its Shapovalov pairing gives the left-hand side of the relation (1.8). On
the other hand, we prove that its restriction to summands Fql0⊗· · ·⊗Fq n−1
n +ln−1
is the Whittaker
vector for the algebra Diffq . So taking the Shapovalov pairing we get the right-hand side of the
relation (1.8).
In the conformal limit q → 1 the analogue of the relation (1.8) in case n = 2 was proven in [8]
by a similar method. The conformal limit of the decomposition (1.4) was studied in [4].
Discussion of q 6= t case. As we mentioned above, Diffq is a specialization of quantum
toroidal algebra Uq,t
(
g̈l1
)
for q = t. It is much more interesting to study the algebra without
the constrain. Let us discuss our expectations on generalizations of the results from this paper.
It is likely that fermionic construction (see Theorem 4.1) will be generalized after the replace-
ment of the fermions by vertex operators of quantum affine gln. Hence we have bosonization,
expressing the currents in terms of exponents dressed by screenings. We also expect that rep-
resentations of twisted and non-twisted Wn-algebras can be realized via these vertex operators
(see [6] for the n = 2 case). It is not clear how one can generalize strange bosonization and
connection with isomonodromy/CFT correspondence for q 6= t.
Plan of the paper. The paper is organized as follows.
In Section 2 we recall basic definitions and properties on the algebra Diffq.
In Section 3 we recall basic constructions of the Fock module Fu.
In Section 4 we present three constructions of the twisted Fock module Fσu : the fermionic
construction in Theorem 4.1, the bosonic construction in Theorem 4.3, and the strange bosonic
construction in Theorem 4.4.
In Section 5 we study restriction of the Fock module to a subalgebra DiffΛq . Using these
restrictions we prove Theorems 4.1, 4.3 and 4.4.
In Section 6 we give an independent proof of Theorem 4.1 using the semi-infinite construction.
In Section 7 we study twisted q-deformed W -algebras. We define Wq(sln, ntw) by generators
and relations. Then we show in Theorem 7.7 that the tensor product Wq(sln, ntw) ⊗ U(Heis)
is isomorphic to the certain quotient of U(Diffq); we denote this quotient by Wq(gln, ntw). We
show that Wq(slnd, n
′d) acts on the tensor product of twisted Fock modules Fσu1 ⊗ · · · ⊗ F
σ
ud
.
At the end of the section we study relation between these modules and the Verma modules for
Wq(glnd, n
′d) and Wq(slnd, n
′d).
In Section 8 we prove decomposition (1.4). Then we study the strange bosonization of W -
algebra modules arising from the restriction of Fock module on DiffΛq .
In Section 9 we recall definitions and properties of Whittaker vector, Shapovalov pairing, and
conformal blocks. Then we prove (1.8), see Theorem 9.30.
In Appendix A we give a definition and study necessary properties of regular product of
currents A(z)B(az) for a ∈ C.
Appendices B and C consist of calculations which are used in Section 7.
In Appendix D we study the Whittaker vector for Diffq in the completion of the tensor
product Fu1 ⊗ · · · ⊗ Fun . We prove its existence and uniqueness (we use this in Section 9).
To prove existence we present a construction of Whittaker vector via an intertwiner operator
from [1]. We also relate this Whittaker vector to the Whittaker vector of Wq(sln) introduced
in [46].
2 q-difference operators
In this section we introduce notation and recall basic facts about algebra Diffq, see [14, 27, 33].
Definition 2.1. The associative algebra of q-difference operators Diff A
q is an associative algebra
generated by D±1 and x±1 with the relation Dx = qxD.
8 M. Bershtein and R. Gonin
Definition 2.2. The algebra of q-difference operators Diffq is a Lie algebra with a basis Ek,l
(where (k, l) ∈ Z2\{(0, 0)}), c and c′. The elements c and c′ are central. All other commutators
are given by
[Ek,l, Er,s] =
(
q(sk−lr)/2 − q(lr−sk)/2
)
Ek+r,l+s + δk,−r δl,−s(c
′k + cl). (2.1)
Remark 2.3. Note that the vector subspace of Diff A
q spanned by xlDk (for (l, k) 6= (0, 0)) is
closed under commutation, i.e., has a natural structure of Lie algebra (denote this Lie algebra
by DiffL
q ). Consider a basis of this Lie algebra Ek,l := qkl/2xlDk. Finally, Diffq is a central
extension of DiffL
q by two-dimensional abelian Lie algebra spanned by c and c′.
2.1 SL2(Z) action
In this section we will define action SL2(Z) on Diffq. Let σ be an element of SL2(Z) corresponding
to a matrix
σ =
(
m′ m
n′ n
)
.
Then σ acts as follows
σ(Ek,l) = Em′k+ml,n′k+nl, σ(c′) = m′c′ + n′c, σ(c) = mc′ + nc. (2.2)
Proposition 2.4. Formula (2.2) defines SL2(Z) action on Diffq by Lie algebra automorphisms.
Proof. Note that (2.1) is SL2(Z) covariant. �
For any Diffq-module M denote by ρM : Diffq → gl(M) the corresponding homomorphism.
Definition 2.5. For any Diffq-module M and σ ∈ SL(2,Z) let us define the representation Mσ
as follows. M and Mσ are the same vector space with different actions, namely ρMσ = ρM ◦ σ.
We will refer to Mσ as a twisted representation. More precisely, Mσ is the representation M ,
twisted by σ.
2.2 Chevalley generators and relations
The Lie algebra Diffq is generated by Ek := E1,k, Fk := E−1,k and Hk := E0,k. We will call
them the Chevalley generators of Diffq. Define the following currents (i.e., formal power series
with coefficients in Diffq)
E(z) =
∑
k∈Z
E1,kz
−k =
∑
k∈Z
Ekz
−k,
F (z) =
∑
k∈Z
E−1,kz
−k =
∑
n∈Z
Fkz
−k,
H(z) =
∑
k 6=0
E0,kz
−k =
∑
k 6=0
Hkz
−k.
Let us also define the formal delta function
δ(x) =
∑
k∈Z
xk.
Twisted Representations of Algebra of q-Difference Operators 9
Proposition 2.6. Lie algebra Diffq is presented by the generators Ek, Fk (for all k ∈ Z), Hl
(for l ∈ Z\{0}), c, c′ and the following relations
[Hk, Hl] = kcδk+l,0, (2.3)
[Hk, E(z)] =
(
q−k/2 − qk/2
)
zkE(z), [Hk, F (z)] =
(
qk/2 − q−k/2
)
zkF (z), (2.4)
(z − qw)
(
z − q−1w
)
[E(z), E(w)] = 0, (z − qw)
(
z − q−1w
)
[F (z), F (w)] = 0, (2.5)
[E(z), F (w)] =
(
H
(
q−1/2w
)
−H
(
q1/2w
)
+ c′
)
δ(w/z) + c
w
z
δ′(w/z), (2.6)
z2z
−1
3 [E(z1), [E(z2), E(z3)]] + cyclic = 0, (2.7)
z2z
−1
3 [F (z1), [F (z2), F (z3)]] + cyclic = 0. (2.8)
One can find a proof of Proposition 2.6 in [38, Theorem 2.1] or [47, Theorem 5.5].
3 Fock module
In this section we review basic constructions of representations of Diffq with c = 1 and c′ = 0.
These construction were studied in [27].
3.1 Free boson realization
Introduce the Heisenberg algebra generated by ak (for k ∈ Z) with relation [ak, al] = kδk+l,0.
Consider the Fock module F aα generated by |α〉 such that ak|α〉 = 0 for k > 0, a0|α〉 = α|α〉.
Proposition 3.1. The following formulas determine an action of Diffq on F aα :
c 7→ 1, c′ 7→ 0, Hk 7→ ak, (3.1)
E(z) 7→ u
1− q
exp
(∑
k>0
q−k/2 − qk/2
k
a−kz
k
)
exp
(∑
k<0
q−k/2 − qk/2
k
a−kz
k
)
, (3.2)
F (z) 7→ u−1
1− q−1
exp
(∑
k>0
qk/2 − q−k/2
k
a−kz
k
)
exp
(∑
k<0
qk/2 − q−k/2
k
a−kz
k
)
. (3.3)
We will denote this representation by Fu.
Remark 3.2. Note that α does not appear in formulas (3.1)–(3.3). But we will need operator
a0 later (see the proof of Proposition 3.12) for the boson-fermion correspondence. Heuristically,
one can think that u = q−α.
Remark 3.3 (on our notation). In this paper, we consider several algebras and their action on
the corresponding Fock modules. We choose the following notation. All these representations
are denoted by the letter F (for Fock) with some superscript to mention an algebra. Since Diffq
is the most important algebra in our paper, we use no superscript for its representation. Also,
let us remark that we consider several copies of the Heisenberg algebra. To distinguish their
Fock modules, we write a letter for generators as a superscript.
The standard bilinear form on F aα is defined by the following conditions: operator a−k is dual
of ak, the pairing of |α〉 with itself equals 1. We will use the bra-ket notation for this scalar
product. For an operator A we denote by 〈α|A|α〉 the scalar product of A|α〉 with |α〉.
Proposition 3.4. Suppose the algebra Diffq acts on F aα so that Hk 7→ ak and 〈α|E(z)|α〉 = u
1−q ;
〈α|F (z)|α〉 = u−1
1−q−1 . Then this representation is isomorphic to Fu.
10 M. Bershtein and R. Gonin
Proof. Consider the current
T (z) = exp
(
−
∑
k>0
q−k/2 − qk/2
k
a−kz
k
)
E(z) exp
(
−
∑
k<0
q−k/2 − qk/2
k
a−kz
k
)
.
It is easy to verify that [ak, T (z)] = 0. Since Fα is irreducible, T (z) = f(z) for some formal
power series f(z) with C-coefficients. On the other hand, f(z) = 〈α|E(z)|α〉 = u
1−q . This
implies (3.2). The proof of (3.3) is analogous. �
Proposition 3.5. Denote El(z) = El,kz
−k. The action of El(z) on Fock representation Fu is
given by the following formula
El(z)→
ul
1− ql
exp
(∑
k>0
q−kl/2 − qkl/2
k
a−kz
k
)
exp
(∑
k<0
q−kl/2 − qkl/2
k
a−kz
k
)
. (3.4)
Proof. The commutation relation (2.1) implies that formula (3.4) holds up to a pre-exponential
factor. Also, we see from (2.1) that
E(z)El(w) =
q−1w
z − q−1w
El+1
(
q−1w
)
− qlw
z − qlw
El+1(w) + reg. (3.5)
The factor can be found inductively from (3.5). �
3.2 Free fermion realization
In this section we give another construction for the Fock representation of Diffq. To do this, let
us consider the Clifford algebra, generated by ψi and ψ∗j for i, j ∈ Z subject to the relations
{ψi, ψj} = 0, {ψ∗i , ψ∗j } = 0, {ψi, ψ∗j } = δi+j,0.
Consider the currents
ψ(z) =
∑
i
ψiz
−i−1, ψ∗(z) =
∑
i
ψ∗i z
−i.
Consider a module Fψ with a cyclic vector |l〉 and relation
ψi|l〉 = 0 for i > l, ψ∗j |l〉 = 0 for j > −l.
The module Fψ is independent of l. The isomorphism can be seen from the formulas ψ∗−l|l〉 =
|l + 1〉 and ψl−1|l〉 = |l − 1〉. Let us define the l-dependent normal ordered product (to be
compatible with |l〉) by the following formulas
:ψiψ
∗
j :(l)= −ψ∗jψi for i > l, (3.6)
:ψiψ
∗
j :(l)= ψiψ
∗
j for i < l. (3.7)
Proposition 3.6. The following formulas determine an action of Diffq on Fψ:
c 7→ 1, c′ 7→ 0, Hk 7→
∑
i+j=k
ψiψ
∗
j , (3.8)
E(z) 7→ qlu
1− q
+ uq−1/2z :ψ
(
q−1/2z
)
ψ∗
(
q1/2z
)
:(l)= uq−1/2zψ
(
q−1/2z
)
ψ∗
(
q1/2z
)
, (3.9)
F (z) 7→ q−lu−1
1− q−1
+ u−1q1/2z :ψ
(
q1/2z
)
ψ∗
(
q−1/2z
)
:(l)= u−1q1/2zψ
(
q1/2z
)
ψ∗
(
q−1/2z
)
.(3.10)
Twisted Representations of Algebra of q-Difference Operators 11
Let us denote this representation by Mu.
Remark 3.7. The Products ψ
(
q−1/2z
)
ψ∗
(
q1/2z
)
and ψ
(
q1/2z
)
ψ∗
(
q−1/2z
)
from formulas (3.9)–
(3.10) are not normally ordered (see Appendix A for a formal definition and some other technical
details on the regular product). In particular, this reformulation implies that Mu does not
depend on l.
3.3 Semi-infinite construction
Definition 3.8. The evaluation representation Vu of the algebra Diffq is a vector space with
the basis xk for k ∈ Z and the action
Ea,bx
k = uaq
ab
2
+akxk+b, c = c′ = 0.
Remark 3.9. The associative algebra Diff A
q acts on Vu. The representation of Diffq is obtained
via evaluation homomorphism ev : Diffq → Diff A
q .
Remark 3.10. Informally, one can consider xk ∈ Vu as xk−α for u = q−α. Define the action of
Diff A
q as follows. The generator x acts by multiplication and Dxk−α = qk−αxk−α = uqkxk−α.
However, q−α is not well defined for arbitrary complex α. So we consider u as a parameter of
representation instead of α.
Let us consider the semi-infinite exterior power of the evaluation representation Λ
∞/2
Vu. It
is spanned by |λ, l〉 = xl−λ1 ∧ xl+1−λ2 ∧ · · · ∧ xl+N ∧ xl+N+1 ∧ xl+N+2 ∧ · · · , where λ is a Young
diagram and l ∈ Z. Let p1 > · · · > pi and q1 > · · · > qi be Frobenius coordinates of λ.
Proposition 3.11. There is a Diffq-modules isomorphism Λ
∞/2
Vu ∼−→Mu given by
|λ, l〉 7→ (−1)
∑
k(qk−1)ψ−p1+l · · ·ψ−pi+l ψ
∗
−qi−l+1 · · ·ψ∗−q1−l+1|l〉. (3.11)
Proposition 3.12. There is an isomorphism of Diffq-modules Mu
∼=
⊕
l∈ZFqlu. The sub-
module Fqlu is spanned by |λ, l〉.
Proof. Recall the ordinary boson-fermion correspondence (see [34]). The coefficients of
a(z) =
∑
n
anz
−n−1 =:ψ(z)ψ∗(z) :(0)
are indeed generators of the Heisenberg algebra. Moreover, Fψ = ⊕l∈ZF a−l. The highest vector
of F a−l is |l〉 (in particular, a0|l〉 = −l|l〉). Note that this is the decomposition of Diffq-modules
as well. Also, note that
〈l|E(z)|l〉 =
qlu
1− q
, 〈l|F (z)|l〉 =
q−lu−1
1− q−1
.
Therefore one can use Proposition 3.4 for each summand F a−l. �
There is a basis in the Fock module Fu given by semi-infinite monomials
|λ〉 = x−λ1 ∧ x1−λ2 ∧ · · · ∧ xi−λi+1 ∧ · · · .
To write the action of Diffq in this basis, let us remind the standard notation. Let l(λ) be the
number of non-zero rows. We will write s = (i, j) for the jth box in the ith row (i.e., j 6 λi).
The content of a box c(s) := i−j. For the diagram µ ⊂ λ, we define a skew Young diagram λ\µ,
being a set of boxes in λ which are not in µ. Ribbon is a skew Young diagram without 2 × 2
squares. The height ht(λ\µ) of a ribbon is one less than the number of its rows.
12 M. Bershtein and R. Gonin
Proposition 3.13. The action of Diffq on Fu is given by the following formulas
Ea,−b |λ〉 = q−
a
2 ua
∑
µ\λ=b−ribbon
(−1)ht(µ\λ)q
a
b
∑
s∈µ\λ
c(s)
|µ〉, (3.12)
Ea,b |λ〉 = q−
a
2 ua
∑
λ\µ=b−ribbon
(−1)ht(λ\µ)q
a
b
∑
s∈µ\λ
c(s)
|µ〉, (3.13)
Ea,0 |λ〉 = ua
1
1− qa
+
l(λ)−1∑
i=0
(
qa(i−λi+1) − qai
) |λ〉, (3.14)
here b > 0.
In particular,
Ea,0 |0〉 =
ua
1− qa
|0〉. (3.15)
Let us introduce the notation c(λ) =
∑
s∈λ
c(s). Define an operator Iτ ∈ End(Fu) by the
following formula
Iτ |λ〉 = u|λ|q−
1
2
|λ|+c(λ) |λ〉. (3.16)
The operator was introduced in [3] and is well known nowadays.
Proposition 3.14. The operator Iτ enjoys the property IτEa,bI
−1
τ = Ea−b,b.
Proof. Follows from (3.12)–(3.14). �
Corollary 3.15. Fτu ∼= Fu for τ = ( 1 1
0 1 ).
Remark 3.16. Also, Corollary 3.15 follows from Proposition 3.4: we will use this approach to
prove Proposition 5.2.
Corollary 3.17. The twisted representation Fσu is determined up to isomorphism by n and n′.
Proof. Corollary 3.15 implies that F τkσ
u
∼= Fσu . Note that
τkσ =
(
1 k
0 1
)(
m′ m
n′ n
)
=
(
m′ + kn′ m+ kn
n′ n
)
.
For the fixed n and n′, all the possible choices of m and m′ appear for the appropriate k. �
4 Explicit formulas for twisted representation
In this section we provide three explicit constructions of twisted Fock module Fσu for
σ =
(
m′ m
n′ n
)
. (4.1)
Constructions are called fermionic, bosonic, and strange bosonic. This section contains no
proofs. We will give proofs in Sections 5. In Section 6 we will provide an independent proof of
Theorem 4.1.
Twisted Representations of Algebra of q-Difference Operators 13
4.1 Fermionic construction
We need to consider the Z/2Z-graded nth tensor power of the Clifford algebra defined above.
More precisely, consider an algebra generated by ψ(a)[i] and ψ∗(b)[j], for i, j ∈ Z; a, b = 0, . . . , n−1,
subject to relations{
ψ(a)[i], ψ(b)[j]
}
= 0,
{
ψ∗(a)[i], ψ
∗
(b)[j]
}
= 0, (4.2){
ψ(a)[i], ψ
∗
(b)[j]
}
= δa,bδi+j,0. (4.3)
Consider the currents
ψ(a)(z) =
∑
i
ψ(a)[i]z
−i−1, ψ∗(b)(z) =
∑
i
ψ∗(b)[i]z
−i.
Consider a module Fnψ with a cyclic vector |l0, . . . , ln−1〉 and the relations
ψ(a)[i] |l0, . . . , ln−1〉 = 0 for i > la,
ψ∗(a)[j] |l0, . . . , ln−1〉 = 0 for j > −la.
The module Fnψ does not depend on l0, . . . , ln−1. The isomorphism can be seen from the
following formulas:
ψ∗(a)[−la] |l0, . . . , la, . . . , ln−1〉 = |l0, . . . , la + 1, . . . , ln−1〉,
ψ(a)[la − 1] |l0, . . . , la, . . . , ln−1〉 = |l0, . . . , la − 1, . . . , ln−1〉.
Theorem 4.1. The formulas below determine an action of Diffq on Fnψ
c′ = n′, c = n,
Htw
k =
∑
a
∑
i+j=k
ψa[i]ψ
∗
a[j],
Etw(z) =
∑
b−a≡−n′ mod n
u
1
n q−1/2zψ(a)
(
q−1/2z
)
ψ∗(b)
(
q1/2z
)
z
n′−a+b
n q(a+b)/2n, (4.4)
F tw(z) =
∑
b−a≡n′ mod n
u−
1
n q1/2zψ(a)
(
q1/2z
)
ψ∗(b)
(
q−1/2z
)
z
−n′−a+b
n q−(a+b)/2n.
The module obtained is isomorphic to Mσ
u.
Since Fσu ⊂Mσ
u, we have obtained a fermionic construction for Fσu .
4.2 Bosonic construction
Let us consider the nth tensor power of the Heisenberg algebra. More precisely, this algebra is
generated by ab[i] for b = 0, . . . , n−1 and i ∈ Z with the relation [ab1 [i], ab2 [j]] = iδb1,b2δi+j,0. Let
us extend the algebra by adding the operators eQb , obeying the following commutation relations.
The operator eQb commutes with all the generators except for ab[0] and satisfy ab[0]eQb =
eQb(ab[0] + 1). Denote
φb(z) =
∑
j 6=0
1
j
ab[j]z
−j +Qb + ab[0] log z.
Remark 4.2. Informally, one can think that there exists an operator Qb satisfying [ab[0], Qb]
= 1. However, this operator will not act on our representation. We will use Qb as a formal
symbol. Our final answer will consist only of eQb , but not of Qb without the exponent.
14 M. Bershtein and R. Gonin
We need a notion of a normally ordered exponent : exp(. . . ) :. The argument of a normally
ordered exponent is a linear combination of ab[i] and Qb. Let a+, a−, a0, and Q denote a linear
combination of ab[i] for i > 0, ab[i] for i < 0, ab[0] and Qb, correspondingly (b is not fixed).
Define
:exp(a+ + a− + a0 + Q) :
def
= exp(a+) exp(a−) exp(Q) exp(a0).
Also, note that a0 will have the coefficient log z. We shall understand it formally; the action of
the operator exp(ab[0] log z) = zab[0] is well defined, since in the representation to be considered
below, ab[0] acts as multiplication by an integer at each Fock module.
Let Q(n) be a lattice with the basis Q0−Q1, . . . , Qn−2−Qn−1. Consider the group alge-
bra C
[
Q(n)
]
. This algebra is spanned by eλ for λ =
∑
i λiQi ∈ Q(n). Let us define the action of
the commutative algebra generated by ab[0] on C
[
Q(n)
]
:
ab[0]e
∑
λiQi = λbe
∑
λiQi .
Let Fna be the Fock representation of the algebra generated by ab[i] for i 6= 0; i.e., there is
a cyclic vector |0〉 ∈ Fna such that ab[i] |0〉 = 0 for i > 0.
Finally, we can consider Fna⊗C
[
Q(n)
]
as representation of the whole Heisenberg algebra as
follows: ab[i] for i 6= 0 acts on the first factor, ab[0] acts on the second factor. Also, C
[
Q(n)
]
acts on Fna ⊗ C
[
Q(n)
]
.
Theorem 4.3. There is an action of Diffq on Fna ⊗ C
[
Q(n)
]
determined by the formulas
Htw[k] =
∑
b
ab[k], c′ = n′, c = n,
Etw(z) =
∑
b−a≡−n′ mod n
u
1
n q
a+b−n
2n z
n′−a+b
n
+1 : exp
(
φb
(
q1/2z
)
− φa
(
q−1/2z
))
: εa,b, (4.5)
F tw(z) =
∑
b−a≡n′ mod n
u−
1
n q
−a−b+n
2n z
−n′−a+b
n
+1 : exp
(
φb
(
q−1/2z
)
− φa
(
q1/2z
))
: εa,b, (4.6)
here εa,b =
∏
r(−1)ar[0] (we consider the product over such r that a − 1 > r > b for a > b and
b− 1 > r > a for b > a).
The representation obtained is isomorphic to Fσu .
4.3 Strange bosonic construction
We will use notation of Section 3.1. Let ζ be a nth primitive root of unity, e.g., ζ = e
2πi
n .
Theorem 4.4. There is an action of Diffq on F aα determined by the formulas
Htw
k = ank, c = n, c′ = n′,
Etw(z) = zn
′/n u
1
n
n
(
1− q1/n
) n−1∑
l=0
ζ ln
′
: exp
(∑
k
q−k/2n − qk/2n
k
akζ
−klz−k/n
)
:, (4.7)
F tw(z) = z−n
′/n u−
1
n
n
(
1− q−1/n
) n−1∑
l=0
ζ−ln
′
: exp
(∑
k
qk/2n − q−k/2n
k
akζ
−klz−k/n
)
: . (4.8)
The representation obtained is isomorphic to Fσu .
As before the representation does not depend on α, see Remark 3.2.
Twisted Representations of Algebra of q-Difference Operators 15
5 Twisted representation via a sublattice
5.1 Sublattices and subalgebras
Consider a full rank sublattice Λ ⊂ Z2 of index n (i.e., Z2/Λ is a finite group of order n). Let
us define a Lie subalgebra DiffΛq ⊂ Diffq which is spanned by Ea,b for (a, b) ∈ Λ and central
elements c, c′.
Denote by E
[n]
a,b, c[n], c
′
[n] standard generators of Diffqn . Let v1 = (k1, l1) and v2 = (k2, l2) be
a basis of Λ. Define a map ϕv1,v2 : Diffqn → Diffq
ϕv1,v2E
[n]
a,b = Eak1+bk2,al1+bl2 , ϕv1,v2c
′
[n] = k1c
′ + l1c, ϕv1,v2c[n] = k2c
′ + l2c.
Proposition 5.1. Let v1, v2 be a positively oriented basis (i.e., k1l2 − k2l1 = n). Then the
map ϕv1,v2 is a Lie algebra isomorphism Diffqn
∼= DiffΛq .
Proof. It follows from (2.1) directly. �
Slightly abusing notation, denote the Fock representation of Diffqn by F [n]
u .
Proposition 5.2. Let v1 = (n, 0) and v2 = (−m, 1). Then F [n]
un
∼= Fu|φv1,v2(Diffqn) as Diffqn-
modules.
Proof. Note that the Fock module Fu is Z-graded with grading given by
deg(|α〉) = 0, deg(Ea,b) = −b.
Recall that a character of a Z-graded module is the generating function of dimensions of the
graded components. Then the character of Fock module chFu = 1/(q)∞ :=
∞∏
k=1
1/
(
1− qk
)
.
Consider a subalgebra Heis0 in Diffqn spanned by E0,k and c. Note that Heis0 is isomor-
phic to the Heisenberg algebra. Since deg(φv1,v2E0,−k) = deg(Ekm,−k) = k, the character of the
Heis0-Fock module is also 1/(q)∞; i.e., it coincides with chFu. This implies that Fu|φv1,v2(Diffqn)
restricted to Heis0 is isomorphic to the Heis0-Fock module. To finish the proof, we use Proposi-
tions 3.4 and 3.5. �
5.2 Twisted Fock vs restricted Fock
From now on we change q → q1/n. Our goal is to construct an action of Diffq on the Fock
module twisted by σ ∈ SL2(Z) as in (4.1) for n 6= 0. Consider a sublattice Λσ ⊂ Z2 spanned by
v1 = (n, 0) and v2 = (−m, 1). Consider another basis of Λσ obtained by σ
w1 = m′v1 + n′v2 = (m′n− n′m,n′) = (1, n′), (5.1)
w2 = mv1 + nv2 = (0, n). (5.2)
Remark 5.3. The construction of the sublattice Λσ ⊂ Z2 naturally appears, if one require σ
to be a transition matrix from vi to wi and assume v1 = (n, 0), w2 = (0, n).
Denote the Fock module of Diffq1/n by F [1/n]
u .
Theorem 5.4. There is an isomorphism of Diffq-modules (Fu)σ ∼= F [1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
.
Proof. Proposition 5.2 implies F [1/n]
u1/n
∣∣∣
φv1,v2 (Diffq)
∼= Fu. On the other hand, relations (5.1)
and (5.2) yield that σ is the transition matrix from w1, w2 to v1, v2. �
16 M. Bershtein and R. Gonin
w1
w2
v1
v2
Figure 1. Lattice Λσ for n = 3, m = 1.
Corollary 5.5. There is an isomorphism of Diffq-modules (Mu)σ ∼=M[1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
.
Theorem 5.4 combined with results from Section 3 enables us to find explicit formulas for
action on Fσu . We will do this below.
5.3 Explicit formulas for restricted Fock
5.3.1 Fermionic construction via sublattice
Denote fermionic representation of Diffq1/n by M[1/n]
u . To be more specific, let us rewrite
formulas from Section 3.2 for M[1/n]
u1/n
.
c 7→ 1, c′ 7→ 0, Hm →
∑
i+j=m
ψiψ
∗
j ,
E(z) 7→ ql/nu1/n
1− q1/n
+ u1/nq−1/2nz :ψ
(
q−1/2nz
)
ψ∗
(
q1/2nz
)
:(l), (5.3)
F (z) 7→ q−l/nu−1/n
1− q−1/n
+ u−1/nq1/2nz :ψ
(
q1/2nz
)
ψ∗
(
q−1/2nz
)
:(l) . (5.4)
Proposition 5.6. The following formulas below determine an action of Diffq on Fnψ
c = n, c′ = ntw, Htw
k =
∑
a
∑
i+j=k
ψa[i]ψ
∗
a[j], (5.5)
Etw(z) =
∑
b−a≡−ntw mod n
u
1
n q−1/2zψ(a)
(
q−1/2z
)
ψ∗(b)
(
q1/2z
)
z
ntw−a+b
n q(a+b)/2n, (5.6)
F tw(z) =
∑
b−a≡ntw mod n
u−
1
n q1/2zψ(a)
(
q1/2z
)
ψ∗(b)
(
q−1/2z
)
z
−ntw−a+b
n q−(a+b)/2n. (5.7)
The module obtained is isomorphic to M[1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
for w1 = (1, ntw), w2 = (0, n).
Remark 5.7. Below we will substitute ntw = n′ to prove Theorem 4.1. However, Proposi-
tion 5.6 is more general, than it is necessary for the proof, since we do not assume here that
gcd(n, ntw) = 1. We will need the case of arbitrary ntw in Section 8.
Twisted Representations of Algebra of q-Difference Operators 17
Proof. We use the notation E(z), F (z) and H(z) for the Chevalley generators of Diffq1/n . The
generators of Diffq
∼= DiffΛ
q1/n
(identified by ϕw1,w2) will be denoted by Etw(z), F tw(z) and
Htw(z). Let us write the identification φw1,w2 explicitly for the Chevalley generators
Htw(z) =
∑
k
H0,nkz
−k, (5.8)
Etw(z) = zntw/n
∑
k≡ntw mod n
Ekz
−k/n, (5.9)
F tw(z) = z−ntw/n
∑
k≡−ntw mod n
Fkz
−k/n. (5.10)
Let us consider currents ψ(a)(z) and ψ∗(b)(z) for a, b = 0, 1, . . . , n − 1. These currents are
defined by following equality
zψ(z) =
n−1∑
a=0
zn−aψ(a)(z
n), ψ∗(z) =
n−1∑
b=0
zbψ∗(b)(z
n). (5.11)
Let us denote the modes of ψ(a)(z) and ψ∗(b)(z) as in equality (4.1). It is easy to see that these
modes satisfy Clifford algebra relations (4.2), (4.3). So we have identified the Clifford algebra
and the nth power of Clifford algebra. This leads to an identification Fψ = Fnψ.
Substituting (5.11) into (5.3) and (5.4), we obtain
E(z) =
ql/nu1/n
1− q1/n
+
n−1∑
a=0
n−1∑
b=0
u1/nq
a+b
2n q−1/2zn−a+b :ψ(a)
(
q−1/2zn
)
ψ∗(b)
(
q1/2zn
)
:(l),
F (z) =
q−l/nu−1/n
1− q−1/n
+
n−1∑
a=0
n−1∑
b=0
u−1/nq−
a+b
2n q1/2zn−a+b :ψ(a)
(
q1/2zn
)
ψ∗(b)
(
q−1/2zn
)
:(l) .
For technical reasons, we need to treat the cases ntw 6= 0 and ntw = 0 separately. Let us first
consider the case ntw 6= 0. Using formulas (5.8)–(5.10), we see that
Etw(z) =
∑
a−b≡ntw mod n
q
a+b
2n u1/nz
ntw−a+b
n q−1/2zψ(a)
(
q−1/2z
)
ψ∗(b)
(
q1/2z
)
,
F tw(z) =
∑
a−b≡−ntw mod n
q
−(a+b)
2n u−1/nz
−ntw−a+b
n q1/2zψ(a)
(
q1/2z
)
ψ∗(b)
(
q−1/2z
)
.
For ntw = 0 we obtain
Etw(z) =
ql/nu1/n
1− q1/n
+
n−1∑
a=0
u1/nqa/nq−1/2z :ψ(a)
(
q−1/2z
)
ψ∗(a)
(
q1/2z
)
:(l),
F tw(z) =
q−l/nu−1/n
1− q−1/n
+
n−1∑
a=0
u−1/nq−a/nq1/2z :ψ(a)
(
q1/2z
)
ψ∗(a)
(
q−1/2z
)
:(l) .
This can be rewritten as
Etw(z) =
n−1∑
a=0
u1/nqa/n
(
qd
l−a
n
e
1− q
+ q−1/2z :ψ(a)
(
q−1/2z
)
ψ∗(a)
(
q1/2z
)
:(l)
)
,
F tw(z) =
n−1∑
a=0
u−1/nq−a/n
(
q−d
l−a
n
e
1− q−1
+ q1/2z :ψ(a)
(
q1/2z
)
ψ∗(a)
(
q−1/2z
)
:(l)
)
.
18 M. Bershtein and R. Gonin
Note that the l-dependent normal ordering is defined in terms of ψi and ψ∗j . One can check
(cf. (A.3))
ψ(a)(z)ψ
∗
(a)(w) =
(w/z)d
l−a
n
e
z(1− w/z)
+ :ψ(a)(z)ψ
∗
(a)(w) :(l) . (5.12)
Hence
q−1/2zψ(a)
(
q−1/2z
)
ψ∗(a)
(
q1/2z
)
=
qd
l−a
n
e
1− q
+ q−1/2z :ψ(a)
(
q−1/2z
)
ψ∗(a)
(
q1/2z
)
:(l) . �
Proof of Theorem 4.1. Follows from Theorem 5.4 and Proposition 5.6. �
5.3.2 Bosonic construction via sublattices
Proposition 5.8. There is an action of Diffq on Fna ⊗ C[Q(n)] determined by the following
formulas
Htw[k] =
∑
b
ab[k], c′ = ntw, c = n, (5.13)
Etw(z) =
∑
b−a≡−ntw mod n
u
1
n q
a+b−n
2n z
ntw−a+b
n
+1 : exp
(
φb(q
1/2z)− φa(q−1/2z)
)
: εa,b, (5.14)
F tw(z) =
∑
b−a≡ntw mod n
u−
1
n q
−a−b+n
2n z
−ntw−a+b
n
+1 : exp
(
φb(q
−1/2z)− φa(q1/2z)
)
: εa,b, (5.15)
here εa,b =
∏
r(−1)ar[0] (we consider the product over such r that a − 1 > r > b for a > b and
b− 1 > r > a for b > a).
The representation obtained is isomorphic to F [1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
for w1 = (1, ntw), w2 = (0, n).
Proof Proposition 5.8. We need an upgraded version boson-fermion correspondence for the
proof. Namely, there is an action of nth tensor power of the Heisenberg algebra on Fnψ given
by
∂φb(z) =:ψ(b)(z)ψ
∗
(b)(z) :(0) .
Let P(n) be a lattice spanned Qb. According to boson-fermion correspondence Fnψ ∼=
⊕
Fna ⊗
C[P(n)].
Lemma 5.9. Vector subspace Fna⊗C[Q(n)] ⊂ Fna⊗C[P(n)] = Fnψ is a Diffq-submodule (with
respect to action, defined in Proposition 5.6). The action of Diffq on the subrepresentation is
given by (5.13)–(5.15).
Proof. One should substitute
ψ(b)(z) =:exp (−φb(z)) : (−1)
b−1∑
k=0
ak[0]
, (5.16)
ψ∗(b)(z) =:exp (φb(z)) : (−1)
b−1∑
k=0
ak[0]
(5.17)
into fermionic formulas (5.5)–(5.7). �
Recall that decomposition of M[1/n]
u is given by eigenvalues of a[0]; more precisely, opera-
tor a[0] acts by −j on F [1/n]
qj/nu
.
Twisted Representations of Algebra of q-Difference Operators 19
Lemma 5.10. Using identification Fψ = Fnψ (cf. (5.11)), we obtain a[0] = a0[0]+ · · ·+an−1[0].
Proof. This follows from
⌈
l−b
n
⌉
= 0 for l = 0 and b = 0, . . . , n− 1 (cf. (5.12)). �
Lemma 5.10 implies that the identification of vector spaces Fψ = Fnψ leads to identification
of subspaces F a0 = Fna ⊗ C[Q(n)]. Let us package identifications of vector subspaces into
a commutative diagram
F [1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
F [1/n]
u F a0 Fna ⊗ C[Q(n)]
M[1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
M[1/n]
u Fψ Fnψ.
Proposition 5.6 states that formulas (5.5)–(5.7) gives an action of Diffq with respect to identifi-
cation of bottom line of the diagram. Therefore, Lemma 5.9 implies that formulas (5.13)–(5.15)
describes the action of Diffq with respect to identification of top line of the diagram. �
Proof of Theorem 4.3. Follows from Theorem 5.4 and Proposition 5.8. �
5.3.3 Strange bosonic construction via sublattices
Proposition 5.11. There is an action of Diffq on F aα defined by formulas
c = n, c′ = ntw, Htw
k = ank,
Etw(z) = zntw/n
u
1
n
n
(
1− q1/n
) n−1∑
l=0
ζ lntw : exp
(∑
k
q−k/2n − qk/2n
k
akζ
−klz−k/n
)
:,
F tw(z) = z−ntw/n
u−
1
n
n
(
1− q−1/n
) n−1∑
l=0
ζ−lntw : exp
(∑
k
qk/2n − q−k/2n
k
akζ
−klz−k/n
)
: .
Obtained module is isomorphic to F [1/n]
u1/n
∣∣∣
φw1,w2 (Diffq)
for w1 = (1, ntw), w2 = (0, n).
Proof. Formulas (5.8)–(5.10) imply
Htw
k = Hnk, (5.18)
Etw(z) =
1
n
zntw/n
n−1∑
l=0
ζ lntwE
(
ζ lz1/n
)
, (5.19)
F tw(z) =
1
n
z−ntw/n
n−1∑
l=0
ζ−lntwF
(
ζ lz1/n
)
. (5.20)
Substitution of Diffq1/n-version of (3.1)–(3.3) to (5.18)–(5.20) finishes the proof. �
Proof of Theorem 4.4. Follows from Theorem 5.4 and Proposition 5.11. �
20 M. Bershtein and R. Gonin
6 Twisted representation via a Semi-infinite construction
This section is devoted to another proof of the Theorem 4.1. So we use the same notation
σ =
(
m′ m
n′ n
)
.
Twisted evaluation representation. Let ea,b be a matrix unit (all entries are 0 except
for one cell, where it is 1; this cell is in bth column and ath row).
Consider a homomorphism tu,σ : Diff A
q → Diff A
q ⊗Matn×n defined by
E0,k 7→ E0,k ⊗ 1,
E1,k 7→ u
1
n
∑
b−a≡−n′ mod n
q
a+b
2n E
1,k+ b−a+n′
n
⊗ ea,b, (6.1)
E−1,k 7→ u−
1
n
∑
b−a≡n′ mod n
q−
a+b
2n E−1,k+ b−a−n′
n
⊗ ea,b.
Algebra Diff A
q ⊗Matn×n tautologically acts on Cn
[
z, z−1
]
. Therefore, homomorphism tu,σ
induces an action of Diff A
q on Cn[z, z−1].
Proposition 6.1. Obtained representation of Diff A
q is isomorphic to V σ
u .
Proof. Consider a basis vl := q
ml2
2n u
ml
n xl of evaluation representation C
[
x, x−1
]σ
. Action with
respect to this basis looks like
E0,kvl = vk+l, (6.2)
E1,kvl = u
1
n q
n′+kn+2l
2n vl+nk+n′ , (6.3)
E−1,kvl = u−
1
n q
n′−kn−2l
2n vl+nk−n′ .
Let a, b = 0, . . . , n− 1 be such numbers that l = nj + b and a ≡ b+ n′ mod n. Substituting
l = nj + b into (6.3) we obtain
E1,kvnj+b = u
1
n q
a+b
2n q
n′+b−a
2n q
k
2 qjv
n(k+j+n′+b−a
n
)+a
. (6.4)
Let us identify Cn
[
z, z−1
] ∼−→ C
[
x, x−1
]
by zjeb 7→ vnj+b. Then formula (6.4) will be rewritten
E1,k(z
jeb) = u
1
n q
a+b
2n
(
E
1,k+ b−a+n′
n
⊗ ea,b
)(
zjeb
)
.
To be compared with formula (6.1) this proves the proposition for E1,k. The proof for E−1,k
is analogous. For E0,k proposition is obvious from (6.2). �
Semi-infinite construction. To apply semi-infinite construction we need to pass from
associative algebras to Lie algebras.
Definition 6.2. Algebra Diffq(gln) is a Lie algebra with basis Ek,l⊗ea,b (where (k, l) ∈ Z2\(0, 0)
and a, b = 0, . . . , n − 1), c and c′. Elements c and c′ are central. All other commutators are
given by
[Ek1,l1 ⊗ ea1,b1 , Ek2,l2 ⊗ ea2,b2 ] = Ek1+k2,l1+l2
⊗
(
q
l2k1−l1k2
2 δb1,a2ea1,b2 − q
l1k2−l2k1
2 δb2,a1ea2,b1
)
+ δk1,−k2δl1,−l2δa2,b1δa1,b2(cl1 + c′k1).
Twisted Representations of Algebra of q-Difference Operators 21
Proposition 6.3. There is an action of Diffq(gln) on Fnψ given by formulas
c 7→ 1, c′ 7→ 0,
E(z)⊗ ea,b 7→ q−
1
2 zψ(a)
(
q−
1
2 z
)
ψ∗(b)
(
q
1
2 z
)
,
F (z)⊗ ea,b 7→ q
1
2 zψ(a)
(
q
1
2 z
)
ψ∗(b)
(
q−
1
2 z
)
.
Obtained representation is isomorphic to Λ
∞/2 Cn
[
z, z−1
]
.
Proof of Theorem 4.1. According to Proposition 3.11,Mu
∼= Λ
∞/2
Vu. ThenMσ
u
∼= Λ
∞/2
V σ
u .
Therefore, Propositions 6.1 and 6.3 imply Theorem 4.1. �
7 q-W -algebras
7.1 Definitions
Topological algebras and completions. In this section we will work with topological al-
gebras. Let us define topological algebra appearing as a completion of Diffq. It is given by
projective limit of U(Diffq)/Jk where Jk is the left ideal generated by non-commutative poly-
nomials in Ej1,j2 of degree −k (with respect to grading degEj1,j2 = −j2). Although each
U(Diffq)/Jk does not have a structure of algebra, so does the projective limit. Moreover, the
projective limit has natural topology.
Below we will ignore all corresponding technical problems concerning completions and topo-
logy. We will use term ‘generators’ instead of ‘topological generators’, the same notation
for Diffq, and its completion and so on.
7.1.1 Non-twisted W -algebras
Let us introduce a notation
∞∑
l=0
fk,n[l]xl = fk,n(x) =
(1− qx)
n−k
n
(
1− q−1x
)n−k
n
(1− x)
2(n−k)
n
.
Definition 7.1. Algebra Wq(sln) is generated by Tk[r] for r ∈ Z and k = 1, . . . , n − 1. It is
convenient to add generators T0[r] = Tn[r] = δr,0. The defining relations are
∞∑
l=0
fk,n[l]
(
T1[r−l]Tk[s+l]− Tk[s−l]T1[r+l]
)
= −
(
q
1
2−q−
1
2
)2
(kr − s)Tk+1[r+s], (7.1)
∞∑
l=0
fn−k,n[l]
(
Tn−1[r−l]Tk[s+l]− Tk[s−l]Tn−1[r+l]
)
= −
(
q
1
2−q−
1
2
)2
((n−k)r − s)Tk−1[r+s]. (7.2)
Introduce currents Tk(z) =
∑
r∈Z
Tk[r]z
−r. Then relations (7.1)–(7.2) can be rewritten in
current form
fk,n(w/z)T1(z)Tk(w)− fk,n(z/w)Tk(w)T1(z)
= −(q
1
2 − q−
1
2 )2
(
(k + 1)
w
z
δ′
(w
z
)
Tk+1(w) + wδ
(w
z
)
∂wTk+1(w)
)
, (7.3)
fn−k,n(w/z)Tn−1(z)Tk(w)− fn−k,n(z/w)Tk(w)Tn−1(z)
= −
(
q
1
2 − q−
1
2
)2 (
(n− k + 1)
w
z
δ′
(w
z
)
Tk−1(w) + wδ
(w
z
)
∂wTk−1(w)
)
. (7.4)
Also note that T0(z) = Tn(z) = 1.
22 M. Bershtein and R. Gonin
Remark 7.2. There are different approaches to definition of q-W -algebra. For example, in [17]
algebra Wq,p(sln) was defined via bosonization. The currents Tk(z) satisfy relation [17, Theo-
rem 2]
fk,n(w/z)T1(z)Tk(w)− fk,n(z/w)Tk(w)T1(z)
=
(1− q)(1− p/q)
1− p
(
δ(w/zp)Tk+1(z)− δ
(
wpk/z
)
Tk+1(w)
)
,
where
fk,n(x) =
(
x|pm−1q, pmq−1, pn, pn−1; pn
)(
x|pm−1, pm, pn−1q, pnq−1; pn
) .
One can check that limit p→ 1 gives relation (7.3). However [17] do not provide presentation
of Wq,p(sln) in terms of generators and relations.
In the paper [43] relation (2.62) defines algebraWq,p(gln) which (non-essentially) differs from
Wq,p(sln) mentioned above (and from Wq(sln) defined above).
7.1.2 Twisted q-W -algebras
Twisted q-W -algebra depends on remainder of ntw modulo n. If ntw = 0, then we get definition
of non-twisted q-W -algebra from last section. One can find definition of Wq,p(sl2, 1) in [44,
equations (37)–(38)].
Definition 7.3. Algebra Wq(sln, ntw) is generated by T twk [r] for r ∈ ntwk/n + Z and k =
1, . . . , n− 1. It is convenient to add T tw0 [r] = T twn [r] = δr,0. The defining relations are
∞∑
l=0
fk,n[l]
(
T tw1 [r−l]T twk [s+l]− T twk [s−l]T tw1 [r+l]
)
= −
(
q
1
2 − q−
1
2
)2
(kr − s)T twk+1[r+s], (7.5)
∞∑
l=0
fn−k,n[l]
(
T twn−1[r−l]T twk [s+l]− T twk [s−l]T twn−1[r+l]
)
= −
(
q
1
2 − q−
1
2
)2
((n−k)r − s)T twk−1[r+s]. (7.6)
Let us rewrite relations (7.5)–(7.6) in the current form. Define currents
T twk (z) :=
∑
T twk [r]z−r, Tk(z) := z
kntw
n T twk (z),
T ◦k (z) := z−
(n−k)ntw
n T twk (z) = z−ntwTk(z).
Note that
T0(z) = T ◦n(z) = 1, Tn(z) = T ◦0 (z) = zntw . (7.7)
Proposition 7.4. Relation (7.5) is equivalent to
fk,n(w/z)T1(z)Tk(w)− fk,n(z/w)Tk(w)T1(z)
= −
(
q
1
2 − q−
1
2
)2 (
(k + 1)
w
z
δ′(w/z)Tk+1(w) + wδ(w/z)∂wTk+1(w)
)
. (7.8)
Relation (7.6) is equivalent to
fn−k,n(w/z)T ◦n−1(z)T ◦k (w)− fn−k,n(z/w)T ◦k (w)T ◦n−1(z)
= −(q
1
2 − q−
1
2 )2
(
(n− k + 1)
w
z
δ′(w/z)T ◦k−1(w) + wδ(w/z)∂wT
◦
k−1(w)
)
.
Remark 7.5. In non-twisted case we have relations (7.3) and (7.4) for currents Tk(z). In twisted
case we have the same relations, but for two different sets of currents Tk(z) and T ◦k (z). One
should also keep in mind (7.7).
Twisted Representations of Algebra of q-Difference Operators 23
7.2 Connection of Wq(sln, ntw) with Diffq
Connection between Wq(sln) and Diffq is known (see [19, Proposition 2.14] or [43, Proposi-
tion 2.25]). In this section we generalize it for arbitrary ntw.
Let Heis be a Heisenberg algebra generated by H̃j with relation
[
H̃i, H̃j
]
= niδi+j,0. We
will prove that there is a surjective homomorphism Diffq � Wq(sln, ntw) ⊗ U(Heis). Secretly,
generators Hj are mapped to H̃j under the homomorphism. Let us introduce a notation to
describe this homomorphism more precisely.
Define
ϕ−(z) =
∑
j>0
q−j/2 − qj/2
j
H−jz
j , ϕ+(z) = −
∑
j>0
qj/2 − q−j/2
j
Hjz
−j , (7.9)
ϕ̃−(z) =
∑
j>0
q−j/2 − qj/2
j
H̃−jz
j , ϕ̃+(z) = −
∑
j>0
qj/2 − q−j/2
j
H̃jz
−j .
Also, let introduce notation
ϕ(z) = ϕ−(z) + ϕ+(z), ϕ̃(z) = ϕ̃−(z) + ϕ̃+(z). (7.10)
Define
T̃k(z) =
1
k!
exp
(
−k
n
ϕ−(z)
)
Ek(z) exp
(
−k
n
ϕ+(z)
)
.
Note that T̃k(z) commute with Hj .
Let Jµ,n,ntw be two sided ideal in Diffq generated by c−n, c′−ntw and T̃n(z)−µnzntw (here
µ ∈ C\{0}). Parameter µ is not essential since automorphism Ea,b 7→ µ−aEa,b maps Jµ,n,ntw to
J1,n,ntw . So we will abbreviate Jn,ntw = Jµ,n,ntw .
Lemma 7.6. T̃k(z) ∈ Jn,n′ for k > n.
Proof. It holds in U(Diffq)/Jn,ntw
Ek(z) = n!µnzntwEk−n(z) :expϕ(z) : .
On the other hand,
Ek−n(z) :expϕ(w) :=
(z − w)2(k−n)
(z − qw)k−n
(
z − q−1w
)k−n expϕ−(w)Ek−n(z) expϕ+(w).
Hence, Ek−n(z) :expϕ(z) := 0. �
Theorem 7.7. There is an algebra isomorphisms S : Wq(sln, ntw)⊗U(Heis)
∼−→ U(Diffq)/Jn,ntw
such that
Tk(z) 7→ µ−kT̃k(z), H̃j 7→ Hj . (7.11)
The map P in opposite direction is given by
Hj 7→ H̃j , c 7→ n, c′ 7→ ntw, (7.12)
E(z) 7→ µ exp
(
1
n
ϕ̃−(z)
)
T1(z) exp
(
1
n
ϕ̃+(z)
)
, (7.13)
F (z) 7→ − µ−1z−ntw(
q
1
2 − q−
1
2
)2 exp
(
− 1
n
ϕ̃−(z)
)
Tn−1(z) exp
(
− 1
n
ϕ̃+(z)
)
. (7.14)
24 M. Bershtein and R. Gonin
The rest of this section is devoted to proof of Theorem 7.7. First of all, we will prove that
formula (7.11) indeed defines a homomorphism S : Wq(sln, ntw) ⊗ U(Heis) → U(Diffq)/Jn,ntw
(see Proposition 7.12). Then we prove that formulas (7.12)–(7.14) defines a homomorphism in
opposite direction (see Proposition 7.13). Finally, we note that maps P and S are mutually
inverse.
Proposition 7.8. Currents T̃k(z) (as power series with coefficients in U(Diffq)/Jn,ntw) satisfy
relation (7.8).
Proof. Let us define power series in two variables
E(k+1)(z, w) = (z − qw)
(
z − q−1w
)
E(z)Ek(w).
According Corollary B.6, E(k+1)(z, w) is regular in sense of Definition A.2. Following relations
follows from results of Appendix A
E(k+1)(z, w) = (z − qw)
(
z − q−1w
)
Ek(w)E(z), (7.15)
E(k+1)(w,w) = (1− q)
(
1− q−1
)
w2Ek+1(w), (7.16)
∂zE(k+1)(z, w)
∣∣
z=w
= (1− q)
(
1− q−1
)
w2 1
k + 1
∂wE
k+1(w)
+ (1− q)
(
1− q−1
)
wEk+1(w). (7.17)
More precisely, (7.15)–(7.16) easily follows from Propositions A.8. One can find a proof of (7.17)
at the end of Appendix A.
It is straightforward to check that
fk,n(w/z)T̃1(z)T̃k(w) =
(
1− qwz
)(
1− q−1w
z
)
k!
(
1− w
z
)2 exp
(
− 1
n
(ϕ−(z) + kϕ−(w))
)
× E(z)Ek(w) exp
(
− 1
n
(ϕ+(z) + kϕ+(w))
)
. (7.18)
Formulas (7.18) and (7.15)–(7.17) implies that
fk,n(w/z)T̃1(z)T̃k(w)− fk,n(z/w)T̃k(w)T̃1(z)
=
1
k!
exp
(
− 1
n
(ϕ−(z) + kϕ−(w))
)
E(k+1)(z, w)
× exp
(
− 1
n
(ϕ+(z) + kϕ+(w))
)
∂w
(
w−1δ(w/z)
)
= (1− q)
(
1− q−1
)
(k + 1)T̃k+1(w)
w
z
δ′(w/z) + (1− q)
(
1− q−1
)
wT̃ ′k+1(w)δ(w/z). �
Lemma 7.9. The following OPE holds in Diffq
F (z)Ek(w) = k(c− k + 1)
w
z E
k−1(w)(
1− w
z
)2
+ k
w∂wE
k−1(w)− w :ϕ′(w)Ek−1(w) : −c′Ek−1(w)
1− w
z
+ reg; (7.19)
or, equivalently[
F (z), Ek(w)
]
= k(c− k + 1)Ek−1(w)
w
z
δ′(w/z)
+ k
(
w∂wE
k−1(w)− w :ϕ′(w)Ek−1(w) : −c′Ek−1(w)
)
δ(w/z). (7.20)
Twisted Representations of Algebra of q-Difference Operators 25
Proof. Denote by E(w) = E(w1) · · ·E(wk). We will write :F (z)E(w) : = F+(z)E(w) +
E(w)F−(z); this definition reminds standard Definition A.3, but is applied in different situ-
ation (E(w) is not a current in one variable). Note that
F (z)E(w) = [F−(z), E(w)] + :F (z)E(w) : =
∑
j
E(w1) · · ·E(wj−1) (7.21)
×
(
H
(
q
1
2wj
)
−H
(
q−
1
2wj
)
− c′
1− wj
z
+ c
wj
z
1(
1− wj
z
)2
)
E(wj+1) · · ·E(wk)+ :F (z)E(w) : .
Recall that wϕ′(w) = H(q−
1
2w)−H(q
1
2w). It follows from (2.4)that
(
H(q
1
2wj)−H(q−
1
2wj)
)
E(wl) = −2E(wl)
1− wl
wj
+
E(wl)
1− q wlwj
+
E(wl)
1− q−1 wl
wj
− wj :ϕ′(wj)E(wl) :,
E(wj)
(
H
(
q
1
2wl
)
−H
(
q−
1
2wl
))
=
2E(wj)
1− wl
wj
− E(wj)
1− q wlwj
− E(wj)
1− q−1 wl
wj
− wl :ϕ′(wl)E(wj) : .
Using identity
1(
1− wj
z
)(
1− wj
wl
) − 1(
1− wl
z
)(
1− wj
wl
) = −
wl
z(
1− wl
z
)(
1− wj
z
)
we obtain
E(wl)(
1− wj
z
)(
1− wj
wl
) − E(wj)(
1− wl
z
)(
1− wj
wl
)∣∣∣∣∣
w=wj=wl
= −
w
z E(w)(
1− w
z
)2 +
w∂wE(w)
1− w
z
.
Finally, we conclude that
∑
j
E(w1) · · ·E(wj−1)
H
(
q
1
2wj
)
−H
(
q−
1
2wj
)
1− wj
z
E(wj+1) · · ·E(wk)
∣∣∣∣∣∣
w=w1=···=wk
= −
(
k
2
)
2wz E
k−1(w)(
1− w
z
)2 +
(
k
2
)
2wEk−2(w)∂wE(w)
1− w
z
− kw :ϕ′(w)Ek−1(w) :
1− w
z
. (7.22)
Relation (7.19) follows from (7.21) and (7.22). �
Proposition 7.10. In algebra U(Diffq)/Jn,ntw holds
F (z) = − µ−nz−ntw(
q
1
2 − q−
1
2
)2 exp
(
− 1
n
ϕ−(z)
)
T̃n−1(z) exp
(
− 1
n
ϕ+(z)
)
.
Proof. Using relation T̃n(z)− µnzntw ∈ Jn,ntw , we find a relation in U(Diffq)/Jn,ntw
F (z)En(w) = n!µnwntwF (z) :exp(φ(w)) :
= n!µnwntw
(1− qw/z)
(
1− q−1w/z
)
(1− w/z)2
exp(φ−(w))F (z) exp(φ+(w)).
Comparing coefficient of (z − w)−2 with (7.19), we obtain
nEn−1(w) = n!µnwntw(1− q)
(
1− q−1
)
exp(φ−(w))F (w) exp(φ+(w)). �
26 M. Bershtein and R. Gonin
Denote by T̃ ◦k (z) = µ−nz−ntw T̃k(z).
Proposition 7.11. Following relation holds in U(Diffq)/Jn,ntw
fn−k,n(w/z)T̃ ◦n−1(z)T̃ ◦k (w)− fn−k,n(z/w)T̃ ◦k (w)T̃ ◦n−1(z)
= −
(
q
1
2 − q−
1
2
)2 (
(n− k + 1)
w
z
δ′
(w
z
)
T̃ ◦k−1(w) + wδ
(w
z
)
∂wT̃
◦
k−1(w)
)
.
Proof. Proposition 7.10 imply
fn−k,n(w/z)T̃ ◦n−1(z)T̃ ◦k (w)− fn−k,n(z/w)T̃ ◦k (w)T̃ ◦n−1(z) = −
(
q
1
2 − q−
1
2
)2µ−nw−ntw
k!
× exp
(
1
n
ϕ−(z)− k
n
ϕ−(w)
)[
F (z), Ek(w)
]
exp
(
1
n
ϕ+(z)− k
n
ϕ+(w)
)
.
It is straightforward to finish the proof using (7.20). �
Proposition 7.12. Formula (7.11) defines a homomorphism S from Wq(sln, ntw)⊗U(Heis) to
the algebra U(Diffq)/Jn,ntw .
Proof. Evidently, Hj and T̃k(z) commute, and Hj form a Heisenberg algebra. We only have
to check that 1
µk
T̃k(z) form Wq(sln, ntw) algebra. Relation of Wq(sln, ntw) algebra follows from
Propositions 7.8 and 7.11. �
Proposition 7.13. Formulas (7.12)–(7.14) defines a homomorphism P from U(Diffq)/Jn,ntw
to the algebra Wq(sln, ntw)⊗ U(Heis).
Proof. Let us check that these formulas define morphism from Diffq. According to Propo-
sition 2.6, it is enough to prove relations (2.3)–(2.8). It is done in Appendix C. Evidently,
P annihilates Jn,ntw . �
Proposition 7.14. Maps P and S are mutually inverse.
Proof. Let us prove PS = idWq(sln,ntw)⊗U(Heis) first. The algebra Wq(sln, ntw) is generated by
modes of T1(z). Hence it is sufficient to check PS(H̃n) = H̃n and PS
(
T1(z)
)
= T1(z). Both of
them are straightforward.
The algebra U(Diffq)/Jn,ntw is generated by modes of E(z) and F (z). Evidently, SP
(
E(z)
)
= E(z). Proposition 7.10 implies SP
(
F (z)
)
= F (z). �
Proof of Theorem 7.7. Follows from Propositions 7.12, 7.13 and 7.14 �
7.3 Bosonization of Wq(sln, ntw)
Let σ be as in (4.1). Corollary 3.17 states that representation Fσu actually does not depend
on m′ and m; it is determined by n′ and n. Let us denote the representation by F (n′,n)
u .
7.3.1 Fock representation via Diffq
In this section we will discuss connection of twisted q-W algebras and twisted representa-
tions F (n′,n)
u .
Lemma 7.15. In representation F (n′,n)
u operator T̃n(z) acts by −q−1/2uzn
′ 1
(q1/2−q−1/2)n
.
Twisted Representations of Algebra of q-Difference Operators 27
Proof. We will use formula (4.4) to calculate En(z).
By Proposition A.8
ψ(a)
(
q−1/2z
)
ψ(b)
(
q−1/2z
)
= −ψ(b)
(
q−1/2z
)
ψ(a)
(
q−1/2z
)
,
ψ(a)
(
q−1/2z
)
ψ∗(b)
(
q1/2z
)
= −ψ∗(b)
(
q1/2z
)
ψ(a)
(
q−1/2z
)
for any a, b (even for a = b).
Consider a sequence of numbers 0 6 a1, . . . , an 6 n − 1 such that ai+1 − ai ≡ −n′ mod n.
Thus,
En(z) = n!uq−1/2zn
′+nψ(a1)
(
q−1/2z
)
ψ∗(a2)
(
q1/2z
)
ψ(a2)
(
q−1/2z
)
ψ∗(a3)
(
q1/2z
)
· · ·
× ψ(an)
(
q−1/2z
)
ψ∗(a1)
(
q1/2z
)
.
Using bosonization (5.16), (5.17) we obtain (cf. (7.9) and (7.10))
En(z) = −n!uq−1/2zn
′ 1(
q1/2 − q−1/2
)n : exp(ϕ(z)) : .
Consequently,
T̃n(z) = −q−1/2uzn
′ 1(
q1/2 − q−1/2
)n . �
Proposition 7.16. Suppose, Mi are representation of Diffq such that ideal Jµi,ni,n′i acts by zero
(for i = 1, . . . , k). Then Jµ,n,n′ acts by zero on M1 ⊗ · · · ⊗Mk for n =
k∑
i=1
ni, n
′ =
k∑
i=1
n′i and
µn = µn1
1 · · ·µ
nk
k .
Proof. Recall that Eni+1(z) ∈ Jni,n′i by Lemma 7.6. Thus, En(z) act on M1 ⊗ · · · ⊗Mk as
n!
n1! · · ·nk!
En1(z)⊗ · · · ⊗ Enk(z) = n!
k∏
i=1
µnii z
n′i : exp(ϕ(z)) : . �
Proposition 7.17. Ideal Jµ,nd,n′d acts by zero on F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud for
µ = (−1)
1
n
q−
1
2n
q
1
2 − q−
1
2
(u1 · · ·ud)
1
nd . (7.23)
Proof. Follows from Lemma 7.15 and Proposition 7.16. �
Theorem 7.18. There is an action of W(slnd, n
′d) ⊗ U(Heis) on F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud such
that action of Diffq factors through W(slnd, n
′d)⊗ U(Heis).
Proof. According to Proposition 7.17, algebra U(Diffq)/Jnd,n′d acts on F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud .
By Theorem 7.7, algebra U(Diffq)/Jnd,n′d is isomorphic to W(slnd, n
′d)⊗ U(Heis). �
Remark 7.19. One can consider tensor product of Fock modules with different twists Fσ1u1 ⊗· · ·
⊗Fσdud . According to Proposition 7.16 algebraW
(
sl∑ni ,
∑
n′i
)
⊗U(Heis) acts on this space. Ob-
tained representation is ‘irregular’ (cf. [39]). In Section D.2 we consider an intertwiner between
irregular and (graded completion of) regular representation.
28 M. Bershtein and R. Gonin
7.3.2 Explicit formula for bosonization
Below we will write explicit formula for bosonization of Wq(slnd, n
′d). This bosonization comes
from action of W(slnd, n
′d) on F (n′,n)
u1 ⊗· · ·⊗F (n′,n)
ud . Recall that realization of F (n′,n)
u is written
via ab[k] for b = 0, . . . , n − 1.Denote by aib[k] (for i = 1, . . . , d and b = 0, . . . , n − 1) generators
of Heisenberg algebra action on ith factor of the tensor product F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud .
To write action of Wq(slnd, n
′d) we need to introduce slightly different version of Heisenberg
algebra. Namely, consider an algebra, generated by ηib[k] for b = 0, . . . , n − 1; i = 1, . . . , d and
k ∈ Z. Relations are given by linear dependence and commutation relations
d∑
i=1
n−1∑
b=0
ηib[k] = 0,
[
ηi1b1 [k1], ηi2b2 [k2]
]
= k1δk1+k2,0
(
δi1,i2δb1,b2 −
1
nd
)
.
Let us define representation F η ⊗C
[
Qd
(n)
]
(cf. Section 4.2). Lattice Qd
(n) consist of elements∑
λibQ
i
b such that λib ∈ Z and for any i it holds
∑
b λ
i
b = 0. Define an action
ηib[0]e
∑
λibQ
i
b = λibe
∑
λibQ
i
b . (7.24)
First factor F η is a Fock space for subalgebra ηib[k] for k 6= 0. We can consider F η ⊗ C
[
Qd
(n)
]
as representation of whole Heisenberg algebra as follows: ηib[k] for k 6= 0 acts on first factor,
ηia[0] acts on the second factor by (7.24). Note, that also C
[
Qd
(n)
]
acts on F η ⊗C
[
Qd
(n)
]
. Let us
introduce notation
ηib(z) =
∑
k 6=0
1
k
ηib[k]z−k +Qib + ηib[0] log z.
Proposition 7.20. There is an action of Wq(slnd, n
′d) on F η ⊗ C
[
Qd
(n)
]
given by formulas
T1(z) 7→ (−1)−
1
n (u1 · · ·ud)−
1
nd
(
q
1
2 − q−
1
2
)
q
1
2n
×
d∑
i=1
∑
b−a≡−n′ mod n
u
1
n
i q
a+b−n
2n z
n′−a+b
n
+1 : exp
(
ηib
(
q1/2z
)
− ηia
(
q−1/2z
))
: ε
(i)
a,b,
Tnd−1(z) 7→ −(−1)
1
n (u1 · · ·ud)
1
nd
(
q
1
2 − q−
1
2
)
q−
1
2n zn
′d
×
d∑
i=1
∑
b−a≡n′ mod n
u
− 1
n
i q
−a−b+n
2n z
−n′−a+b
n
+1 : exp
(
ηib
(
q−1/2z
)
− ηia
(
q1/2z
))
: ε
(i)
a,b,
here ε
(i)
a,b =
∏
r(−1)η
i
r[0] (we consider product over such r that a − 1 > r > b for a > b and
b− 1 > r > a for b > a).
Proof. Denote by
[
F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud
]
h
subspace of such v ∈ F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud that
Hkv = 0 for k > 0. According to Theorem 7.18, the algebra Wq(slnd, n
′d) acts on
[
F (n′,n)
u1 ⊗ · · ·
⊗ F (n′,n)
ud
]
h
.
On the other hand map ηib[k] 7→ aib[k] −
∑
b,i a
i
b[k] is a homomorphism. Therefore, one can
identify F η ⊗ C
[
Qd
(n)
]
and
[
F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud
]
h
.
Substitution of (4.5), (4.6), and (7.23) to (7.13), (7.14) finishes the proof. �
Denote obtained representation by FWq(slnd,n
′d)
u1,...,ud .
Twisted Representations of Algebra of q-Difference Operators 29
Remark 7.21. The parameter µ is determined by u1, . . . , ud only up to nd-th root of unity. The
modules FWq(slnd,n
′d)
u1,...,ud with different µ are non-isomorphic in general (so notation FWq(slnd,n
′d)
u1,...,ud
is ambiguous). For example, one can see this from the highest weights λs defined in the next
section.
The modules FWq(slnd,n
′d)
u1,...,ud with different µ are related by an external automorphism of
Wq(sln, ntw).
Example 7.22. Let us consider case of twisted Virasoro algebra, i.e., n = 2, n′ = 1, d = 1.
Then everything is expressed via one boson η(z) with relation [η[k1], η[k2]] = 1
2k1δk1+k2,0 and
[η[0], Q] = 1
2 . So there is an action of Wq(sl2, 1) on F η ⊗ C[Z] given by
T1(z) 7→ (−1)−
1
2
(
q
1
2 − q−
1
2
)
(−1)η[0]
×
[
z : exp
(
η
(
q1/2z
)
+ η
(
q−1/2z
))
: +z2 : exp
(
−η(q1/2z
)
− η
(
q−1/2z
))
:
]
.
We can simplify the formula using conjugation by (−1)η[0]2/2
T1(z) 7→
(
q
1
2 − q−
1
2
)[
z : exp
(
η(q1/2z
)
+ η
(
q−1/2z
))
: +z2 : exp
(
−η
(
q1/2z
)
− η
(
q−1/2z
))
:
]
.
7.3.3 Explicit formulas for strange bosonization
To write formulas for strange bosonization we need to consider Heisenberg algebra generated
by ξi[k] for i = 1, . . . , d and k ∈ Z. Relations are given by linear dependence and commutation
relations
d∑
i=1
ξi[nk] = 0,
[ξi1 [k1], ξi2 [k2]] = k1δk1+k2,0δi1,i2 for either n - k1 or n - k2,
[ξi1 [nj1], ξi2 [nj2]] = nj1δj1+j2,0
(
δi1,i2 −
1
d
)
.
Denote corresponding Fock module by F ξ.
Proposition 7.23. There is an action of Wq(slnd, n
′d) on F ξ given by
T1(z) 7→ −(−1)−
1
n
q
1
2 − q−
1
2
n
(
q
1
2n − q−
1
2n
)(u1 · · ·ud)−
1
nd z
n′
n
×
d∑
i=1
u
1/n
i
n−1∑
l=0
ζ ln
′
: exp
(∑
k
q−k/2n − qk/2n
k
ξi[k]ζ−klz−k/n
)
:,
Tnd−1(z) 7→ −(−1)
1
n
q
1
2 − q−
1
2
n
(
q
1
2n − q−
1
2n
)(u1 · · ·ud)
1
nd z
nd−1
n
n′
×
d∑
i=1
u
−1/n
i
n−1∑
l=0
ζ−ln
′
: exp
(∑
k
qk/2n − q−k/2n
k
ξi[k]ζ−klz−k/n
)
: .
Obtained representation is isomorphic to FWq(slnd,n
′d)
u1,...,ud .
Proof. The proof is analogous to proof of Proposition 7.20. The only difference is that we have
to use (4.7), (4.8) instead of (4.5), (4.6). This representation is isomorphic to FWq(slnd,n
′d)
u1,...,ud since
it also corresponds to F (n′,n)
u1 ⊗ · · · ⊗ F (n′,n)
ud . �
30 M. Bershtein and R. Gonin
7.4 Verma modules vs Fock modules
Connection of Fock module and Verma module is known in non-twisted case. In this Subsection
we will generalize it for Wq(sln, ntw). Denote d = gcd(ntw, n).
Definition 7.24. Wq(gln, ntw) = U(Diffq)/Jn,ntw .
Denote modes of Ek(z) by Ek[j]. Namely, Ek(z) =
∑
dE
k[j]z−j .
Definition 7.25. Verma module VWq(gln,ntw)
λ1,...,λd
is a module over Wq(gln, ntw) with cyclic vec-
tor
∣∣λ̄〉
gl
and relations
Hk
∣∣λ̄〉
gl
= 0 for k > 0,
Ek[j]
∣∣λ̄〉
gl
= 0 for j +
kntw
n
> 0,
E
ns
d [−sntw/d]
∣∣λ̄〉
gl
=
ns
d
!λs
∣∣λ̄〉
gl
.
Consider a grading onWq(gln, ntw) given by degEk[j]=−j− kntw
n . Verma module VWq(gln,ntw)
λ1,...,λd
is a graded module with grading defined by requirement deg
∣∣λ̄〉
gl
= 0.
Proposition 7.26. VWq(gln,ntw)
λ1,...,λd
is spanned by Ek1 [j1] · · ·Ekt [jt]
∣∣λ̄〉
gl
for j1
k1
6 j2
k2
6 · · · 6 jt
kt
<
−ntw
n and 1 6 ki 6 n.
Proof. Recall that En(z) = n!zntw : exp(ϕ(z)) :. Therefore En[j] acts on FH .
Lemma 7.27. Module FH is spanned by En[−j1] · · ·En[−jk]|0〉.
Sketch of the proof. Let us consider operator En−[−j] defined by
∑
j E
n
−[−j]zj = exp(ϕ−(z)).
On one hand En−[−j1] · · ·En−[−jk]|0〉 is a basis of FH (this basis coincide with a basis of complete
homogeneous polynomials up to renormalization of Heisenberg algebra generators). One the
other hand,
En[−j1] · · ·En[−jk]|0〉 = En−[−j1] · · ·En−[−jk]|0〉+ lower terms,
here lower terms are taken with respect to lexicographical order. �
Remark 7.28. Lemma 7.27 holds for any exponent : exp
(∑
αjHjz
−j) : such that α−i 6= 0 for
i > 0. The proof does not use any other properties of coefficients αj . One can find the proof as
the last part of proof of [43, Proposition 2.29].
Define Diff >0
q and Diff >0
q as subalgebras of Diffq spanned by Ek,j with k > 0 and k > 0
correspondingly.
Lemma 7.29. Vector
∣∣λ̄〉
gl
∈ VWq(gln,ntw)
λ1,...,λd
is cyclic with respect to action of Diff>0
q .
Proof. Theorem 7.7 implies that the natural map U
(
Diff >0
q
)
→ U(Diffq)/Jn,ntw is surjec-
tive. Hence Verma module is generated by non-commutative monomials in Ei and Hj applied
to
∣∣λ̄〉
gl
. Using relation [Hj , Ei] =
(
q−j/2 − qj/2
)
Ei+j , we see that the module is spaned by
Ei1 · · ·EikHj1 · · ·Hjl
∣∣λ̄〉
gl
. Lemma 7.27 implies that the module is spanned by Ei1 · · ·EikEn[j1]
· · ·En[jl]
∣∣λ̄〉
gl
�
Twisted Representations of Algebra of q-Difference Operators 31
In [42, (3.48)] author states, that algebra Diff >0
q has a PBW-like basis Ek1 [j1] · · ·Ekt [jt] for
j1
k1
6 j2
k2
6 · · · 6 jt
kt
. Thus VWq(gln,ntw)
λ1,...,λd
is spanned by Ek1 [j1] · · ·Ekt [jt]
∣∣λ̄〉
gl
(with the same
condition on ki and ji).
Note that if jt
kt
> −ntw
n then such vector is 0; moreover if jt
kt
= −ntw
n then Ekt [jt] acts by
multiplication on a constant, hence it can be excluded.Also note that Ek[j] ∈ Jn,ntw for k > n+1,
therefore we assume ki 6 n. The Proposition 7.26 is proven. �
Corollary 7.30. Coefficients of chVWq(gln,ntw)
λ1,...,λd
are less or equal to coefficients of
∞∏
k=1
(
1−q
kd
n
)−d
.
Theorem 7.31. If F (ntw/d,n/d)
u1 ⊗· · ·⊗F (ntw/d,n/d)
ud is irreducible then natural map p : VWq(gln,ntw)
λ1,...,λd
→ F (ntw/d,n/d)
u1 ⊗ · · · ⊗ F (ntw/d,n/d)
ud is an isomorphism for
λs =
(
−q−
1
2
)s 1(
q
1
2 − q−
1
2
)ns
d
es(u1, . . . , ud). (7.25)
Proof. Let us first prove existence of the map p. This will follow automatically from the
following lemma.
Lemma 7.32. The highest vector |ū〉 ∈ F (ntw/d,n/d)
u1 ⊗ · · · ⊗ F (ntw/d,n/d)
ud satisfy following condi-
tions
Hk |ū〉 = 0 for k > 0, (7.26)
Ek[j] |ū〉 = 0 for j +
kntw
n
> 0, (7.27)
E
ns
d [−sntw/d] |ū〉 =
ns
d
!λs |ū〉 . (7.28)
Proof. Assertions (7.26)–(7.27) are evident. Let us check (7.28). Denote by εj(z) action of E(z)
on jth tensor multiple; in particular E(z) 7→ ε1(z) + · · ·+εd(z). Recall that εj(z)
n
d
+1 = 0. Thus
E
ns
d (z) 7→
ns
d !
(nd !)s
∑
k1<k2<···<kj
ε
n
d
k1
(z)ε
n
d
k2
(z) · · · ε
n
d
ks
(z) + · · · ,
here dots denote summands with a power which is not a multiple of n
d (thus, this summands do
not contribute to E
ns
d [−sntw/d] |ū〉). To finish the proof we note that
ns
d !
(nd !)s
〈ū|ε
n
d
k1
(z)ε
n
d
k2
(z) · · · ε
n
d
ks
(z) |ū〉 =
ns
d
!uk1uk2 · · ·uks
(
−q−1/2
)s
z
ntw
d
s 1
(q1/2 − q−1/2)
ns
d
. �
Image of p is whole F (ntw/d,n/d)
u1 ⊗ · · · ⊗ F (ntw/d,n/d)
ud , since F (ntw/d,n/d)
u1 ⊗ · · · ⊗ F (ntw/d,n/d)
ud
is irreducible. Note that ch
(
F (ntw/d,n/d)
u1 ⊗ · · · ⊗ F (ntw/d,n/d)
ud
)
=
∞∏
k=1
1(
1−q
kd
n
)d . Corollary 7.30
implies that p is injective and chVWq(gln,ntw)
λ1,...,λd
=
∞∏
k=1
1(
1−q
kd
n
)d . �
Remark 7.33. Note that representation FWq(sln,ntw)
u1,...,ud is irreducible iff Fu1 ⊗ · · · ⊗ Fud is ir-
reducible. In particular, for d = 1 then FWq(sln,ntw)
u is irreducible automatically. Generally,
criterion of irreducibility of Fu1 ⊗ · · · ⊗ Fud is given by Lemma 9.1.
32 M. Bershtein and R. Gonin
Definition 7.34. Verma module VWq(sln,ntw)
λ1,...,λd
is a module over Wq(sln, ntw) with cyclic vec-
tor
∣∣λ̄〉
sl
and relations
T twk [r]
∣∣λ̄〉
sl
= 0 for r > 0,
T twns/d[0]
∣∣λ̄〉
sl
= λs
∣∣λ̄〉
sl
.
Introduce grading onWq(sln, ntw) by deg T twk [r] = −r. Verma module VWq(gln,ntw)
λ1,...,λd
is a graded
module with grading defined by deg
∣∣λ̄〉
sl
= 0.
To simplify notation for comparison of sln and gln cases, we will assume below that µ = 1
(cf. Remark 7.21).
Proposition 7.35. VWq(gln,ntw)
λ1,...,λd
∼= VWq(sln,ntw)
λ1,...,λd
⊗FH with respect to identificationWq(gln, ntw) =
Wq(sln, ntw)⊗ U(Heis).
Proof. The existence of maps in both directions can be checked directly using universal property
of the Verma module. Evidently, these maps are mutually inverse. �
Corollary 7.36. If FWq(sln,ntw)
u1,...,ud is irreducible then natural map p̃ : VWq(sln,ntw)
λ1,...,λd
→ FWq(sln,ntw)
u1,...,ud
is an isomorphism for λs as in (7.25).
8 Restriction on DiffΛq for general sublattice
We generalize results of Section 5 for arbitrary sublattice. For applications in Section 9 we will
need only case of sublattice Λ0 = span(e1, ne2) ⊂ Z2.
8.1 Decomposition of restriction
Let Λ ⊂ Z2 be a sublattice of finite index. Let us choose basis w1, w2 of lattice Λ so that
w1 = (r, ntw) and w2 = (0, n). Let d be the greatest common divisor of n and ntw.
Theorem 8.1. There is an isomorphism of Diffq-modules
F [1/nr]
ud/nr
∣∣∣
φw1,w2 (Diffq)
∼=
⊕
l∈Q(d)
F (ntw/d, n/d)
uqrl0
⊗ · · · ⊗ F (ntw/d, n/d)
uqr(
α
d
+lα) ⊗ · · · ⊗ F (ntw/d, n/d)
uqr(
d−1
d
+ld−1)
. (8.1)
Proof. Proposition 3.5 implies that F [1/nr]
ud/nr
∣∣∣
φre1,e2
∼= F [1/n]
ud/n
. Hence it is enough to consider case
r = 1. We will use realization of F [1/n]
ud/n
∣∣∣
φw1,w2 (Diffq)
constructed in Proposition 5.8. Strategy of
our proof is as follows; first we will construct decomposition on the level of vector spaces and
then study action on each direct summand.
For each α = 0, . . . , d− 1 let Qα,lα
(n/d) be a subset of lattice P(n) consisting of elements∑
a≡α mod d
l̃aQa such that
∑
a≡α mod d
l̃a = lα.
Note that
Q(n) =
∐
l0+···+ld−1=0
Q0,l0
(n/d) ⊕ · · · ⊕Q
d−1,ld−1
(n/d) .
Twisted Representations of Algebra of q-Difference Operators 33
Or equivalently
C[Q(n)] =
⊕
l0+···+ld−1=0
C
[
Q0,l0
(n/d)
]
⊗ · · · ⊗ C
[
Q
d−1,ld−1
(n/d)
]
.
Let F
n
d
a,α be Fock module for Heisenberg algebra generated by ab[k] for b ≡ α mod d. Then
Fna ⊗ C[Q(n)] =
⊕
l0+···+ld−1=0
(
F
n
d
a,0 ⊗ C
[
Q0,l0
(n/d)
])
⊗ · · · ⊗
(
F
n
d
a,d−1 ⊗ C
[
Q
d−1,ld−1
(n/d)
])
. (8.2)
Let us show that (8.2) is a decomposition of Diffq-modules (moreover, that it leads to decom-
position (8.1)).Let us define
Htw
α [k] =
∑
b≡α mod d
ab[k], (8.3)
Etwα (z) =
∑
b−a≡−ntw mod n
a≡α mod d
u
d
n q
a+b−n
2n z
ntw−a+b
n
+1 : exp
(
φb
(
q1/2z
)
− φa
(
q−1/2z
))
: εa,b, (8.4)
F twα (z) =
∑
b−a≡ntw mod n
a≡α mod d
u−
d
n q
−a−b+n
2n z
−ntw−a+b
n
+1 : exp
(
φb
(
q−1/2z
)
− φa
(
q1/2z
))
: εa,b, (8.5)
here εa,b =
∏
r(−1)ar[0] (product over such r that a− 1 > r > b for a > b and b− 1 > r > a for
b > a).
Lemma 8.2. Formulas (8.3)–(8.5) defines an action of Diffq on F
n
d
a,α ⊗ C
[
Qα,lα
(n/d)
]
; obtained
representation is isomorphic to F (ntw/d,n/d)
uq
α
d
+lα
.
Sketch of the proof. Let us define ε̃a,b =
∏
r(−1)ar[0] (product over r satisfying above in-
equalities and condition r ≡ α mod d. One can check that there exists an index set I such
that conjugation of Etwα (z) and F twα (z) by
∏
(i,j)∈I
(−1)ai[0]aj [0] will turn εa,b to ε̃a,b. Theorem 4.3
finishes the proof. �
On the other hand, formulas (5.13)–(5.15) implies
Htw[k] =
∑
α
Htw
α [k], Etw(z) =
∑
α
Etwα (z), F tw(z) =
∑
α
F twα (z).
Therefore embedding of vector space from first row of following commutative diagram leads to
second row embedding of Diffq-modules(
F
n
d
a,0 ⊗ C
[
Q0,l0
(n/d)
])
⊗ · · · ⊗
(
F
n
d
a,d−1 ⊗ C
[
Q
d−1,ld−1
(n/d)
])
Fna ⊗ C[Q(n)]
F (ntw/d,n/d)
uql0
⊗ · · · ⊗ F (ntw/d,n/d)
uq
α
d
+lα
⊗ · · · ⊗ F (ntw/d,n/d)
uq
d−1
d
+ld−1
F [1/n]
ud/n
∣∣∣
φw1,w2 (Diffq)
.
�
Corollary 8.3. Following Diffq-modules are isomorphic
F [1/n]
u
∣∣∣
φe1,ne2 (Diffq)
∼=
⊕
l∈Q(n)
Fuql0 ⊗ · · · ⊗ Fuq αn+lα ⊗ · · · ⊗ F
uq
n−1
n +ln−1
. (8.6)
Remark 8.4. Lattice Λ admits another basis v1 = (N, 0), v2 = (R, d). There is an isomorphism
F [1/nr]
ud/nr
∣∣∣
φv1,v2 (Diffq)
∼=
⊕
l∈Q(d)
Fuqrl0 ⊗ · · · ⊗ Fuqr(αn+lα) ⊗ · · · ⊗ F
uq
r
(
d−1
d
+ld−1
) .
34 M. Bershtein and R. Gonin
8.2 Strange bosonization and odd bosonization
Representation F [1/n]
u
∣∣∣
φw1,w2Diffq
admits fermionic, bosonic, and strange bosonic realizations;
formulas are given in Propositions 5.6, 5.8 and 5.11 correspondingly. This Section is devoted
to study of corresponding Wq(sln, ntw) algebra action on F [1/n]
u
∣∣∣
φw1,w2Diffq
. We will consider
strange bosonization and its classical limit.
Let us introduce following notation (cf. with (3.2))
e(z) = :exp
∑
k 6=0
q−k/2n − qk/2n
k
akz
−k
:.
Proposition 8.5. For w1 = e1+ntwe2, w2 = ne2 ideal Jµ,n,ntw acts by zero on F [1/n]
u
∣∣∣
φw1,w2 (Diffq)
for
µ =
u
n
(
1− q
1
n
) × (−1)
d
n
q−
1
2n − q
1
2n
q
1
2 − q−
1
2
n = (−1)
d
n
q−
1
2nu(
q
1
2 − q−
1
2
) . (8.7)
Proof. We will use strange bosonization of F [1/n]
u
∣∣∣
φw1,w2 (Diffq)
. Recall that we denote by ζ
a primitive root of unity of degree n. We have
E(z) 7→ u
n
(
1− q
1
n
)z ntwn (e(z1/n
)
+ ζntwe
(
ζz1/n
)
+ · · ·+ ζ(n−1)ntwe
(
ζn−1z1/n
))
.
Let us calculate(
n
(
1− q
1
n
)
u
E(z)
)n
= zntw
(
e
(
z1/n
)
+ ζntwe
(
ζz1/n
)
+ · · ·+ ζ(n−1)ntwe
(
ζn−1z1/n
))n
= n!zntw(−1)(n−1)ntwe
(
z1/n
)
e
(
ζz1/n
)
· · · e
(
ζn−1z1/n
)
= (−1)(n−1)ntwn!zntw
∏
i<j
(
1− ζj−i
)2((
1− q
1
n ζj−i
)(
1− q−
1
n ζj−i
)) :e(z1/n)e
(
ζz1/n
)
· · · e
(
ζn−1z1/n
)
:.
We need to compute
∏
i<j
(
1− ζj−i
)2(
1− q
1
n ζj−i
)(
1− q−
1
n ζj−i
) =
∏
i<j
(
1− ζj−i
)(
1− ζi−j
)
q−
1
n
(
1− q
1
n ζj−i
)(
1− q
1
n ζi−j
)
= q
1
n(n2)
∏
k 6=0
(
1− ζk
)n(
1− q
1
n ζk
)n = q
n−1
2
(
1− q
1
n
)n
(1− q)n
nn =
(
q
1
2n − q−
1
2n
)n(
q
1
2 − q−
1
2
)n nn.
So (
n
(
1− q
1
n
)
u
E(z)
)n
= n!zntw(−1)(n−1)ntw
×
((
q
1
2n − q−
1
2n
)(
q
1
2 − q−
1
2
) n)n :e
(
z1/n
)
e
(
ζz1/n
)
· · · e
(
ζn−1z1/n
)
: .
Twisted Representations of Algebra of q-Difference Operators 35
Note that :e
(
z1/n
)
e
(
ζz1/n
)
· · · e
(
ζn−1z1/n
)
:=:expϕ(z) :. Hence
µn = (−1)(n−1)ntw
(
u
n
(
1− q
1
n
) (q 1
2n − q−
1
2n
)(
q
1
2 − q−
1
2
) n)n = (−1)(n−1)ntw+n
(
uq−
1
2n
q
1
2 − q−
1
2
)n
.
Finally note that (n− 1)ntw + n ≡ d mod 2. �
Remark 8.6. Another way to prove Proposition 8.5 is to derive it from Proposition 7.17
(since isomorphism (8.1)). Beware inconsistency of our notation in (7.23) and (8.7). Let us
rewrite (7.23)
µ = (−1)
d
n
q−
d
2n
q
1
2 − q−
1
2
(u1 · · ·ud)
1
n = (−1)
d
n
q−
d
2n
q
1
2 − q−
1
2
((
u
n
d
)d
q
d−1
2
) 1
n = (−1)
d
n
q−
1
2n
q
1
2 − q−
1
2
u.
Let us consider subalgebra of Heisenberg algebra generated by Jk = ak for n - k. Denote
corresponding Fock module by F J .
Corollary 8.7. There is an action of Wq(sln, ntw) on F J given by
T1(z) = −(−1)
d
n
1
n
q
1
2 − q−
1
2
q
1
2n − q−
1
2n
z
ntw
n
n−1∑
l=0
ζ lntw : exp
∑
n-k
q−
k
2n − q
k
2n
k
Jkζ
−klz−
k
n
: .
Obtained representation corresponds to F [1/n]
u
∣∣∣
φw1,w2 (Diffq)
.
Proof. Follows from Theorem 7.7 and Proposition 8.5 �
Remark 8.8. Let us consider non-twisted case ntw = 0. Then d = n and total sign −(−1)
d
n = 1.
More accurately, the coefficient is a root of unity of degree n (cf. Remark 7.21). Nevertheless,
this freedom disappears if we require T1(z) = n + o(~) for ~ = log q (this is a standard setting
for classical limit).
Example 8.9. Odd bosonization is a particular case of strange bosonization for n = 2 and
ntw = 0,
T1(z) =
q
1
4 + q−
1
4
2
: exp
∑
2-r
q−
r
4 − q
r
4
r
Jrz
− r
2
: + :exp
−∑
2-r
q−
r
4 − q
r
4
r
Jrz
− r
2
:
. (8.8)
Consider classical limit q → 1. It is convenient to assume q = e~ and ~ → 0. If there exists an
expansion
T1(z) = 2 + z2L(z)~2 + o
(
~2
)
,
then modes of current L(z) = Lnz
−n−2 form ‘not q-deformed’ Virasoro algebra. Note that
T1(z)→ 2 + z2
(
z−2
16
+
1
4
∑
:Jodd(z)2:
)
~2 + o
(
~2
)
,
where Jodd(z) =
∑
2-r Jrz
− r
2
−1. Hence
L(z) =
z−2
16
+
1
4
∑
:Jodd(z)2: .
Or equivalently
Lk =
1
4
∑
1
2
(r+s)=k
:JrJs : +
1
16
δk,0. (8.9)
Formula (8.9) is well-known; it coincides with [48, equation (2.16)] after substitution Jr = 2I 1
2
r.
Let us emphasize that formula (8.8) is a q-deformation of (8.9).
36 M. Bershtein and R. Gonin
9 Relations on conformal blocks
9.1 Whittaker vector
In this section we define and study basic properties of Whittaker vector W (z|u1, . . . , uN ) ∈
Fu1 ⊗ · · · ⊗ FuN . We will restrict ourself to case when Fu1 ⊗ · · · ⊗ FuN is irreducible.
Lemma 9.1. Fu1 ⊗ · · · ⊗ FuN is irreducible if ui/uj 6= qk for any k ∈ Z.
Proof. Follows from the proof of [16, Lemma 3.1]. �
In papers [41, 47] Whittaker vector is defined geometrically. We will define Whittaker vector by
algebraic properties (cf. [41, Proposition 4.15]). Then we will prove that these properties define
Whittaker vector uniquely up to normalization if the module Fu1 ⊗ · · · ⊗ FuN is irreducible.
Definition 9.2. Whittaker vector W (z|u1, . . . , uN ) ∈ Fu1 ⊗ · · · ⊗ FuN is an eigenvector of
operators Ea,b for Nb > a > 0 with eigenvalues
E0,kW (z|u1, . . . , uN ) =
zk
qk/2 − q−k/2
W (z|u1, . . . , uN ), (9.1)
ENk,kW (z|u1, . . . , uN ) =
((
−q−
1
2
)N
u1 · · ·uNz
)k
q−k/2 − qk/2
W (z|u1, . . . , uN ) (9.2)
for k > 0;
Ek1,k2W (z|u1, . . . , uN ) = 0 (9.3)
for Nk2 > k1 > 0.We require W (z|u1, . . . , uN ) = |0〉 ⊗ · · · ⊗ |0〉 + · · · to fix normalization (by
dots we mean lower vectors).
Remark 9.3. Whittaker vector is an element of graded completion of Fu1⊗· · ·⊗FuN . Abusing
notation, we use the same symbols for modules and their completions.
Remark 9.4. Whittaker vector is an eigenvector for surprisingly big algebra. This explains
why we have to consider specific eigenvalues (for general eigenvalues there is no eigenvector in
corresponding representation). Theorem D.10 clarify origin of this eigenvalues.
Theorem 9.5. If Fu1 ⊗ · · · ⊗ FuN is irreducible, then there exists unique Whittaker vector.
One can find a proof of Theorem 9.5 in Appendix D. This statement can be considered as
a part of folklore; unfortunately, we do not know a precise reference for the theorem.
Proposition 9.6. Decomposition of Whittaker vector W
(
z1/n|u
)
∈ F [1/n]
u with respect to (8.6)
is given by Whittaker vectors W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
up to normalization.
Proof. The idea of the proof is that relations (9.1) and (9.2) for N = 1 implies these relations
for N = n. Let us work out conditions (9.2) for W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
Enk,kW
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
=
((
−q−
1
2
)n∏
k
(
uq
k
n
+lk
)
z
)k(
q−k/2 − qk/2
) W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
.
Let us calculate
(
−q−
1
2
)n n−1∏
k=0
(
uq
k
n
+lk
)
=
(
−q−
1
2
)n
unq
n−1
2 =
(
−q−
1
2nu
)n
.
Twisted Representations of Algebra of q-Difference Operators 37
So W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
is defined (up to normalization) by conditions
E0,kW
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
=
zk
qk/2 − q−k/2
W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
, (9.4)
Enk,kW
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
=
(
−q−
1
2n z
1
nu
)nk
q−k/2 − qk/2
W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
, (9.5)
Ek1,k2W
(
z|uql0 , . . . , uq
n−1
n
+ln−1
)
= 0 (9.6)
for k > 0 and nk2 > k1 > 0. Denote by E
[1/n]
a,b generators of Diffq1/n . Then
E0,kW
(
z1/n|u
)
= E
[1/n]
0,nk W
(
z1/n|u
)
=
(
z1/n
)nk(
q1/n
)nk/2 − (q1/n
)−nk/2W (z1/n|u
)
, (9.7)
Enk,kW
(
z1/n|u
)
= E
[1/n]
nk,nkW
(
z1/n|u
)
=
(
−q−
1
2n z
1
nu
)nk(
q1/n
)−nk/2 − (q1/n
)nk/2W (z1/n|u
)
, (9.8)
Ek1,k2W
(
z1/n|u
)
= E
[1/n]
k1,nk2
W
(
z1/n|u
)
= 0. (9.9)
Note that conditions (9.7)–(9.9) and conditions (9.4)–(9.6) coincide. Hence each component
of W
(
z1/n|u
)
also satisfy those conditions, i.e., coincide with Whittaker vector up to normaliza-
tion. �
9.1.1 Whittaker vector for Fu
Recall that we use notation c(λ) =
∑
s∈λ
c(s).
Proposition 9.7. We have an expansion of vector W (z|u) in the basis |λ〉
W (z|u) =
∑ q−
1
2
c(λ)∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ||λ〉. (9.10)
To prove the proposition we need the following lemmas.
Lemma 9.8. Following vectors in Fu coincide
exp
( ∞∑
k=1
zk
k
(
qk/2 − q−k/2
)a−k
)
|∅〉 =
∑ q−
1
2
c(λ)∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ||λ〉. (9.11)
Proof. Recall Cauchy identity
exp
(
−
∑
k
1
k
pk(x)pk(y)
)
=
∏
i,j
(1− xiyj) =
∑
λ
(−1)|λ|sλ′(x)sλ(y).
Let us use specialization of Cauchy identity (see [36, Section 1.4, Example 2])
pk(x) 7→ −zk
qk/2 − q−k/2
, sλ′(x) 7→ (−1)|λ|
q−
1
2
c(λ)z|λ|∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
) .
To finish the proof let us recall that there is an identification of space of symmetric polynomials
and Fock module F aα given by sλ 7→ |λ〉 and pk 7→ a−k (see [34]). �
38 M. Bershtein and R. Gonin
Remark 9.9. For |q| < 1 this specialization comes from substitution pk
(
zq1/2, zq3/2, . . .
)
.
Lemma 9.10. Following vectors in Fu coincide
exp
(
−
∞∑
k=1
(
−q−1/2uz
)k
k
(
qk/2 − q−k/2
)ρ(E−k,−k)
)
|∅〉 =
∑ q−
1
2
c(λ)∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ||λ〉.
Proof. Recall that we have defined an operator Iτ by (3.16). Let us calculate
Iτ
∑ q−
1
2
c(λ)∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ||λ〉
=
∑ u|λ|q
1
2
c(λ)− 1
2
|λ|∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ||λ〉. (9.12)
Proposition 3.14 implies that
Iτ
(
exp
(
−
∞∑
k=1
(
−q−1/2uz
)k
k
(
qk/2 − q−k/2
)ρ(E−k,−k)
)
|∅〉
)
= exp
(
−
∞∑
k=1
(
−q−1/2uz
)k
k
(
qk/2 − q−k/2
)a−k
)
|∅〉. (9.13)
Using Cauchy identity for another specialization
pk(x) 7→
(
−q−1/2uz
)k
qk/2 − q−k/2
, sλ′(x) 7→ (−1)|λ|
u|λ|q
1
2
c(λ)− 1
2
|λ|∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)z|λ|,
we see that r.h.s. of (9.12) and (9.13) coincide. �
Remark 9.11. For |q| > 1 this specialization comes from substitution pk
(
−zuq−1,−zuq−2, . . .
)
.
Proof of Proposition 9.7. To prove the Proposition let us check that r.h.s. of (9.10) satisfy
condition (9.1)–(9.3). Conditions (9.1) and (9.2) are equivalent to Lemmas 9.8 and 9.10 corre-
spondingly. To finish the proof we note that for N = 1 conditions (9.1) and (9.2) imply (9.3). �
Remark 9.12. Note that we did not use Theorem 9.5 in the proof of Proposition 9.7. Moreover,
we have proven a particular case of the Theorem for W (z|u).
9.1.2 Whittaker vector and restriction on sublattice
Let us recall interpretation of decomposition (8.6) in terms of boson-fermion correspondence.
One can identify Fnψ = Fψ. Embedding F a0 ⊂ Fψ corresponded to embedding Fna⊗C
[
Q(n)
]
⊂
Fnψ. Hence we have decomposition
F a =
⊕
l∈Q(n)
Fna ⊗ e
∑
i liQi . (9.14)
We argue by construction that decomposition (8.6) correspond to (9.14).
Proposition 9.13. Decomposition (9.14) identifies |0〉 ⊗ e−
∑
i liQi with |λ〉 for some λ. More-
over, partition λ satisfy following properties.
(i) Hooks of λ are in bijection with tuples {(i, j, ki, kj) | ki < li; kj > lj ; nki + i > nkj + j}.
Length of a hook corresponding to a tuple (i, j, ki, kj) equals to n(ki − kj) + i− j.
Twisted Representations of Algebra of q-Difference Operators 39
(ii) 1
2
n−1∑
i=0
( in + li)
2 = |λ|
n + 1
2
n−1∑
i=0
i2
n2 .
Proof. The n-fermion Fock space Fnψ is isomorphic to tensor product Fψ ⊗ · · · ⊗ Fψ. The
n-Heisenberg highest vectors are products |l0〉 ⊗ · · · ⊗ |ln−1〉. After identification of Fnψ with
one Fψ, these products becomes (3.11) for special λ. Such diagrams λ are called n-cores.
Combinatorially boson-fermion correspondence is a correspondence between Maya diagrams
and charged partitions (λ, l), see, e.g., [40, Section 6.4] or [22]. Boxes of a partition correspond
to pairs of white and black points in Maya diagram such that the coordinate of white point is
greater than the coordinate of the black point. The hook length equals the difference between
the coordinates of white and black points (cf. [40, Section 6.4]). This proves (i).
For formula (ii) see, e.g., [22, Proposition 2.30]. �
Lemma 9.14. Let li > lj. Let us consider hooks with fixed i and j (see Proposition 9.13).
� If i > j, then possible lengths of hooks are nk + i− j for k = 0, 1, . . . , li − lj − 1.
� If i < j then possible lengths of hooks are nk + i− j for k = 1, 2, . . . , li − lj − 1.
There are exactly li − lj − k such hooks of length nk + i− j for all possible k.
For each l ∈ Q(n) we will use notation
∏
(i,j,k)
for product over triples (i, j, k) satisfying following
conditions. Numbers i, j run over 0, . . . , n − 1 with condition li > lj . If i > j, then k =
0, 1, . . . , li − lj − 1; if i < j then k = 1, 2, . . . , li − lj − 1.
Corollary 9.15. If diagram λ corresponds to |0〉 ⊗ e−
∑
i liQi, then∏
s∈λ
(
q
1
2
h(s) − q−
1
2
h(s)
)
=
∏
(i,j,k)
(
q
1
2
(nk+i−j) − q−
1
2
(nk+i−j))li−lj−k.
Theorem 9.16. Decomposition of Whittaker vector W
(
z1/n|u1/n
)
∈ F [1/n]
u1/n
with respect to (8.6)
is given by
q−
1
2n
c(λ)z
1
2
∑((
li+
i
n
)2
−
(
i
n
)2)
∏
(i,j,k)
(
q
1
2
(
k+ i−j
n
)
− q−
1
2
(
k+ i−j
n
))li−lj−kW (z|uql0 , . . . , uq n−1
n
+ln−1
)
.
Proof. Recall that according to Proposition 9.6 we just have to verify the coefficient. This
coefficient can be found as coefficient of W
(
z1/n|u1/n
)
at highest vector of Fuql0⊗· · ·⊗Fuq kn+lk
⊗
· · · ⊗ F
uq
n−1
n +ln−1
.
By Proposition 9.7, coefficient of W
(
z1/n|u1/n
)
at |λ〉 is
q−
1
2n
c(λ)z
|λ|
n∏
s∈λ
(
q
1
2n
h(s) − q−
1
2n
h(s)
) =
q−
1
2n
c(λ)z
1
2
∑(
(li+
i
n
)2−( i
n
)2
)
∏
(i,j,k)
(
q
1
2
(
k+ i−j
n
)
− q−
1
2
(
k+ i−j
n
))li−lj−k . (9.15)
Equality (9.15) follows from Corollary 9.15 and Proposition 9.13(ii). �
40 M. Bershtein and R. Gonin
9.2 Shapovalov form
Definition 9.17. Let M1, M2 be two representations of Diffq. A pairing 〈−,−〉s : M1⊗M2 → C
is called Shapovalov if 〈v,Ea,bw〉s = −〈E−a,−bv, w〉s.
Proposition 9.18. There exists a unique Shapovalov pairing Fu ⊗Fqu−1 → C such that 〈0|0〉s
= 1.
Proof. There exists a unique pairing on Fock space such that ak is dual to −a−k. Since algebra
Diffq is generated by modes of E(z) and F (z), it remains to check Shapovalov property for
them. Formulas (3.2) and (3.3) implies 〈v,E(z)w〉 = −
〈
F
(
z−1
)
v, w
〉
. �
Remark 9.19. Note that this pairing differs from the pairing defined in Section 3.1. More
precisely, in Section 3.1 we required ak to be dual to a−k, not −a−k.
Proposition 9.20.
〈
W
(
1|qu−1
)
,W (z|u)
〉
= (qz; q, q)∞.
Proof. Formulas (9.10) and (9.11) imply
W (z|u) = exp
( ∞∑
k=1
zk
k
(
qk/2 − q−k/2
)a−k
)
|∅〉. (9.16)
Using (9.16) one can finish the proof by straightforward computation. �
Proposition 9.21. There exists a unique Shapovalov pairing Mu ⊗Mu−1 → C such that
〈1|0〉s = 1, 〈v, ψiw〉 = 〈ψ−iv, w〉s, 〈v, ψ∗iw〉 = 〈ψ∗−iv, w〉s. (9.17)
Proof. There exists unique pairing satisfying (9.17). The Shapovalov property can be checked
directly using (3.8)–(3.10). �
Proposition 9.22. Shapovalov pairing for Fock modules for basis |λ〉 has form
〈λ|µ〉s = (−1)|λ|δλ,µ′ .
Proof. Let p1, . . . , pi and q1, . . . , qi be Frobenius coordinates of λ; analogously, p̃1, . . . , p̃j and
q̃1, . . . , q̃j be Frobenius coordinates of µ. Using identification given by (3.11) we obtain
〈µ, 1|λ, 0〉s = (−1)
∑
k(qk−1)(−1)
∑
k(q̃k−1)
× 〈1|ψ∗q̃1 · · ·ψ
∗
q̃iψp̃i−1 · · ·ψp̃1−1ψ−p1 · · ·ψ−pi ψ∗−qi+1 · · ·ψ∗−q1+1|0〉s. (9.18)
Evidently, if this product is non-zero, then i = j, qk = p̃k, and pl = q̃l; this exactly means that
µ = λ′. It remains to calculate r.h.s. of (9.18) in this case; it equals (−1)
∑
k qk+
∑
l pl+i
2
= (−1)|λ|
since |λ| =
∑
k qk +
∑
l pl − i. �
Definition 9.23. Standard Shapovalov pairing 〈−,−〉ss : M1⊗M2 → C for M1 = Fu1⊗· · ·⊗Fun
and M2 = Fq/un ⊗ · · · ⊗ Fq/u1 is defined by
〈x1 ⊗ · · · ⊗ xn, yn ⊗ · · · ⊗ y1〉ss =
∏
i
〈xi, yi〉i.
Here 〈−,−〉i stands for Shapovalov pairing Fui ⊗Fqu−1
i
→ C as in Proposition 9.18.
Proposition 9.24. Shapovalov pairing on F [1/n]
u ⊗ F [1/n]
q
1
n /u
restricts to (−1)|λ|〈−,−〉ss : M1 ⊗
M2 → C for M1 = Fuql0 ⊗ · · · ⊗Fuq n−1
n +ln−1
and M2 = F
q
1
n−ln−1/u
⊗ · · · ⊗Fq1−l0/u (with respect
to decomposition (8.6)). Other pairs of direct summands are orthogonal.
Twisted Representations of Algebra of q-Difference Operators 41
Proof. Property 〈v,Ea,bw〉s = −〈E−a,−bv, w〉s is preserved under restriction. Since module M1
and M2 are irreducible, there exists unique Shapovalov pairing M1 ⊗M2 → C. So restriction of
pairing coincides with the standard pairing up to multiplicative constant. The constant equals
to 〈λ′|λ〉 = (−1)|λ| due to Proposition 9.22 (and Proposition 9.13).
Orthogonality with all other summands also follows from Proposition 9.22 and irreducibi-
lity. �
Remark 9.25. Let us comment on another way to prove orthogonality mentioned in Proposi-
tion 9.24. All direct summands are pairwise non-isomorphic. Hence there is no non-zero pairing
for all other pairs of direct summands.
9.3 Conformal blocks
Definition 9.26. Pochhammer and double Pochhammer symbols are defined by
(u; q1)∞ =
∞∏
i=0
(
1− qi1u
)
, (u; q1, q2)∞ =
∞∏
i,j=0
(
1− qi1q
j
2u
)
.
Remark 9.27. Standard definition works for |q1|, |q2| < 1 and any u. For sufficiently small u
double Pochhammer symbol can be presented as
(u; q1, q2)∞ = exp
(
−
∞∑
k=1
uk
k
(
1− qk1
)(
1− qk2
)) .
The series in u has non-zero radius of convergence for |q1|, |q2| 6= 1. Moreover the series enjoys
property
(
u; q−1
1 , q2
)
∞ = 1/(q1u; q1, q2)∞, hence we can define double Pochhammer symbol for
any |q1|, |q2| 6= 1.
In particular, new definition implies
(
u; q, q−1
)
∞ = 1/(qu; q, q)∞; this is important to compare
our formulas with [5]. Below we assume |q| 6= 1.
Definition 9.28. Let us define q-deformed conformal block
Z(u1, . . . , un; z) = z
∑
(log ui)
2
2(log q)2
×
∏
i 6=j
1(
quiu
−1
j ; q, q
)
∞
〈
Wu
(
1|qu−1
n , . . . , qu−1
1
)
,W (z|u1, . . . , un)
〉
ss
. (9.19)
Remark 9.29. AGT statement claims that function Z(u1, . . . , un; z) is equal to Nekrasov par-
tition function for pure supersymmetric SU(n) 5d theory. This was conjectured in [2], the proof
follows from the geometric construction of the Whittaker vector given in the [41] and [47].
Theorem 9.30.
z
1
2
∑ i2
n2
∏
i 6=j
1(
q1+ i−j
n ; q, q
)
∞
(
q
1
n z
1
n ; q
1
n , q
1
n
)
∞
=
∑
(l0,...,ln−1)∈Q(n)
Z
(
ql0 , q
1
n
+l1 , . . . , q
n−1
n
+ln−1 ; z
)
.
The idea of the proof is to find two different expressions for
〈
W
(
1|q1/n
)
,W (z1/n|1)
〉
using
Theorem 9.16. To do this we need to simplify Z(u1, . . . , un; z) after substitution ui−1 = q
i
n
+li .
Let us concentrate on the second factor of (9.19).
42 M. Bershtein and R. Gonin
Proposition 9.31.
∏
i 6=j
(
q1+ i−j
n ; q, q
)
∞(
qli−lj+1+ i−j
n ; q, q
)
∞
= (−1)|λ|
∏
(i,j,k)
1(
q
k
2
+ i−j
2n − q−
k
2
− i−j
2n
)2(li−lj−k)
.
Proof. Let li − lj > 0. It is straightforward to check that(
q1+ i−j
n ; q, q
)
∞(
qli−lj+1+ i−j
n ; q, q
)
∞
=
(
q1+ i−j
n ; q
)li−lj
∞
li−lj−1∏
k=1
1(
1− qk+ i−j
n
)li−lj−k , (9.20)
(
q1+ j−i
n ; q, q
)
∞(
qlj−li+1+ j−i
n ; q, q
)
∞
=
(
q1+ j−i
n ; q
)lj−li
∞
li−lj−1∏
k=0
1(
1− q−k−
i−j
n
)li−lj−k . (9.21)
Denote by vij = q
i−j
n for i > j and vij = q
n+i−j
n for i < j. Formulas (9.20)–(9.21) implies the
following assertions. For i > j(
q1+ i−j
n ; q, q
)
∞(
qli−lj+1+ i−j
n ; q, q
)
∞
×
(
q1+ j−i
n ; q, q
)
∞(
qlj−li+1+ j−i
n ; q, q
)
∞
= (vij ; q)
li−lj
∞ (vji; q)
lj−li
∞
li−lj−1∏
k=0
(−1)li−lj−k(
q
k
2
+ i−j
2n − q−
k
2
− i−j
2n
)2(li−lj−k)
. (9.22)
For j > i(
q1+ i−j
n ; q, q
)
∞(
qli−lj+1+ i−j
n ; q, q
)
∞
×
(
q1+ j−i
n ; q, q
)
∞(
qlj−li+1+ j−i
n ; q, q
)
∞
= (vij ; q)
li−lj
∞ (vji; q)
lj−li
∞
li−lj−1∏
k=1
(−1)li−lj−k(
q
k
2
+ i−j
2n − q−
k
2
− i−j
2n
)2(li−lj−k)
. (9.23)
Using identities (9.22)–(9.23), we obtain
∏
i 6=j
(
q1+ i−j
n ; q, q
)
∞(
qli−lj+1+ i−j
n ; q, q
)
∞
=
∏
(i,j,k)
(−1)li−lj−k(
q
k
2
+ i−j
2n − q−
k
2
− i−j
2n
)2(li−lj−k)
.
To finish the proof, it remains to clarify the sign. This product already appeared as the product
over all hooks. For diagram λ the number of hooks is |λ|. �
Proof of Theorem 9.30. We will provide two different expressions for
〈
W
(
1|q1/n
)
,W (z1/n|1)
〉
to prove the theorem. On one hand (by Proposition 9.20)〈
W
(
1|q1/n
)
,W
(
z1/n|1
)〉
=
(
q
1
n z
1
n ; q
1
n , q
1
n
)
∞. (9.24)
On the other hand (by Theorem 9.16 and Proposition 9.24)
〈
W
(
1|q1/n
)
,W
(
z1/n|1
)〉
=
∑
(l0,...,ln−1)∈Q(n)
z
1
2
∑((
li+
i
n
)2
−
(
i
n
)2)
∏
(i,j,k)
(
q
1
2
(
k+ i−j
n
)
− q−
1
2
(
k+ i−j
n
))2(li−lj−k)
× (−1)|λ|
〈
W
(
1|q1−n−1
n
−ln−1 , . . . , q1−l0),W (z|ql0 , . . . , q n−1
n
+ln−1
)〉
ss
. (9.25)
Twisted Representations of Algebra of q-Difference Operators 43
Note that to prove (9.25) we also used following observations: lengths of hooks in λ and λ′
coincides, and c(λ) + c(λ′) = 0.
Multiplying r.h.s. of (9.24) and (9.25) by z
1
2
∑ i2
n2
∏
i 6=j
1(
q1+
i−j
n ;q,q
)
∞
we obtain
z
1
2
∑ i2
n2
∏
i 6=j
1(
q1+ i−j
n ; q, q
)
∞
(
q
1
n z
1
n ; q
1
n , q
1
n
)
∞ =
∑
(l0,...,ln−1)∈Q(n)
Z
(
ql0 , q
1
n
+l1 , . . . , q
n−1
n
+ln−1 ; z
)
.
Note that here we applied Proposition 9.31. �
A Regular product
In this section, we develop general theory of regular product. Term ‘regular product’ should be
considered as an opposite to regularized (i.e., normally ordered) product.
Let A(z) =
∑
k∈Z
Akz
−k be a formal power series with coefficients in End(V ) for a vector
space V .
Definition A.1. The series A(z) is called smooth if for any vector v ∈ V there exists N such
that Akv = 0 for k > N .
Let G(z, w) =
∑
k,l∈Z
Gk,lz
−kw−l be a formal power series in two variables with operator coef-
ficients. The operators Gk,l acts on a vector space V .
Definition A.2. We will call G(z, w) regular if for any N and for any v ∈ V there are only
finitely many Gk,l such that k + l = N and Gk,lv 6= 0.
If a current G(z, w) in two variables is regular one can substitute w = az and obtain well-
defined power series G(z, az) for any a ∈ C.
Let A(z) and B(w) be two smooth formal power series with operator coefficients. Recall
definition of normal ordering. Denote A+(z) =
∑
k>0
A−kz
k and A−(z) =
∑
k<0
A−kz
k.
Definition A.3. Normal ordered product is defined as
:A(z)B(w) : = A+(z)B(w) + (−1)εB(w)A−(z).
The sign (−1)ε depends on parity of A(z) and B(z) in the standard way. Note that smooth
formal power series in two variables :A(z)B(w) : is regular. Formal power series A(z) and B(z)
are called local (in weaker sense) if
A(z)B(w)− (−1)εB(w)A(z) =
N∑
i=1
si∑
j=0
C
(i)
j (w)∂jwδ(aiz, w), (A.1)
where s1, . . . , sN ∈ Z≥0, a1, . . . , aN ∈ C and C
(i)
j (w) are operator valued power series.
Then one has the following OPE
A(z)B(w) =
∑ C
(i)
j (w)
(aiz)j
(
1− w
aiz
)j + :A(z)B(w) : . (A.2)
Proposition A.4. If currents A(z) and B(w) are smooth and satisfy (A.1), then the following
product (a1z − w)s1 · · · (aNz − w)sNA(z)B(w) is regular.
44 M. Bershtein and R. Gonin
Definition A.5. For a 6= ai define regular product
A(z)B(az) :=
(
(a1z − w)s1 · · · (aNz − w)sNA(z)B(w)
)∣∣
w=az
(a1z − az)s1 · · · (aNz − az)sN
.
From (A.2) one obtains that normally ordered product and regular product are connected by
the following relation
A(z)B(az) =
∑ C
(i)
j (az)
(aiz)j(1− a/ai)j
+ :A(z)B(az) : .
Example A.6. Let us consider case of fermions A(z) = ψ(z), B(z) = ψ∗(z), introduced in Sec-
tion 3.2. Beware, that we use notation A(z) = Akz
−k, but ψ(z) = ψiz
−i−1 (hence Ak = ψk−1).
Comparing formulas (3.6)–(3.7) with Definition A.3, we conclude
:ψ(z)ψ∗(w) : =: ψ(z)ψ∗(w) :(0) .
Using l-depended normal ordering, we obtain
ψ(z)ψ∗(w) =
wlz−l−1
1− w/z
+ :ψ(z)ψ∗(w) :(l) . (A.3)
Hence
ψ(z)ψ∗(qz) =
ql
1− q
z−1+ :ψ(z)ψ(qz) :(l) .
This relation was used in formula (3.9).
Example A.7. Let A(z) = B(z) = E(z). Then
E(z)E(w) =
q−1w
z − q−1w
E2
(
q−1w
)
− qw
z − qw
E2(w)+ : E(z)E(w) : .
Therefore,
E2(w) =
1
q − 1
E2
(
q−1w
)
+
q
q − 1
E2(w)+ : E(w)E(w) : . (A.4)
Let us comment on deep meaning of formula (A.4). One can present algebra Diffq using cur-
rents Ek(z) (currents of Lie algebra type) or Ek(z) (currents of q-W algebra type). This two
series of currents are connected in non-trivial way starting from k = 2. For k = 2 they are
related by (A.4). For general k see formula (7.17) in [43].
Proposition A.8. Regular product is (super) commutative and associative
A(z)B(az) = (−1)εB(az)A(z),
A(a1z)(B(a2z)C(a3z)) = (A(a1z)B(a2z))C(a3z)
(of course we assume that these regular products are well defined).
Proposition A.9. Let A(z), B(z), and a ∈ C be as in Definition A.5. Then
∂z(A(z)B(az)) = A′(z)B(az) + aA(z)B′(az).
Twisted Representations of Algebra of q-Difference Operators 45
Proof. Let f(z, w) be a polynomial such that following power series in two variables are regular
(∂zf(z, w))A(z)B(w), (∂wf(z, w))A(z)B(w),
f(z, w)(∂zA(z))B(w), f(z, w)A(z)(∂wB(w)).
Moreover, assume that f(z, az) 6= 0. It is easy to see that such f(z, w) exists.
∂z(A(z)B(az)) = (∂z + a∂w)
f(z, w)A(z)B(w)
f(z, az)
∣∣∣∣
w=az
.
One should differentiate this expression by application of Leibniz rule (and obtain six sum-
mands). Due to our assumptions, each of these summands is regular in the sense of Defini-
tion A.2. Hence, one can substitute w = az to each summand separately. The proof is finished
by straightforward computation. �
As a corollary we prove formula (7.17).
Proof of (7.17).
∂zE(k+1)(z, w)
∣∣
z=w
= (z − qw)
(
z − q−1w
)
E′(z)Ek(w)
+
(
2z − qw − q−1w
)
E(z)Ek(w)
∣∣
z=w
,
(z − qw)2
(
z − q−1w
)2
(1− q)
(
1− q−1
)
z2
E′(z)Ek(w)
+
(z − qw)
(
z − q−1w
)
(1− q)
(
1− q−1
)
z2
(
2z − qw − q−1w
)
E(z)Ek(w)
∣∣∣∣
z=w
.
Note that each summand is regular. Hence, we are allowed substitute z = w to each of them
separately
∂zE(k+1)(z, w)
∣∣
z=w
= (1− q)
(
1− q−1
)
w2E′(w)Ek(w) +
(
2− q − q−1
)
wEk+1(w). (A.5)
Using Propositions A.8 and A.9, we can prove inductively that ∂wE
k+1(w) = (k+1)E′(w)Ek(w).
To finish the proof, we substitute last formula into (A.5). �
B Serre relation
This appendix is devoted to detailed study of Serre relation.
z2z
−1
3 [E(z1), [E(z2), E(z3)]] + cyclic = 0.
Let Ē(z) be a current satisfying
(z − qw)
(
z − q−1w
)
[Ē(z), Ē(w)] = 0. (B.1)
Remark B.1. Let us emphasize the difference between E(z) and Ē(z). Current E(z) is a current
from Diffq, but current Ē(z) is just a current satisfying (B.1). We need Ē(z) to formulate
equivalent conditions to Serre relations.
Define δ(z1/z2/z3) =
∑
a+b+c=0
za1z
b
2z
c
3 for a, b, c ∈ Z.
46 M. Bershtein and R. Gonin
Proposition B.2. There exist three currents R1(z), R2(z) and Ē3(z) such that triple commu-
tator [Ē(z1), [Ē(z2), Ē(z3)]] equals to
Ē3(z1)δ
(
q2z1/qz2/z3
)
− Ē3(z2)δ
(
q2z2/qz3/z1
)
− Ē3(z1)δ
(
q2z1/qz3/z2
)
+ Ē3(z3)δ
(
q2z3/qz2/z1
)
+R1(z1)δ(z1/qz2/z3)−R1(z1)δ(z1/z2/qz3)
+R2(z1)δ
(
z1/z2/q
−1z3
)
−R2(z1)
(
z1/q
−1z2/z3
)
. (B.2)
Proof. First of all, note that condition (B.1) is equivalent to existence of currents Ē2(z)
and Ē◦2(w) such that
[Ē(z), Ē(w)] = Ē2(z)δ(w/qz)− Ē◦2(w)δ(z/qw). (B.3)
Commutator [Ē(z), Ē(w)] is skew symmetric on z and w, hence Ē2(z) = Ē◦2(z). Now consider
triple commutator [Ē(z1), [Ē(z2), Ē(z3)]. Jacobi identity and relation (B.1) imply
(z1 − qz2)
(
z1 − q−1z2
)
(z1 − qz3)
(
z1 − q−1z3
)
[Ē(z1), [Ē(z2), Ē(z3)] = 0. (B.4)
Substituting (B.3) to (B.4), we obtain
(z1 − qz2)
(
z1 − q−1z2
)(
z1 − q2z2
)
(z1 − z2)[Ē(z1), Ē2(z2)]δ(z3/qz2)
+
(
z1 − q2z3
)
(z1 − z3)(z1 − qz3)
(
z1 − q−1z3
)
[Ē(z1), Ē2(z3)]δ(z2/qz3) = 0.
This implies that [Ē(z1)[Ē(z2), Ē(z3)]] is indeed a sum of triple delta functions with some oper-
ator coefficients as in (B.2); it remains to prove proposed relations on the coefficients.
Note that triple commutator [Ē(z1), [Ē(z2), Ē(z3)]] is skew symmetric on z2, z3. Also note
that the sum over cyclic permutations is zero. This implies relation (B.2). �
Proposition B.3. Serre relation for Ē(z) is equivalent to R1(z) = R2(z) = 0.
Proof. Straightforward computation. �
B.1 Operator product expansion for E(w1) · · ·E(wk)
One can find reformulation of Serre relation in terms of OPE in [20, Section 3.3].
Proposition B.4. Formal power series in three variables Ē(z1)Ē(z2)Ē(z3) can be presented as
sum some of regular part and singular part. Regular part is some regular power series in three
variables, singular part has a form
∑
ε1,ε2=±1
((
1− qε1 z2
z1
)−1(
1− qε2 z3
z1
)−1
R(1)
ε1,ε2(z3)
+
(
1− qε1 z2
z1
)−1(
1− qε2 z3
z2
)−1
R(2)
ε1,ε2(z3)
+
(
1− qε1 z3
z1
)−1(
1− qε2 z3
z2
)−1
R(3)
ε1,ε2(z3)
)
+
∑
ε=±1
((
1− qε z3
z2
)−1
R(1)
ε (z1, z3)
+
(
1− qε z3
z1
)−1
R(2)
ε (z2, z3) +
(
1− qε z2
z1
)−1
R(3)
ε (z1, z3)
)
+ reg.
Proof. Denote G(z1, z2, z3) :=
∏
i<j
(zi− qzj)
(
zi− q−1zj
)
E(z1)E(z2)E(z3). Relation (B.1) yields
G(z1, z2, z3) to be regular. �
Twisted Representations of Algebra of q-Difference Operators 47
Proposition B.5. Serre relation for Ē(z) is equivalent to condition that singular part of
E(z1)E(z2)E(z3) restricted to z1 = z3 has no poles of order greater than 1.
Proof. Note that second order pole can appear only from terms of form(
1− qε z2
z1
)−1(
1− q−ε z3
z2
)−1
R
(2)
ε,−ε(z3)
for ε = ±1. Note that these two poles can not cancel because they are at the different points z1 =
qεz2. On the other hand,R1(z) = R
(2)
1,−1(z) and R2(z) = R
(2)
−1,1(z). Application of Proposition B.3
completes the proof. �
Corollary B.6. E(z)Ek(w) has no poles of order greater than one. Poles may appear only at
points z = q±1w.
Proof. To study E(z)E(w)k we will consider OPE E(z)E(w1) · · ·E(wk) and substitute wi = w.
Only term
(
z − qεwi
)−1(
z − qεwj
)−1 · · · can give poles of order higher than 1 after substi-
tution. OPE is symmetric on z, w1, . . . , wk as a rational function. We will consider order(
E(wi)E(z)E(wj)
)
· · · . According to Proposition B.5, the term
(
z − qεwi
)−1(
z − qεwj
)−1 · · ·
does not appear. �
C Homomorphism from Diffq to W -algebra
This appendix is devoted to proof of Proposition 7.13. The proof is a straightforward check of
relation from Proposition 2.6. The relations will be checked for operators
H̃(z) =
∑
j 6=0
H̃jz
−j , c = n, c′ = ntw,
Ẽ(z) = µ exp
(
1
n
ϕ̃−(z)
)
T1(z) exp
(
1
n
ϕ̃+(z)
)
,
F̃ (z) = − µ−1z−ntw(
q
1
2 − q−
1
2
)2 exp
(
− 1
n
ϕ̃−(z)
)
Tn−1(z) exp
(
− 1
n
ϕ̃+(z)
)
.
Proposition C.1. Relations (2.3), (2.4) are satisfied.
Proof. Straightforward. �
Proposition C.2. Currents Ẽ(z) and F̃ (z) satisfy relation (2.5).
Proof. It is easy to see that
Ẽ(z)Ẽ(w) =
(1− w
z )2
(1− qwz )(1− q−1w
z )
f1,n(w/z)µ2
× exp
(
1
n
(ϕ̃−(z) + ϕ̃−(w))
)
T1(z)T1(w) exp
(
1
n
(ϕ̃+(z) + ϕ̃+(w))
)
.
Thus,
(z − qw)
(
z − q−1w
)(
Ẽ(z)Ẽ(w)− Ẽ(w)Ẽ(z)
)
= (z − w)2 exp
(
1
n
(ϕ̃−(z) + ϕ̃−(w))
)
×
(
f1,n(w/z)T1(z)T1(w)− f1,n(z/w)T1(w)T1(z)
)
exp
(
1
n
(ϕ̃+(z) + ϕ̃+(w))
)
= 0.
Proof for F̃ (z) is analogous. �
48 M. Bershtein and R. Gonin
Proposition C.3. Currents Ẽ(z) and F̃ (z) satisfy relation (2.6) for c = n and c′ = ntw.
Proof. It is easy to see that
Ẽ(z)F̃ (w) = − 1(
q
1
2 − q−
1
2
)2 fn−1,n(w/z) exp
(
1
n
(ϕ̃−(z)− ϕ̃−(w))
)
× T1(z)Tn−1(w) exp
(
1
n
(ϕ̃+(z)− ϕ̃+(w))
)
w−ntw .
Thus,
[
Ẽ(z), F̃ (w)
]
= exp
(
1
n
(ϕ̃−(z)− ϕ̃−(w))
)
×
(
n
w
z
δ′
(w
z
)
wntw + wδ
(w
z
)
∂ww
ntw
)
exp
(
1
n
(ϕ̃+(z)− ϕ̃+(w))
)
w−ntw .
Consequently,
[
Ẽ(z), F̃ (w)
]
= n
w
z
δ′
(w
z
)
+
(
H̃
(
q−
1
2 z
)
− H̃
(
q
1
2 z
)
+ ntw
)
δ
(w
z
)
. �
Denote by Ẽ(2)(z, w) = (z − qw)(z − q−1w)Ẽ(z)Ẽ(w).
Lemma C.4. Ẽ2(z) = 2µ2 exp
(
2
n ϕ̃−(z)
)
T2(z) exp
(
2
n ϕ̃+(z)
)
.
Proof.
f1,n(w/z)T1(z)T1(w)− f1,n(z/w)T1(w)T1(z) = µ−2 exp
(
− 1
n
(ϕ̃−(z) + ϕ̃−(w))
)
× Ẽ(2)(z, w) exp
(
− 1
n
(ϕ̃+(z) + ϕ̃+(w))
)
∂w
(
w−1δ(w/z)
)
= −µ−2
(
q
1
2 − q−
1
2
)2
exp
(
− 2
n
ϕ̃−(w)
)
Ẽ2(w) exp
(
− 2
n
ϕ̃+(w)
)
w
z
δ′(w/z) + · · · δ(w/z).
On the other hand, l.h.s. can be found from relation (7.8). Comparing coefficients of δ′(w/z)
completes the proof. �
Proposition C.5. Ẽ(z) and F̃ (z) satisfy Serre relation (2.7).
Proof. Using Lemma C.4, we see that
Ẽ(z)Ẽ2(w) = 2µ3
(
1− w
z
)2(
1− qwz
)(
1− q−1w
z
) exp
(
1
n
(ϕ̃−(z) + 2ϕ̃−(w))
)
× f2,n(w/z)T1(z)T2(w) exp
(
1
n
(ϕ̃+(z) + 2ϕ̃+(w))
)
.
Note that (z − w)2f2,n(w/z)T1(z)T2(w) is regular. Proposition B.5 completes the proof of
Serre relation for Ẽ(z). Proof for F̃ (z) is analogous. �
Twisted Representations of Algebra of q-Difference Operators 49
D Whittaker vector
D.1 Uniqueness of Whittaker vector
Recall that operators Ek[d] are defined by Ek(z) =
∑
dE
k[d]z−d.
Proposition D.1. Whittaker vector W (z|u1, . . . , un) is annihilated by Ek[d] for d > 0 and
k = 1, . . . , n− 1.
Proof. Actually we will prove that Whittaker vector is annihilated by Ek[d] for nd > k. To do
this we will need [43, equation (7.17)]. Let us rewrite this with respect to our notation
Ek[d] =
∑
t>1
(−1)k−t
ki∈N, di∈Z;
k1+···+kt=k
d1+···+dt=d∑
v={ d1
k1
6 d2
k2
6···6 dt
kt
}
cvEk1,d1 · · ·Ekt,dt . (D.1)
Here cv denotes some combinatorially defined coefficient, which is not quite important for us.
Inequality nd > k implies ndt > kt; therefore Ekt,dtW (z|u1, . . . , un) = 0. So, any summand of
r.h.s. annihilates Whittaker vector. �
Let us denote Wq(gln) :=Wq(gln, 0) = U(Diffq)/Jn,0. Denote Verma module for Wq(gln) by
VWq(gln)
λ1,...,λd
(cf. Definition 7.25).
Definition D.2. For each graded Diffq module M let us define Shapovalov dual module M∨.
As a vector space M∨ is graded dual to M . Action is defined by requirement that canonical
pairing M∨ ⊗M → C is Shapovalov.
Finally note that involution Ea,b 7→ E−a,−b maps ideal Jn,0 to Jn,0 (maybe with different µ).
Hence if M is a Wq(gln)-module then so is M∨.
Proposition D.3. Let ui/uj 6= qk for any k ∈ Z (cf. Lemma 9.1). There is no more than one
Whittaker vector W (z|u1, . . . , uN ) ∈ Fu1 ⊗ · · · ⊗ FuN .
Proof. Denote by n a subalgebra of Wq(gln) generated by Ek[d] and Hj for k = 1, . . . , n − 1,
d > 0 and j > 0. Analogously, let n∨ be a subalgebra of Wq(gln) generated by F k[−d] and H−j
for k = 1, . . . , n−1, d > 0 and j > 0. Note that involution Ea,b 7→ E−a,−b induces an involoution
on Wq(gln) which swaps n and n∨.
Consider Fu1 ⊗ · · · ⊗ Fun as a Wq(gln)-module. Whittaker vector is an eigenvector for n.
Hence, it is enough to show that Fu1⊗· · ·⊗Fun is cocyclic for n. Equivalently, we need to prove
that Shapovalov dual module (Fu1 ⊗ · · · ⊗ Fun)∨ ∼= Fq/un ⊗ · · · ⊗ Fq/u1 is cyclic for n∨.
Fock module Fq/un⊗· · ·⊗Fq/u1 is isomorphic to Verma module VWq(gln)
λ1,...,λd
for corresponding λj
by Theorem 7.31. Verma module VWq(gln)
λ1,...,λd
is cyclic for n∨ by an analogue of Proposition 7.26
for F k[d]. �
D.2 Construction of Whittaker vector
Let (n′1, n1) and (n′2, n2) be a basis of Z2.
Theorem D.4 ([1]). There exist homomorphisms
Φ: F (n′1,n1)
u ⊗F (n′2,n2)
v → F (n′1+n′2,n1+n2)
−q−1/2uv
,
Φ∗ : F (n′1+n′2,n1+n2)
−q−1/2uv
→ F (n′1,n1)
u ⊗F (n′2,n2)
v .
These homomorphisms are defined uniquely up to normalization.
50 M. Bershtein and R. Gonin
Remark D.5. Actually operators Φ and Φ∗ maps to graded completion of F (n′1+n′2,n1+n2)
−q−1/2uv
and
F (n′1,n1)
u ⊗F (n′2,n2)
v correspondingly. Abusing notation, we will use the same symbol for a module
and its completion. Moreover, we are going to consider a composition of such Φ∗; there appear
an infinite sum as a result of such composition (a priori this sum does not make sense). We
will use a calculus approach to infinite sums; below we will provide a sufficient condition for
convergence of the series.
Denote by Φ∗µ(u) component of Φ∗ corresponding to |µ〉 ∈ F (n′2,n2)
u . More precisely, for any x
in F (n′1+n′2,n1+n2)
−q−1/2uv
Φ∗ · x =
∑
µ
|µ〉 ⊗ (Φ∗µ(u) · x).
To simplify our notation we will consider particular case
Φ∗ : F (1,−k)
−q−1/2uv
→ F (0,1)
u ⊗F (1,−k−1)
v .
Note that both F (1,−k)
−q−1/2uv
and F (1,−k−1)
v are Fock modules for Heisenberg algebra generated by
ak = Ek,0.
Proposition D.6 ([1]). Operator Φ∗µ(u) is defined by following explicit formulas
Φ∗∅(u) = :exp
∑
k 6=0
u−k
k
(
1− q−k
)ak
:,
Φ∗µ(u) ∼ :Φ∗∅(u)
∏
s∈λ
F
(
qc(s)−
1
2u
)
:.
Here sign ∼ means up to multiplication by a number. Recall
F (z) ∼ : exp
(∑
k
qk/2 − q−k/2
k
akz
−k
)
: .
Corollary D.7. Heisenberg normal ordering is given by
Φ∗λ(u)Φ∗µ(v) =
fλ,µ(v/u)
(qv/u; q, q)∞
:Φ∗λ(u)Φ∗µ(v) :
for some rational function fλ,µ(v/u) =
∏
i(1−qkiv/u)∏
j(1−q
lj v/u)
; here lj and ki are integer numbers.
Consider a homomorphism
Φ̃0 : F (1,0)
q1/2z−1 ⊗F
(−1,n)
q1/2z(−q−1/2)nu1···un
→
(
F (0,1)
u1 ⊗ · · · ⊗ F (0,1)
un
)
⊗F (1,−n)
q1/2(−q1/2)n(zu1...un)−1 ⊗F
(−1,n)
q1/2z(−q−1/2)nu1···un
obtained as composition of
id⊗k⊗Φ∗ ⊗ id :
(
F (0,1)
u1 ⊗ · · · ⊗ F (0,1)
uk
)
⊗F (1,−k)
q1/2(−q1/2)k(zu1...uk)−1 ⊗F
(−1,n)
q1/2z(−q−1/2)nu1···un
→
(
F (0,1)
u1 ⊗ · · · ⊗ F (0,1)
uk+1
)
⊗F (1,−k−1)
q1/2(−q1/2)k+1(zu1...uk+1)−1 ⊗F
(−1,n)
q1/2z(−q−1/2)nu1···un
→ .
Twisted Representations of Algebra of q-Difference Operators 51
Lemma D.8. There exists a unique invariant pairing Fu ⊗Fqu−1 → C such that 〈0|0〉 = 1.
Proof. This is equivalent to Proposition 9.18. �
Composition of Φ̃0 and the pairing gives a homomorphism
Φ̃1 : F (1,0)
q
1
2 z−1
⊗F (−1,n)
q
1
2 u1···unz(−q−1/2)n
→ F (0,1)
u1 ⊗ · · · ⊗ F (0,1)
un .
Let us reformulate above inductive procedure via an explicit formula
Φ̃1(|λ1〉 ⊗ |λ2〉) =
∑
〈λ2|Φ∗µn(un) · · ·Φ∗µ1(u1)|λ1〉 |µ1〉 ⊗ · · · ⊗ |µn〉.
As we warned in Remark D.5, operator Φ̃1 is not a priori well defined. However, the series
(appearing from the composition) converges in a domain |u1| � |u2| � · · · � |un|. This
assertion follows from a formula
〈λ2|Φ∗µn(un) . . .Φ∗µ1(u1)|λ1〉 =
∏
i<j
fµi,µj (ui/uj)
(qui/uj ; q, q)∞
〈λ1| :Φ∗µn(un) . . .Φ∗µ1(u1) : |λ2〉.
Moreover, one can consider analytic continuation of obtained function given by r.h.s. of the
formula. Corollary D.7 implies that we can extend the domain to ui/uj 6= qk for any k ∈ Z.
Evidently, analytic continuation also enjoys intertwiner property. Hence we obtained following
proposition
Proposition D.9. If ui/uj 6= qk for any k ∈ Z, then there is an intertwiner
Φ̃ : F (1,0)
q1/2z−1 ⊗F
(−1,n)
q1/2u1···unz(−q−1/2)n
→ F (0,1)
u1 ⊗ · · · ⊗ F (0,1)
un . (D.2)
Denote the highest vector of F (n1,n2)
u by |n1, n2〉.
Theorem D.10. Whittaker vector W (z|u1, . . . , un) ∈ F (0,1)
u1 ⊗· · ·⊗F (0,1)
un can be constructed via
homomorphism Φ̃ (as in (D.2))
W (z|u1, . . . , un) :=
∏
i<j
(qui/uj ; q, q)∞Φ̃(|1, 0〉 ⊗ | − 1, n〉).
Proof. Follows from (3.15). �
Remark D.11. Recall that existence of Whittaker vector can be seen from geometric construc-
tion (see [41] and [47]).
Proof of Theorem 9.5. Existence and uniqueness follows from Theorem D.10 and Proposi-
tion D.3 correspondingly. �
D.3 Whittaker vector for Wq(sln) algebra
Let us define coefficient c̄1/m by (cf. (D.1))
Em[1] = c̄1/mEm,1 + · · · . (D.3)
Also recall that for Fu1 ⊗ · · · ⊗ Fun
µ =
1
1− q
(u1 · · ·un)
1
n .
52 M. Bershtein and R. Gonin
Definition D.12. For m = 1, . . . , n − 1 Whittaker vector W sln
m (z|u1, . . . , un) ∈ FWq(sln)
u1,...,ud with
respect to Wq(sln) is an eigenvector for Tk[r] (for k = 1, . . . , n − 1 and r > 0) with eigenvalues
given by
Tm[1]W sln
m (z|u1, . . . , un) =
(
−q
1
2
)n
u1 · · ·unz
q−1/2 − q1/2
c̄1/mµ
−m
m!
W sln
m (z|u1, . . . , un),
Tk[1]W sln
m (z|u1, . . . , un) = 0 for k 6= m,
Tk[r]W
sln
m (z|u1, . . . , un) = 0 for r > 2.
We require W sln
m (z|u1, . . . , un) = |ū〉+ · · · to fix normalization (by dots we mean lower vectors).
One can find notion of Whittaker vector for Wq(sln) in the literature (see [46]). In this
section we will explain connection between notion of Whittaker vector W sln
m (z|u1, . . . , un) and
Whittaker vector W (z|u1, . . . , un) for Diffq (see Definition 9.2). Our plan to explain this con-
nection is as follows. First we define Whittaker vector with respect to Wq(gln) (we denote it by
W
gln
m (z|u1, . . . , un)). Then we will see, that on the one hand, the vector W
gln
m (z|u1, . . . , un) is
connected with W sln
m (z|u1, . . . , un); on the other hand it is connected with W (z|u1, . . . , un).
Recall, thatWq(gln) = Diffq/Jn,0; ideal Jn,0 annihilates Fu1⊗· · ·⊗Fun . Hence Fu1⊗· · ·⊗Fun
is a representation of Wq(gln).
Definition D.13. Form = 1, . . . , n−1 Whittaker vectorW
gln
m (z|u1, . . . , un) is a vector belonging
to Fu1 ⊗ · · · ⊗ Fun and satisfying following conditions
HkW
gln
m (z|u1, . . . , un) = 0 for k > 0,
Em[1]W gln
m (z|u1, . . . , un) =
(
−q
1
2
)n
u1 · · ·unz
q−1/2 − q1/2
c̄1/mW
gln
m (z|u1, . . . , un),
Ek[1]W gln
m (z|u1, . . . , un) = 0 for k < m,
Ek[r]W gln
m (z|u1, . . . , un) = 0 for r > 2 and k 6 m,
F k[r]W gln
m (z|u1, . . . , un) = 0 for r > 1 and k < n−m.
We require W
gln
m (z|u1, . . . , un) = |ū〉+ · · · to fix normalization (by dots we mean lower vectors).
Recall that Fu1 ⊗ · · · ⊗ Fun ∼= F
Wq(sln)
u1,...,un ⊗ FH with respect algebra identification Wq(gln) ∼=
Wq(sln)⊗ U(Heis).
Lemma D.14. Vector satisfies properties of W
gln
m (z|u1, . . . , un) iff it is W sln
m (z|u1, . . . , un) ⊗
|0〉H .
Proof. Note that
Ek(t)
(
W sln
m (z|u1, . . . , un)⊗ |0〉H
)
=
k!
µ−k
Tk(t)W
sln
m (z|u1, . . . , un)⊗ exp
(
k
n
ϕ−(t)
)
|0〉H ,
F k(t)
(
W sln
m (z|u1, . . . , un)⊗ |0〉H
)
=
k!
µk
Tn−k(t)W
sln
m (z|u1, . . . , un)⊗ exp
(
−k
n
ϕ−(t)
)
|0〉H .
Moreover ϕ−(t) has only terms of positive degree in t. Hence we expressed action of Ek[l] via
T̃k[s] for s > l. Therefore we have proven that W sln
m (z|u1, . . . , un) ⊗ |0〉H satisfies property of
W
gln
m (z|u1, . . . , un).
The implication in opposite direction is analogous. �
Proposition D.15. There exists at most one vector W
gln
m (z|u1, . . . , un) ∈ Fu1 ⊗ · · · ⊗ Fun.
Twisted Representations of Algebra of q-Difference Operators 53
Proof. Analogous to proof of Proposition D.3. The only difference is that we consider a different
character of subalgebra n. �
We need to generalize notion of Whittaker vector for Diffq to compare it with W sln
m (z|u1,
. . . , un).
Definition D.16. For any m ∈ Z, Whittaker vector Wm(z|u1, . . . , un) ∈ Fu1 ⊗ · · · ⊗ Fun is an
eigenvector of operators Ea,b for mb > a > −(n−m)b and b > 0. More precisely,
E−(n−m)k,kWm(z|u1, . . . , un) =
zk
qk/2 − q−k/2
Wm(z|u1, . . . , un),
Emk,kWm(z|u1, . . . , un) =
((
−q
1
2
)n
u1 · · ·unz
)k
q−k/2 − qk/2
Wm(z|u1, . . . , un)
for k > 0,
Ek1,k2Wm(z) = 0
for (n−m)k2 > k1 > −mk2 and k2 > 0.We require Wm(z|u1, . . . , un) = |0〉 ⊗ · · · ⊗ |0〉+ · · · to
fix normalization (by dots we mean lower vectors).
Recall that we have defined operator Iτ ∈ End(Fu) by (3.16). By Proposition 3.14 the
operator enjoys intertwiner property Iτρ(Ea,b)I
−1
τ = ρ(Ea−b,b). Denote Iτ,n = Iτ ⊗ · · · ⊗ Iτ ∈
End(Fu1⊗· · ·⊗Fun). Note that Iτ,n also enjoys intertwiner property Iτρn(Ea,b)I
−1
τ = ρn(Ea−b,b)
(here ρn denotes the homomorphism of the representation ρn : Diffq → End(Fu1 ⊗ · · · ⊗ Fun)).
Proposition D.17. Wm(z|u1, . . . , un) = In−mτ,n W (z|u1, . . . , un).
Corollary D.18. There exists unique Wm(z|u1, . . . , un) if ui/uj 6= qk.
Proposition D.19. There exists unique vector W
gln
m (z|u1, . . . , un). Moreover, W
gln
m (z|u1, . . . ,
un) = Wm(z|u1, . . . , un).
Proof. We already know uniqueness of W
gln
m (z|u1, . . . , un) and existence of Wm(z|u1, . . . , un)
from Proposition D.15 and Corollary D.18 correspondingly. So it is sufficient to show that
Wm(z|u1, . . . , un) satisfies properties of W
gln
m (z|u1, . . . , un). Last assertion follows from for-
mula (D.1) (also see (D.3)). �
Theorem D.20. There exists unique vector W sln
m (z|u1, . . . , un). Moreover, W sln
m (z|u1, . . . , un)⊗
|0〉H = Wm(z|u1, . . . , un) = In−mσ,n W (z|u1, . . . , un).
Proof. Follows from Lemma D.14 and Proposition D.19. �
Acknowledgments
We are grateful to B. Feigin, P. Gavrylenko, E. Gorsky, A. Neguţ, J. Shiraishi, for interest to our
work and discussions. The work is partially supported by Russian Foundation of Basic Research
under grant mol a ved 18-31-20062 and by the HSE University Basic Research Program jointly
with Russian Academic Excellence Project ‘5-100’. R.G. was also supported in part by Young
Russian Mathematics award. The results of Section 9 are obtained under the support of the
Russian Science Foundation under grant 19-11-00275.
54 M. Bershtein and R. Gonin
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1 Introduction
2 q-difference operators
2.1 SL2(Z ) action
2.2 Chevalley generators and relations
3 Fock module
3.1 Free boson realization
3.2 Free fermion realization
3.3 Semi-infinite construction
4 Explicit formulas for twisted representation
4.1 Fermionic construction
4.2 Bosonic construction
4.3 Strange bosonic construction
5 Twisted representation via a sublattice
5.1 Sublattices and subalgebras
5.2 Twisted Fock vs restricted Fock
5.3 Explicit formulas for restricted Fock
5.3.1 Fermionic construction via sublattice
5.3.2 Bosonic construction via sublattices
5.3.3 Strange bosonic construction via sublattices
6 Twisted representation via a Semi-infinite construction
7 q-W-algebras
7.1 Definitions
7.1.1 Non-twisted W-algebras
7.1.2 Twisted q-W-algebras
7.2 Connection of Wq ( sln, ntw ) with Diffq
7.3 Bosonization of Wq ( sln, ntw )
7.3.1 Fock representation via Diffq
7.3.2 Explicit formula for bosonization
7.3.3 Explicit formulas for strange bosonization
7.4 Verma modules vs Fock modules
8 Restriction on Diffq for general sublattice
8.1 Decomposition of restriction
8.2 Strange bosonization and odd bosonization
9 Relations on conformal blocks
9.1 Whittaker vector
9.1.1 Whittaker vector for Fu
9.1.2 Whittaker vector and restriction on sublattice
9.2 Shapovalov form
9.3 Conformal blocks
A Regular product
B Serre relation
B.1 Operator product expansion for E(w1) @let@token E(wk)
C Homomorphism from Diffq to W-algebra
D Whittaker vector
D.1 Uniqueness of Whittaker vector
D.2 Construction of Whittaker vector
D.3 Whittaker vector for Wq ( sln ) algebra
References
|
| id | nasplib_isofts_kiev_ua-123456789-210771 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T16:09:49Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bershtein, Mikhail Gonin, Roman 2025-12-17T14:32:12Z 2020 Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks. Mikhail Bershtein and Roman Gonin. SIGMA 16 (2020), 077, 55 pages 1815-0659 2020 Mathematics Subject Classification: 17B67; 17B69; 81R10 arXiv:1906.00600 https://nasplib.isofts.kiev.ua/handle/123456789/210771 https://doi.org/10.3842/SIGMA.2020.077 We study certain representations of the quantum toroidal 𝖌𝔩₁ algebra for 𝑞 = 𝘵. We construct explicit bosonization of the Fock modules 𝓕⁽ⁿ′'ⁿ⁾ᵤ with a nontrivial slope 𝑛′/𝑛. As a vector space, it is naturally identified with the basic level 1 representation of affine 𝖌𝔩ₙ. We also study twisted 𝑊-algebras of s𝔩n acting on these Fock modules. As an application, we prove the relation on 𝑞-deformed conformal blocks, which was conjectured in the study of 𝑞-deformation of isomonodromy/CFT correspondence. We are grateful to B. Feigin, P. Gavrylenko, E. Gorsky, A. Negut, and J. Shiraishi for their interest in our work and discussions. The work is partially supported by the Russian Foundation of Basic Research under grant mol_a_ved 18-31-20062 and by the HSE University Basic Research Program jointly with the Russian Academic Excellence Project "5-100". R.G. was also supported in part by the Young Russian Mathematics Award. The results of Section 9 are obtained under the support of the Russian Science Foundation under grant19-11-00275. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks Article published earlier |
| spellingShingle | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks Bershtein, Mikhail Gonin, Roman |
| title | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks |
| title_full | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks |
| title_fullStr | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks |
| title_full_unstemmed | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks |
| title_short | Twisted Representations of Algebra of 𝑞-Difference Operators, Twisted 𝑞-𝑊 Algebras and Conformal Blocks |
| title_sort | twisted representations of algebra of 𝑞-difference operators, twisted 𝑞-𝑊 algebras and conformal blocks |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210771 |
| work_keys_str_mv | AT bershteinmikhail twistedrepresentationsofalgebraofqdifferenceoperatorstwistedqwalgebrasandconformalblocks AT goninroman twistedrepresentationsofalgebraofqdifferenceoperatorstwistedqwalgebrasandconformalblocks |