Quantum Toroidal Comodule Algebra of Type $A_{n-1}$ and Integrals of Motion

Quantum Toroidal Comodule Algebra of Type $\boldsymbol{A_{n-1}}$
and Integrals of Motion

Boris FEIGIN ab, Michio JIMBO c and Evgeny MUKHIN d

B. Feigin, M. Jimbo and E. Mukhin

a) National Research University Higher School of Economics,
a) 20 Myasnitskaya Str., Moscow, 101000, Russia b) Landau Institute for
Theoretical Physics, 1a Akademika Semenova Ave.,
b) Chernogolovka, 142432, Russia<EMAIL_ADDRESS>
c) Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501,
Japan<EMAIL_ADDRESS>
d) Department of Mathematics, Indiana University Purdue University
Indianapolis,
d) 402 N. Blackford St., LD 270, Indianapolis, IN 46202, USA<EMAIL_ADDRESS>
Received March 02, 2022, in final form June 27, 2022; Published online July
07, 2022

We introduce an algebra $\mathcal{K}_{n}$ which has a structure of a left
comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra
$\mathcal{K}_{n}$ is a higher rank generalization of $\mathcal{K}_{1}$, which
provides a uniform description of deformed $W$ algebras associated with Lie
(super)algebras of types BCD. We show that $\mathcal{K}_{n}$ possesses a
family of commutative subalgebras.

quantum toroidal algebras; comodule; integrals of motion

81R10; 81R12; 17B69; 17B80

##  1 Introduction

Integrable systems originated in quantum toroidal algebras have been drawing a
lot of attention in recent years, see [1, 4, 5, 8, 9, 10, 11, 14, 15, 16], to
name a few. In this paper we continue the study launched in [7] concerning
deformed $W$ algebras and a quantum toroidal version of integrable models with
reflections in the spirit of [21].

The work [7] is based on a new algebra $\mathcal{K}_{1}$ generated by current
$E(z)$, Heisenberg half-currents $K^{\pm}(z)$, and a central element $C$. In
various representations the current $E(z)$ recovers the fundamental current of
known deformed $W$ algebras of types BCD as well as their supersymmetric
analogs. It was shown that $\mathcal{K}_{1}$ has a family of commuting
elements $\\{\mathbf{I}_{N}\\}_{N=1}^{\infty}$ called integrals of motion. The
element $\mathbf{I}_{N}$ is written as an $N$-fold integral of the product
$\mathbf{E}(z_{N})\cdots\mathbf{E}(z_{1})$ with the explicit elliptic kernel
of [19]. Here $\mathbf{E}(z)$ is the dressed current of the form
$\mathbf{E}(z)=E(z)\mathbf{K}^{+}(z)^{-1}$, where $\mathbf{K}^{+}(z)$ is given
in terms of $K^{+}(z)$. In the representations corresponding to deformed $W$
algebras, the integrals of motion $\mathbf{I}_{N}$ are deformations of local
integrals of motion in the $W$ algebras, see [2].

In the present paper we generalize this construction by introducing algebras
$\mathcal{K}_{n}$ for $n>1$. The algebra $\mathcal{K}_{n}$ depends on
parameters $q_{1},q_{2}=q^{2}$, and is generated by currents
$E_{1}(z),\dots,E_{n}(z)$, Heisenberg half-currents
$K_{1}^{\pm}(z),\dots,K_{n}^{\pm}(z)$, and a central element $C$. We show that
it has the following key properties:

  * •

Algebra $\mathcal{K}_{n}$ has essentially the same size as the Borel
subalgebra of quantum toroidal $\mathfrak{gl}_{n}$ algebra $\mathcal{E}_{n}$;
see Remark 3.1.

  * •

For $m<n$, algebra $\mathcal{K}_{m}$ can be obtained as a subalgebra of
$\mathcal{K}_{n}$ using the fusion procedure in the same way as
$\mathcal{E}_{m}$ is found as a subalgebra of $\mathcal{E}_{n}$ in [3]; see
Proposition 6.2.

  * •

Algebra $\mathcal{K}_{n}$ is a left comodule over quantum toroidal algebra
$\mathcal{E}_{n}$; see Theorem 3.4.

  * •

Algebra $\mathcal{K}_{n}$ has two “boundary” modules $\mathbb{F}^{D}$ of level
$C=q^{-1}$ and $\mathbb{F}^{B}$ of level $C=q^{1/2}$; see Lemmas 3.5 and 3.6.

  * •

Algebra $\mathcal{K}_{n}$ has a family of commuting integrals of motion
$\\{\mathbf{I}_{N}\\}_{N=1}^{\infty}$, such that $\mathbf{I}_{N}$ is an
$nN$-fold integral of a current given by an ordered product of the form
$\prod_{1\leq i\leq n}^{\curvearrowleft}\prod_{1\leq a\leq
N}^{\curvearrowleft}\mathbf{E}_{i}(z_{a})$ together with the kernel of [12];
see (4.9) and Theorem 4.2. Here the dressed currents have the form
$\mathbf{E}_{i}(z)=\mathbf{K}^{-}_{i}(z)^{-1}E_{i}(z)$, where
$\mathbf{K}^{-}_{i}(z)$ is given in terms of $K^{-}_{i}(z)$.111We change the
convention slightly from [7].

Here we should make a disclaimer: the relations of $\mathcal{K}_{n}$ involve
infinite sums, so that an appropriate completion is necessary. The same is
true with the fusion procedure. Also the comodule map is defined at the level
of a class of representations called admissible ones.

These properties ensure that $\mathcal{K}_{n}$ has a rich representation
theory. At least, we have representations of the form $V\otimes W$, where $W$
is one of the two boundary modules and $V$ is any admissible $\mathcal{E}_{n}$
module. Here by an admissible module we mean a homogeneously graded module
with the degree bounded from above; for example any tensor product of Fock
modules (or, more generally, modules with highest weight with respect to the
“rotated” set of generators) is admissible. Then, under a mild restriction on
parameters $q_{1}$, $q_{2}$ and the level $C$, in any $\mathcal{K}_{n}$ module
we have an integrable system given by action of $\\{\mathbf{I}_{N}\\}$. This
is the principal result of this paper. For $n=1$, this system is related to
the system with reflections, see [7, Section 5.1], and the Bethe ansatz for it
is studied in [16] in the setting of affine Yangians. Although the expressions
of $\\{\mathbf{I}_{N}\\}$ are non-local, we shall sometimes refer to them as
“local” integrals of motion, since in the case $n=1$ they are deformations of
local ones as already mentioned.

We do not completely understand the nature of algebras $\mathcal{K}_{n}$. The
$\mathcal{K}_{n}$ algebras we discuss here are of A type. Similar algebras can
be defined in the same way for arbitrary simply laced types, though the
construction of integrals of motion is more challenging. At first glance,
algebra $\mathcal{K}_{n}$ bears resemblance to $\imath$quantum groups; formula
(3.16) for comodule structure is reminiscent of the Chevalley generators [13,
formula (5.1)], and the defining relations (3.1)–(3.6) of $\mathcal{K}_{n}$
look similar to the Drinfeld realization given in [17, Theorem 5.1]. However,
the details seem to be very different. For example, the relations in
$\mathcal{K}_{n}$ are invariant under scaling $z\to cz$ but such homogeneity
does not exist in those of [17]. The current appearing in the residues in the
right-hand side of (5.5) in [17] is commutative, whereas our current
$K_{i}^{-}(z)K_{i}^{+}\big{(}C^{2}z\big{)}$ in (3.4) is not. The
$\imath$quantum groups are coideals in the corresponding quantum groups, while
we are not aware of inclusion of $\mathcal{K}_{n}$ to $\mathcal{E}_{n}$. We do
not know if $\imath$quantum groups have interesting families of integrals of
motion in general. Natural candidates which could be compared to
$\mathcal{K}_{n}$ are the reflection algebras and, in particular, quantum
twisted Yangians, see [18]. The quantum twisted Yangians are coideal
subalgebras behind integrable systems with reflections. However, the exact
relation to $\mathcal{K}_{n}$ algebras is unclear. The quantum twisted
Yangians are given in terms of the $R$-matrix realization (as opposed to the
Drinfeld-type realization of $\mathcal{K}_{n}$) and the formulas for the
generators of the coideal are quadratic in generators of quantum group.

In quantum toroidal algebras $\mathcal{E}_{n}$, the integrals of motion
originate from the standard construction of transfer matrices and depend on
$n$ arbitrary parameters. They commute for a simple reason following from the
Yang–Baxter equation. Then a computation provides the expression for the
integrals of motion in the form of integrals. The integrals of motion in the
$\mathcal{K}_{n}$ algebra are given virtually by the same formula as in
$\mathcal{E}_{n}$, except that one of the parameters (the elliptic nome) is
fixed by the central element $C$. But we do not have any general reason for
their commutativity; we simply check it directly.

It is known that in type A there is a natural duality of such models of
$(\mathfrak{gl}_{m},\mathfrak{gl}_{n})$ type, see [6]. Under that duality, the
algebras $\mathcal{E}_{m}$ and $\mathcal{E}_{n}$ act on the same space in such
a way that the screening operators for one algebra are given in terms of
generating currents of the other. Furthermore the integrals of motion of both
algebras commute with each other. For $\mathcal{K}_{1}$ acting in a
representation, we have a system of screening operators, see [7]. We expect
that the algebras generated by the corresponding screening currents also
possess a commutative family which should be called “non-local” integrals of
motion, and that “local” and “non-local” integrals of motion commute.

Similarly, we expect that algebras $\mathcal{K}_{n}$ acting in representations
commute with a family of screening operators. Moreover the algebra generated
by the screening currents does not depend on $n$. For the case of the tensor
product of $\ell$ Fock $\mathcal{E}_{n}$ modules with a boundary
$\mathcal{K}_{n}$ module, we anticipate $\ell+1$ screening currents each given
as a sums of $n$ vertex operators. The corresponding “non-local” integrals of
motion should commute with the “local” integrals of motion
$\\{\mathbf{I}_{N}\\}$ constructed in this paper. We plan to address this
issue in future publications.

The text is organized as follows.

In Section 2 we introduce our convention about the quantum toroidal algebra
${\mathcal{E}}_{n}$ of type $\mathfrak{gl}_{n}$. In Section 3 we define
algebra $\mathcal{K}_{n}$. Working with admissible representations, we
rephrase the Serre relations as zero conditions (called wheel conditions) on
matrix coefficients of products of generating currents. We show that on
admissible representations, $\mathcal{K}_{n}$ may be viewed as a left comodule
over ${\mathcal{E}}_{n}$. We use this fact to construct representations
starting with elementary ones. In Section 4 we introduce integrals of motion.
They are integrals of products of generating currents of $\mathcal{K}_{n}$
obtained by a dressing procedure. The kernel functions entering integrals of
motion are precisely the same as those used for those of ${\mathcal{E}}_{n}$.
We then prove that integrals of motion mutually commute by direct computation.

For simplicity of presentation, we treat the case $n\geq 3$ in the body of the
text. We shall mention the modifications necessary for $n=1,2$ in Section 5.

In Section 6 we show that, for any $k=1,\dots,n-1$, algebra
$\mathcal{K}_{n}=\mathcal{K}_{n}(q_{1},q_{2},q_{3})$ with parameters $q_{1}$,
$q_{2}$, $q_{3}$ contains mutually commuting subalgebras
$\mathcal{K}_{k}(\bar{q}_{1},\bar{q}_{2},\bar{q}_{3})$ and
$\mathcal{K}_{n-k}(\bar{\bar{q}}_{1},\bar{\bar{q}}_{2},\bar{\bar{q}}_{3})$
with appropriate parameters $\bar{q}_{i}$, $\bar{\bar{q}}_{i}$.

In Appendix A, we give a proof of the theta function identities used in the
commutativity of integrals of motion.

######  Notation 1.1.

Throughout the text we fix parameters
$q^{1/2},d^{1/2}\in{\mathbb{C}}^{\times}$ and define

$\displaystyle q_{1}=q^{-1}d,\qquad q_{2}=q^{2},\qquad q_{3}=q^{-1}d^{-1},$

so that $q_{1}q_{2}q_{3}=1$. We assume that $q_{1}^{i}q_{2}^{j}=1$ for
$i,j\in{\mathbb{Z}}$ implies $i=j=0$.

For a positive integer $N$, we write $\delta^{(N)}_{i,k}=1$ if $i\equiv k\bmod
N$ and $\delta^{(N)}_{i,k}=0$ otherwise.

We use the standard symbols for infinite products

$\displaystyle(z_{1},\dots,z_{r};p)_{\infty}=\prod_{i=1}^{r}\prod_{k=0}^{\infty}\big{(}1-z_{i}p^{k}\big{)},\qquad\Theta_{p}(z)=\big{(}z,pz^{-1},p;p\big{)}_{\infty}.$

We use also the formal series $\delta(z)=\sum_{k\in{\mathbb{Z}}}z^{k}$.

##  2 Preliminaries

In this section we fix our convention regarding the quantum toroidal algebra
${\mathcal{E}}_{n}$ of type $\mathfrak{gl}_{n}$.

###  2.1 Quantum toroidal algebra $\boldsymbol{{\mathcal{E}}_{n}}$

Fix a positive integer $n\geq 3$. We define $g_{i,j}(z,w)$, $G_{i,j}(x)$ for
$i,j\in{\mathbb{Z}}/n{\mathbb{Z}}$ as follows

$\displaystyle g_{i,i}(z,w)=z-q^{2}w,\qquad
G_{i,i}(x)=q^{2}\frac{1-q^{-2}x}{1-q^{2}x},$ $\displaystyle g_{i,i\pm
1}(z,w)=z-q^{-1}d^{\pm 1}w,\qquad G_{i,i\pm 1}(x)=q^{-1}\frac{1-qd^{\pm
1}x}{1-q^{-1}d^{\pm 1}x},$ $\displaystyle g_{i,j}(z,w)=z-w,\qquad
G_{i,j}(x)=1\qquad\text{if}\quad j\not\equiv i,i\pm 1.$

Here $G_{i,j}(x)$ should be understood as power series expansion in $x$. As
rational function we have

$\displaystyle
G_{i,j}(w/z)=-\frac{g_{j,i}(w,z)}{d_{i,j}g_{i,j}(z,w)}=G_{j,i}(z/w)^{-1},$

where $d_{i,j}=d^{\pm 1}$, $i\equiv j\pm 1$, and $d_{i,j}=1$ otherwise.

Let $P$ be a free ${\mathbb{Z}}$ module with basis $\\{\varepsilon_{i}\mid
i\in{\mathbb{Z}}/n{\mathbb{Z}}\\}$, equipped with a symmetric bilinear form
$(~{},~{})\colon P\times P\to{\mathbb{Z}}$ such that $\varepsilon_{i}$ are
orthonormal. We write $\bar{\alpha}_{i}=\varepsilon_{i-1}-\varepsilon_{i}$.

The quantum toroidal algebra ${\mathcal{E}}_{n}$ of type $\mathfrak{gl}_{n}$
is a unital associative algebra generated by

$\displaystyle e_{i,k},\quad f_{i,k},\quad h_{i,r},\quad
i\in{\mathbb{Z}}/n{\mathbb{Z}},\ k\in{\mathbb{Z}},\
r\in{\mathbb{Z}}\backslash\\{0\\},\qquad q^{h},\quad h\in P,\qquad C^{\pm 1}.$

In terms of the generating series

$\displaystyle e_{i}(z)=\sum_{k\in{\mathbb{Z}}}e_{i,k}z^{-k},\qquad
f_{i}(z)=\sum_{k\in{\mathbb{Z}}}f_{i,k}z^{-k},$
$\displaystyle\psi^{\pm}_{i}(z)=\psi_{i,0}^{\pm}\exp\bigg{(}{\pm}\big{(}q-q^{-1}\big{)}\sum_{r>0}h_{i,\pm
r}z^{\mp r}\bigg{)},\qquad\psi_{i,0}^{\pm}=q^{\pm\bar{\alpha}_{i}},$

the defining relations read as follows

$\displaystyle q^{h}q^{h^{\prime}}=q^{h+h^{\prime}},\quad h,h^{\prime}\in
P,\qquad q^{0}=1,\qquad\text{$C$ is central},$ $\displaystyle
q^{h}e_{i}(z)q^{-h}=q^{(h,\bar{\alpha}_{i})}e_{i}(z),\quad
q^{h}f_{i}(z)q^{-h}=q^{-(h,\bar{\alpha}_{i})}f_{i}(z),\quad
q^{h}\psi^{\pm}_{i}(z)q^{-h}=\psi_{i}^{\pm}(z),\quad h\in P,$
$\displaystyle\psi^{\pm}_{i}(z)\psi^{\pm}_{j}(w)=\psi^{\pm}_{j}(w)\psi^{\pm}_{i}(z),$
$\displaystyle\psi^{+}_{i}(z)\psi^{-}_{j}(w)=\psi^{-}_{j}(w)\psi^{+}_{i}(z)G_{i,j}(Cw/z)^{-1}G_{i,j}\big{(}C^{-1}w/z\big{)},$
$\displaystyle\psi_{i}^{+}(z)e_{j}(w)=e_{j}(w)\psi_{i}^{+}(z)G_{i,j}\big{(}C^{-1}w/z\big{)},\qquad
e_{j}(w)\psi_{i}^{-}(z)=\psi_{i}^{-}(z)e_{j}(w)G_{j,i}(z/w),$
$\displaystyle\psi_{i}^{+}(z)f_{j}(w)=f_{j}(w)\psi_{i}^{+}(z)G_{i,j}(w/z)^{-1},\qquad
f_{j}(w)\psi_{i}^{-}(z)=\psi_{i}^{-}(z)f_{j}(w)G_{j,i}(Cz/w)^{-1},$
$\displaystyle[e_{i}(z),f_{j}(w)]=\frac{\delta_{i,j}}{q-q^{-1}}(\delta\bigl{(}C{w}/{z}\bigr{)}\psi_{i}^{+}(w)-\delta\bigl{(}C{z}/{w}\bigr{)}\psi_{i}^{-}(z)),$
$\displaystyle[e_{i}(z),e_{j}(w)]=0,\qquad[f_{i}(z),f_{j}(w)]=0,\qquad
i\not\equiv j,j\pm 1,$ $\displaystyle
d_{i,j}g_{i,j}(z,w)e_{i}(z)e_{j}(w)+g_{j,i}(w,z)e_{j}(w)e_{i}(z)=0,$
$\displaystyle
d_{j,i}g_{j,i}(w,z)f_{i}(z)f_{j}(w)+g_{i,j}(z,w)f_{j}(w)f_{i}(z)=0,$
$\displaystyle\mathop{\mathrm{Sym}}_{{z_{1},z_{2}}}[e_{i}(z_{1}),[e_{i}(z_{2}),e_{i\pm
1}(w)]_{q}]_{q^{-1}}=0,$
$\displaystyle\mathop{\mathrm{Sym}}_{{z_{1},z_{2}}}[f_{i}(z_{1}),[f_{i}(z_{2}),f_{i\pm
1}(w)]_{q}]_{q^{-1}}=0.$

Here $[A,B]_{p}=AB-pBA$, and the symbol $\mathrm{Sym}$ stands for
symmetrization:

$\displaystyle\mathop{\mathrm{Sym}}_{x_{1},\dots,x_{N}}\
f(x_{1},\dots,x_{N})=\frac{1}{N!}\sum_{\sigma\in\mathfrak{S}_{N}}f(x_{\sigma(1)},\dots,x_{\sigma{(N)}}).$

The relations above imply in particular

$\displaystyle[h_{i,r},h_{j,s}]=\delta_{r+s,0}\cdot
a_{i,j}(r)\frac{C^{r}-C^{-r}}{q-q^{-1}},\qquad r,s\neq 0,$ $\displaystyle
a_{i,j}(r)=\frac{1}{r}\frac{q^{r}-q^{-r}}{q-q^{-1}}\big{(}(q^{r}+q^{-r})\delta^{(n)}_{i,j}-d^{r}\delta^{(n)}_{i,j-1}-d^{-r}\delta^{(n)}_{i,j+1}\big{)}.$
(2.1)

Algebra ${\mathcal{E}}_{n}$ has a ${\mathbb{Z}}$ grading which we call
homogeneous grading, given by

$\displaystyle\deg x_{i,r}=r,\quad x=e,f,h,\qquad\deg q^{h}=\deg C=0.$

###  2.2 Coproduct

Quite generally, let $\mathcal{A}$ be a ${\mathbb{Z}}$ graded algebra with a
central element $C$. The completion of $\mathcal{A}$ in the positive direction
is the algebra $\tilde{\mathcal{A}}$, linearly spanned by products of series
of the form $\sum_{k=M}^{\infty}f_{k}g_{k}$, where $M\in{\mathbb{Z}}$,
$f_{k},g_{k}\in\mathcal{A}$ and $\deg g_{k}=k$. We call an $\mathcal{A}$
module $V$ admissible if $C$ is diagonalizable, and if $V$ is ${\mathbb{Z}}$
graded with finite dimensional components of degree bounded from above:
$V=\oplus_{k=-\infty}^{N}V_{k}$, where $V_{k}=\\{v\in V\mid\deg v=k\\}$, $\dim
V_{k}<\infty$. The completion $\tilde{\mathcal{A}}$ acts on all admissible
modules.

Let ${\mathcal{E}}_{n}\tilde{\otimes}{\mathcal{E}}_{n}$ be the tensor algebra
${\mathcal{E}}_{n}\otimes{\mathcal{E}}_{n}$ completed in the positive
direction. We use the topological coproduct
$\Delta\colon{\mathcal{E}}_{n}\to{\mathcal{E}}_{n}\tilde{\otimes}{\mathcal{E}}_{n}$

$\displaystyle\Delta e_{i}(z)=e_{i}(C_{2}z)\otimes\psi^{-}_{i}(z)+1\otimes
e_{i}(z),$ $\displaystyle\Delta f_{i}(z)=f_{i}(z)\otimes
1+\psi^{+}_{i}(z)\otimes f_{i}(C_{1}z),$
$\displaystyle\Delta\psi^{+}_{i}(z)=\psi^{+}_{i}(z)\otimes\psi^{+}_{i}(C_{1}z),$
$\displaystyle\Delta\psi^{-}_{i}(z)=\psi^{-}_{i}(C_{2}z)\otimes\psi^{-}_{i}(z),$
$\displaystyle\Delta x=x\otimes x,\qquad x=q^{h},C.$

Here $C_{1}=C\otimes 1$ and $C_{2}=1\otimes C$.

Later on we shall use the elements in the completed algebra
$\tilde{\mathcal{E}}_{n}$

$\displaystyle\tilde{f}_{i}(z)=S^{-1}\bigl{(}f_{i}(z)\bigr{)}=-f_{i}\big{(}C^{-1}z\big{)}\psi_{i}^{+}\big{(}C^{-1}z\big{)}^{-1},$

where $S$ denotes the antipode. The $\tilde{f}_{i}(z)$’s satisfy the same
quadratic relations and the Serre relations as do $e_{i}(z)$’s. In addition

$\displaystyle
d_{j,i}g_{j,i}(w,z)e_{i}(z)\tilde{f}_{j}(w)+g_{i,j}(z,w)\tilde{f}_{j}(w)e_{i}(z)$
$\displaystyle\qquad{}=-\delta_{i,j}\frac{g_{i,i}(w,z)}{q-q^{-1}}\bigl{(}\delta(w/z)-\delta\big{(}C^{2}z/w\big{)}\psi_{i}^{-}(z){\psi_{i}^{+}(Cz)}^{-1}\bigr{)}.$
(2.2)

###  2.3 Fock modules

It is well known [20] that ${\mathcal{E}}_{n}$ has admissible representations
given in terms of vertex operators. Consider a vector space with basis
${|\mathbf{m}\rangle}$ labeled by $n$ tuples of integers
$\mathbf{m}=(m_{0},m_{1},\dots,m_{n-1})\allowbreak\in{\mathbb{Z}}^{n}$. We
define linear operators ${\rm e}^{\pm\epsilon_{i}}$ and $\partial_{i}$, $0\leq
i\leq n-1$, by

$\displaystyle{\rm
e}^{\pm\epsilon_{i}}{|\mathbf{m}\rangle}=(-1)^{\sum_{s=0}^{i-1}m_{s}}{|\mathbf{m}\pm\mathbf{1}_{i}\rangle},\qquad\partial_{i}{|\mathbf{m}\rangle}=m_{i}{|\mathbf{m}\rangle},$

with $\mathbf{1}_{i}=(\delta_{i,j})_{0\leq j\leq n-1}$. We have ${\rm
e}^{\epsilon_{i}}{\rm e}^{\epsilon_{j}}=-{\rm e}^{\epsilon_{j}}{\rm
e}^{\epsilon_{i}}$ for $i\neq j$. We set

$\displaystyle\mathbb{F}={\mathbb{C}}[\\{h_{i,-r}\\}_{r>0,0\leq i\leq
n-1}]\otimes\bigg{(}\bigoplus_{\mathbf{m}\in{\mathbb{Z}}^{n}}{\mathbb{C}}{|\mathbf{m}\rangle}\bigg{)}.$

We write operators $h_{i,\pm r}\otimes{\rm id}$, ${\rm id}\otimes{\rm
e}^{\pm\epsilon_{i}}$ and ${\rm id}\otimes\partial_{i}$ simply as $h_{i,\pm
r}$, ${\rm e}^{\pm\epsilon_{i}}$ and $\partial_{i}$. We extend the range of
the suffix $i$ by periodicity, e.g., $\partial_{i+n}=\partial_{i}$. Define a
vertex operator which acts on $\mathbb{F}$

$\displaystyle V_{i}(z)={\rm e}^{-\epsilon_{i}}{\rm
e}^{\epsilon_{i-1}}z^{\partial_{i-1}-\partial_{i}}d^{(\partial_{i-1}+\partial_{i})/2}\exp\bigg{(}\sum_{r>0}\frac{q^{-r}}{[r]}h_{i,-r}z^{r}\bigg{)}\exp\bigg{(}{-}\sum_{r>0}\frac{1}{[r]}h_{i,r}z^{-r}\bigg{)}.$

For $u\in{\mathbb{C}}^{\times}$, the following assignment gives $\mathbb{F}$ a
structure of an admissible ${\mathcal{E}}_{n}$ module denoted by
$\mathbb{F}(u)$,

$\displaystyle e_{i}(z)\mapsto
u^{-\delta_{i,0}}zV_{i}(z),\qquad\tilde{f}_{i}(z)\mapsto-q^{-1}u^{\delta_{i,0}}z\,{:}V_{i}(z)^{-1}{:},$
$\displaystyle h_{i,r}\mapsto h_{i,r},\quad r\neq 0,\qquad
q^{\varepsilon_{i}}\mapsto q^{\partial_{i}},\qquad C\mapsto q.$

Here we use the usual normal ordering symbol to bring ${\rm
e}^{\pm\epsilon_{i}}$, $h_{i,-r}$, $r>0$, to the left and $\partial_{i}$,
$h_{i,r}$, $r>0$, to the right, except that the order between ${\rm
e}^{\pm\epsilon_{i}}$’s is kept unchanged. The ${\mathcal{E}}_{n}$ module
$\mathbb{F}(u)$ is a direct sum of irreducible submodules, see [6, Lemma 3.2].

##  3 Comodule algebra $\boldsymbol{\mathcal{K}_{n}}$

In this section we introduce an algebra $\mathcal{K}_{n}$, and show that it
has the structure of an ${\mathcal{E}}_{n}$ comodule in the sense to be made
precise below. Algebra $\mathcal{K}_{1}$ was first introduced and studied in
[7] in order to give a uniform description of deformed $W$ algebras associated
with simple Lie (super)algebras of types BCD.

###  3.1 Algebra $\boldsymbol{\mathcal{K}_{n}}$

We introduce an algebra $\mathcal{K}_{n}$ through generating series

$\displaystyle E_{i}(z)=\sum_{k\in{\mathbb{Z}}}E_{i,k}z^{-k},\qquad
K^{\pm}_{i}(z)=K^{\pm}_{i,0}\exp\biggl{(}\pm\big{(}q-q^{-1}\big{)}\sum_{\pm
r>0}H_{i,r}z^{-r}\biggr{)},\qquad i\in{\mathbb{Z}}/n{\mathbb{Z}},$

and an invertible central element $C$. The defining relations read as follows

$\displaystyle K_{i}^{\pm}(z)K_{j}^{\pm}(w)=K_{j}^{\pm}(w)K_{i}^{\pm}(z),$
(3.1) $\displaystyle
K_{i}^{+}(z)K^{-}_{j}(w)=K^{-}_{j}(w)K_{i}^{+}(z)G_{i,j}(w/z)G_{i,j}\big{(}C^{2}w/z\big{)},$
(3.2) $\displaystyle
K_{i}^{+}(z)E_{j}(w)=E_{j}(w)K_{i}^{+}(z)G_{i,j}(w/z),\quad
E_{j}(w)K_{i}^{-}(z)=K_{i}^{-}(z)E_{j}(w)G_{j,i}(z/w),$ (3.3) $\displaystyle
d_{i,j}g_{i,j}(z,w)E_{i}(z)E_{j}(w)+g_{j,i}(w,z)E_{j}(w)E_{i}(z)$ (3.4)
$\displaystyle\quad{}=\\!\frac{\delta_{i,j}}{q\\!-\\!q^{-1}}\big{(}g_{i,i}(z,w)\delta\bigl{(}C^{2}z/w\bigr{)}K^{-}_{i}(z)K^{+}_{i}\big{(}C^{2}z\big{)}\\!+\\!g_{i,i}(w,z)\delta\bigl{(}C^{2}w/z\bigr{)}K^{-}_{i}(w)K^{+}_{i}\big{(}C^{2}w\big{)}\big{)},$
$\displaystyle E_{i}(z)E_{j}(w)=E_{j}(w)E_{i}(z),\qquad i\not\equiv j,\,j\pm
1.$ (3.5)

In addition, we impose the Serre relations

$\displaystyle\mathop{\mathop{\mathrm{Sym}}}_{z_{1},z_{2}}\,[E_{i}(z_{1}),[E_{i}(z_{2}),E_{i\pm
1}(w)]_{q}]_{q^{-1}}=-\mathop{\mathop{\mathrm{Sym}}}_{z_{1},z_{2}}$
$\displaystyle\bigg{\\{}\bigg{(}\frac{1}{1-q^{-1}d^{\mp
1}z_{1}/w}+\frac{q^{-1}d^{\pm 1}w/z_{2}}{1-q^{-1}d^{\pm 1}w/z_{2}}\bigg{)}$
$\displaystyle\times\delta\big{(}C^{2}z_{1}/z_{2}\big{)}K_{i}^{-}(z_{1})E_{i\pm
1}(w)K_{i}^{+}(z_{2})\bigg{\\}}.$ (3.6)

In the right-hand side, the rational functions of the form $1/(1-az_{1}/w)$ or
$1/(1-bw/z_{2})$ stand for their expansions in non-negative powers of
$z_{1}/w$ or $w/z_{2}$, respectively.

In the right-hand sides of (3.4) and (3.6), the Fourier coefficients of
$K_{i}^{-}(z)K_{i}^{+}\big{(}C^{2}z\big{)}$ are infinite series. Therefore the
relations require some justification. We proceed as follows. We define a
${\mathbb{Z}}$ grading of generators by

$\displaystyle\deg E_{i,k}=k,\qquad\deg H_{i,\pm r}=\pm r,\qquad\deg
K_{i,0}^{\pm}=\deg C=0.$

We call it the homogeneous grading. Consider the free algebra $\mathcal{A}$
generated by $E_{i,k}$, $H_{i,\pm r}$, $K^{\pm}_{i,0}$, $C^{\pm 1}$. Let
$\widetilde{\mathcal{A}}$ be the completion of $\mathcal{A}$ with respect to
the grading in the positive direction. Then we define $\mathcal{K}_{n}$ to be
the graded quotient algebra of $\widetilde{\mathcal{A}}$ under the relations
(3.1)–(3.6).

The quadratic relations (3.4) imply

$\displaystyle
d_{i,j}\hat{g}_{i,j}(z,w)E_{i}(z)E_{j}(w)+\hat{g}_{j,i}(w,z)E_{j}(w)E_{i}(z)=0,$
(3.7)

where

$\displaystyle\hat{g}_{i,j}(z,w)=\begin{cases}\big{(}z-C^{2}w\big{)}\big{(}z-C^{-2}w\big{)}\big{(}z-q^{2}w\big{)},&i=j,\\\
g_{i,j}(z,w),&i\neq j.\end{cases}$

The factor $\big{(}z-C^{2}w\big{)}\big{(}z-C^{-2}w\big{)}$ for $i=j$ is chosen
to kill the delta functions in the right-hand side of (3.4).

######  Remark 3.1.

Algebra $\mathcal{K}_{n}$ has a filtration defined by $\mathop{\rm
pdeg}E_{i}(z)=1$, $\mathop{\rm pdeg}K_{i}^{\pm}(z)=0$ for all $i$, which we
call the principal filtration. In the associated graded algebra, the
subalgebra generated by $\\{E_{i}(z)\\}$ is isomorphic to the subalgebra of
the quantum toroidal algebra ${\mathcal{E}}_{n}$ generated by
$\\{e_{i}(z)\\}$.

The relations for $K^{\pm}_{i}(z)$ are slightly different from those of
$\psi^{\pm}_{i}(z)$ in ${\mathcal{E}}_{n}$, since the right-hand side of (3.2)
contains a product of $G_{i,j}$’s as opposed to the ratio. In particular,
while the elements $\psi^{\pm}_{i,0}\in{\mathcal{E}}_{n}$ are inverse to each
other, the elements $K_{i,0}^{\pm}\in\mathcal{K}_{n}$ are not mutually
commutative,

$\displaystyle
K_{i,0}^{+}K_{j,0}^{-}=q^{2(\bar{\alpha}_{i},\bar{\alpha}_{j})}K_{j,0}^{-}K_{i,0}^{+}.$

We have the relations

$\displaystyle[H_{i,r},H_{i,s}]=-\delta_{r+s,0}\cdot
a_{i,j}(r)\frac{1+C^{2r}}{q-q^{-1}},\qquad r>0,$

which are to be compared with (2.1). In the limit $q\to 1$ with $C=q^{c}$,
algebra ${\mathcal{E}}_{n}$ reduces to the enveloping algebra of a Lie algebra
of matrix valued difference operators, see [3, Section 3.7]. In contrast,
algebra $\mathcal{K}_{n}$ does not seem to have a reasonable limit due to the
presence of the sum $1+C^{2r}$.

###  3.2 Wheel conditions

All representations of ${\mathcal{E}}_{n}$ or $\mathcal{K}_{n}$ considered in
this paper are admissible representations. In this setting, we rewrite the
formal series relations of generating currents in the language of matrix
coefficients.

Let $V=\\!\oplus_{k=-\infty}^{k_{0}}\\!V_{k}$ be a graded vector space with
degrees bounded from above and $\dim V_{k}\\!<\\!\infty$. We denote by
$V^{*}=\oplus_{k=-\infty}^{k_{0}}V^{*}_{k}$ the restricted dual space. Suppose
that we are given a set of formal series
$\big{\\{}E_{i}(z)=\sum_{k\in{\mathbb{Z}}}E_{i,k}z^{-k}\mid
i\in{\mathbb{Z}}/n{\mathbb{Z}}\big{\\}}$, such that $E_{i,k}\in\mathop{\rm
End}V$, $E_{i,k}V_{l}\subset V_{k+l}$, satisfying the relations (3.5) and
(3.7) with some $C\in{\mathbb{C}}^{\times}$. We assume that the zeros of
$\hat{g}_{i,i}(z,w)$ are distinct:

$\displaystyle C^{2}\neq\pm 1,q^{\pm 2}.$

It follows from (3.5) and (3.7) that all matrix coefficients

$\displaystyle\langle w,E_{i_{1}}(z_{1})\cdots E_{i_{N}}(z_{N})v\rangle,\qquad
v\in V,\quad w\in V^{*},$

converge in the region $|z_{1}|\gg\cdots\gg|z_{N}|$ to rational functions
which have at most simple poles at $\hat{g}_{i_{r},i_{s}}(z_{r},z_{s})=0$,
where $r<s$ and $i_{r}\equiv i_{s},i_{s}\pm 1$.

Quite generally, let $A_{1}(z)$, $A_{2}(z)$ be operator-valued formal series
and $r_{i,j}(x)$ be rational functions. We shall say that an exchange relation

$\displaystyle
r_{1,2}(z_{2}/z_{1})A_{1}(z_{1})A_{2}(z_{2})=r_{2,1}(z_{1}/z_{2})A_{2}(z_{2})A_{1}(z_{1})$

holds as rational functions, if an arbitrary matrix coefficient of each side
converges (in regions $|z_{1}|\gg|z_{2}|$ and $|z_{2}|\gg|z_{1}|$
respectively) to the same rational function. Similarly we shall use the term
“exchange relation as meromorphic functions”.

Equations (3.5) and (3.7) imply that we have relations as rational functions

$\displaystyle\lambda^{0}_{i,j}(w/z)E_{i}(z)E_{j}(w)=\lambda^{0}_{j,i}(z/w)E_{j}(w)E_{i}(z),$
(3.8)

where we set

$\displaystyle\lambda_{i,i}^{0}(x)=\frac{1-C^{2}x}{1-x}\frac{1-C^{-2}x}{1-x}\frac{1-q^{2}x}{1-x},$
$\displaystyle\lambda_{i,i\pm 1}^{0}(x)=d^{\mp 1/2}\frac{1-q^{-1}d^{\pm
1}x}{1-x},$

and $\lambda^{0}_{i,j}(x)=1$ for $j\not\equiv i,i\pm 1$.

Set

$\displaystyle E_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})=\prod_{1\leq r<s\leq
N}\lambda^{0}_{i_{r},i_{s}}(z_{s}/z_{r})E_{i_{1}}(z_{1})\cdots
E_{i_{N}}(z_{N}).$ (3.9)

It follows from (3.8) that

$\displaystyle
E_{i_{\sigma(1)},\dots,i_{\sigma(N)}}(z_{\sigma(1)},\dots,z_{\sigma(N)})=E_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N}),\qquad\sigma\in\mathfrak{S}_{N}.$
(3.10)

In particular (3.9) is symmetric with respect to $z_{r}$ and $z_{s}$ when they
have the same “color” $i_{r}=i_{s}$. Moreover all matrix coefficients of (3.9)
have the form

$\displaystyle\frac{P(z_{1},\dots,z_{N})}{\prod_{r<s}(z_{r}-z_{s})^{N_{r,s}}},\qquad
P(z_{1},\dots,z_{N})\in{\mathbb{C}}\big{[}z_{1}^{\pm 1},\dots,z_{N}^{\pm
1}\big{]},$ (3.11)

where $N_{r,s}=2$ if $i_{r}=i_{s}$, $N_{r,s}=1$ if $i_{r}\equiv i_{s}\pm 1$,
and $N_{r,s}=0$ otherwise. Note that, when $i_{r}=i_{s}$, third order poles
are absent due to the symmetry in variables of the same color.

The following is easy to see.

######  Proposition 3.2.

Under the conditions (3.5) and (3.7), the condition (3.4) holds if and only if

$\displaystyle
E_{i,i}\big{(}z,C^{2}z\big{)}=\frac{1}{q-q^{-1}}\frac{1+C^{2}}{1-C^{2}}\frac{1-q^{2}C^{2}}{1-C^{2}}\,K^{-}_{i}(z)K_{i}^{+}\big{(}C^{2}z\big{)}.$
(3.12)

We have also

######  Proposition 3.3.

Under the conditions (3.5) and (3.7), the Serre relations (3.6) are satisfied
if and only if the following zero conditions hold:

$\displaystyle E_{i,i,i\pm 1}\big{(}z,q^{2}z,qd^{\mp 1}z\big{)}=0,$ (3.13)
$\displaystyle E_{i,i,i\pm 1}\big{(}z,C^{2}z,C^{2}q^{-1}d^{\mp 1}z\big{)}=0.$
(3.14)

If it is the case then we have also

$\displaystyle E_{i,i,i\pm 1}\big{(}z,C^{-2}z,C^{-2}qd^{\mp 1}z\big{)}=0.$
(3.15)

We call (3.13)–(3.15) “wheel conditions”.

###### Proof.

The left-hand side and the right-hand side of the Serre relations (3.6) read
respectively as

$\displaystyle{\rm LHS}=\\!{}$
$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2}}\bigl{(}E_{i}(z_{1})E_{i}(z_{2})E_{i+1}(w)\\!-\\!\big{(}q\\!+\\!q^{-1}\\!\big{)}E_{i}(z_{1})E_{i+1}(w)E_{i}(z_{2})\\!+\\!E_{i+1}(w)E_{i}(z_{1})E_{i}(z_{2})\bigr{)},$
$\displaystyle{\rm RHS}=\\!{}$
$\displaystyle-\mathop{\mathrm{Sym}}_{z_{1},z_{2}}\biggl{(}-\frac{q_{3}^{-1}w/z_{1}}{1-q_{3}^{-1}w/z_{1}}+\frac{q_{1}w/z_{2}}{1-q_{1}w/z_{2}}+\delta(q_{3}z_{1}/w)\biggr{)}\delta\big{(}C^{2}z_{1}/z_{2}\big{)}K_{i}^{-}(z_{1})$
$\displaystyle\times E_{i+1}(w)K^{+}_{i}(z_{2}).$

The case involving $E_{i}(z_{1})$, $E_{i}(z_{2})$ and $E_{i-1}(w)$ can be
treated by interchanging $q_{3}$ with $q_{1}$.

Using the definition (3.9), LHS can be rewritten as

$\displaystyle{\rm LHS}={}$ $\displaystyle
E_{i,i,i+1}(z_{1},z_{2},w)\bigg{(}\frac{1}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{2}}\big{)}}-\frac{q+q^{-1}}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{1}}\big{)}\underline{\lambda^{0}_{i+1,i}\big{(}\frac{z_{2}}{w}\big{)}}}$
$\displaystyle+\frac{1}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\underline{\lambda^{0}_{i+1,i}\big{(}\frac{z_{1}}{w}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{2}}{w}\big{)}}}+\frac{1}{\underline{\lambda^{0}_{i,i}\big{(}\frac{z_{1}}{z_{2}}\big{)}}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{2}}\big{)}}$
$\displaystyle-\frac{q+q^{-1}}{\underline{\lambda^{0}_{i,i}\big{(}\frac{z_{1}}{z_{2}}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{1}}{w}\big{)}}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{2}}\big{)}}+\frac{1}{\underline{\lambda^{0}_{i,i}(\frac{z_{1}}{z_{2}})\lambda^{0}_{i+1,i}\big{(}\frac{z_{1}}{w}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{2}}{w}\big{)}}}\bigg{)}.$

The matrix coefficients of $E_{i,i,i+1}(z_{1},z_{2},w)$ are Laurent
polynomials up to poles on $z_{1}=z_{2}$ or $z_{i}=w$, which are cancelled by
the zeros of $1/\lambda^{0}_{i,j}$.

This is a sum of six formal series, each expanded in a different region. In
order to bring them to expansions in a common region, we use the identity
$1/(1-z)=\delta(z)-z^{-1}/\big{(}1-z^{-1}\big{)}$, where $1/(1-z)$ stands for
its expansion in non-negative powers of $z$, while
$z^{-1}/\big{(}1-z^{-1}\big{)}$ means the one in negative powers of $z$. In
the underlined factors, we substitute

$\displaystyle\frac{1}{\lambda^{0}_{i+1,i}\big{(}\frac{z}{w}\big{)}}=\text{(expansion
in $|z|\gg|w|$)}+d^{-1/2}\big{(}1-q_{3}^{-1}\big{)}\delta(q_{3}{z}/{w}),$
$\displaystyle\frac{1}{\lambda^{0}_{i,i}\big{(}\frac{z_{1}}{z_{2}}\big{)}}=\text{(expansion
in
$|z_{1}|\gg|z_{2}|$)}-\frac{q^{-2}\big{(}1-q^{2}\big{)}^{3}}{\big{(}1-C^{-2}q^{2}\big{)}\big{(}1-C^{2}q^{2}\big{)}}\delta\big{(}q^{2}{z_{1}}/{z_{2}}\big{)}$
$\displaystyle\hphantom{\frac{1}{\lambda^{0}_{i,i}(\frac{z_{1}}{z_{2}})}=}{}+\frac{1-C^{2}}{1+C^{2}}\frac{1-C^{2}}{1-q^{2}C^{2}}\delta\big{(}C^{2}{z_{2}}/{z_{1}}\big{)}+\frac{1-C^{-2}}{1+C^{-2}}\frac{1-C^{-2}}{1-q^{2}C^{-2}}\delta\big{(}C^{-2}{z_{2}}/{z_{1}}\big{)}$

and bring all terms into the sum of their expansions in
$|z_{1}|\gg|z_{2}|\gg|w|$ and additional delta functions. We then compare the
coefficients of each product of delta functions.

Up to the symmetry $z_{1}\leftrightarrow z_{2}$, there are the following cases
to consider:

$\displaystyle(1)\quad$ $\displaystyle 1,$ $\displaystyle(2)\quad$
$\displaystyle\delta(q_{3}z_{1}/w),$ $\displaystyle(3)\quad$
$\displaystyle\delta\big{(}q^{2}z_{1}/z_{2}\big{)},$ $\displaystyle(4)\quad$
$\displaystyle\delta\big{(}C^{2}z_{2}/z_{1}\big{)},$ $\displaystyle(5)\quad$
$\displaystyle\delta(q_{3}z_{1}/w)\delta\big{(}q^{2}z_{1}/z_{2}\big{)},$
$\displaystyle(6)\quad$
$\displaystyle\delta(q_{3}z_{1}/w)\delta\big{(}C^{2}z_{2}/z_{1}\big{)},$
$\displaystyle(7)\quad$
$\displaystyle\delta(q_{3}z_{1}/w)\delta\big{(}C^{-2}z_{2}/z_{1}\big{)},$
$\displaystyle(8)\quad$
$\displaystyle\delta(q_{3}z_{1}/w)\delta(q_{3}z_{2}/w).$

In cases (1), (2) and (3), the coefficients vanish due to the identities of
rational functions

$\displaystyle 0={}$
$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2}}\bigg{\\{}\frac{1}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{2}}\big{)}}-\frac{q+q^{-1}}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\lambda^{0}_{i,i+1}\big{(}\frac{w}{z_{1}}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{2}}{w}\big{)}}$
$\displaystyle+\frac{1}{\lambda^{0}_{i,i}\big{(}\frac{z_{2}}{z_{1}}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{1}}{w}\big{)}\lambda^{0}_{i+1,i}\big{(}\frac{z_{2}}{w}\big{)}}\bigg{\\}},$
$\displaystyle 0={}$
$\displaystyle\frac{1}{\lambda^{0}_{i,i}(z_{2}/z_{1})}\frac{1}{\lambda^{0}_{i+1,i}(z_{2}/w)}\\!-\\!\big{(}q\\!+\\!q^{-1}\big{)}\frac{1}{\lambda_{i,i}^{0}(z_{1}/z_{2})}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{2})}\\!+\\!\frac{1}{\lambda^{0}_{i,i}(z_{1}/z_{2})}\frac{1}{\lambda^{0}_{i+1,i}(z_{2}/w)},$
$\displaystyle 0={}$
$\displaystyle\frac{1}{\lambda^{0}_{i,i+1}(w/z_{1})}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{2})}-\big{(}q+q^{-1}\big{)}\frac{1}{\lambda^{0}_{i+1,i}(z_{1}/w)}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{2})}$
$\displaystyle+\frac{1}{\lambda^{0}_{i+1,i}(z_{1}/w)}\frac{1}{\lambda^{0}_{i+1,i}(z_{2}/w)}.$

In case (8) the coefficient vanishes on $z_{1}=z_{2}$ due to the triple zero
of $\lambda^{0}_{i,i}(z_{1}/z_{2})^{-1}$ there.

In case (4), the relevant terms are the 3 terms containing
${\lambda^{0}_{i,i}(z_{1}/z_{2})}^{-1}$. Using the relation (3.12), we find
that the coefficient of $\delta\big{(}C^{2}z_{2}/z_{1}\big{)}$ is

$\displaystyle\frac{1-C^{2}}{1+C^{2}}\frac{1-C^{2}}{1-q^{2}C^{2}}E_{i,i}\big{(}C^{2}z_{2},z_{2}\big{)}E_{i+1}(w)\lambda^{0}_{i,i+1}(w/z_{1})\lambda^{0}_{i,i+1}(w/z_{2})$
$\displaystyle\qquad{}\times\bigg{(}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{1})}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{2})}-\big{(}q+q^{-1}\big{)}\frac{1}{\lambda^{0}_{i+1,i}(z_{1}/w)}\frac{1}{\lambda^{0}_{i,i+1}(w/z_{2})}$
$\displaystyle\qquad\hphantom{\times\bigg{(}}+\frac{1}{\lambda^{0}_{i+1,i}(z_{1}/w)}\frac{1}{\lambda^{0}_{i+1,i}(z_{2}/w)}\bigg{)}$
$\displaystyle\qquad{}=-\frac{d(w/z_{1}-q^{2}w/z_{2})}{\big{(}1-q_{3}^{-1}w/z_{1}\big{)}\big{(}1-q_{3}^{-1}w/z_{2}\big{)}}K_{i}^{-}(z_{2})K^{+}_{i}\big{(}C^{2}z_{2}\big{)}E_{i+1}(w),$

which coincides with the corresponding term in RHS.

Likewise, in case (7) the coefficient of
$\delta\big{(}C^{2}z_{2}/z_{1}\big{)}\delta\big{(}q_{3}^{-1}w/z_{1}\big{)}$
comes from

$\displaystyle
E_{i,i,i+1}(z_{1},z_{2},w)\frac{1}{\lambda^{0}_{i,i}(z_{1}/z_{2})}\frac{1}{\lambda^{0}_{i+1,i}(z_{2}/w)}\bigl{(}{-}\big{(}q+q^{-1}\big{)}\lambda^{0}_{i,i+1}(w/z_{2})+\lambda^{0}_{i+1,i}(z_{2}/w)\bigr{)}.$

Computing similarly we find that it matches with the corresponding term in
RHS.

It remains to consider (5) and (6). The coefficients of the product of delta
functions are proportional (with non-zero multipliers) to the left-hand side
of (3.13) and (3.14), respectively. Since there are no terms that come from
RHS, we have shown that the Serre relations are equivalent to (3.13) and
(3.14).

Instead of rewriting terms into expansions in $|z_{1}|\gg|z_{2}|\gg|w|$, one
can equally well proceed to the opposite region $|z_{1}|\ll|z_{2}|\ll|w|$.
Computing in the same way we obtain (3.15) in place of (3.14). ∎

###  3.3 Comodule structure

In this section we show that, when we restrict to admissible modules,
$\mathcal{K}_{n}$ may be viewed as a left comodule of ${\mathcal{E}}_{n}$. To
be precise we prove the following.222Formula for $\Delta$ is slightly changed
from the one used in [7]. The ${\mathcal{E}}_{1}$ comodule structure discussed
there should be understood at the level of admissible representations.

######  Theorem 3.4.

Let $V$ be an admissible ${\mathcal{E}}_{n}$ module and $W$ an admissible
$\mathcal{K}_{n}$ module, such that $C$ acts as scalar $C_{1}$, $C_{2}$
respectively. We set $C_{12}=C_{1}C_{2}$ and assume that

$\displaystyle C_{2}^{2}\neq\pm 1,q^{\pm 2},\qquad C_{12}^{2}\neq\pm 1,q^{\pm
2}.$

Then $V\otimes W$ is given the structure of an admissible $\mathcal{K}_{n}$
module by the action of the following currents:

$\displaystyle\Delta E_{i}(z)=e_{i}(C_{2}z)\otimes K^{-}_{i}(z)+1\otimes
E_{i}(z)+\tilde{f}_{i}\big{(}C_{2}^{-1}z\big{)}\otimes K^{+}_{i}(z),$ (3.16)
$\displaystyle\Delta
K_{i}^{+}(z)=\psi_{i}^{+}\big{(}C_{1}^{-1}C_{2}^{-1}z\big{)}^{-1}\otimes
K_{i}^{+}(z),$ (3.17) $\displaystyle\Delta
K_{i}^{-}(z)=\psi_{i}^{-}(C_{2}z)\otimes K_{i}^{-}(z),$ (3.18)
$\displaystyle\Delta C=C_{12}.$ (3.19)

Moreover the following “coassociativity” and “counit property” hold:

$\displaystyle(V_{1}\otimes V_{2})\otimes W=V_{1}\otimes(V_{2}\otimes W),$
$\displaystyle{\mathbb{C}}\otimes W=W.$

Here $V_{1}$, $V_{2}$ are admissible ${\mathcal{E}}_{n}$ modules and
${\mathbb{C}}$ denotes the trivial ${\mathcal{E}}_{n}$ module.

###### Proof.

On each vector of an admissible module, the currents of ${\mathcal{E}}_{n}$ or
$\mathcal{K}_{n}$ comprise only a finite number of negative powers in $z$.
Hence each Fourier coefficient of (3.16)–(3.18) is a well-defined operator on
$V\otimes W$.

It is easy to verify the relations (3.1)–(3.3) and (3.5) by direct
calculation. With the aid of (2.2), one can also check (3.4) for $i\neq j$.

Let us check (3.4) for $i=j$. We shall write $A_{1}(z,w)\equiv A_{2}(z,w)$ if
each matrix coefficient of $A_{1}(z,w)-A_{2}(z,w)$ is a Laurent polynomial.
Then we have on $V$

$\displaystyle g_{i,j}(z,w)e_{i}(z)e_{j}(w)\equiv 0,\qquad
g_{i,j}(z,w)\tilde{f}_{i}(z)\tilde{f}_{j}(w)\equiv 0,$ (3.20) $\displaystyle
e_{i}(z)\tilde{f}_{j}(w)\equiv\frac{\delta_{i,j}}{q-q^{-1}}\biggl{(}{-}\frac{1}{1-w/z}+\frac{1}{1-C^{-2}w/z}\psi_{i}^{-}(z)\psi_{i}^{+}\big{(}C^{-1}w\big{)}^{-1}\biggr{)},$
(3.21)

and on $W$

$\displaystyle g_{i,j}(z,w)E_{i}(z)E_{j}(w)$
$\displaystyle\qquad{}\equiv\frac{\delta_{i,j}}{q-q^{-1}}\bigg{(}\frac{z-q^{2}w}{1-C^{-2}w/z}K_{i}^{-}(z)K_{i}^{+}(w)+\frac{w-q^{2}z}{1-C^{2}w/z}K_{i}^{-}(w)K_{i}^{+}(z)\bigg{)}.$
(3.22)

Using these we obtain

$\displaystyle\big{(}q-q^{-1}\big{)}\big{(}z-q^{2}w\big{)}\Delta
E_{i}(z)\Delta E_{i}(w)$
$\displaystyle\qquad{}\equiv\big{(}q-q^{-1}\big{)}\big{(}z-q^{2}w\big{)}\bigl{(}e_{i}(C_{2}z)\tilde{f}_{i}\big{(}C_{2}^{-1}w\big{)}\otimes
K_{i}^{-}(z)K_{i}^{+}(w)$
$\displaystyle\qquad\hphantom{\equiv}{}+\tilde{f}_{i}\big{(}C_{2}^{-1}z\big{)}e_{i}(C_{2}w)\otimes
K_{i}^{+}(z)K_{i}^{-}(w)+1\otimes E_{i}(z)E_{i}(w)\bigr{)}$
$\displaystyle\qquad{}\equiv-\biggl{(}\frac{z-q^{2}w}{1-C_{2}^{-2}w/z}-\frac{z-q^{2}w}{1-C_{12}^{-2}w/z}\psi^{-}_{i}(C_{2}z)\psi_{i}^{+}\big{(}C_{12}^{-1}w\big{)}^{-1}\biggr{)}\otimes
K_{i}^{-}(z)K_{i}^{+}(w)$
$\displaystyle\qquad\hphantom{\equiv}{}+\bigg{(}\frac{w-q^{2}z}{1-C_{2}^{-2}z/w}-\frac{w-q^{2}z}{1-C_{12}^{-2}z/w}\psi^{-}_{i}(C_{2}w)\psi_{i}^{+}\big{(}C_{12}^{-1}z\big{)}^{-1}\bigg{)}\otimes
K_{i}^{-}(w)K_{i}^{+}(z)$
$\displaystyle\qquad\hphantom{\equiv}{}+\frac{z-q^{2}w}{1-C_{2}^{-2}w/z}1\otimes
K_{i}^{-}(z)K_{i}^{+}(w)+\frac{w-q^{2}z}{1-C_{2}^{2}w/z}1\otimes
K_{i}^{-}(w)K_{i}^{+}(z)$
$\displaystyle\qquad{}\equiv\frac{z-q^{2}w}{1-C_{12}^{-2}w/z}\psi_{i}^{-}(C_{2}z)\psi_{i}^{+}\big{(}C_{12}^{-1}w\big{)}^{-1}\otimes
K_{i}^{-}(z)K_{i}^{+}(w)$
$\displaystyle\qquad\hphantom{\equiv}{}-\frac{w-q^{2}z}{1-C_{12}^{-2}z/w}\psi_{i}^{-}(C_{2}w)\psi_{i}^{+}\big{(}C_{12}^{-1}z\big{)}^{-1}\otimes
K_{i}^{-}(w)K_{i}^{+}(z).$

Interchanging $z$ with $w$ and summing these, we arrive at (3.4) for $i=j$.

Next let us verify the Serre relations (3.6). By Proposition 3.3, it suffices
to show that the current $\Delta E_{i,i,i\pm 1}(z_{1},z_{1},w)$ satisfies the
wheel conditions (3.13)–(3.14). It is comprised of 27 terms which we group
together according to the principal grading in the first component, defined by
$\mathop{\rm pdeg}e_{i}(z)=1$, $\mathop{\rm pdeg}f_{i}(z)=-1$, $\mathop{\rm
pdeg}\psi^{\pm}_{i}(z)=\mathop{\rm pdeg}C=0$. For the terms of principal
degrees $\pm 3$ and $\pm 2$, the wheel conditions are easily checked.

Let us verify the case of principal degree $1$. There are 6 terms coming from
$\Delta E_{i,i,i\pm 1}(z_{1},z_{2},w)$:

$\displaystyle A_{1}=\phi\,e_{i}(C_{2}z_{1})e_{i}(C_{2}z_{2})\tilde{f}_{i\pm
1}\big{(}C_{2}^{-1}w\big{)}\otimes K_{i}^{-}(z_{1})K_{i}^{-}(z_{2})K_{i\pm
1}^{+}(w),$ $\displaystyle A_{2}=\phi\,e_{i}(C_{2}z_{1})e_{i\pm
1}(C_{2}w)\tilde{f}_{i}\big{(}C_{2}^{-1}z_{2}\big{)}\otimes
K_{i}^{-}(z_{1})K_{i\pm 1}^{-}(w)K_{i}^{+}(z_{2})G_{i,i\pm 1}(w/z_{2}),$
$\displaystyle A_{3}=\phi\,e_{i}(C_{2}z_{2})e_{i\pm
1}(C_{2}w)\tilde{f}_{i}\big{(}C_{2}^{-1}z_{1}\big{)}\otimes
K_{i}^{-}(z_{2})K_{i\pm 1}^{-}(w)K_{i}^{+}(z_{1})G_{i,i}(z_{2}/z_{1})G_{i,i\pm
1}(w/z_{1}),$ $\displaystyle A_{4}=\phi\,e_{i}(C_{2}z_{1})\otimes
K_{i}^{-}(z_{1})E_{i}(z_{2})E_{i\pm 1}(w),$ $\displaystyle
A_{5}=\phi\,e_{i}(C_{2}z_{2})\otimes K^{-}_{i}(z_{2})E_{i}(z_{1})E_{i\pm
1}(w)G_{i,i}(z_{2}/z_{1}),$ $\displaystyle A_{6}=\phi\,e_{i\pm
1}(C_{2}w)\otimes K^{-}_{i\pm 1}(w)E_{i}(z_{1})E_{i}(z_{2})G_{i,i\pm
1}(w/z_{1})G_{i,i\pm 1}(w/z_{2}),$

where

$\displaystyle\phi=\frac{z_{1}-C_{12}^{2}z_{2}}{z_{1}-z_{2}}\frac{z_{1}-C_{12}^{-2}z_{2}}{z_{1}-z_{2}}\frac{z_{1}-q^{2}z_{2}}{z_{1}-z_{2}}\frac{z_{1}-q^{-1}d^{\pm
1}w}{z_{1}-w}\frac{z_{2}-q^{-1}d^{\pm 1}w}{z_{1}-w}.$

From (3.20)–(3.22), we see that the sum $\sum_{i=1}^{6}A_{i}$ has no poles
other than $z_{1}=z_{2}$, $z_{i}=w$. In fact, the only possible poles at
$z_{2}=C_{2}^{2}z_{1}$ (resp. $z_{2}=C_{2}^{-2}z_{1}$) arise from $A_{2}$ and
$A_{6}$ (resp., $A_{3}$ with $A_{6}$), and they cancel by virtue of (3.21) and
(3.22).

Let us check (3.13). Under the specialization $z_{2}=q^{2}z_{1}$, $w=qd^{\mp
1}z_{1}=q^{-1}d^{\mp 1}z_{2}$, each term has a zero due to the vanishing
factors

$\displaystyle\big{(}z_{1}-q^{2}z_{2}\big{)}G_{i,i}(z_{2}/z_{1}),\qquad
z_{1}-q^{-1}d^{\pm 1}w,\qquad\big{(}z_{2}-q^{-1}d^{\pm 1}w\big{)}G_{i,i\pm
1}(w/z_{2}).$

To check (3.14), set $z_{2}=C_{12}^{2}z_{1}$. All $A_{i}$’s vanish at
$z_{2}=C_{12}^{2}z_{1}$ with the exception of $A_{2}$. Since the latter has
the factor $\big{(}z_{2}-q^{-1}d^{\pm 1}w\big{)}G_{i,i\pm 1}(w/z_{2})$, it
also vanishes by setting further $w=q^{-1}d^{\mp 1}z_{2}$.

In the case of principal degree $0$, there are 7 terms:

$\displaystyle B_{0}=\phi\,1\otimes E_{i}(z_{1})E_{i}(z_{2})E_{i\pm 1}(w),$
$\displaystyle
B_{1}=\phi\,e_{i}(C_{2}z_{1})\tilde{f}_{i}\big{(}C_{2}^{-1}z_{2}\big{)}\otimes
K_{i}^{-}(z_{1})E_{i\pm 1}(w)K^{+}_{i}(z_{2})G_{i,i\pm 1}(w/z_{2}),$
$\displaystyle
B_{2}=\phi\,e_{i}(C_{2}z_{2})\tilde{f}_{i}\big{(}C_{2}^{-1}z_{1}\big{)}\otimes
K^{-}_{i}(z_{2})E_{i\pm 1}(w)K^{+}_{i}(z_{1})G_{i,i}(z_{2}/z_{1})G_{i,i\pm
1}(w/z_{1}),$ $\displaystyle B_{3}=\phi\,e_{i}(C_{2}z_{1})\tilde{f}_{i\pm
1}\big{(}C_{2}^{-1}w\big{)}\otimes K_{i}^{-}(z_{1})E_{i}(z_{2})K^{+}_{i\pm
1}(w),$ $\displaystyle B_{4}=\phi\,e_{i}(C_{2}z_{2})\tilde{f}_{i\pm
1}\big{(}C_{2}^{-1}w\big{)}\otimes K_{i}^{-}(z_{2})E_{i}(z_{1})K^{+}_{i\pm
1}(w)G_{i,i}(z_{2}/z_{1}),$ $\displaystyle B_{5}=\phi\,e_{i\pm
1}(C_{2}w)\tilde{f}_{i}\big{(}C_{2}^{-1}z_{2}\big{)}\otimes K^{-}_{i\pm
1}(w)E_{i}(z_{1})K^{+}_{i}(z_{2})G_{i,i\pm 1}(w/z_{1})G_{i,i\pm 1}(w/z_{2}),$
$\displaystyle B_{6}=\phi\,e_{i\pm
1}(C_{2}w)\tilde{f}_{i}\big{(}C_{2}^{-1}z_{1}\big{)}\otimes K^{-}_{i\pm
1}(w)E_{i}(z_{2})K^{+}_{i}(z_{1})G_{i,i}(z_{2}/z_{1})G_{i,i\pm
1}(w/z_{1})G_{i,i\pm 1}(w/z_{2}).$

Again possible poles at $z_{2}=C_{2}^{\pm 2}z_{1}$ are cancelled among
$B_{0}$, $B_{1}$, $B_{2}$ and the sum $\sum_{i=0}^{6}B_{i}$ has no poles other
than $z_{1}=z_{2}$, $z_{i}=w$.

Consider $z_{2}=q^{2}z_{1}$, $w=qd^{\mp 1}z_{1}=q^{-1}d^{\mp 1}z_{2}$. By the
assumption on $C_{2}^{2}$, $B_{0}$ vanishes. The rest of the terms are zero
for the same reason as in principal degree $1$. Next let
$z_{2}=C_{12}^{2}z_{1}$. The only term which may survive is $B_{1}$. Due to
the factor $\big{(}z_{2}-q^{-1}d^{\pm 1}w\big{)}G_{i,i\pm 1}(w/z_{2})$ it also
vanishes by setting further $w=q^{-1}d^{\mp 1}z_{2}$.

The case of principal degree $-1$ being similar to principal degree $1$, we
omit the details. Finally the coassociativity and counit property can be
readily checked by using (3.16)–(3.19). ∎

###  3.4 Representations

Theorem 3.4 allows us to construct a large family of admissible
representations of $\mathcal{K}_{n}$ starting from simple ones.

Algebra $\mathcal{K}_{n}$ has a Heisenberg subalgebra $\mathcal{H}_{n}$
generated by $H_{i,r}$, $i\in{\mathbb{Z}}/n{\mathbb{Z}}$, $r\neq 0$,
$K^{\pm}_{i,0}$, and $C^{\pm 1}$. Let $\mathbb{F}^{\textsf{D}}$ be an
irreducible representation of $\mathcal{H}_{n}$ on which $C$ acts as $q^{-1}$.

######  Lemma 3.5.

With the following action, $\mathbb{F}^{\textsf{D}}$ is an admissible
$\mathcal{K}_{n}$ module:

$\displaystyle E_{i}(z)\mapsto 0,\qquad K_{i}^{\pm}(z)\mapsto
K_{i}^{\pm}(z),\qquad C\mapsto q^{-1}.$

###### Proof.

The defining relations are obviously satisfied except (3.4). To see the latter
it suffices to note that $g_{i,i}\big{(}z,C^{2}z\big{)}=0$ when $C=q^{-1}$. ∎

In the next case we need to adjoin $\big{(}K^{\pm}_{i,0}\big{)}^{1/2}$ to
$\mathcal{H}_{n}$ and $\mathcal{K}_{n}$. We define $\tilde{K}_{i}^{\pm}(z)$ by

$\displaystyle\tilde{K}_{i}^{\pm}(z)\tilde{K}_{i}^{\pm}(qz)=K^{\pm}_{i}(z).$

Let $\mathbb{F}^{\textsf{B}}$ be an irreducible representation of
$\mathcal{H}_{n}$ on which $C$ acts as $q^{1/2}$.

######  Lemma 3.6.

With the following action, $\mathbb{F}^{\textsf{B}}$ is an admissible
$\mathcal{K}_{n}$ module:

$\displaystyle E_{i}(z)\mapsto
k\tilde{K}^{-}_{i}(z)\tilde{K}^{+}_{i}(qz),\qquad K_{i}^{\pm}(z)\mapsto
K_{i}^{\pm}(z),\qquad C\mapsto q^{1/2},$

where $k=1/(q^{1/2}-q^{-1/2})$.

###### Proof.

We can write $G_{i,j}(x)=\tilde{G}_{i,j}(x)\tilde{G}_{i,j}(q^{-1}x)$, where

$\displaystyle\tilde{G}_{i,i}(x)=q\frac{1-q^{-1}x}{1-x}\frac{1-qx}{1-q^{2}x},\qquad\tilde{G}_{i,i\pm
1}(x)=q^{-1/2}\frac{1-qd^{\pm 1}x}{1-d^{\pm 1}x},$

and $\tilde{G}_{i,j}(x)=1$ otherwise. The assertion can be checked by noting

$\displaystyle\tilde{K}^{+}_{i}(z)\tilde{K}_{j}^{-}(w)=\tilde{K}_{j}^{-}(w)\tilde{K}^{+}_{i}(z)\tilde{G}_{i,j}(w/z).$

We use also

$\displaystyle E_{i,i}(z,w)\mapsto
qk^{2}\tilde{K}^{-}_{i}(z)\tilde{K}^{-}_{i}(w)\tilde{K}^{+}_{i}(qz)\tilde{K}^{+}_{i}(qw)\frac{1-q^{-2}w/z}{1-w/z}\frac{1-q^{2}w/z}{1-w/z}\,$

and Proposition 3.2 to check the coefficients of delta functions in (3.4), and
use Proposition 3.3 along with

$\displaystyle E_{i,i,i\pm 1}(z_{1},z_{2},w)\mapsto{}$ $\displaystyle
k^{3}\tilde{K}^{-}_{i}(z_{1})\tilde{K}^{-}_{i}(z_{2})\tilde{K}^{-}_{i\pm
1}(w)\tilde{K}^{+}_{i}(qz_{1})\tilde{K}^{+}_{i}(qz_{2})\tilde{K}^{+}_{i\pm
1}(qw)$
$\displaystyle\times\frac{1-q^{-2}z_{2}/z_{1}}{1-z_{2}/z_{1}}\frac{1-q^{2}z_{2}/z_{1}}{1-z_{2}/z_{1}}\frac{1-d^{\pm
1}w/z_{1}}{1-w/z_{1}}\frac{1-d^{\pm 1}w/z_{2}}{1-w/z_{2}}\,$

for the Serre relations (3.6). ∎

Together with Theorem 3.4, the above lemmas imply

######  Corollary 3.7.

Let $V$ be an admissible ${\mathcal{E}}_{n}$ module, where $C$ acts as scalar
$C_{1}$ with $q^{-2}C_{1}^{2}\neq\pm 1,q^{\pm 2}$. Then
$V\otimes\mathbb{F}^{\textsf{D}}$ is an admissible $\mathcal{K}_{n}$ module
with the action

$\displaystyle\Delta E_{i}(z)=e_{i}\big{(}q^{-1}z\big{)}\otimes
K_{i}^{-}(z)+\tilde{f}_{i}(qz)\otimes K^{+}_{i}(z),$ $\displaystyle\Delta
K_{i}^{+}(z)=\psi_{i}^{+}\big{(}qC_{1}^{-1}z\big{)}^{-1}\otimes
K_{i}^{+}(z),\qquad\Delta
K_{i}^{-}(z)=\psi_{i}^{-}\big{(}q^{-1}z\big{)}\otimes K_{i}^{-}(z),$
$\displaystyle\Delta C=C_{1}q^{-1}.$

###### Proof.

Since $C_{2}^{2}=q^{-2}$, Theorem 3.4 does not literally apply. Nevertheless,
in the proof there, the only place where this matters is the term $B_{0}$ in
degree $0$. Since this term is identically zero in $\mathbb{F}^{\textsf{D}}$,
the same proof works. ∎

######  Corollary 3.8.

Let $V$ be an admissible ${\mathcal{E}}_{n}$ module, where $C$ acts as scalar
$C_{1}$ with $qC_{1}^{2}\neq\pm 1,q^{\pm 2}$. Then
$V\otimes\mathbb{F}^{\textsf{B}}$ is an admissible $\mathcal{K}_{n}$ module
with the action

$\displaystyle\Delta E_{i}(z)=e_{i}\big{(}q^{1/2}z\big{)}\otimes
K_{i}^{-}(z)+1\otimes
k\tilde{K}^{-}_{i}(z)\tilde{K}^{+}_{i}(qz)+\tilde{f}_{i}\big{(}q^{-1/2}z\big{)}\otimes
K^{+}_{i}(z),$ $\displaystyle\Delta
K_{i}^{+}(z)=\psi_{i}^{+}\big{(}q^{-1/2}C_{1}^{-1}z\big{)}^{-1}\otimes
K_{i}^{+}(z),\qquad\Delta
K_{i}^{-}(z)=\psi_{i}^{-}\big{(}q^{1/2}z\big{)}\otimes K_{i}^{-}(z),$
$\displaystyle\Delta C=C_{1}q^{1/2},$

where $\tilde{K}^{\pm}_{i}(z)$ and $k$ are as in Lemma 3.6.

######  Example 3.9.

Consider the $\mathcal{K}_{n}$ module

$\displaystyle\mathbb{F}(u_{1})\otimes\cdots\otimes\mathbb{F}(u_{\ell})\otimes\mathbb{F}^{\textsf{D}},\qquad\ell\geq
3.$

The central element acts as $C^{2}=q^{2\ell-2}$. The current $E_{i}(z)$ is
represented as a sum of $2\ell$ vertex operators:

$\displaystyle
z^{-1}E_{i}(z)=-\sum_{k=1}^{\ell}q^{k-\ell}u_{k}\Lambda_{i,k}(z)+\sum_{k=1}^{\ell}q^{-k+\ell-1}u_{k}^{-1}\Lambda_{i,\bar{k}}(z),$
(3.23)

where

$\displaystyle\Lambda_{i,k}(z)=\overbrace{1\otimes\cdots\otimes
1}^{k-1}\otimes{:}V_{i}\big{(}q^{k-\ell+1}z\big{)}^{-1}{:}\otimes\psi_{i}^{+}\big{(}q^{k-\ell+1}z\big{)}^{-1}\otimes\cdots\otimes\psi^{+}_{i}(z)^{-1}\otimes
K_{i}^{+}(z),$
$\displaystyle\Lambda_{i,\bar{k}}(z)=\overbrace{1\otimes\cdots\otimes
1}^{k-1}\otimes
V_{i}\big{(}q^{-k+\ell-1}z\big{)}\otimes\psi_{i}^{-}\big{(}q^{-k+\ell-2}z\big{)}\otimes\cdots\otimes\psi^{-}_{i}\big{(}q^{-1}z\big{)}\otimes
K_{i}^{-}(z).$

In the right-hand side, $\psi^{\pm}_{i}(z)$ or $K^{\pm}_{i}(z)$ stand for
their action on $\mathbb{F}(u_{i})$ or $\mathbb{F}^{\textsf{D}}$,
respectively.

In what follows we introduce an ordering
$1\\!\prec\cdots\\!\prec\ell\\!\prec\bar{\ell}\\!\prec\cdots\\!\prec\bar{1}$
to the set $\\{1,\dots,\ell,\bar{\ell},\dots,\bar{1}\\}$. We use letters
$a,b,\dots$ for elements of the latter, with the convention that
$\bar{\bar{a}}=a$. Each $\Lambda_{i,a}(z)$ is a product of oscillator part
$\Lambda^{\rm osc}_{i,a}(z)$ and the zero mode part. The contractions of the
former are given by the following table.

Table 1: Contractions $\contraction[1ex]{}{\Lambda^{\rm osc}_{i,a}(z)}{}{\Lambda^{\rm osc}_{j,b}(w)}\Lambda^{\rm osc}_{i,a}(z)\Lambda^{\rm osc}_{j,b}(w)$ with $x=w/z$ | $\begin{array}[]{c}a\prec b\\\\[-7.74997pt] b\neq\bar{a}\end{array}$ | $a=b$ |  $\begin{array}[]{c}a\succ b\\\\[-7.74997pt] b\neq\bar{a}\end{array}$ | $\begin{array}[]{c}a\prec b\\\\[-7.74997pt] b=\bar{a}\end{array}$ | $\begin{array}[]{c}a\succ b\\\\[-7.74997pt] b=\bar{a}\end{array}$
---|---|---|---|---|---
$i\equiv j$ | $\dfrac{1\\!-\\!q_{2}^{-1}x}{1-q_{2}x}$ | $(1\\!-\\!x)(1\\!-\\!q_{2}^{-1}x)$ | $1$ |  $\begin{array}[]{c}\dfrac{1-q_{2}^{-1}x}{1-q_{2}x}\times\\\\[3.87498pt] \dfrac{1}{(1!-\\!C^{2}q_{2}^{1-a}x)(1\\!-\\!C^{2}q_{2}^{2-a}x)}\end{array}$ | $\dfrac{1}{(1\\!-\\!C^{-2}q_{2}^{b}x)(1\\!-\\!C^{-2}q_{2}^{b-1}x)}$
$i\\!+\\!1\\!\equiv\\!j$ | ${\dfrac{1\\!-\\!q_{3}^{-1}x}{1-q_{1}x}}$ | $\dfrac{1}{1-q_{1}x}$ | $1$ |  $\dfrac{1-q_{3}^{-1}x}{1-q_{1}x}$ $(1-C^{2}q_{2}^{1-a}q_{1}x)$ | $1-C^{-2}q_{2}^{b-1}q_{3}^{-1}x$

The contractions in the case $i-1\equiv j$ can be obtained by switching
$q_{1}$ and $q_{3}$.

Using this table one can check directly the wheel conditions, namely that
$E_{i,i,i+1}(z_{1},z_{2},w)$ represented by

$\displaystyle\frac{1-C^{2}z_{2}/z_{1}}{1-z_{2}/z_{1}}\frac{1-C^{-2}z_{2}/z_{1}}{1-z_{2}/z_{1}}\frac{1-q_{2}z_{2}/z_{1}}{1-z_{2}/z_{1}}\frac{1-q_{1}w/z_{1}}{1-w/z_{1}}\frac{1-q_{1}w/z_{2}}{1-w/z_{2}}\,\Lambda_{i,a}(z_{1})\Lambda_{i,b}(z_{2})\Lambda_{i+1,c}(w)$

vanishes for all $a,b,c\in\\{1,\dots,\ell,\overline{\ell},\dots,\bar{1}\\}$
under the following specializations:

  1. $(i)$

$(z_{1},z_{2},w)=\big{(}z,q_{2}z,q_{1}^{-1}z\big{)}$,

  2. $(ii)$

$(z_{1},z_{2},w)=\big{(}z,C^{2}z,C^{2}q_{3}z\big{)}$,

  3. $(iii)$

$(z_{1},z_{2},w)=\big{(}z,C^{-2}z,C^{-2}q_{1}^{-1}z\big{)}$.

Case $\boldsymbol{(i)}$. This follows from

$\displaystyle\contraction[1ex]{}{\Lambda_{i,a}(z_{1})}{}{\Lambda_{i,b}(z_{2})}\Lambda_{i,a}(z_{1})\Lambda_{i,b}(z_{2})\bigl{|}_{z_{2}=q_{2}z_{1}}=0\qquad\text{if}\quad
a\preceq b,$
$\displaystyle\contraction[1ex]{}{\Lambda_{i,b}(z_{2})}{}{\Lambda_{i+1,c}(w)}\Lambda_{i,b}(z_{2})\Lambda_{i+1,c}(w)\bigl{|}_{w=q_{3}z_{2}}=0\qquad\text{if}\quad
b\prec c,$
$\displaystyle(1-q_{1}w/z_{1})\contraction[1ex]{}{\Lambda_{i,a}(z_{1})}{}{\Lambda_{i+1,c}(w)}\Lambda_{i,a}(z_{1})\Lambda_{i+1,c}(w)\bigl{|}_{w=q_{1}^{-1}z_{1}}=0\qquad\text{if}\quad
a\succ c.$

Case $\boldsymbol{(ii)}$. This follows from

$\displaystyle\big{(}1-C^{-2}z_{2}/z_{1}\big{)}\contraction[1ex]{}{\Lambda_{i,a}(z_{1})}{}{\Lambda_{i,b}(z_{2})}\Lambda_{i,a}(z_{1})\Lambda_{i,b}(z_{2})\bigl{|}_{z_{2}=C^{2}z_{1}}=0\qquad\text{if}\quad(a,b)\neq(\bar{1},1),$
$\displaystyle\contraction[1ex]{}{\Lambda_{i,1}(z_{2})}{}{\Lambda_{i+1,c}(w)}\Lambda_{i,1}(z_{2})\Lambda_{i+1,c}(w)\bigl{|}_{w=q_{3}z_{2}}=0\qquad\text{if}\quad
c\neq 1,$
$\displaystyle\contraction[1ex]{}{\Lambda_{i,\bar{1}}(z_{1})}{}{\Lambda_{i+1,1}(w)}\Lambda_{i,\bar{1}}(z_{1})\Lambda_{i+1,1}(w)\bigl{|}_{w=C^{2}q_{3}z_{1}}=0.$

Case $\boldsymbol{(iii)}$. This follows from

$\displaystyle\big{(}1-C^{2}z_{2}/z_{1}\big{)}\contraction[1ex]{}{\Lambda_{i,a}(z_{1})}{}{\Lambda_{i,b}(z_{2})}\Lambda_{i,a}(z_{1})\Lambda_{i,b}(z_{2})\bigl{|}_{z_{2}=C^{-2}z_{1}}=0\qquad\text{if}\quad(a,b)\neq(1,\bar{1}),$
$\displaystyle(1-q_{1}w/z_{2})\contraction[1ex]{}{\Lambda_{i,\bar{1}}(z_{2})}{}{\Lambda_{i+1,c}(w)}\Lambda_{i,\bar{1}}(z_{2})\Lambda_{i+1,c}(w)\bigl{|}_{w=q_{1}^{-1}z_{2}}=0\qquad\text{if}\quad
c\neq\bar{1},$
$\displaystyle\contraction[1ex]{}{\Lambda_{i,1}(z_{1})}{}{\Lambda_{i+1,\bar{1}}(w)}\Lambda_{i,1}(z_{1})\Lambda_{i+1,\bar{1}}(w)\bigl{|}_{w=C^{-2}q_{1}^{-1}z_{1}}=0.$

Similarly for $E_{i,i,i-1}(z_{1},z_{2},w)$.

######  Remark 3.10.

As it turns out, the currents (3.23) commute with a system of screening
operators derived from quantum toroidal algebra of type $D$. We hope to
address this point elsewhere.

##  4 Integrals of motion

We continue to work with admissible representations of $\mathcal{K}_{n}$ on
which $C$ acts as a scalar. Our goal in this section is to introduce a family
of commuting operators acting on each such representation (with mild
restrictions on parameters), which we call integrals of motion (IMs). Loosely
speaking, we may think of them as generators of a commutative subalgebra of
$\mathcal{K}_{n}$.

The IMs will depend on $q$, $d$, $C$ as well as additional arbitrary
parameters

$\displaystyle\mu_{1},\dots,\mu_{n}\in{\mathbb{C}},\qquad\sum_{i=1}^{n}\mu_{i}=0.$
(4.1)

In addition, they depend also on an “elliptic” parameter $p$. However, our
experience with the algebra $\mathcal{K}_{1}$ tells that the IMs exist only
when $p$ and $C$ are related in a special way. Guided by the result in [7], we
shall impose the relation

$\displaystyle C^{2}=pq^{2}$

and make the following assumption:

$\displaystyle|p|<1,\qquad\big{|}qd^{\pm
1}\big{|}<1,\qquad\big{|}pq^{-2}\big{|}<1,\qquad\big{|}pq^{-1}d^{\pm
1}\big{|}<1.$ (4.2)

In particular $|q^{2}|<1$. We use parameters $\tau\in{\mathbb{C}}$ with
$\mathop{\rm Im}\tau>0$ and $\beta,\gamma\in{\mathbb{C}}$ given by

$\displaystyle p={\rm e}^{-2\pi{\rm i}/\tau},\qquad q^{2}=p^{\gamma},\qquad
d=p^{\beta}.$ (4.3)

We use also $\theta(u)=p^{u^{2}/2-u/2}\Theta_{p}(p^{u})$, which satisfies

$\displaystyle\theta(u+1)=-\theta(u),\qquad\theta(u+\tau)=-{\rm e}^{-2\pi{\rm
i}u-\pi{\rm i}\tau}\theta(u).$ (4.4)

###  4.1 Dressed currents

A basic constituent of IMs is the dressed currents, obtained by modifying the
currents $E_{i}(z)$ of $\mathcal{K}_{n}$.

In the following we define operators $H_{i,0}$ by
$(K_{i,0}^{-})^{-1}K_{i,0}^{+}=q^{2H_{i,0}}$. The dressed currents of
$\mathcal{K}_{n}$ are defined by

$\displaystyle\mathbf{E}_{i}(z)=\mathbf{K}_{i}^{-}(z)^{-1}E_{i}(z)z^{-\gamma
H_{i,0}-\gamma+\mu_{i}},$
$\displaystyle\mathbf{K}_{i}^{-}(z)=\exp\bigg{(}{-}\big{(}q-q^{-1}\big{)}\sum_{r>0}\frac{H_{i,-r}}{1-p^{r}}z^{r}\bigg{)},$

where $\gamma$ and $\mu_{i}$ are as in (4.3) and (4.1).

The dressed currents have well-defined matrix coefficients on admissible
representations. They satisfy the exchange relations as meromorphic functions

$\displaystyle\lambda_{i,j}(z_{2}/z_{1})\mathbf{E}_{i}(z_{1})\mathbf{E}_{j}(z_{2})=\lambda_{j,i}(z_{1}/z_{2})\mathbf{E}_{j}(z_{2})\mathbf{E}_{i}(z_{1}),$

with

$\displaystyle\lambda_{i,i}(x)=x^{-\gamma}\frac{1-C^{2}x}{1-x}\frac{1-C^{-2}x}{1-x}\frac{1-q^{2}x}{1-x}\frac{\big{(}q^{-2}x;p\big{)}_{\infty}}{(q^{2}x;p)_{\infty}},$
$\displaystyle\lambda_{i,i\pm 1}(x)=x^{\gamma/2}d^{\mp
1/2}\frac{1-q^{-1}d^{\pm 1}x}{1-x}\frac{\big{(}qd^{\pm
1}x;p\big{)}_{\infty}}{\big{(}q^{-1}d^{\pm 1}x;p\big{)}_{\infty}},$

and $\lambda_{i,j}(x)=1$ in all other cases. Up to a power of $x$,
$\lambda_{i,j}(x)$ specializes to $\lambda^{0}_{i,j}(x)$ at $p=0$. Each matrix
coefficient of products of dressed currents is a meromorphic function (up to
an overall power function). We have “elliptic” exchange relations as
meromorphic functions

$\displaystyle\mathbf{E}_{i}(z)\mathbf{E}_{i}(w)=\mathbf{E}_{i}(w)\mathbf{E}_{i}(z)\frac{\theta(v-u+\gamma)}{\theta(v-u-\gamma)},$
$\displaystyle\mathbf{E}_{i}(z)\mathbf{E}_{i\pm 1}(w)=\mathbf{E}_{i\pm
1}(w)\mathbf{E}_{i}(z)q^{\pm
2\beta}\frac{\theta(v-u\pm\beta-\gamma/2)}{\theta(v-u\pm\beta+\gamma/2)},$

where $z=p^{u}$, $w=p^{v}$.

As before we define

$\displaystyle\mathbf{E}_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})=\prod_{1\leq
r<s\leq
N}\lambda_{i_{r},i_{s}}(z_{s}/z_{r})\mathbf{E}_{i_{1}}(z_{1})\cdots\mathbf{E}_{i_{N}}(z_{N}).$
(4.5)

Then it has the same symmetry as (3.10), and we have

$\displaystyle\mathbf{E}_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})=\prod_{r<s}z_{s}^{-\gamma(\bar{\alpha}_{i_{r}},\bar{\alpha}_{i_{s}})}\prod_{s=1}^{N}\mathbf{K}_{i_{s}}^{-}(z_{s})^{-1}E_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})\prod_{s=1}^{N}z_{s}^{-\gamma
H_{i_{s},0}-\gamma+\mu_{i_{s}}}.$

The product $\prod_{s=1}^{N}\mathbf{K}_{i_{s}}^{-}(z_{s})^{-1}$ contains only
$\\{H_{i,-r}\\}$ with $r>0$. Its contribution to any matrix coefficient of
$\mathbf{E}_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})$ is a Laurent polynomial in
$z_{s}$’s for degree reasons. Hence the equality above shows that, up to an
overall power of $z_{i}$’s, each matrix coefficient of operator (4.5) has the
same functional form (3.11) as that of
$E_{i_{1},\dots,i_{N}}(z_{1},\dots,z_{N})$. In particular the wheel conditions
(3.13)–(3.15) are valid also for (4.5).

###  4.2 Integrals of motion

Given parameters (4.1), let ${X}_{n}={X}_{n}(\mu_{1},\dots,\mu_{n})$ denote
the space of entire functions $\vartheta(u_{1},\dots,u_{n})$ satisfying

$\displaystyle\vartheta(u_{1},\dots,u_{i}+1,\dots,u_{n})=\vartheta(u_{1},\dots,u_{i},\dots,u_{n}),$
(4.6) $\displaystyle\vartheta(u_{1},\dots,u_{i}+\tau,\dots,u_{n})={\rm
e}^{-2\pi{\rm
i}(2u_{i}-u_{i-1}-u_{i+1}-\mu_{i}+\tau)}\,\vartheta(u_{1},\dots,u_{i},\dots,u_{n}),$
(4.7)

where $u_{i+n}=u_{i}$. We have $\dim X_{n}=n$.

For $M\geq 1$ and $\vartheta\in X_{n}$, we introduce the kernel functions

$\displaystyle h_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta)$
$\displaystyle\qquad{}=\frac{\vartheta(\overline{u}_{1},\dots,\overline{u}_{n}){\prod_{i=1}^{n}\prod_{1\leq
a<b\leq
M}\theta(u_{i,a}-u_{i,b})\theta(u_{i,a}-u_{i,b}-\gamma)}}{{\prod_{i=1}^{n-1}\prod_{1\leq
a,b\leq
M}\theta\big{(}u_{i,a}-u_{i+1,b}-\beta-\frac{\gamma}{2}\big{)}\prod_{1\leq
a,b\leq M}\theta\big{(}u_{n,a}-u_{1,b}-\beta+\frac{\gamma}{2}\big{)}}},$ (4.8)

where

$\displaystyle\mathbf{u}_{i}=(u_{i,1},\dots,u_{i,M}),\qquad\overline{u}_{i}=\sum_{a=1}^{M}u_{i,a}.$

We define the integrals of motion $\mathbf{G}_{M}(\vartheta)$ of
$\mathcal{K}_{n}$ by

$\displaystyle\mathbf{G}_{M}(\vartheta)=\int\\!\\!\cdots\\!\\!\int\prod_{1\leq
i\leq n}^{\curvearrowleft}\prod_{1\leq a\leq
M}^{\curvearrowleft}\mathbf{E}_{i}(x_{i,a})\cdot
h_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta)\
\prod_{i=1}^{n}\prod_{a=1}^{M}\frac{{\rm d}x_{i,a}}{x_{i,a}}.$ (4.9)

Here $x_{i,a}=p^{u_{i,a}}$, and the symbol $\prod_{1\leq i\leq
N}^{\curvearrowleft}$ stands for ordered product

$\prod_{1\leq i\leq N}^{\curvearrowleft}A_{i}=A_{N}A_{N-1}\cdots A_{1}.$

Even though individual factors contain fractional powers, the integrand of
(4.9) comprises only integer powers with respect to each $x_{i,a}$. The
integrals are taken over a common circle $|x_{i,a}|=R$. The result is
independent of the choice of $R>0$.

Formally, (4.9) looks identical to that of the integrals of motion of type A
[5, 8, 12]. In particular the kernel functions (4.8) are the same as those
used there. The difference is hidden in the pole structure of products of
dressed currents.

For later use we rewrite the integrand in terms of the current in (4.5),

$\displaystyle\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})=\mathbf{E}_{1,\dots,1,\dots,n,\dots,n}(\mathbf{x}_{1},\dots,\mathbf{x}_{n}),$

where $\mathbf{x}_{i}=(x_{i,1},\dots,x_{i,M})$. In the right-hand side, each
index $i$ occurs $M$ times.

######  Proposition 4.1.

Set $x=p^{u}$ and

$\displaystyle\xi(u)=p^{u^{2}/2}(1-x)\big{(}x,p^{2}q^{2}x,p;p\big{)}_{\infty},$
$\displaystyle\eta(u)=p^{u^{2}/2-(\beta+1/2)u}\frac{d^{-1/2}}{1-x^{-1}}\big{(}qd^{-1}x,qdx^{-1},p;p\big{)}_{\infty}.$

Then the IM’s can be expressed as

$\displaystyle\mathbf{G}_{M}(\vartheta)=c_{M}\int\\!\\!\cdots\\!\\!\int\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\,k_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta)\prod_{i=1}^{n}\prod_{a=1}^{M}\frac{{\rm
d}x_{i,a}}{x_{i,a}},$ (4.10)

where $c_{M}$ is a constant, and

$\displaystyle
k_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta)=\vartheta(\overline{u}_{1},\dots,\overline{u}_{n})\,\frac{\prod_{i=1}^{n}\prod_{1\leq
a\neq b\leq M}\xi(u_{i,a}-u_{i,b})}{\prod_{i=1}^{n}\prod_{1\leq a,b\leq
M}\eta(u_{i,a}-u_{i+1,b})},$ $\displaystyle
x_{i,a}=p^{u_{i,a}},\qquad\overline{u}_{i}=\sum_{a=1}^{M}u_{i,a}.$

Here and after, we set $u_{n+1,a}=u_{1,a}$.

###### Proof.

This follows from the relations

$\displaystyle\frac{\lambda_{i,i}(p^{-u})}{\lambda_{i,i}(p^{u})}=\frac{\theta(u+\gamma)}{\theta(u-\gamma)},\qquad\frac{\lambda_{i,i\mp
1}(p^{-u})}{\lambda_{i,i\pm 1}(p^{u})}=q^{\pm
2\beta}\frac{\theta(u\pm\beta-\gamma/2)}{\theta(u\pm\beta+\gamma/2)},$
$\displaystyle\lambda_{i,i}(p^{u})\xi(u)\xi(-u)=-p^{-\gamma^{2}/2-3\gamma/2-1}\theta(u)\theta(u-\gamma),$
$\displaystyle\lambda_{i,i\pm 1}\big{(}p^{\mp
u}\big{)}\theta(u\pm\gamma/2-\beta)=p^{(\gamma/2\mp\beta)^{2}/2-(\gamma/2-\beta)/2}\,\eta(u).$
∎

###  4.3 Commutativity

We are now in a position to state the commutativity of IMs.

######  Theorem 4.2.

For all $M,N\geq 1$ and $\vartheta_{1},\vartheta_{2}\in X_{n}$, the integrals
of motion mutually commute:

$\displaystyle[\mathbf{G}_{M}(\vartheta_{1}),\mathbf{G}_{N}(\vartheta_{2})]=0.$

Theorem 4.2 is the main result of this paper. The rest of this section is
devoted to its proof.

###### Proof.

We start by examining the product of operators, dropping irrelevant constants

$\displaystyle\mathbf{G}_{M}(\vartheta_{1})=\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{1}}\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\,k_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta_{1})\prod_{i=1}^{n}\prod_{a=1}^{M}\frac{{\rm
d}x_{i,a}}{x_{i,a}},$
$\displaystyle\mathbf{G}_{N}(\vartheta_{2})=\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{R}}\mathbf{E}^{(N)}(\mathbf{y}_{1},\dots,\mathbf{y}_{n})\,k_{N}(\mathbf{v}_{1},\dots,\mathbf{v}_{n};\vartheta_{2})\prod_{i=1}^{n}\prod_{b=1}^{N}\frac{{\rm
d}y_{i,b}}{y_{i,b}}.$

Here $x_{i,a}=p^{u_{i,a}}$, $y_{i,b}=p^{v_{i,b}}$. We choose the contours to
be the unit circle ${\mathcal{C}}_{1}\colon|x_{i,a}|=1$ for
$\mathbf{G}_{M}(\vartheta_{1})$, and a circle
${\mathcal{C}}_{R}\colon|y_{i,b}|=R$ of radius $R$ for
$\mathbf{G}_{N}(\vartheta_{2})$.

Consider first the product

$\displaystyle\mathbf{G}_{N}(\vartheta_{2})\mathbf{G}_{M}(\vartheta_{1})=\\!\\!\\!{}$
$\displaystyle\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{R}}\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{1}}\\!\\!\mathbf{E}^{(N)}(\mathbf{y}_{1},\dots,\mathbf{y}_{n})\,\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})$
$\displaystyle\\!\\!\times
k_{N}(\mathbf{v}_{1},\dots,\mathbf{v}_{n};\vartheta_{2})\,k_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta_{1})\prod_{i=1}^{n}\prod_{b=1}^{N}\frac{{\rm
d}y_{i,b}}{y_{i,b}}\prod_{i=1}^{n}\prod_{a=1}^{M}\frac{{\rm
d}x_{i,a}}{x_{i,a}}.$ (4.11)

We choose $R\gg 1$ to ensure that the product of operators in the integrand
converges absolutely. Introducing the combined variables
$\mathbf{z}_{i}=(z_{i,1},\dots,z_{i,M+N})$, where $z_{i,a}=x_{i,a}$, $1\leq
a\leq M$, $z_{i,a+M}=y_{i,a}$, and $u_{i,M+a}=v_{i,a}$, $1\leq a\leq N$, and
substituting

$\displaystyle\mathbf{E}^{(N)}(\mathbf{y}_{1},\dots,\mathbf{y}_{n})\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})$
$\displaystyle\qquad{}=\mathbf{E}^{(M+N)}(\mathbf{z}_{1},\dots,\mathbf{z}_{n})\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\lambda_{i,i}(x_{i,a}/y_{i,b})^{-1}\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\lambda_{i+1,i}(x_{i,a}/y_{i+1,b})^{-1}$
$\displaystyle\qquad\hphantom{=}{}\times\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\lambda_{i-1,i}(x_{i,a}/y_{i-1,b})^{-1},$

we find that the integrand of (4.11) can be written as a product of two
factors $ST$, with

$\displaystyle S(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$
$\displaystyle\qquad{}=\mathbf{E}^{(M+N)}(\mathbf{z}_{1},\dots,\mathbf{z}_{n})\prod_{i=1}^{n}\prod_{1\leq
a\neq b\leq M+N}\xi(u_{i,a}-u_{i,b})\,\prod_{i=1}^{n}\prod_{1\leq a,b\leq
M+N}\eta(u_{i,a}-u_{i+1,b})^{-1},$ $\displaystyle
T(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})$
$\displaystyle\qquad{}=\vartheta_{1}(\overline{u}_{1},\dots,\overline{u}_{n})\vartheta_{2}(\overline{v}_{1},\dots,\overline{v}_{n})$
$\displaystyle\qquad\hphantom{=}{}\times\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\frac{\theta(u_{i,a}-v_{i+1,b}-\gamma/2-\beta)\theta(v_{i,b}-u_{i+1,a}+\gamma/2-\beta)}{\theta(u_{i,a}-v_{i,b})\theta(u_{i,a}-v_{i,b}-\gamma)}.$
(4.12)

Starting from $R\gg 1$, we bring the contours ${\mathcal{C}}_{R}$ for
$y_{i,b}$ to the unit circle. For that purpose we need to locate the position
of poles between the two circles ${\mathcal{C}}_{R}$ and ${\mathcal{C}}_{1}$.
The poles between $y$’s and $x$’s come from $\xi$, $\eta$ and $\theta$:

$\displaystyle\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq a\leq M\\\ 1\leq
b\leq
N\end{subarray}}(1-x_{i,a}/y_{i,b})(1-y_{i,b}/x_{i,a})\big{(}\cancel{x_{i,a}/y_{i,b}},p^{2}q^{2}x_{i,a}/y_{i,b},\overset{\scriptstyle
1-y_{i,b}/x_{i,a}}{\cancel{y_{i,b}/x_{i,a}}},p^{2}q^{2}y_{i,b}/x_{i,a};p\big{)}_{\infty}$
$\displaystyle\qquad{}\times\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\frac{(1-y_{i+1,b}/x_{i,a})(1-x_{i+1,a}/y_{i,b})}{\big{(}qd^{-1}x_{i,a}/y_{i+1,b},\cancel{qdy_{i+1,b}/x_{i,a}},\cancel{qd^{-1}y_{i,b}/x_{i+1,a}},qdx_{i+1,a}/y_{i,b};p\big{)}_{\infty}}$
$\displaystyle\qquad{}\times\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq
a\leq M\\\ 1\leq b\leq
N\end{subarray}}\frac{\big{(}pq^{-1}d^{-1}x_{i,a}/y_{i+1,b},\cancel{qdy_{i+1,b}/x_{i,a}},\cancel{qd^{-1}y_{i,b}/x_{i+1,a}},pq^{-1}dx_{i+1,a}/y_{i,b};p\big{)}_{\infty}}{\big{(}\cancel{x_{i,a}/y_{i,b}},\cancel{py_{i,b}/x_{i,a}},q^{-2}x_{i,a}/y_{i,b},pq^{2}y_{i,b}/x_{i,a};p\big{)}_{\infty}}.$

Under our assumption on parameters (4.2), we see that the only poles between
${\mathcal{C}}_{R}$ and ${\mathcal{C}}_{1}$ are

$\displaystyle y_{i,b}=q_{2}^{-1}x_{i,a},p^{-1}q_{2}^{-1}x_{i,a},\qquad 1\leq
i\leq n,\quad 1\leq a\leq M,\quad 1\leq b\leq N.$ (4.13)

Including the poles between $y$’s which come from $\eta$, poles of the
integrand with respect to each $y_{i,b}$ are as follows (see the figure
below):

$\displaystyle y_{i,b}=p^{-k}q_{1}y_{i+1,c},\qquad p^{-k}q_{3}y_{i-1,c},\qquad
k\geq 0,$ $\displaystyle y_{i,b}=p^{k}q_{3}^{-1}y_{i+1,c},\qquad
p^{k}q_{1}^{-1}y_{i-1,c},\qquad k\geq 0,$ $\displaystyle
y_{i,b}=p^{k}q_{2}^{-1}x_{i,a},\qquad k\geq-1,$ $\displaystyle
y_{i,b}=p^{k}q_{1}^{-1}x_{i-1,a},\qquad p^{k}q_{3}^{-1}x_{i+1,a},\qquad k\geq
0.$

$\bullet$$\bullet$$pq_{1}^{-1}y_{i-1,c}$$q_{1}^{-1}y_{i-1,c}$$\bullet$$\bullet$$q_{3}y_{i-1,c}$$p^{-1}q_{3}y_{i-1,c}$$\bullet$$\bullet$$pq_{3}^{-1}y_{i+1,c}$$q_{3}^{-1}y_{i+1,c}$$\bullet$$\bullet$$q_{1}y_{i+1,c}$$p^{-1}q_{1}y_{i+1,c}$$\bullet$$\bullet$$\bullet$$pq_{2}^{-1}x_{i,a}$$q_{2}^{-1}x_{i,a}$$p^{-1}q_{2}^{-1}x_{i,a}$$\bullet$$\bullet$$q_{1}^{-1}x_{i-1,a}$$pq_{1}^{-1}x_{i-1,a}$$\bullet$$\bullet$$pq_{3}^{-1}x_{i+1,a}$$q_{3}^{-1}x_{i+1,a}$$\mathcal{C}_{R}$$\mathcal{C}_{1}$
Figure 1: The contour in the $y_{i,b}$ plane.

We move the contour to the unit circle, picking up residues of poles (4.13)
along the way. At first glance, taking residues in one variable
$y_{i,b}=p^{-1}q_{2}^{-1}x_{i,a}$ or $y_{i,b}=q_{2}^{-1}x_{i,a}$ seems to
produce new poles in other variables,

$\displaystyle y_{i+1,c}=q_{2}^{-1}q_{1}^{-1}x_{i,a},\quad
p^{-1}q_{2}^{-1}q_{1}^{-1}x_{i,a},$ $\displaystyle
y_{i-1,c}=q_{2}^{-1}q_{3}^{-1}x_{i,a},\quad
p^{-1}q_{2}^{-1}q_{3}^{-1}x_{i,a}.$

An important point is that these poles are cancelled by the zeros of
$\mathbf{E}^{(M+N)}(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$ due to the wheel
conditions (3.13)–(3.15).

For the product in the opposite order

$\displaystyle\mathbf{G}_{M}(\vartheta_{1})\mathbf{G}_{N}(\vartheta_{2})=\\!\\!\\!{}$
$\displaystyle\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{1}}\int\\!\\!\cdots\\!\\!\int_{{\mathcal{C}}_{R}}\mathbf{E}^{(M)}(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\,\mathbf{E}^{(N)}(\mathbf{y}_{1},\dots,\mathbf{y}_{n})$
$\displaystyle\times
k_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta_{1})k_{N}(\mathbf{v}_{1},\dots,\mathbf{v}_{n};\vartheta_{2})\prod_{i=1}^{n}\prod_{a=1}^{M}\frac{{\rm
d}x_{i,a}}{x_{i,a}}\prod_{i=1}^{n}\prod_{b=1}^{N}\frac{{\rm
d}y_{i,b}}{y_{i,b}},\\!$ (4.14)

we choose $R\ll 1$ for convergence. The integrand can be obtained from the one
for (4.11) by switching the roles of $\mathbf{x}$, $M$ and $\mathbf{y}$, $N$.
It amounts to changing (4.12) to

$\displaystyle
T^{\prime}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})={}$
$\displaystyle\vartheta_{1}(\overline{u}_{1},\dots,\overline{u}_{n})\vartheta_{2}(\overline{v}_{1},\dots,\overline{v}_{n})$
$\displaystyle\times\prod_{i=1}^{n}\prod_{\begin{subarray}{c}1\leq a\leq M\\\
1\leq b\leq
N\end{subarray}}\frac{\theta(v_{i,b}\\!-\\!u_{i+1,a}\\!-\\!\gamma/2\\!-\\!\beta)\theta(u_{i,a}\\!-\\!v_{i+1,b}\\!+\\!\gamma/2\\!-\\!\beta)}{\theta(v_{i,b}-u_{i,a})\theta(v_{i,b}-u_{i,a}-\gamma)}.$

Similarly as above, we start from $R\ll 1$ and bring ${\mathcal{C}}_{R}$ to
the unit circle. The relevant poles in between are

$\displaystyle y_{i,b}=q_{2}x_{i,a},\,pq_{2}x_{i,a},\qquad 1\leq i\leq n,\quad
1\leq a\leq M,\quad 1\leq b\leq N.$

Moving the contour we pick up residues of these poles. Again the wheel
conditions ensure that no new poles arise for the remaining variables.

In order to prove the commutativity, we must show that, after moving contours
to the unit circle, (4.11) and (4.14) give the same result.

First, let us compare terms that arise without taking residues. Since all
integrals are taken on the unit circle, we can symmetrize the integrand.
Taking into account the symmetry of $S(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$,
we see that the equality of integrals reduces to the identity of theta
functions

$\displaystyle\mathop{\mathrm{Sym}}\,T(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})=\mathop{\mathrm{Sym}}\,T^{\prime}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n}),$
(4.15)

where $\mathop{\mathrm{Sym}}$ stands for symmetrization in each group of
variables $\\{u_{i,a}\\}_{a=1}^{M}\cup\\{v_{i,b}\\}_{b=1}^{N}$, $1\leq i\leq
n$. This identity was stated in [12]. Since their proof contains some gaps, we
prove it in Appendix; see Theorem A.1 and Remark A.6 below.

In general, we compare terms obtained by taking residues with respect to some
group of variables. First consider terms coming from (4.11). In view of
symmetry and the zeros of $S(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$ at
$y_{i,a}=y_{i,b}$, it is enough to consider the case

$\displaystyle y_{i,a}=p^{-1}q_{2}^{-1}x_{i,a},\qquad 1\leq a\leq k_{i},$
(4.16) $\displaystyle y_{i,b}=q_{2}^{-1}x_{i,b},\qquad k_{i}+1\leq b\leq
l_{i},$ (4.17)

with some $0\leq k_{i}\leq l_{i}\leq\min(M,N)$, $1\leq i\leq n$. We compare it
with the corresponding residue coming from (4.14)

$\displaystyle y_{i,a}=pq_{2}x_{i,a},\qquad 1\leq a\leq k_{i},$ (4.18)
$\displaystyle y_{i,b}=q_{2}x_{i,b},\qquad k_{i}+1\leq b\leq l_{i},$ (4.19)

for $1\leq i\leq n$. We further rename variables $x_{i,a}$ to
$p^{-1}q^{-1}_{2}x_{i,a}$ for (4.18) and to $q^{-1}_{2}x_{i,a}$ for (4.19).
The contours for them change to $\big{|}p^{-1}q^{-1}_{2}x_{i,a}\big{|}=1$ and
$\big{|}q^{-1}_{2}x_{i,a}\big{|}=1$, respectively. We shift them back to the
unit circle $|x_{i,a}|=1$, noting that the wheel conditions ensure there are
no poles which hinder the shift.

The factor $S(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$ has no poles at (4.16) and
(4.17) and specializes to

$\displaystyle
S\big{(}\dots,z_{i,a},\dots,z_{i,b},\dots,p^{-1}q_{2}^{-1}z_{i,a},\dots,q_{2}^{-1}z_{i,b},\dots\big{)}.$
(4.20)

Similarly, at (4.18) and (4.19) it specializes to

$\displaystyle
S(\dots,z_{i,a},\dots,z_{i,b},\dots,pq_{2}z_{i,a},\dots,q_{2}z_{i,b},\dots).$
(4.21)

After the renaming, (4.21) is brought to (4.20) because
$S(\mathbf{z}_{1},\dots,\mathbf{z}_{n})$ is symmetric in $\\{z_{i,a}\\}_{1\leq
a\leq M+N}$ for each $i$.

It remains to compare the residues of the functions $T$, $T^{\prime}$. Since
they are periodic with period $1$, their residues at
$v_{i,a}=u_{i,a}-\gamma-1$ are the same as those at $v_{i,a}=u_{i,a}-\gamma$.
Hence we are to show the equality

$\displaystyle\mathrm{Sym}^{*}\mathop{\rm
res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}-\gamma,\\\ 1\leq a\leq l_{i},\,1\leq
i\leq
n\end{subarray}}T(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})$
$\displaystyle\qquad{}=\mathrm{Sym}^{*}\Big{\\{}(-1)^{\sum_{i=1}^{n}l_{i}}\mathop{\rm
res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}+\gamma,\\\ 1\leq a\leq l_{i},\,1\leq
i\leq
n\end{subarray}}T^{\prime}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})\Big{\\}}\Big{|}_{\begin{subarray}{c}u_{i,a}\to
u_{i,a}-\gamma,\\\ 1\leq a\leq l_{i},\,1\leq i\leq n\end{subarray}},$

where $\mathrm{Sym}^{*}$ stands for the symmetrization with respect to the
remaining variables
$\\{u_{i,a}\\}_{a=l_{i}+1}^{M}\cup\\{v_{i,b}\\}_{b=l_{i}+1}^{N}$, $1\leq i\leq
n$. Note that moving the contours $\mathcal{C}_{R}$ from $R<1$ to
$\mathcal{C}_{1}$ we obtain a sign factor.

To check the equality, we start from the residues of the identity (4.15)

$\displaystyle\mathop{\rm res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}-\gamma,\\\
1\leq a\leq l_{i},\,1\leq i\leq
n\end{subarray}}\\!\\!\\!\\!\\!\mathop{\mathrm{Sym}}\,T(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})\\!=\\!\\!\\!\\!\mathop{\rm
res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}-\gamma,\\\ 1\leq a\leq l_{i},\,1\leq
i\leq
n\end{subarray}}\\!\\!\\!\\!\\!\mathop{\mathrm{Sym}}\,T^{\prime}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n}).\\!$
(4.22)

Symmetrization $\mathop{\mathrm{Sym}}$ amounts to replacing variables
$\mathbf{u}_{i}$, $\mathbf{v}_{i}$ as

$\displaystyle\mathbf{u}_{i}\to\\{u_{i,a}\\}_{a\in
I_{i}}\cup\\{v_{i,b}\\}_{b\in
J^{\prime}_{i}},\qquad\mathbf{v}_{i}\to\\{u_{i,a}\\}_{a\in
I^{\prime}_{i}}\cup\\{v_{i,b}\\}_{b\in J_{i}},$

and sum over partitions of indices

$\displaystyle I_{i}\sqcup I_{i}^{\prime}=\\{1,\dots,M\\},\qquad J_{i}\sqcup
J_{i}^{\prime}=\\{1,\dots,N\\},\qquad|I_{i}^{\prime}|=|J_{i}^{\prime}|.$

In order to have non-zero residues at $v_{i,a}=u_{i,a}-\gamma$ ($1\leq a\leq
l_{i}$) in the left-hand side of (4.22), $I_{i}$ and $J_{i}$ must contain
$\\{1,\dots,l_{i}\\}$. Therefore the left-hand side of (4.22) reduces to

$\displaystyle\mathrm{Sym}^{*}\mathop{\rm
res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}-\gamma,\\\ 1\leq a\leq l_{i},\,1\leq
i\leq
n\end{subarray}}T(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n}).$

Similarly, in the right-hand side of (4.22), non-zero residues appear only
when $J_{i}^{\prime}$ and $I_{i}^{\prime}$ contain $\\{1,\dots,l_{i}\\}$. For
each $i$ we have

$\displaystyle\mathop{\rm res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}-\gamma,\\\
1\leq a\leq
l_{i}\end{subarray}}T^{\prime}(\dots,v_{i,1},\dots,v_{i,l_{i}},\dots|\dots,u_{i,1},\dots,u_{i,l_{i}},\dots)$
$\displaystyle\qquad{}=(-1)^{l_{i}}\mathop{\rm
res}_{\begin{subarray}{c}u^{\prime}_{i,a}=v^{\prime}_{i,a}+\gamma,\\\ 1\leq
a\leq
l_{i}\end{subarray}}T^{\prime}(\dots,v^{\prime}_{i,1},\dots,v^{\prime}_{i,l_{i}},\dots|\dots,u^{\prime}_{i,1},\dots,u^{\prime}_{i,l_{i}},\dots)\Big{|}_{\begin{subarray}{c}v^{\prime}_{i,a}\to
u_{i,a}-\gamma\\\ 1\leq a\leq l_{i}\end{subarray}}.$

Altogether we can rewrite the right-hand side of (4.22) as

$\displaystyle\mathrm{Sym}^{*}\Bigl{\\{}(-1)^{\sum_{i=1}^{n}l_{i}}\mathop{\rm
res}_{\begin{subarray}{c}v_{i,a}=u_{i,a}+\gamma,\\\ 1\leq a\leq l_{i},1\leq
i\leq
n\end{subarray}}T^{\prime}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\mathbf{v}_{1},\dots,\mathbf{v}_{n})\Bigr{\\}}\Bigl{|}_{\begin{subarray}{c}u_{i,a}\to
u_{i,a}-\gamma,\\\ 1\leq a\leq l_{i},1\leq i\leq n\end{subarray}}.$

Proof of Theorem 4.2 is now complete. ∎

##  5 The cases $\boldsymbol{n=1,2}$

All results discussed so far are valid also in the case $n=1,2$ with suitable
modifications. In this section we indicate the necessary changes.

###  5.1 Case $\boldsymbol{n=2}$

Dealing with algebras ${\mathcal{E}}_{n}$ and $\mathcal{K}_{n}$ for $n=2$,
there are two points to be taken care of.

First, the structure functions are changed. We keep $g_{i,i}(z,w)$ unchanged
but replace $g_{i,j}(z,w)$ for $i\neq j$ with $g^{n\geq 3}_{i,j}(z,w)g^{n\geq
3}_{j,i}(z,w)$. Similar changes are due for $G_{i,j}(x)$,
$\lambda^{0}_{i,j}(x)$ and $\lambda_{i,j}(x)$. Namely we use

$\displaystyle g_{i,1-i}(z,w)=(z-q_{1}w)(z-q_{3}w),$ $\displaystyle
G_{i,1-i}(x)=q^{-2}\frac{\big{(}1-q_{1}^{-1}x\big{)}\big{(}1-q_{3}^{-1}x\big{)}}{(1-q_{1}x)(1-q_{3}x)},$
$\displaystyle\lambda^{0}_{i,1-i}(x)=\frac{1-q_{1}x}{1-x}\frac{1-q_{3}x}{1-x},$
$\displaystyle\lambda_{i,1-i}(x)=x^{\gamma}\frac{1-q_{1}x}{1-x}\frac{1-q_{3}x}{1-x}\frac{\big{(}q_{1}^{-1}x,q_{3}^{-1}x;p\big{)}_{\infty}}{(q_{1}x,q_{3}x;p)_{\infty}}.$

We set also $d_{i,i}=1$, $d_{i,1-i}=-1$.

Second, the Serre relations are modified.

The defining relations of ${\mathcal{E}}_{2}$ are the same as those for $n\geq
3$ except for the Serre relations

$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}[e_{i}(z_{1}),[e_{i}(z_{2}),[e_{i}(z_{3}),e_{1-i}(w)]_{q^{2}}]]_{q^{-2}}=0,$
(5.1)
$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}[f_{i}(z_{1}),[f_{i}(z_{2}),[f_{i}(z_{3}),f_{1-i}(w)]_{q^{2}}]]_{q^{-2}}=0.$
(5.2)

The defining relations of $\mathcal{K}_{2}$ are (3.1)–(3.4), together with the
Serre relations

$\displaystyle\mathop{\mathop{\mathrm{Sym}}}_{z_{1},z_{2}}\,\bigl{(}q_{1}(z_{1}-q_{3}w)(z_{2}-q_{3}w)E_{i}(z_{1})E_{i}(z_{2})E_{1-i}(w)-\big{(}1+q_{2}^{-1}\big{)}(z_{1}-q_{3}w)(q_{1}z_{2}-w)$
$\displaystyle\qquad{}\times
E_{i}(z_{1})E_{1-i}(w)E_{i}(z_{2})+q_{3}(q_{1}z_{1}-w)(q_{1}z_{2}-w)E_{1-i}(w)E_{i}(z_{1})E_{i}(z_{2})\bigr{)}$
$\displaystyle\qquad{}=\mathop{\mathop{\mathrm{Sym}}}_{z_{1},z_{2}}\,\biggl{\\{}q^{-3}w\bigg{(}\big{(}z_{2}-d^{-2}z_{1}\big{)}\frac{1-q_{3}^{-1}z_{1}/w}{1-q_{3}z_{1}/w}+\big{(}z_{2}-d^{2}z_{1}\big{)}\frac{1-q_{1}^{-1}w/z_{2}}{1-q_{1}w/z_{2}}-z_{2}+q^{2}z_{1}\biggr{)}$
$\displaystyle\qquad\hphantom{=}{}\times\delta\big{(}C^{2}z_{1}/z_{2}\big{)}K^{-}_{i}(z_{1})E_{1-i}(w)K_{i}^{+}(z_{2})\bigg{\\}},$
(5.3)

and the same relation with $q_{1}$ and $q_{3}$ interchanged.

At first glance, the quartic Serre relations (5.1) and (5.2) for
${\mathcal{E}}_{2}$ and the cubic Serre relations (5.3) for $\mathcal{K}_{2}$
look very different. Actually the former are equivalent (under the rest of the
relations) to cubic relations which are the left-hand sides of (5.3) and their
“$f$-version”. See [3, Lemma 2.1] and remark after that.

With the above changes, formula for comodule structure (Theorem 3.4), formulas
for integrals of motion (4.9) and (4.10) and their commutativity (Theorem 4.2)
remain valid.

###  5.2 Case $\boldsymbol{n=1}$

Algebra $\mathcal{K}_{1}$ has been discussed in [7]. For reader’s reference we
mention the necessary changes.

We drop suffixes $i$, $j$ from structure functions and set

$\displaystyle g(z,w)=\prod_{s=1}^{3}(z-q_{s}w),$ $\displaystyle
G(x)=\prod_{s=1}^{3}\frac{1-q_{s}^{-1}x}{1-q_{s}x},$
$\displaystyle\lambda^{0}(x)=\frac{1-C^{2}x}{1-x}\frac{1-C^{-2}x}{1-x}\prod_{s=1}^{3}\frac{1-q_{s}x}{1-x},$
$\displaystyle\lambda(x)=\frac{1-C^{2}x}{1-x}\frac{1-C^{-2}x}{1-x}\prod_{s=1}^{3}\frac{1-q_{s}x}{1-x}\frac{\big{(}q_{s}^{-1}x;p\big{)}_{\infty}}{(q_{s}x;p)_{\infty}}.$

We change also the normalization of generators of ${\mathcal{E}}_{1}$
slightly, so that

$\displaystyle\psi^{\pm}(z)=\exp\bigg{(}\sum_{\pm r>0}\kappa_{r}h_{\pm
r}z^{\mp r}\bigg{)},$
$\displaystyle[e(z),f(w)]=\frac{1}{\kappa_{1}}(\delta\bigl{(}C{w}/{z}\bigr{)}\psi^{+}(w)-\delta\bigl{(}C{z}/{w}\bigr{)}\psi^{-}(z)),$

where $\kappa_{r}=\prod_{s=1}^{3}(1-q_{s}^{r})$.

The Serre relations for ${\mathcal{E}}_{1}$ are

$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}z_{2}z_{3}^{-1}[e(z_{1}),[e(z_{2}),e(z_{3})]]=0,$
$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}z_{2}z_{3}^{-1}[f(z_{1}),[f(z_{2}),f(z_{3})]]=0.$

The Serre relation for $\mathcal{K}_{1}$ reads

$\displaystyle\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}\frac{z_{2}}{z_{3}}[E(z_{1}),[E(z_{2}),E(z_{3})]]$
$\displaystyle\qquad{}=\mathop{\mathrm{Sym}}_{z_{1},z_{2},z_{3}}\bigg{\\{}X(z_{1},z_{2},z_{3})\frac{1}{q-q^{-1}}\delta\big{(}C^{2}z_{1}/z_{3}\big{)}K^{-}(z_{1})E(z_{2})K^{+}(z_{3})\bigg{\\}},$

where

$\displaystyle X(z_{1},z_{2},z_{3})={}$
$\displaystyle\frac{(z_{1}+z_{2})\big{(}z_{3}^{2}-z_{1}z_{2}\big{)}}{z_{1}z_{2}z_{3}}G(z_{2}/z_{3})+\frac{(z_{2}+z_{3})\big{(}z_{1}^{2}-z_{2}z_{3}\big{)}}{z_{1}z_{2}z_{3}}G(z_{1}/z_{2})$
$\displaystyle+\frac{(z_{3}+z_{1})\big{(}z_{2}^{2}-z_{3}z_{1}\big{)}}{z_{1}z_{2}z_{3}}.$

The space of theta functions $X_{1}$ is defined to be that of constant
functions ${\mathbb{C}}$. Replacing
$h_{M}(\mathbf{u}_{1},\dots,\mathbf{u}_{n};\vartheta)$ by

$\displaystyle h_{M}(\mathbf{u})=\prod_{1\leq a<b\leq
M}\frac{\theta(u_{a}-u_{b})\theta(u_{a}-u_{b}-\gamma)}{\theta(u_{a}-u_{b}-\beta-\gamma/2)\theta(u_{a}-u_{b}+\beta-\gamma/2)},$

we define integrals of motion by (4.9). Then the commutativity Theorem 4.2
holds true.

##  6 Fusion

It is known [3] that the completed quantum toroidal algebra
$\tilde{{\mathcal{E}}}_{n}$ with parameters $q_{1}$, $q_{2}$, $q_{3}$ contains
various subalgebras isomorphic to $\tilde{{\mathcal{E}}}_{k}$, with $1\leq
k\leq n-1$ and suitable parameters $\bar{q}_{1}$, $\bar{q}_{2}$,
$\bar{q}_{3}$. In this section we discuss an analogue of this construction for
$\mathcal{K}_{n}$. Throughout this section we take $n\geq 2$, and fix an
admissible $\mathcal{K}_{n}$ module $V$, where $C$ acts as a scalar such that
$C^{2}\neq\pm 1,q^{\pm 2}$.

Consider the following set of operators
$\big{\\{}\bar{E}_{i}(z),\bar{K}^{\pm}_{i}(z)\big{\\}}_{0\leq i\leq n-2}$
acting on $V$:

$\displaystyle\bar{E}_{0}(z)=\bar{c}\,E_{n-1,0}(q_{1}z,z),\qquad\bar{K}^{\pm}_{0}(z)=K^{\pm}_{n-1}(q_{1}z)K^{\pm}_{0}(z),$
(6.1)
$\displaystyle\bar{E}_{i}(z)=E_{i}\big{(}q_{1}^{\frac{i}{n-1}}z\big{)},\qquad\bar{K}^{\pm}_{i}(z)=K^{\pm}_{i}\big{(}q_{1}^{\frac{i}{n-1}}z\big{)},\qquad
1\leq i\leq n-2.$ (6.2)

Here $\bar{c}=(qq_{1})^{1/2}\big{(}1-q_{1}^{-1}\big{)}$ for $n\geq 3$ and
$\bar{c}=\big{(}1-q_{1}^{-1}\big{)}^{2}$ for $n=2$.

######  Proposition 6.1.

The operators (6.1) and (6.2) together with the same $C$ give an action of
$\mathcal{K}_{n-1}$ on $V$ with parameters

$\displaystyle\bar{q}_{1}=q_{1}q_{1}^{\frac{1}{n-1}},\qquad\bar{q}_{2}=q_{2},\qquad\bar{q}_{3}=q_{3}q_{1}^{-\frac{1}{n-1}}.$

###### Proof.

Denote by $\bar{G}_{i,j}(x)$, $0\leq i,j\leq n-2$ the functions $G_{i,j}(x)$
for $\mathcal{K}_{n-1}$ with the parameters given above. Then we have

$\displaystyle\bar{G}_{i,i}(x)=G_{i,i}(x)^{2}G_{i+1,i}(q_{1}x)G_{i,i+1}\big{(}q_{1}^{-1}x\big{)},\qquad
n\geq 2,$

and

$\displaystyle\bar{G}_{i,i\pm 1}(x)=\begin{cases}G_{i,i\pm
1}\big{(}q_{1}^{\pm\frac{1}{n-1}}x\big{)},&n\geq 4,\\\ G_{i,i\pm
1}\big{(}q_{1}^{\pm\frac{1}{n-1}}x\big{)}G_{i\pm
1,i}\big{(}q_{1}^{\mp\frac{1}{n-1}}x\big{)},&n=3.\end{cases}$

Using these one can verify the relations (3.1)–(3.4) as rational functions.

For simplicity we assume $n\geq 3$ below. The case $n=2$ can be treated with
minor modifications. Let us check the condition (3.12) for $\bar{E}_{0}(z)$.
We have

$\displaystyle\bar{E}_{0}(z_{1})\bar{E}_{0}(z_{2})={}$ $\displaystyle
E_{n-1,n-1,0,0}(q_{1}z_{1},q_{1}z_{2},z_{1},z_{2})\bar{c}d\big{(}1-q_{1}^{-1}\big{)}^{2}\\!\frac{(1-z_{2}/z_{1})^{2}}{(1-C^{2}z_{2}/z_{1})^{2}}\frac{(1-z_{2}/z_{1})^{2}}{(1-C^{-2}z_{2}/z_{1})^{2}}$
$\displaystyle\times\frac{(1-z_{2}/z_{1})^{2}}{\big{(}1-q^{2}z_{2}/z_{1}\big{)}^{2}}\frac{\big{(}1-q_{1}^{-1}z_{2}/z_{1}\big{)}(1-q_{1}z_{2}/z_{1})}{\big{(}1-q^{-2}z_{2}/z_{1}\big{)}(1-z_{2}/z_{1})}.$

The wheel conditions (3.13)–(3.15) ensure that the apparent double poles in
the right-hand side are actually all simple. To compute the residue at
$z_{2}=C^{2}z_{1}$, it suffices to consider

$\displaystyle\frac{1}{1-C^{-2}z_{2}/z_{1}}E_{n-1,n-1,0,0}(q_{1}z_{1},q_{1}z_{2},z_{1},z_{2})\bigg{|}_{z_{2}=C^{2}z_{1}}$
$\displaystyle\quad{}=\\!w\frac{\partial}{\partial
w}E_{n-1,n-1,0,0}(q_{1}z_{1},q_{1}z_{2},w,z_{2})\bigg{|}_{\begin{subarray}{c}w=z_{1}\\\
z_{2}=C^{2}z_{1}\end{subarray}}\\!\\!\\!\\!\\!\\!+\\!E_{n-1,n-1,0,0}\big{(}q_{1}w,C^{2}q_{1}z_{1},z_{1},C^{2}z_{1}\big{)}\bigg{|}_{\begin{subarray}{c}w=z_{1}\\\
z_{2}=C^{2}z_{1}\end{subarray}}\\!\\!.$

The second term in the right-hand side vanishes due to the wheel condition.
Substituting

$\displaystyle E_{n-1,n-1,0,0}(q_{1}z_{1},q_{1}z_{2},w,z_{2})=$
$\displaystyle{}K_{n-1}^{-}(q_{1}z_{1})E_{0,0}(w,q_{1}z_{1})K_{n-1}^{+}\big{(}C^{2}q_{1}z_{1}\big{)}$
$\displaystyle\times\frac{1-w/z_{1}}{1\\!-\\!q_{1}^{-1}w/z_{1}}\frac{1-q^{2}C^{-2}w/z_{1}}{1\\!-\\!q_{1}^{-1}C^{-2}w/z_{1}}\frac{-d^{-2}q^{-1}}{1-q_{1}^{-1}}\frac{1+C^{2}}{1-C^{2}}\frac{1-q^{2}C^{2}}{1\\!-\\!q_{1}^{-1}C^{2}},$

we arrive at

$\displaystyle\lambda^{0}_{0,0}(z_{2}/z_{1})\bar{E}_{0}(z_{1})\bar{E}_{0}(z_{2})\bigl{|}_{z_{2}=C^{2}z_{1}}=\frac{1}{q-q^{-1}}\frac{1+C^{2}}{1-C^{2}}\frac{1-q^{2}C^{2}}{1-C^{2}}\bar{K}_{0}^{-}(z_{1})\bar{K}_{0}^{+}\big{(}C^{2}z_{1}\big{)}.$

The wheel conditions for $\bar{E}_{i}(z)$ can be verified similarly. As an
example, consider $\bar{E}_{0,0,1}(z_{1},z_{2},w)$. Up to an irrelevant factor
it can be written as

$\displaystyle\bar{E}_{0,0,1}(z_{1},z_{2},w)\propto\frac{1}{\lambda^{0}_{0,0}(z_{2}/z_{1})}E_{n-1,n-1,0,0,1}\big{(}q_{1}z_{1},q_{1}z_{2},z_{1},z_{2},q_{1}^{\frac{1}{n-1}}w\big{)}.$
(6.3)

Since

$E_{n-1,n-1,0}(q_{1}z,q_{1}q_{2}z,z)=0\qquad\text{and}\qquad\text{E}_{0,0,1}(z,q_{2}z,q_{2}q_{3}z)=0,$

$E_{n-1,n-1,0,0,1}\big{(}q_{1}z_{1},q_{1}z_{2},z_{1},z_{2},q_{1}^{-1}z_{1}\big{)}$
is divisible by $(z_{2}-q_{2}z_{1})^{2}$. Therefore (6.3) vanishes at
$z_{2}=q_{2}z_{1}$, $w=\bar{q}_{1}^{-1}z_{1}$. Similarly, the condition
$E_{n-1,n-1,0}(q_{1}z,q_{1}C^{2}z,z)=0$ and
$E_{0,0,1}\big{(}z,C^{2}z,C^{2}q_{3}z\big{)}\allowbreak=0$ imply that (6.3)
vanishes at $z_{2}=C^{2}z_{1}$, $w=C^{2}\bar{q}_{3}z_{1}$. ∎

Interchanging the roles of $q_{1}$ and $q_{3}$, one can equally well consider
action of $\mathcal{K}_{n-1}$ with parameters

$\displaystyle\bar{\bar{q}}_{1}=q_{1}q_{3}^{-\frac{1}{n-1}},\qquad\bar{\bar{q}}_{2}=q_{2},\qquad\bar{\bar{q}}_{3}=q_{3}q_{3}^{\frac{1}{n-1}}.$

Note that in either case the values of $q$, $C$ are unchanged.

Iterating this construction we obtain in general

######  Proposition 6.2.

Let $1\leq k\leq n-1$. Then the following hold with an appropriate choice of
constants $\bar{c}_{n,k}$, $\bar{\bar{c}}_{n,n-k}$:

  1. $(i)$

The operators

$\displaystyle\bar{E}_{0}(z)=\bar{c}_{n,k}\,E_{k,k+1,\dots,n-1,0}\big{(}q_{1}^{n-k}z,q_{1}^{n-k-1}z,\dots,q_{1}z,z\big{)},$
$\displaystyle\bar{K}^{\pm}_{0}(z)=K^{\pm}_{k}\big{(}q_{1}^{n-k}z\big{)}K^{\pm}_{k+1}\big{(}q_{1}^{n-k-1}z\big{)}\cdots
K^{\pm}_{n-1}(q_{1}z)K^{\pm}_{0}(z),$
$\displaystyle\bar{E}_{i}(z)=E_{i}\big{(}q_{1}^{\frac{n-k}{k}i}z\big{)},\qquad\bar{K}^{\pm}_{i}(z)=K^{\pm}_{i}\big{(}q_{1}^{\frac{n-k}{k}i}z\big{)},\qquad
1\leq i\leq k-1,$

together with the same $C$ give an action of $\mathcal{K}_{k}$ on $V$ with
parameters

$\displaystyle\bar{q}_{1}=q_{1}q_{1}^{\frac{n-k}{k}},\qquad\bar{q}_{2}=q_{2},\qquad\bar{q}_{3}=q_{3}q_{1}^{-\frac{n-k}{k}}.$

  2. $(ii)$

The operators

$\displaystyle\bar{\bar{E}}_{0}(z)=\bar{\bar{c}}_{n,n-k}E_{k,k-1,\dots,1,0}\big{(}q_{3}^{k}z,q_{3}^{k-1}z,\dots,q_{3}z,z\big{)},$
$\displaystyle\bar{\bar{K}}^{\pm}_{0}(z)=K^{\pm}_{k}\big{(}q_{3}^{k}z\big{)}K^{\pm}_{k-1}\big{(}q_{3}^{k-1}z\big{)}\cdots
K^{\pm}_{1}(q_{3}z)K^{\pm}_{0}(z),$
$\displaystyle\bar{\bar{E}}_{i}(z)=E_{i+k}\big{(}q_{3}^{\frac{k}{n-k}i}z\big{)},\qquad\bar{\bar{K}}^{\pm}_{i}(z)=K^{\pm}_{i+k}\big{(}q_{3}^{\frac{k}{n-k}i}z\big{)},\qquad
1\leq i\leq n-k-1,$

together with the same $C$ give an action of $\mathcal{K}_{n-k}$ on $V$ with
parameters

$\displaystyle\bar{\bar{q}}_{1}=q_{1}q_{3}^{-\frac{k}{n-k}},\qquad\bar{\bar{q}}_{2}=q_{2},\qquad\bar{\bar{q}}_{3}=q_{3}q_{3}^{\frac{k}{n-k}}.$

  3. $(iii)$

The actions $(i)$ and $(ii)$ mutually commute.

###### Proof.

We need only to prove the commutativity $(iii).$ For simplicity of
presentation we assume $n\geq 3$. Evidently
$\big{\\{}\bar{E}_{i}(z),\bar{K}^{\pm}_{i}(z)\big{\\}}_{1\leq i\leq k-1}$
commute with
$\big{\\{}\bar{\bar{E}}_{i}(z),\bar{\bar{K}}^{\pm}_{i}(z)\big{\\}}_{1\leq
i\leq n-k-1}$.

To show that $\bar{\bar{K}}^{+}_{i}(z)$, $1\leq i\leq n-k-1$, commutes with
$\bar{E}_{0}(w)$, it suffices to note that

$\displaystyle
K^{+}_{j}(z)E_{k,k+1,\dots,n-1,0}\big{(}q_{1}^{n-k}w,\dots,w\big{)}={}$
$\displaystyle
E_{k,k+1,\dots,n-1,0}\big{(}q_{1}^{n-k}w,\dots,w\big{)}K^{+}_{j}(z)$
$\displaystyle\times
G_{j,j-1}\big{(}q_{1}^{n-j+1}w/z\big{)}G_{j,j}\big{(}q_{1}^{n-j}w/z\big{)}$
$\displaystyle\times G_{j,j+1}\big{(}q_{1}^{n-j-1}w/z\big{)},\qquad k+1\leq
j\leq n-1,$

and apply the identity

$\displaystyle
G_{i,i-1}(q_{1}x)G_{i,i}(x)G_{i,i+1}\big{(}q_{1}^{-1}x\big{)}=1.$

Commutativity of
$\big{\\{}\bar{\bar{E}}_{i}(z),\bar{\bar{K}}^{\pm}_{i}(z)\big{\\}}_{1\leq
i\leq n-k-1}$ with $\big{\\{}\bar{E}_{0}(w),\bar{K}^{\pm}_{0}(w)\big{\\}}$
(for $\bar{\bar{E}}_{i}(z)$ with $\bar{E}_{0}(w)$ as rational functions) can
be shown similarly.

We have further

$\displaystyle\bar{\bar{K}}_{0}(z)\bar{E}_{0}(w)={}$
$\displaystyle\bar{E}_{0}(w)\bar{\bar{K}}_{0}(z)G_{k-1,k}\big{(}q_{1}^{n}q_{2}^{k}q_{3}w/z\big{)}G_{k,k}\big{(}q_{1}^{n}q_{2}^{k}w/z\big{)}G_{k,k+1}\big{(}q_{1}^{n}q_{2}^{k}q_{1}^{-1}w/z\big{)}$
$\displaystyle\times
G_{1,0}\big{(}q_{3}^{-1}w/z\big{)}G_{0,0}(w/z)G_{0,n-1}(q_{1}w/z),$

so that we can apply

$\displaystyle
G_{i-1,i}(q_{3}x)G_{i,i}(x)G_{i,i+1}\big{(}q_{1}^{-1}x\big{)}=1.$

By the same token
$\big{\\{}\bar{\bar{E}}_{0}(z),\bar{\bar{K}}^{\pm}_{0}(z)\big{\\}}$ commute
with $\big{\\{}\bar{E}_{0}(w),\bar{K}^{\pm}_{0}(w)\big{\\}}$ (for
$\bar{\bar{E}}_{0}(z)$ with $\bar{E}_{0}(w)$ as rational functions).

To finish the proof, it remains to examine the singularities of
$\bar{\bar{E}}_{i}(z)\bar{E}_{0}(w)$. For $1\leq i\leq n-k-1$, we use

$\displaystyle E_{j}(z)E_{j-1,j,j+1}\big{(}q_{1}w,w,q_{1}^{-1}w\big{)}={}$
$\displaystyle
E_{j-1,j,j,j+1}\big{(}q_{1}w,z,w,q_{1}^{-1}w\big{)}\frac{1-q_{1}w/z}{1-q_{2}^{-1}w/z}\frac{1-w/z}{1-C^{2}w/z}$
$\displaystyle\times\frac{1-w/z}{1-q^{2}w/z}\frac{1-w/z}{1-C^{-2}w/z}\frac{1-q_{1}^{-1}w/z}{1-w/z},$

and observe that all poles are cancelled due to the wheel conditions. Finally
for $i=0$ we have

$\displaystyle
E_{k,k-1,\dots,1,0}\big{(}q_{3}^{k}z,\dots,z\big{)}E_{k,k+1,\dots,n-1,0}\big{(}q_{1}^{n-k}w,\dots,w\big{)}$
$\displaystyle\qquad{}=E_{0,0,1,\dots,k-1,k,k,k+1,\dots,n-1}\big{(}z,w,q_{3}z,\dots,q_{3}^{k-1}z,q_{3}z,q_{1}^{n-k}w,\dots,q_{1}w\big{)}$
$\displaystyle\qquad\hphantom{=}{}\times\bigl{(}\lambda^{0}_{k-1,k}\big{(}q_{1}^{n}q_{2}^{k}q_{3}w/z\big{)}\lambda^{0}_{k,k}\big{(}q_{1}^{n}q_{2}^{k}w/z\big{)}\lambda^{0}_{k,k+1}\big{(}q_{1}^{n}q_{2}^{k}q_{1}^{-1}w/z\big{)}\bigr{)}^{-1}$
$\displaystyle\qquad\hphantom{=}{}\times\bigl{(}\lambda^{0}_{0,n-1}(q_{1}w/z)\lambda^{0}_{0,0}(w/z)\lambda^{0}_{1,0}\big{(}q_{3}^{-1}w/z\big{)}\bigr{)}^{-1},$

so the cancellation works similarly as above. ∎

##  Appendix A Theta function identities

As in Section 4.2 we use the space of theta functions
$X_{n}(\mu_{1},\dots,\mu_{n})$ given by (4.6) and (4.7) for $n\geq 2$. For
$n=1$, we define ${X}_{1}={\mathbb{C}}$ to be the space of constant functions.
In this section we prove the following identity.

######  Theorem A.1.

The following identities hold for all $n,M,N\geq 1$,
$\alpha,\gamma\in{\mathbb{C}}$ and $\vartheta_{1},\vartheta_{2}\in{X}_{n}$:

$\displaystyle\sum_{I_{1},J_{1}}\cdots\sum_{I_{n},J_{n}}\vartheta_{1}(\overline{u}_{1,I_{1}},\dots,\overline{u}_{n,I_{n}})\vartheta_{2}(\overline{u}_{1,J_{1}},\dots,\overline{u}_{n,J_{n}})$
$\displaystyle\qquad{}\times\prod_{i=1}^{n}\frac{\prod_{\begin{subarray}{c}a\in
I_{i}\\\ b\in
J_{i+1}\end{subarray}}\theta(u_{i,a}-u_{i+1,b}-\alpha)\prod_{\begin{subarray}{c}b\in
J_{i}\\\ a\in
I_{i+1}\end{subarray}}\theta(u_{i,b}-u_{i+1,a}-\alpha+\gamma)}{\prod_{\begin{subarray}{c}a\in
I_{i}\\\ b\in
J_{i}\end{subarray}}\theta(u_{i,a}-u_{i,b})\theta(u_{i,a}-u_{i,b}-\gamma)}$
$\displaystyle\qquad{}=\sum_{I_{1},J_{1}}\cdots\sum_{I_{n},J_{n}}\vartheta_{1}(\overline{u}_{1,I_{1}},\dots,\overline{u}_{n,I_{n}})\vartheta_{2}(\overline{u}_{1,J_{1}},\dots,\overline{u}_{n,J_{n}})$
$\displaystyle\qquad\hphantom{=}{}\times\prod_{i=1}^{n}\frac{\prod_{\begin{subarray}{c}b\in
J_{i}\\\ a\in
I_{i+1}\end{subarray}}\theta(u_{i,b}-u_{i+1,a}-\alpha)\prod_{\begin{subarray}{c}a\in
I_{i}\\\ b\in
J_{i+1}\end{subarray}}\theta(u_{i,a}-u_{i+1,b}-\alpha+\gamma)}{\prod_{\begin{subarray}{c}a\in
I_{i}\\\ b\in
J_{i}\end{subarray}}\theta(u_{i,b}-u_{i,a})\theta(u_{i,b}-u_{i,a}-\gamma)}.$
(A.1)

Here we set $\overline{u}_{i,I}=\sum_{a\in I}u_{i,a}$ for a subset $I$ of
$\\{1,\dots,M+N\\}$, and the sum is taken over all partitions

$\displaystyle I_{i}\sqcup
J_{i}=\\{1,\dots,M+N\\}\qquad\text{satisfying}\quad|I_{i}|=M,\quad|J_{i}|=N,\quad
i=1,\dots,n.$

For the proof of theorem, we prepare several lemmas.

######  Lemma A.2.

If $\vartheta\in{X}_{n}(\mu_{1},\dots,\mu_{n})$, then

$\displaystyle\vartheta(u_{1}+u,\dots,u_{n}+u)=\vartheta(u_{1},\dots,u_{n}),\qquad
u\in{\mathbb{C}}.$

###### Proof.

Let $\phi(u)=\vartheta(u_{1}+u,\dots,u_{n}+u)$. Then $\phi(u)$ is entire and
satisfies $\phi(u+1)=\phi(u)$, $\phi(u+\tau)=\phi(u)$. Hence it is a constant:
$\phi(u)=\phi(0)$. ∎

######  Lemma A.3.

Let $\phi(v_{1},\dots,v_{L})$ be an entire function with quasi-periodicity
property

$\displaystyle\phi(v_{1},\dots,v_{i}+1,\dots,v_{L})=\phi(v_{1},\dots,v_{i},\dots,v_{L}),$
$\displaystyle\phi(v_{1},\dots,v_{i}+\tau,\dots,v_{L})={\rm e}^{-2\pi{\rm
i}m\sum_{i=1}^{L}v_{i}+k}\phi(v_{1},\dots,v_{i},\dots,v_{L})$

with some $m\in{\mathbb{Z}}_{\geq 0}$ and $k\in{\mathbb{C}}$. Then it is a
function of $\sum_{i=1}^{L}v_{i}$.

###### Proof.

Fix $i\neq j$, and set
$\varphi(u)=\phi(v_{1},\dots,v_{i}+u,\dots,v_{j}-u,\dots,v_{L})$. Then
$\varphi(u)$ is an entire function and satisfies $\varphi(u+1)=\varphi(u)$,
$\varphi(u+\tau)=\varphi(u)$. Hence it is a constant: $\varphi(u)=\varphi(0)$.
Since $i$, $j$, $u$ are arbitrary, this proves the lemma. ∎

Denote both sides of the identity (A.1) by ${\rm LHS}_{n,M,N}$ and ${\rm
RHS}_{n,M,N}$, respectively, and set $\Phi_{n,M,N}={\rm LHS}_{n,M,N}-{\rm
RHS}_{n,M,N}$. Where necessary we exhibit also their dependence on the
variables $\mathbf{u}_{i}=(u_{i,1},\dots,u_{i,M+N})$ and the choice of
$\vartheta_{1},\vartheta_{2}\in{X}_{n}$. We set $\Phi_{n,M,N}=0$ if $M=0$ or
$N=0$.

######  Lemma A.4.

For $n\geq 2$, $\Phi_{n,M,N}$ has the quasi-periodicity in each variable
$u_{i,a}$,

$\displaystyle\Phi_{n,M,N}(\dots,u_{i,a}+1,\dots)=\Phi_{n,M,N}(\dots,u_{i,a},\dots),$
$\displaystyle\Phi_{n,M,N}(\dots,u_{i,a}+\tau,\dots)={\rm e}^{-2\pi{\rm
i}(2\bar{u}_{i}-\bar{u}_{i-1}-\bar{u}_{i+1}-\mu_{i}+\tau)}\Phi_{n,M,N}(\dots,u_{i,a},\dots).$

For $n=1$, $\Phi_{1,M,N}$ has periods $1$ and $\tau$.

Modulo ${\mathbb{Z}}\oplus{\mathbb{Z}}\tau$, the only poles of $\Phi_{n,M,N}$
are $u_{i,a}=u_{i,b}+\gamma$ with some $a$, $b$, $i$.

###### Proof.

The quasi-periodicity follows from (4.4), (4.6) and (4.7). Since
$\Phi_{n,M,N}$ is symmetric in each group of variables
$\\{u_{i,a}\\}_{a=1,\dots,M+N}$, it cannot have a simple pole at
$u_{i,a}=u_{i,b}$. ∎

######  Lemma A.5.

We have

$\displaystyle\mathop{\rm res}_{u_{1,1}=u_{1,2}+\gamma}\cdots\mathop{\rm
res}_{u_{n,1}=u_{n,2}+\gamma}\,\Phi_{n,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\vartheta_{1},\vartheta_{2})$
$\displaystyle\qquad{}=A^{n}\frac{\prod_{i=1}^{n}\prod_{a=2}^{M+N}\theta(u_{i,a}-u_{i+1,2}-\alpha)\theta(u_{i,2}-u_{i+1,a}-\alpha+\gamma)}{\prod_{i=1}^{n}\prod_{a=3}^{M+N}\theta(u_{i,a}-u_{i,2})\theta(u_{i,a}-u_{i,2}-\gamma)}$
$\displaystyle\qquad\hphantom{=}{}\times\Phi_{n,M-1,N-1}(\mathbf{u}^{\prime}_{1},\dots,\mathbf{u}^{\prime}_{n}|\vartheta^{\prime}_{1},\vartheta^{\prime}_{2}),$

where

$\displaystyle\mathbf{u}^{\prime}_{i}=(u_{i,3},\dots,u_{i,M+N}),$
$\displaystyle\vartheta^{\prime}_{s}(v_{1},\dots,v_{n})=\vartheta_{s}(v_{1}+u_{1,2},\dots,v_{n}+u_{n,2}),\qquad
s=1,2,$

and $A=\theta(\gamma)^{-1}\mathop{\rm res}_{u=0}\theta(u)^{-1}$.

###### Proof.

In the sum ${\rm LHS}_{n,M,N}$, the iterated residue is non-zero only for
terms such that $1\in I_{i}$ and $2\in J_{i}$ for all $i=1,\dots,n$. Write
$I_{i}=\\{1\\}\sqcup\bar{I}_{i}$ and $J_{i}=\\{2\\}\sqcup\bar{J}_{i}$.
Similarly, for ${\rm RHS}_{n,M,N}$ one must have $2\in I_{i}$ and $1\in J_{i}$
for $i=1,\dots,n$. Write $I_{i}=\\{2\\}\sqcup\bar{I}_{i}$ and
$J_{i}=\\{1\\}\sqcup\bar{J}_{i}$ in that case.

In either case, the functions $\vartheta_{s}$ become

$\displaystyle\vartheta_{1}(u_{1,2}+\bar{u}_{1,\bar{I}_{1}},\dots,u_{n,2}+\bar{u}_{n,\bar{I}_{n}})\vartheta_{2}(u_{1,2}+\bar{u}_{1,\bar{J}_{1}},\dots,u_{n,2}+\bar{u}_{n,\bar{J}_{n}})$

due to Lemma A.2.

The rest is a simple computation. ∎

######  Remark A.6.

Lemma A.5 is proved in [12]. However the rest of the argument necessary for
completing the proof of Theorem A.1 is missing there.

######  Lemma A.7.

Fix $i\in\\{1,\dots,n\\}$. Then the following hold:

$\displaystyle\Phi_{n,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\vartheta_{1},\vartheta_{2})\bigl{|}_{\begin{subarray}{c}u_{i+1,a}=u_{i,a}-\alpha\\\
1\leq a\leq M+N\end{subarray}}$
$\displaystyle\qquad{}=(-1)^{MN}\Phi_{n-1,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{i},\mathbf{u}_{i+2},\dots,\mathbf{u}_{n}|\vartheta^{\prime\prime}_{1},\vartheta^{\prime\prime}_{2}),$
$\displaystyle\Phi_{n,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{n}|\vartheta_{1},\vartheta_{2})\bigl{|}_{\begin{subarray}{c}u_{i+1,a}=u_{i,a}-\alpha+\gamma\\\
1\leq a\leq M+N\end{subarray}}$
$\displaystyle\qquad{}=(-1)^{MN}\Phi_{n-1,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{i},\mathbf{u}_{i+2},\dots,\mathbf{u}_{n}|\vartheta^{\prime\prime\prime}_{1},\vartheta^{\prime\prime\prime}_{2}),$

where

$\displaystyle\vartheta^{\prime\prime}_{1}(v_{1},\dots,v_{n-1})=\vartheta_{1}(v_{1},\dots,v_{i},v_{i}-M\alpha,v_{i+1},\dots,v_{n-1}),$
$\displaystyle\vartheta^{\prime\prime}_{2}(v_{1},\dots,v_{n-1})=\vartheta_{2}(v_{1},\dots,v_{i},v_{i}-N\alpha,v_{i+1},\dots,v_{n-1}),$

and $\vartheta^{\prime\prime\prime}_{s}$ is given by replacing $\alpha$ by
$\alpha-\gamma$ in $\vartheta^{\prime\prime}_{s}$.

###### Proof.

Since the calculations are similar, we consider the first case. Under this
specialization, terms in ${\rm LHS}_{n,M,N}$ survive only when $I_{i}\cap
J_{i+1}=\varnothing$, which implies $I_{i}=I_{i+1}$ and $J_{i}=J_{i+1}$. Then
the factor $\prod_{\begin{subarray}{c}a\in I_{i}\\\ b\in
J_{i+1}\end{subarray}}\theta(u_{i,a}-u_{i+1,b}-\alpha)\prod_{\begin{subarray}{c}a\in
I_{i+1}\\\ b\in J_{i}\end{subarray}}\theta(u_{i,b}-u_{i+1,a}-\alpha+\gamma)$
from the numerator cancels $\prod_{\begin{subarray}{c}a\in I_{i+1}\\\ b\in
J_{i+1}\end{subarray}}\theta(u_{i+1,a}-u_{i+1,b})\theta(u_{i+1,a}-u_{i+1,b}-\gamma)$
from the denominator, thereby reducing $n$ to $n-1$. For ${\rm RHS}_{n,M,N}$
the situation is entirely similar. ∎

######  Lemma A.8 ([19]).

The identity (A.1) holds for $n=1$.

###### Proof.

For $n=1$ we may assume $\vartheta_{1}=\vartheta_{2}=1$, so the identity takes
the form

$\displaystyle\sum_{I,J}\prod_{\begin{subarray}{c}a\in I\\\ b\in
J\end{subarray}}\frac{\theta(u_{a}-u_{b}-\alpha)\theta(u_{b}-u_{a}-\alpha+\gamma)}{\theta(u_{a}-u_{b})\theta(u_{a}-u_{b}-\gamma)}=\sum_{I,J}\prod_{\begin{subarray}{c}a\in
I\\\ b\in
J\end{subarray}}\frac{\theta(u_{b}-u_{a}-\alpha)\theta(u_{a}-u_{b}-\alpha+\gamma)}{\theta(u_{b}-u_{a})\theta(u_{b}-u_{a}-\gamma)}.$

Lemma A.5 together with the symmetry shows that $\Phi_{1,M,N}$ has no pole at
$u_{a}=u_{b}+\gamma$. Hence $\Phi_{1,M,N}$ is an elliptic function without
poles, therefore it is a constant. Upon specializing $u_{a}=a\alpha$,
$a=1,\dots,M+N$, we obtain

$\displaystyle{\rm LHS}_{1,M,N}$
$\displaystyle=(-1)^{MN}\frac{\prod_{a=1}^{M}\theta((a-1)\alpha+\gamma)/\theta(a\alpha)\prod_{a=1}^{N}\theta((a-1)\alpha+\gamma)/\theta(a\alpha)}{\prod_{a=1}^{M+N}\theta((a-1)\alpha+\gamma)/\theta(a\alpha)}$
$\displaystyle={\rm RHS}_{1,M,N},$

thereby proving $\Phi_{1,M,N}=0$. ∎

###### Proof of Theorem A.1.

We use induction on $n$. The base of induction is proved in Lemma A.8. Assume
that theorem is true for $n-1$.

For $i=1,\dots,n$ we set

$\displaystyle Z_{i}=\mathop{\rm res}_{u_{i,2}=u_{i,1}-\gamma}\mathop{\rm
res}_{u_{i+1,2}=u_{i+1,1}-\gamma}\cdots\mathop{\rm
res}_{u_{n,2}=u_{n,1}-\gamma}\,\Phi_{n,M,N}(\mathbf{u}_{1},\dots,\mathbf{u}_{n},\alpha,\gamma|\vartheta_{1},\vartheta_{2}).$

We show $Z_{i}=0$ by induction on $i$. By Lemma A.5 and the induction
hypothesis $\Phi_{n-1,M,N}=0$, we have $Z_{1}=0$. Suppose $Z_{i}=\mathop{\rm
res}_{u_{i,2}=u_{i,1}-\gamma}Z_{i+1}=0$. Then $Z_{i+1}$ has no poles in
$\mathbf{u}_{i}$.

Since $u_{i+1,2}=u_{i+1,1}-\gamma$, non-zero terms in $Z_{i+1}$ are such that
$1\in I_{i+1}$, $2\in J_{i+1}$ in LHS. Then in the numerator we have factors
$\theta(u_{i,a}-u_{i+1,2}-\alpha)$, $a\in I_{i}$, and
$\theta(u_{i,b}-u_{i+1,1}-\alpha+\gamma)$, $b\in J_{i}$. Therefore each term
has a factor

$\displaystyle\prod_{a=1}^{M+N}\theta(u_{i,a}-u_{i+1,2}-\alpha).$

By the same argument, RHS, and hence $Z_{i+1}$, is shown to have the same
factor. If $Z_{i+1}\neq 0$, this is a contradiction because the quasi-
periodicity property of $Z_{i+1}$ implies that it is a function of
$\bar{u}_{i}=\sum_{a=1}^{M+N}u_{i,a}$, see Lemma A.3. Hence we have
$Z_{i+1}=0$.

It follows that $\Phi_{n,M,N}$ has no poles at $u_{n,a}=u_{n,b}-\gamma$.
Therefore, if non-zero, it is a function of the sum
$\bar{u}_{n}=\sum_{a=1}^{M+N}u_{n,a}$. On the other hand, by Lemma A.7 and the
induction hypothesis, $\Phi_{n,M,N}$ has the following zeros with respect to
$\bar{u}_{n}$:

$\displaystyle\bar{u}_{n}=\bar{u}_{1}+(M+N)\alpha,\qquad\bar{u}_{n}=\bar{u}_{1}+(M+N)(\alpha-\gamma),$
$\displaystyle\bar{u}_{n}=\bar{u}_{n-1}-(M+N)\alpha,\qquad\bar{u}_{n}=\bar{u}_{n-1}-(M+N)(\alpha-\gamma).$

From the quasi-periodicity, if $\Phi_{n,M,N}\neq 0$, then it can have only two
zeros modulo ${\mathbb{Z}}\oplus{\mathbb{Z}}\tau$, which contradicts to the
above zeros.

This completes the proof of Theorem A.1. ∎

### Acknowledgments

The study has been funded within the framework of the HSE University Basic
Research Program. MJ is partially supported by JSPS KAKENHI Grant Number
JP19K03549. EM is partially supported by grants from the Simons Foundation
#353831 and #709444.

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