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# A dynamical analogue of Ding-Iohara quantum algebras

Masamune Hattori, Shintarou Yanagida  Graduate School of Mathematics, Nagoya
University. Furocho, Chikusaku, Nagoya, Japan, 464-8602.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: 2022.10.05)

###### Abstract.

We introduce a family of dynamical Hopf algebroids $U_{q,p}(g,X_{l})$
depending on a complex parameter $q$, a formal parameter $p$, a set $g$ of
structure functions satisfying the so-called Ding-Iohara condition, and a
finite root system of type $X_{l}$. If $g$ is set to be certain theta
functions, then our family recovers the elliptic algebras
$U_{q,p}(\widehat{\mathfrak{g}})$ for untwisted affine Lie algebras
$\widehat{\mathfrak{g}}$ studied by Konno (1998, 2009), Jimbo-Konno-Odake-
Shiraishi (1999) and Farghly-Konno-Oshima (2014). Also, taking the limit $p\to
0$ in the case $X_{l}=A_{l}$, we recover the Hopf algebras
$U_{q}(\overline{g},A_{l})$ of type $A_{l}$ with structure functions
$\overline{g}:=\lim_{p\to 0}g$, introduced by Ding-Iohara (1998) as a
generalization of Drinfeld quantum affine algebras. Thus, our Hopf algebroid
$U_{q,p}(g,X_{l})$ can be regarded as a dynamical analogue of the Ding-Iohara
quantum algebras. As a byproduct, we obtain an extension of the Ding-Iohara
quantum algebras to those of non-simply-laced type.

###### Key words and phrases:

Hopf algebroids, Ding-Iohara algebras, elliptic quantum groups, Drinfeld
comultiplication

###### Contents

  1. 0 Introduction
    1. 0.1 Motivation and backgrounds
    2. 0.2 Results and organization
  2. 1 Recollections on Ding-Iohara quantum algebras and elliptic algebras
    1. 1.1 Hopf algebroid
    2. 1.2 Ding-Iohara quantum algebras
    3. 1.3 Elliptic algebras $U_{q,p}(\widehat{\mathfrak{g}})$
  3. 2 Dynamical Ding-Iohara algebroids of simply-laced root systems
    1. 2.1 Algebra structure
    2. 2.2 $H$-Hopf algebroid structure
    3. 2.3 Hopf algebra limit
  4. 3 Dynamical Ding-Iohara algebroids of non-simply-laced root systems
    1. 3.1 Strange formulas
    2. 3.2 Types $B_{l},C_{l},F_{4}$
    3. 3.3 Type $G_{2}$
    4. 3.4 Hopf algebra limit
  5. 4 Concluding remarks

##  0\. Introduction

###  0.1. Motivation and backgrounds

The motivation of this note is to investigate the Hopf algebroid structure of
elliptic quantum groups and its relation to the Hopf algebra structure of
Ding-Iohara quantum algebras. Let us explain briefly these two objects, Ding-
Iohara quantum algebras and elliptic quantum groups.

#### Ding-Iohara quantum algebras

Ding and Iohara [DI97b] introduced a family of (topological or formal) Hopf
algebras which generalize the Drinfeld realization [D89] of the quantum affine
algebras $U_{q}(\widehat{\mathfrak{sl}}_{l+1})$ with the Drinfeld
comultiplication [DI97a]. This Hopf algebra depends on a quantum parameter $q$
and a set $g$ of structure functions satisfying certain conditions. we denote
it by $U_{q}(g,A_{l})$ and call it _the Ding-Iohara quantum algebra_. A brief
review will be given in § 1.2.

Let us briefly recall the algebra structure of the Ding-Iohara quantum algebra
$U_{q}(g,A_{l})$. See Definition 1.2.1 for the detail. The algebra
$U_{q}(g,A_{l})$ is compactly presented by generating currents

$e_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}e_{i,n}z^{-n},\quad
f_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}f_{i,n}z^{-n},\quad\psi^{+}_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}\psi^{+}_{i,n}z^{-n},\quad\psi^{-}_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}\psi^{-}_{i,n}z^{-n},$

for $i=1,\dotsc,l$ and current relations among them. The genuine generators of
the algebra are the Fourier modes $e_{i,n},f_{i,n},\psi^{\pm}_{i,n}$, and the
genuine relations are obtained by expanding the current relations. See Remark
1.2.2 for the precise treatment. The current relations involve the structure
functions $g=\\{g_{ij}(z)\mid i,j=1,\dotsc,l\\}$. If we set
$g_{ij}(z)\coloneqq q^{-a_{ij}}(1-q^{a_{ij}}z)/(1-q^{-a_{ij}}z)$ with
$(a_{ij})_{i,j=1}^{l}$ being the Cartan matrix of type $A_{l}$, then the
algebra $U_{q}(g,A_{l})$ recovers the Drinfeld realization [D89] of the
quantum affine algebra $U_{q}(\widehat{\mathfrak{sl}}_{l+1})$. See Example
1.2.5 for the detail.

The algebra structure of $U_{q}(g,A_{l})$ is designed to admit the _Drinfeld-
type comultiplication_ , which involves infinite-summation of the genuine
generators. In the case
$g_{ij}(z)=q^{-a_{ij}}(1-q^{a_{ij}}z)/(1-q^{-a_{ij}}z)$, this comultiplication
reduces to the Drinfeld comultiplication [DI97a] of the quantum affine algebra
$U_{q}(\widehat{\mathfrak{sl}}_{l+1})$. To admit such a comultiplication, the
structure functions $g$ must satisfy a certain condition, which we call the
Ding-Iohara condition. See (1.2.2) and (1.2.3) for the detail.

Let us give one more comment on the algebra structure of $U_{q}(g,A_{l})$,
which is the starting point of our study. Among the current relations, we have
the _Serre-type relations_ (1.2.17). For the generating currents $e_{i}(z)$
and $e_{j}(z)$ with $a_{ij}=-1$, the Serre-type relation is given by

$\displaystyle
e_{i}(z_{1})e_{i}(z_{2})e_{j}(w)-h_{ij}(\tfrac{z_{1}}{w},\tfrac{z_{2}}{w})e_{i}(z_{1})e_{j}(w)e_{i}(z_{2})+e_{j}(w)e_{i}(z_{1})e_{i}(z_{2})$
$\displaystyle+\,$ $\displaystyle
e_{i}(z_{2})e_{i}(z_{1})e_{j}(w)-h_{ij}(\tfrac{z_{2}}{w},\tfrac{z_{1}}{w})e_{i}(z_{2})e_{j}(w)e_{i}(z_{1})+e_{j}(w)e_{i}(z_{2})e_{i}(z_{1})=0,$

where $h_{ij}(x,y)$ is given in terms of the structure functions $g_{ij}(z)$
as (1.2.18):

$\displaystyle
h_{ij}(x,y)\coloneqq\frac{(g_{ii}(x/y)+1)(g_{ij}(x)g_{ij}(y)+1)}{g_{ij}(y)+g_{ii}(x/y)g_{ij}(x)}.$
(0.1.1)

If the structure functions are set to be
$g_{ij}(z)=q^{-a_{ij}}(1-q^{a_{ij}}z)/(1-q^{-a_{ij}}z)$, then the function
$h_{ij}$ is equal to the $q$-binomial coefficient
$\genfrac{[}{]}{0.0pt}{}{2}{1}_{q}=(q^{2}-q^{-2})/(q-q^{-1})$, and the above
relation is nothing but the Serre relation of the quantum affine algebra
$U_{q}(\widehat{\mathfrak{sl}}_{l+1})$.

As mentioned in the final paragraph of [DI97b], the argument carries over the
simply-laced types $D_{l}$ and $E_{6,7,8}$, and we have _the Ding-Iohara
quantum algebra $U_{q}(g,X_{l})$ of simply-laced type $X_{l}$_. At present,
this class of quantum algebras appears in various contexts, some of which will
be reviewed in § 1.2.

#### Elliptic quantum groups

Elliptic quantum groups are still under active investigation, and the meaning
of the word “elliptic quantum group” is less definite than the word “quantum
group”, as far as we understand. Roughly speaking, the notion of elliptic
quantum groups refers to an “elliptic deformation” or a “dynamical analogue”
of quantum groups. It originates in the classification of solutions of the
Yang-Baxter equation: the rational, trigonometric, and elliptic solutions, and
also in the (classical or quantum) dynamical Yang-Baxter equation. We refer to
[Ko20, §1.1] for the history and backgrounds of elliptic quantum groups.

At present, there are several objects in literature which are called elliptic
quantum groups. This note focuses on _the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ for the untwisted affine Lie algebra
$\widehat{\mathfrak{g}}$ of a finite-dimensional simple Lie algebra
$\mathfrak{g}$_, which was first introduced by Konno [Ko98] for the case
$\mathfrak{g}=\mathfrak{sl}_{2}$, and by Jimbo-Konno-Odake-Shiraishi [J+99,
Appendix] for general $\mathfrak{g}$. Here $q$ denotes the quantum parameter,
and $p$ denotes the elliptic nome. We will give a review of
$U_{q,p}(\widehat{\mathfrak{g}})$ in § 1.3. For a comprehensive review of the
relation between the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ and
the other “elliptic quantum groups”, we refer to [Ko20, §1.2].

Here we give a brief explanation on the algebra structure of the elliptic
algebra $U_{q,p}(\widehat{\mathfrak{g}})$. See Definition 1.3.2 for the
detail. Let $\mathfrak{g}$ be a simple Lie algebra of rank $l$, and
$\widehat{\mathfrak{g}}$ be the untwisted affine Lie algebra of
$\mathfrak{g}$. The algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is presented by
generators and generating currents

$\displaystyle\operatorname{\mathcal{M}}(\widetilde{H}^{*}),\ q^{\pm c/2},\
d,\ K^{\pm}_{i},\ e_{i}(z),\ f_{i}(z),\ \psi^{\pm}_{i}(z)$ (0.1.2)

for $i=1,\dotsc,l$, subject to the relations given in Definition 1.3.2. Here
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ denotes the field of
meromorphic functions on the affine space $\widetilde{H}$ of dimension $2l$,
given in (1.3.8). There is a distinguished basis (1.3.10) of $\widetilde{H}$,
and the elements $P_{i}$ among them are called _the dynamical parameters_. See
§ 1.3.2 for the detail.

As mentioned in [J+99, p.613], if $\mathfrak{g}$ is of simply-laced type
$X_{l}$, then the subalgebra of $U_{q}(\widehat{\mathfrak{g}})$ generated by
the currents $e_{i}(z),f_{i}(z),\psi^{\pm}_{i}(z)$ is essentially the special
case of the Ding-Iohara quantum algebra $U_{q}(g,X_{l})$ with structure
function $g$ set to be certain theta functions. Note that we should take care
of the Serre-type relations in the case $l\geq 2$.

Although the algebra structure of the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ has a definite sense, the situation seems to
be complicated on the coalgebra structure. As far as we understand, there are
two coalgebra structures in literature.

  * •

The paper [J+99] gave the quasi-Hopf formulation of
$U_{q,p}(\widehat{\mathfrak{g}})$ based on the Drinfeld realization of the
quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$. Hence the quasi-Hopf
algebra $U_{q,p}(\widehat{\mathfrak{g}})$ has a comultiplication induced by
the Drinfeld comultiplication on $U_{q}(\widehat{\mathfrak{g}})$. We call the
induced comultiplication $\Delta$ on $U_{q,p}(\widehat{\mathfrak{g}})$ the
Drinfeld-type comultiplication. Note that this comultiplication $\Delta$ is
not coassociative in terms of the ordinary tensor product $\otimes$.

  * •

After a decade, another coalgebra structure $\Delta^{\prime}$ on
$U_{q,p}(\widehat{\mathfrak{g}})$ was introduced for the case
$\mathfrak{g}=\mathfrak{sl}_{2}$ by Konno [Ko09], and for general
$\mathfrak{g}$ by Farghly-Konno-Oshima [FKO14]. They used the notion of
$\mathfrak{h}$-Hopf algebroids in the sense of Etingof and Varchenko [EV98],
which will be reviewed in § 1.1. Here we only remark that one uses the
modified tensor product $\mathbin{\widetilde{\otimes}}$ in the Hopf algebroid
formulation. The comultiplication $\Delta^{\prime}$ can be written neatly in
terms of the $L$-operators.

One of the motivations of this note is to investigate the Drinfeld-type
coalgebra structure $\Delta$ of the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ in terms of the $\mathfrak{h}$-Hopf
algebroid formulation.

#### The goal of this note

At this point, it is natural to ask the following questions.

  * •

Is there an extension $U_{q}(g,X_{l})$ of the Ding-Iohara quantum algebra for
a simple Lie algebra $\mathfrak{g}$ of _non-simply-laced type_
$X_{l}=B_{l},C_{l},F_{4}$ or $G_{2}$ so that the quantum affine algebra
$U_{q}(\widehat{\mathfrak{g}})$ of the same type $X_{l}$ is a special case of
$U_{q}(g,X_{l})$ with structure functions $g$ set to be some rational
functions?

  * •

Is there an algebra $U_{q,p}(g,X_{l})$ which extends the above
$U_{q}(g,X_{l})$, has the additional elliptic parameter $p$, and gives the
elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ of the same type $X_{l}$ if
the structure functions $g$ set to be some theta functions?

  * •

Does the algebra $U_{q,p}(g,X_{l})$ admit a Hopf algebroid structure with
Drinfeld-type comultiplication?

The goal of this note is to introduce a family of Hopf algebroids
$U_{q,p}(g,X_{l})$ which gives a simultaneous answer to the above questions.

###  0.2. Results and organization

#### Summary of results

As stated above, the goal of this note is to introduce a family of Hopf
algebroids which unifies the elliptic algebras
$U_{q,p}(\widehat{\mathfrak{g}})$ and the Ding-Iohara quantum algebras
$U_{q}(g,A_{l})$. For this, we will construct a Hopf algebroid
$U_{q,p}(g,X_{l})$ depending on

  * •

a complex parameter $q$ (_quantum parameter_),

  * •

a formal parameter $p$ (_elliptic nome_),

  * •

a finite root system of type $X_{l}$,

  * •

a set $g=\\{g_{ij}(z;p)\mid i,j=1,\dotsc,l\\}$ of functions depending on $p$
(_structure functions_),

and require the following conditions.

  1. 1

Specializing the structure functions $g$ appropriately, we recover the
elliptic algebras $U_{q,p}\bigl{(}\widehat{\mathfrak{g}}(X_{l})\bigr{)}$ with
the Drinfeld-type comultiplication $\Delta$.

  2. 2

In the case $X_{l}=A_{l}$, we recover the Ding-Iohara quantum algebra
$U_{q}(\overline{g},A_{l})$ by taking the limit $p\to 0$ in $U_{q,p}(g,A_{l})$
and setting the structure functions to be $\overline{g}\coloneqq\lim_{p\to
0}g$.

Let us give a summary of the results obtained in this note. We will introduce
the following three objects.

  * •

_The dynamical Ding-Iohara algebroid $U_{q,p}(g,X_{l})$_, a Hopf algebroid
satisfying the above conditions 1 and 2. The generators are (0.1.2), the same
as the elliptic algebra $U_{q}(\widehat{\mathfrak{g}})$:

$\operatorname{\mathcal{M}}(\widetilde{H}^{*}),\ q^{\pm c/2},\ d,\
K^{\pm}_{i},\ e_{i}(z),\ f_{i}(z),\ \psi^{\pm}_{i}(z)$

The precise definitions are given in the following places.

    * –

Definitions 2.1.2, 2.2.1, 2.2.2, 2.2.4 and 2.2.7 for simply-laced type
$X_{l}=A_{l},D_{l},E_{6,7,8}$

    * –

Definition 3.2.1 and Theorem 3.2.4 for type $X_{l}=B_{l},C_{l},F_{4}$.

    * –

Definition 3.3.1 and Theorem 3.3.4 for type $G_{2}$.

  * •

A Hopf algebroid structure on the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ whose comultiplication is the Drinfeld-type
comultiplication $\Delta$, given in Proposition 1.3.9.

  * •

The Ding-Iohara quantum algebra $U_{q}(g,X_{l})$ of non-simply-laced type
$X_{l}=B_{l},C_{l},F_{4}$ or $G_{2}$, given in Definition 3.4.1 and
Proposition 3.4.2.

The first and third objects are definitely new, and the first one is the main
object of this note. The second object might be known already, although we
cannot find it in literature. The main statements of this note are the check
of the conditions 1 and 2 for the dynamical Ding-Iohara algebroid
$U_{q,p}(g,X_{l})$. Note that 2 makes sense for general $X_{l}$ after the
extension of Ding-Iohara quantum algebras. The check are given in:

  * •

Corollary 2.2.9, and Proposition 2.3.1 for type $X_{l}=A_{l},D_{l},E_{6,7,8}$.

  * •

Proposition 3.2.3 and Proposition 3.4.3 for type $X_{l}=B_{l},C_{l},F_{4}$.

  * •

Proposition 3.3.3 and Proposition 3.4.3 for type $G_{2}$.

Our Hopf algebroid $U_{q,p}(g,X_{l})$ is an example of $H$-Hopf algebroid in
the sense of Etingof-Varchenko [EV98, Chap. 4], where $H$ is a certain finite-
dimensional commutative Lie algebra constructed from the root system of type
$X_{l}$ (see (2.1.1)). Let us also mention that $H$-Hopf algebroids form a
subclass of _dynamical Hopf algebroids_ introduced in [EV98, Chap. 5]. Thus,
our $U_{q,p}(g,X_{l})$ is a dynamical Hopf algebroid, as indicated in the
title and abstract of this note.

There is a non-trivial point in the algebra structure of our dynamical Ding-
Iohara algebroid $U_{q,p}(g,X_{l})$: the Serre-type relations. To spell out
the point, let us cite from Definition 2.1.2 the Serre-type relations in the
case of simply-laced type $X_{l}=A_{l},D_{l},E_{6,7,8}$. For the generating
currents $e_{i}(z),e_{j}(z)$ with the Cartan matrix entry $a_{ij}=-1$, the
Serre-type relation is given by (2.1.20):

$\displaystyle\begin{split}&\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{1})e_{i}(z_{2})e_{j}(z)-h_{ij}\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})e_{i}(z_{1})e_{j}(z)e_{i}(z_{2})\\\
&+\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})e_{j}(z)e_{i}(z_{1})e_{i}(z_{2})+\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{2})e_{i}(z_{1})e_{j}(z)\\\
&-h_{ij}\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{2})e_{j}(z)e_{i}(z_{1})+e_{j}(z)e_{i}(z_{2})e_{i}(z_{1})=0.\end{split}$
(0.2.1)

Here $\widehat{g}^{*}_{ij}(z)$ is determined from the structure function
$g_{ij}(z;p)$, and the function $h_{ij}$ is given by (2.1.19):

$\displaystyle
h_{ij}\coloneqq\frac{(\overline{g}_{ii}(\frac{z_{1}}{z_{2}})+1)(\overline{g}_{ij}(\frac{z_{1}}{z})\overline{g}_{ij}(\frac{z_{2}}{z})+1)}{\overline{g}_{ij}(\frac{z_{2}}{z})+\overline{g}_{ii}(\frac{z_{1}}{z_{2}})\overline{g}_{ij}(\frac{z_{1}}{z})},$

which is essentially equal to (0.1.1). The Serre-type relation (0.2.1) is
guessed from (1.2.18) of the Ding-Iohara algebra $U_{q}(g,X_{l})$ and (1.3.33)
of the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$. If we specialize
the structure function $g_{ij}(z;p)$ to the theta function as (2.1.22), then
$h_{ij}$ is equal to the $q$-binomial $\genfrac{[}{]}{0.0pt}{}{2}{1}_{q}$, and
the above Serre relation (2.1.20) coincides with (1.3.33). Thus the dynamical
Ding-Iohara algebroid under the specialization coincides with the elliptic
algebra $U_{q,p}(\widehat{\mathfrak{g}})$ (see Proposition 2.1.5).

However, in the case of non-simply-laced type $X_{l}=B_{l},C_{l},F_{4},G_{2}$
explained in § 3, we need special care since there seems no definition of the
Ding-Iohara algebra of type $X_{l}$ in literature. To find out correct Serre-
type relations for non-simply-laced types, we build certain principles to find
an admissible form of the function $h_{ij}$ for the Cartan matrix entry
$a_{ij}=-2,-3$, and the result is summarized in the “strange formulas” in §
3.1. Due to the behavior of the obtained formulas, we need to divide the
argument into the two cases: type $B_{l},C_{l},F_{4}$ and type $G_{2}$. The
former case is treated in § 3.2, and the latter is in § 3.3. The Serre-type
relation of the generating currents $e_{i}(z),e_{j}(z)$ with the Cartan matrix
entry $a_{ij}=-2$ in the type $X_{l}=B_{l},C_{l},F_{4}$ is given by (3.2.3) in
Definition 3.2.1:

$\displaystyle\Phi_{2}(z;z_{1},z_{2},z_{3})+\widetilde{g}^{*}_{ii}(z_{12})\Phi_{2}(z;z_{2},z_{1},z_{3})$
$\displaystyle+\widetilde{g}^{*}_{ii}(z_{12})\widetilde{g}^{*}_{ii}(z_{13})\Phi_{2}(z;z_{2},z_{3},z_{1})+\widetilde{g}^{*}_{ii}(z_{23})\Phi_{2}(z;z_{1},z_{3},z_{2})$
$\displaystyle+\widetilde{g}^{*}_{ii}(z_{13})\widetilde{g}^{*}_{ii}(z_{23})\Phi_{2}(z;z_{3},z_{1},z_{2})+\widetilde{g}^{*}_{ii}(z_{12})\widetilde{g}^{*}_{ii}(z_{13})\widetilde{g}^{*}_{ii}(z_{23})\Phi_{2}(z;z_{3},z_{2},z_{1})=0.$

Here $\Phi_{2}$ is given by (3.2.2):

$\displaystyle\Phi_{2}(z;z_{1},z_{2},z_{3})\coloneqq\ $
$\displaystyle\widetilde{g}^{*}_{ij}(z_{10})\widetilde{g}^{*}_{ij}(z_{20})\widetilde{g}^{*}_{ij}(z_{30})e_{i}(z_{3})e_{i}(z_{2})e_{i}(z_{1})e_{j}(z)$
$\displaystyle-h^{3}_{ij}\widetilde{g}^{*}_{ij}(z_{20})\widetilde{g}^{*}_{ij}(z_{30})e_{i}(z_{3})e_{i}(z_{2})e_{j}(z)e_{i}(z_{1})$
$\displaystyle+h^{3}_{ij}\widetilde{g}^{*}_{ij}(z_{30})e_{i}(z_{3})e_{j}(z)e_{i}(z_{2})e_{i}(z_{1})-e_{j}(z)e_{i}(z_{3})e_{i}(z_{2})e_{i}(z_{1})$

with the function
$h^{3}_{ij}\coloneqq\bigl{(}\widehat{g^{123}_{000}}-\widehat{1}\bigr{)}/\bigl{(}\widehat{g^{23}_{00}}-\widehat{g^{3}_{0}}\bigr{)}$
given in (3.1.13). For the Serre-type relations of type $G_{2}$, there involve
two functions $h^{4,1}_{ij}$ and $h^{4,2}_{ij}$, given in (3.3.1). See
Definition 3.3.1 for the detail.

There is one problem on these Serre-type relations which remains unclear. For
all the type $X_{l}$ except $G_{2}$, our Serre-type relations are “unique” in
the sense that there is no other cubic or quartic relation satisfying the
following two conditions.

  * •

It respects the Drinfeld-type comultiplication in Lemma 2.2.2.

  * •

It reduces to the Serre-type relation of the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ under the specialization of structure
functions $g$ to the theta functions (2.1.22).

See Remarks 2.2.3 and 3.2.5 for the detail. However, for the type $G_{2}$, the
compatibility of the Drinfeld-type comultiplication only yields the vanishing
condition on a certain linear combination of the functions $h^{4,1}_{ij}$ and
$h^{4,2}_{ij}$, and there remains freedom of choice of these functions (the
ideal $J$ in (3.3.1)). This problem will be mentioned in Remark 3.3.5 and §
4.

#### Organization

Let us explain the organization of this note. The starting § 1 is a
preliminary part, consisting of three parts. The first part is § 1.1, where
we explain the notion of $\mathfrak{h}$-Hopf algebroids in the sense of
Etingof and Varchenko [EV98]. We also introduce the basic terminology of Hopf
algebras. The second part is § 1.2, where we review the Ding-Iohara quantum
algebras $U_{q}(g,A_{l})$ [DI97b]. The last part § 1.3 gives a review of the
elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$. We also give in
Proposition 1.3.9 a Hopf algebroid structure on
$U_{q,p}(\widehat{\mathfrak{g}})$ with Drinfeld-type comultiplication
$\Delta$. Some part of the verification of Proposition 1.3.9 will be given in
the following sections.

In § 2, we introduce the dynamical Ding-Iohara algebroid $U_{q,p}(g,X_{l})$
for simply-laced type $X_{l}=A_{l},D_{l}$ or $E_{6,7,8}$. In § 2.1, we give
the algebra structure on $U_{q,p}(g,X_{l})$ in Definition 2.1.2, and show that
the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is a specialization of
$U_{q,p}(g,X_{l})$ in Proposition 2.1.5. Then, in § 2.2, we give the $H$-Hopf
algebroid structure, and check in Corollary 2.2.9 that the special case
$U_{q,p}(\widehat{\mathfrak{g}})$ recovers the Drinfeld-type comultiplication
$\Delta$ mentioned above. Finally, in § 2.3, we check that the $p\to 0$ limit
of $U_{q,p}(g,X_{l})$ recovers the general Ding-Iohara algebra
$U_{q}(\overline{g},X_{l})$ with structure functions
$\overline{g}\coloneqq\lim_{p\to 0}g$.

In § 3, we extend our Hopf algebroid $U_{q,p}(g,X_{l})$ to non-simply-laced
types $X_{l}=B_{l},C_{l},F_{4}$ and $G_{2}$. After the search of possible
Serre-type relations explained in § 3.1, we treat the former case in § 3.2
and the latter in § 3.3. The organization of each subsection is similar to §
2.1 and § 2.2, and we omit the explanation. The limit $p\to 0$ is treated in
§ 3.4, where we first introduce the Ding-Iohara quantum algebra for non-
simply-laced type $X_{l}$ in Definition 3.4.1 and Proposition 3.4.2, and then
check that the condition 2 holds.

We close this note by concluding remarks in § 4

#### Global notations

Here are the notations and terminology used throughout in this paper.

  * •

We denote by $\mathbb{N}=\mathbb{Z}_{\geq 0}\coloneqq\\{0,1,2,\ldots\\}$ the
set of non-negative integers.

  * •

We denote by $\delta_{i,j}$ the Kronecker delta on a set $I\ni i,j$.

  * •

The symbol $\operatorname{diag}(x_{1},\dotsc,x_{n})$ denotes the diagonal
matrix of size $n$ with diagonal entries $x_{1},\dotsc,x_{n}$.

  * •

The word ‘ring’ or ‘algebra’ means a unital associative one unless otherwise
stated. The unit of a ring $R$ is denoted by $1_{R}$.

  * •

For a commutative ring $k$, we use the word “an algebra over $k$” and do not
use the word “a $k$-algebra” to distinguish it from “an
$\mathfrak{h}$-algebra” for a finite-dimensional Lie algebra $\mathfrak{h}$
which will be explained in § 1.1.

  * •

For a commutative ring $k$ and a variable $x$, we denote the Laurent
polynomial ring by $k[x^{\pm 1}]$, the formal series ring by $k\llbracket
x\rrbracket$, the Laurent series ring by $k(\\!(x)\\!)$, and the module of
formal Laurent series by $k\llbracket x^{\pm 1}\rrbracket$.

  * •

We use Gasper and Rahman’s notation [GR04] for $q$-shifted factorials

$\displaystyle(x;q)_{\infty}\coloneqq\prod_{n=0}^{\infty}(1-xq^{n}),\quad(x_{1},\dotsc,x_{n};q)_{\infty}\coloneqq\prod_{i=1}^{n}(x_{i};q)_{\infty}.$
(0.2.2)

  * •

We denote by $\mathfrak{S}_{n}$ the symmetric group of degree $n$.

##  1\. Recollections on Ding-Iohara quantum algebras and elliptic algebras

###  1.1. Hopf algebroid

We begin with the definition of $\mathfrak{h}$-Hopf algebroid for a finite-
dimensional Lie algebra $\mathfrak{h}$ in the sense of Etingof and Varchenko
[EV98, Chap. 4].

Let $\mathfrak{h}$ be a finite-dimensional commutative Lie algebra over the
complex number field $\mathbb{C}$. We denote the field
$\operatorname{\mathcal{M}}(\mathfrak{h}^{*})$ of meromorphic functions on the
linear dual $\mathfrak{h}^{*}$, regarded as a complex affine space, by

$\mathbb{F}\coloneqq\operatorname{\mathcal{M}}(\mathfrak{h}^{*}).$

An algebra $A$ over $\mathbb{C}$ is called an _$\mathfrak{h}$ -algebra_ if it
is equipped with an $\mathfrak{h}^{*}$-bigrading
$A=\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}A_{\alpha,\beta}$, and with two
algebra embeddings $\mu_{l},\mu_{r}\colon\mathbb{F}\to A_{0,0}$, called _the
left_ and _right moment maps_ , such that

$\displaystyle\mu_{l}(F)a=a\mu_{l}(T_{\alpha}F),\quad\mu_{r}(F)a=a\mu_{r}(T_{\beta}F)\quad(F\in\mathbb{F},\,a\in
A_{\alpha,\beta}),$

where $T_{\alpha}$ is the shift operator

$(T_{\alpha}F)(x)\coloneqq F(x+\alpha)\quad(x\in\mathfrak{h}^{*}).$

An _$\mathfrak{h}$ -algebra morphism $A\to B$_ is an algebra homomorphism
consistent with the moment maps. Such a morphism automatically preserves the
$\mathfrak{h}^{*}$-bigrading.

We define _the modified tensor product_ of two $\mathfrak{h}$-algebras $A$ and
$B$ to be the $\mathfrak{h}$-algebra

$\displaystyle\bigl{(}A\mathbin{\widetilde{\otimes}}B,\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}(A\mathbin{\widetilde{\otimes}}B)_{\alpha,\beta},\mu_{l}^{A\mathbin{\widetilde{\otimes}}B},\mu_{r}^{A\mathbin{\widetilde{\otimes}}B}\bigr{)}$
(1.1.1)

as follows. We set

$\displaystyle
A\mathbin{\widetilde{\otimes}}B\coloneqq\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}(A\mathbin{\widetilde{\otimes}}B)_{\alpha,\beta},\quad(A\mathbin{\widetilde{\otimes}}B)_{\alpha,\beta}\coloneqq\bigoplus_{\gamma\in\mathfrak{h}^{*}}A_{\alpha,\gamma}\otimes_{\mathbb{F}}B_{\gamma,\beta},$
(1.1.2)

where $A_{\alpha,\gamma}\otimes_{\mathbb{F}}B_{\gamma,\beta}$ is the quotient
of the standard tensor product
$A_{\alpha,\gamma}\otimes_{\mathbb{C}}B_{\gamma,\beta}$ of linear spaces over
$\mathbb{C}$ by the linear subspace generated by the elements

$\displaystyle\mu^{A}_{r}(F)a\otimes b-a\otimes\mu^{B}_{l}(F)b\quad(a\in
A,\,b\in B,\,F\in\mathbb{F}).$ (1.1.3)

We denote an element of $A\mathbin{\widetilde{\otimes}}B$ by
$a\mathbin{\widetilde{\otimes}}b$ for $a\in A$ and $b\in B$. The modified
tensor product $A\mathbin{\widetilde{\otimes}}B$ is again an
$\mathfrak{h}$-algebra by the multiplication

$(a\mathbin{\widetilde{\otimes}}b)(a^{\prime}\mathbin{\widetilde{\otimes}}b^{\prime})\coloneqq(aa^{\prime})\mathbin{\widetilde{\otimes}}(bb^{\prime})$

and the moment maps

$\displaystyle\mu^{A\mathbin{\widetilde{\otimes}}B}_{l}(F)\coloneqq\mu_{l}^{A}(F)\mathbin{\widetilde{\otimes}}1_{B},\quad\mu^{A\mathbin{\widetilde{\otimes}}B}_{r}(F)\coloneqq
1_{A}\mathbin{\widetilde{\otimes}}\mu_{r}^{B}(F)\quad(F\in\mathbb{F}).$
(1.1.4)

We also have the natural notion of _the modified tensor product
$f\mathbin{\widetilde{\otimes}}g$ of $\mathfrak{h}$-algebra homomorphisms $f$
and $g$_, whose detail is omitted.

The category of $\mathfrak{h}$-algebras has a monoidal structure whose tensor
product is the above $\mathbin{\widetilde{\otimes}}$ and the unit object is
_the $\mathfrak{h}$-algebra $D_{\mathfrak{h}}$ of difference operators on the
meromorphic function field
$\mathbb{F}=\operatorname{\mathcal{M}}(\mathfrak{h}^{*})$_. As a
$\mathbb{C}$-linear space, $D_{\mathfrak{h}}$ is defined to be the subspace of
$\mathbb{C}$-linear endomorphisms
$\operatorname{End}_{\mathbb{C}}(\mathbb{F})$ spanned by the operators of the
form $fT_{\alpha}$ with $f\in\mathbb{F}$ and $\alpha\in\mathfrak{h}^{*}$. It
is a subalgebra of $\operatorname{End}_{\mathbb{C}}(\mathbb{F})$, and has the
$\mathfrak{h}^{*}$-bigrading

$\displaystyle
D_{\mathfrak{h}}=\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}(D_{\mathfrak{h}})_{\alpha,\beta},\quad(D_{\mathfrak{h}})_{\alpha,\beta}\coloneqq\begin{cases}\\{fT_{-\alpha}\mid
f\in\mathbb{F}\\}&(\alpha=\beta)\\\ 0&(\alpha\neq\beta)\end{cases}.$ (1.1.5)

Note that
$(D_{\mathfrak{h}})_{0,0}=\mathbb{F}T_{0}=\mathbb{F}\operatorname{id}_{\mathbb{F}}$.
The moment maps are given by

$\mu_{l}=\mu_{r}\colon\mathbb{F}\longrightarrow(D_{\mathfrak{h}})_{0,0}=\mathbb{F}T_{0},\quad
f\longmapsto fT_{0}.$

These data give $D_{\mathfrak{h}}$ a structure of an $\mathfrak{h}$-algebra,
and it is a monoidal unit with respect to the modified tensor product
$\mathbin{\widetilde{\otimes}}$ [EV98, Lemma 4.1].

An _$\mathfrak{h}$ -bialgebroid_ is a triple $(A,\Delta,\varepsilon)$
consisting of an $\mathfrak{h}$-algebra $A$ and two $\mathfrak{h}$-algebra
homomorphisms $\Delta\colon A\to A\mathbin{\widetilde{\otimes}}A$ and
$\varepsilon\colon A\to D_{\mathfrak{h}}$ satisfying _the coassociativity and
counit axioms_

$\displaystyle\begin{split}(\Delta\mathbin{\widetilde{\otimes}}\operatorname{id}_{A})\Delta=(\operatorname{id}_{A}\mathbin{\widetilde{\otimes}}\Delta)\Delta&:A\longrightarrow(A\mathbin{\widetilde{\otimes}}A)\mathbin{\widetilde{\otimes}}A\cong
A\mathbin{\widetilde{\otimes}}(A\mathbin{\widetilde{\otimes}}A),\\\
(\varepsilon\mathbin{\widetilde{\otimes}}\operatorname{id}_{A})\Delta=\operatorname{id}_{A}=(\operatorname{id}_{A}\mathbin{\widetilde{\otimes}}\varepsilon)\Delta&:A\longrightarrow
D_{\mathfrak{h}}\mathbin{\widetilde{\otimes}}A\cong A\cong
A\mathbin{\widetilde{\otimes}}D_{\mathfrak{h}},\end{split}$ (1.1.6)

where $\mathbin{\widetilde{\otimes}}$ denotes the modified tensor product of
$\mathfrak{h}$-algebra homomorphisms, and $\cong$’s denote the natural
isomorphisms. The map $\Delta$ is called _the comultiplication_ , and
$\varepsilon$ is called _the counit_ of the $\mathfrak{h}$-bialgebroid $A$.

Let $(A,\Delta,\varepsilon)$ be an $\mathfrak{h}$-bialgebroid. A linear map
$S\in\operatorname{End}_{\mathbb{C}}A$ is called an _antipode_ if the
following relations hold for $f\in\mathbb{F}$ and $a\in A_{\alpha,\beta}$.

$\displaystyle\begin{split}S(\mu_{r}(f)a)=S(a)\mu_{l}(f),\quad&S(a\mu_{l}(f))=\mu_{r}(f)S(a),\\\
m(\operatorname{id}_{A}\otimes
S)\Delta(a)=\mu_{l}(\varepsilon(a)1),\quad&m(S\otimes\operatorname{id}_{A})\Delta(a)=\mu_{r}\bigl{(}T_{\alpha}(\varepsilon(a)1)\bigr{)},\end{split}$
(1.1.7)

where $\varepsilon(a)1$ denotes the action of the operator $\varepsilon(a)\in
D_{\mathfrak{h}}$ on the constant function $1$, and $\otimes$ in the second
line denotes the ordinary tensor product of linear maps over $\mathbb{C}$. An
$\mathfrak{h}$-bialgebroid equipped with an antipode is called an
_$\mathfrak{h}$ -Hopf algebroid_.

Note that for $\mathfrak{h}=0$, the zero-dimensional commutative Lie algebra,
we have $\mathbb{F}=\mathbb{C}$ and $\mathfrak{h}$-algebra is nothing but an
algebra $A$ over $\mathbb{C}$ whose moment maps are the unit
$\eta\colon\mathbb{C}\to A$. Then the modified tensor product
$\mathbin{\widetilde{\otimes}}$ is equal to the standard tensor product
$\otimes_{\mathbb{C}}$ and $D_{\mathfrak{h}}=\mathbb{C}$. Moreover, an
$\mathfrak{h}$-bialgebroid $(A,\Delta,\varepsilon)$ is a bialgebra over
$\mathbb{C}$ with comultiplication $\Delta\colon A\to A\otimes A$ and counit
$\varepsilon\colon A\to\mathbb{C}$, and an $\mathfrak{h}$-Hopf algebroid
$(A,\Delta,\varepsilon,S)$ is a Hopf algebra with antipode $S$. Thus, an
ordinary Hopf algebra is a special case of $\mathfrak{h}$-Hopf algebroid.

To close this § 1.1, let us mention the relation of
$\mathfrak{h}$-bialgebroids and Hopf algebroids in the sense of [L96, X01]. As
far as we know, what is today commonly called a Hopf algebroid means the one
introduced in loc. cit. Our presentation follows [X01, §3].

A Hopf algebroid is a tuple $(A,R,\alpha,\beta,m,\Delta,\varepsilon)$
consisting of:

  1. 1

An algebra $(A,m)$.

  2. 2

Another algebra $R$.

  3. 3

An algebra homomorphism $\alpha\colon R\to A$ and an algebra anti-homomorphism
$\beta\colon R\to A$ such that the images of $\alpha$ and $\beta$ commute in
$A$.

At this point, the algebra $A$ has an $R$-$R$-bimodule structure with
$r.a\coloneqq\alpha(r)a$ and $a.r\coloneqq\beta(r)a$ for $a\in A$ and $r\in
R$. In particular, we have the $R$-$R$-bimodule product $A\otimes_{R}A$, which
admits a natural $R$-$R$-bimodule structure.

  1. 4

An $R$-$R$-bimodule homomorphism $\Delta\colon A\to A\otimes_{R}A$.

  2. 5

An $R$-$R$-bimodule homomorphism $\varepsilon\colon A\to R$.

These should satisfy the following conditions.

  1. (1)

The map $\Delta$ satisfies $\Delta(1_{A})=1_{A}\otimes 1_{A}$ and
$(\Delta\otimes_{R}\operatorname{id}_{A})=(\operatorname{id}_{A}\otimes_{R}\Delta)$
under the natural isomorphism $(H\otimes_{R}H)\otimes_{R}H\cong
H\otimes_{R}(H\otimes_{R}H)$.

  2. (2)

The maps $m$ and $\Delta$ are compatible in the sense that
$\Delta(a)\bigl{(}\beta(r)\otimes 1_{A}-1_{A}\otimes\alpha(r)\bigr{)}=0$ for
any $r\in R$ and $a\in A$, and that $\Delta(ab)=\Delta(a)\Delta(b)$ for any
$a,b\in A$.

  3. (3)

The map $\varepsilon$ satisfies $\varepsilon(1_{A})=1_{R}$ and
$(\varepsilon\otimes_{R}\operatorname{id}_{A})\Delta=\operatorname{id}_{A}=(\operatorname{id}_{A}\otimes_{R}\varepsilon)\Delta$
under the natural isomorphisms $R\otimes_{R}A\cong A\cong A\otimes_{R}R$.

An $\mathfrak{h}$-bialgebroid $(A,\Delta,\varepsilon)$ is an Hopf algebroid in
the above sense by setting
$R\coloneqq\mathbb{F}=\operatorname{\mathcal{M}}(\mathfrak{h})$ and
$\alpha\coloneqq\mu_{l}$, $\beta\coloneqq\mu_{R}$. In fact, $\mathbb{F}$ is
commutative so that the commuting condition in (iii) holds. Then we find that
$A\otimes_{\mathbb{F}}A\cong A\mathbin{\widetilde{\otimes}}A$ (see (1.1.3)).
The conditions (1)–(3) are checked immediately.

###  1.2. Ding-Iohara quantum algebras

Here we give an overview of the quantum algebras introduced by Ding and Iohara
in [DI97b].

For $l\in\mathbb{Z}_{>0}$, let $(a_{ij})_{i,j=1}^{l}$ be the Cartan matrix of
type $A_{l}$, i.e.,

$a_{ij}=\begin{cases}2&(i=j)\\\ {-1}&(\left|i-j\right|=1)\\\
0&(\text{otherwise})\end{cases}.$

We will also denote it as $(a_{ij})_{i,j\in I}$ with
$I\coloneqq\\{1,\dotsc,l\\}$.

Also, we denote

$\displaystyle\mathcal{A}\coloneqq\\{\text{analytic functions of $z$ without
poles except $z=0$ or $\infty$}\\},$ (1.2.1)

regarded as a commutative algebra over $\mathbb{C}$. Let

$g=\\{g_{ij}(z)\mid i,j\in I\\}$

be a set of analytic functions satisfying the following condition.

  1. 1

There are some $G_{ij}^{\pm}(z)\in\mathcal{A}$ such that

$\displaystyle g_{ij}(z)=G_{ij}^{+}(z)/G_{ij}^{-}(z).$ (1.2.2)

  2. 2

We have

$\displaystyle g_{ij}(z)=g_{ji}(z^{-1})^{-1}.$ (1.2.3)

We call the set $g$ _the structure functions_ , and call the condition on $g$
_the Ding-Iohara condition_. Note that the condition
$G^{\pm}_{ij}(z)\in\mathcal{A}$ implies that we can expand $G^{\pm}_{ij}(z)$
and $g_{ij}(z)=G_{ij}^{+}(z)/G_{ij}^{-}(z)$ as

$\displaystyle
G^{\pm}_{ij}(z)=\sum_{n=-L^{\pm}}^{M^{\pm}}G^{\pm}_{ij,n}z^{n}\in\mathbb{C}[z^{\pm
1}],\quad g_{ij}(z)^{\pm
1}=\sum_{n=-N^{\pm}}^{\infty}g_{ij,n}z^{n}\in\mathbb{C}(\\!(z)\\!)$ (1.2.4)

for some $L^{\pm},M^{\pm},N^{\pm}\in\mathbb{N}$.

Let also $q$ be a complex number such that $\left|q\right|\neq 0,1$, called
_the quantum parameter_. Using these data $A,g$ and $q$, we define the
topological algebra $U_{q}(g,A_{l})$ as follows. See Remark 1.2.2 for the
meaning of the phrase “topological”.

######  Definition 1.2.1.

Using the Cartan matrix $A=(a_{ij})_{i,j\in I}$ of type $A_{l}$, the structure
functions $g$ and the quantum parameter $q$, we define

$U=U_{q}(g,A_{l}),$

to be the topological algebra over $\mathbb{C}$ generated by the elements

$\displaystyle q^{\pm c/2},\ e_{i,n},\ f_{i,n},\ \psi^{+}_{i,n},\
\psi^{-}_{i,n}\quad(i\in I,\,n\in\mathbb{Z})$ (1.2.5)

with the following defining relations. We will use the _generating currents_

$\displaystyle e_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}e_{i,n}z^{-l},\quad
f_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}f_{i,n}z^{-l},\quad\psi^{\pm}_{i}(z)\coloneqq\sum_{n\in\mathbb{Z}}\psi^{\pm}_{i,n}z^{-n},$
(1.2.6)

where $z$ is a formal variable.

  * •

The generators $q^{\pm c/2}$ satisfy that

$q^{\pm c/2}$ are central and $q^{c/2}q^{-c/2}=1$. (1.2.7)

Below we denote $q^{rc/2}\coloneqq(q^{c/2})^{r}$ for $r\in\mathbb{Z}$.

  * •

The generating currents $\psi^{\pm}(z)$ satisfy that

$\psi^{\pm}(z)$ are invertible as series of $z$, (1.2.8)

which means that $\psi^{\pm}(z)$ are bounded above or bounded below, and the
initial terms are invertible in $U$. Explicitly, in the case $\psi^{+}(z)$ is
bounded above, we have

there is some $N\in\mathbb{Z}$ such that $\psi^{+}_{i,n}=0$ for $n>N$, and
$\psi^{+}_{i,N}$ is invertible in $U$.

  * •

For any $i,j\in I$, we have in terms of the formal variables $z$ and $w$ that

$\displaystyle\psi^{+}_{i}(z)\psi^{+}_{j}(w)=\psi^{+}_{j}(w)\psi^{+}_{i}(z),\quad$
$\displaystyle\psi^{-}_{i}(z)\psi^{-}_{j}(w)=\psi^{-}_{j}(w)\psi^{-}_{i}(z),$
(1.2.9)
$\displaystyle\psi^{+}_{i}(z)\psi^{-}_{j}(w)\psi^{+}_{i}(z)^{-1}\psi^{-}_{j}(w)^{-1}=\
$ $\displaystyle g_{ij}(q^{-c}z/w)g_{ij}(q^{c}z/w)^{-1},$ (1.2.10)
$\displaystyle\psi^{+}_{i}(z)e_{j}(w)=g_{ij}(q^{-c/2}z/w)e_{j}(w)\psi^{+}_{i}(z),\quad$
$\displaystyle\psi^{+}_{i}(z)f_{j}(w)=g_{ij}(q^{c/2}z/w)^{-1}f_{j}(w)\psi^{+}_{i}(z),$
(1.2.11)
$\displaystyle\psi^{-}_{i}(z)e_{j}(w)=g_{ij}(q^{c/2}z/w)e_{j}(w)\psi^{-}_{i}(z),\quad$
$\displaystyle\psi^{-}_{i}(z)f_{j}(w)=g_{ij}(q^{-c/2}z/w)^{-1}f_{j}(w)\psi^{-}_{i}(z),$
(1.2.12) $\displaystyle
G_{ij}^{-}(z/w)e_{i}(z)e_{j}(w)=G_{ij}^{+}(z/w)e_{j}(w)e_{i}(z),\quad$
$\displaystyle
G_{ij}^{+}(z/w)f_{i}(z)f_{j}(w)=G_{ij}^{-}(z/w)f_{j}(w)f_{i}(z),$ (1.2.13)
$\displaystyle[e_{i}(z),f_{j}(w)]=\frac{\delta_{i,j}}{q-q^{-1}}\bigl{(}\delta(q^{-c}z/w)$
$\displaystyle\psi^{-}_{i}(q^{c/2}w)-\delta(q^{c}z/w)\psi^{+}_{i}(q^{c/2}z)\bigr{)}.$
(1.2.14)

In each equality, we expand both sides as series of $z$ and $w$ using (1.2.4)
and (1.2.6), and compare the coefficients of $z^{r}w^{s}$ for each
$(r,s)\in\mathbb{Z}^{2}$ to get genuine relations. Here we should consider an
appropriate topology for general structure functions $g$, which will be
explained in Remark 1.2.2. Also, in (1.2.14), we used the commutator
$[X,Y]\coloneqq XY-YX$ and the formal delta function

$\displaystyle\delta(x)\coloneqq\sum_{n\in\mathbb{Z}}x^{n}.$ (1.2.15)

  * •

For $i,j\in I$ with $a_{ij}=0$, we have in terms of the formal variables $z$
and $w$ that

$\displaystyle[e_{i}(z),e_{j}(w)]=[f_{i}(z),f_{j}(w)]=0.$ (1.2.16)

The meaning of these equalities are the same as the previous item.

  * •

For $i,j\in I$ such that $a_{ij}={-1}$, we have in terms of the formal
variables $z_{1},z_{2}$ and $w$ _the Serre-type_ (or _cubic_) _relations_

$\displaystyle\begin{split}X_{i}(z_{1})X_{i}(z_{2})X_{j}(w)&-h_{ij}(\tfrac{z_{1}}{w},\tfrac{z_{2}}{w})X_{i}(z_{1})X_{j}(w)X_{i}(z_{2})+X_{j}(w)X_{i}(z_{1})X_{i}(z_{2})\\\
&+(\text{three terms given by $z_{1}\leftrightarrow z_{2}$})=0\end{split}$
(1.2.17)

for $X=e$ or $f$. Here we set

$\displaystyle
h_{ij}(x,y)\coloneqq\frac{(g_{ii}(x/y)+1)(g_{ij}(x)g_{ij}(y)+1)}{g_{ij}(y)+g_{ii}(x/y)g_{ij}(x)},$
(1.2.18)

regarded as a series belonging to $\mathbb{C}(\\!(x,y,x/y)\\!)$ by expanding
$g_{ij}$ and $g_{ii}$ as in (1.2.4).

######  Remark 1.2.2.

Let us comment on the topology of $U$. As mentioned after (1.2.14), the
genuine relations are obtained by expanding the current relations. For
example, the first equality of (1.2.13) yields

$\sum_{k=-L^{-}}^{M^{-}}G_{ij,k}^{-}e_{i,m}e_{j,n}=\sum_{k^{\prime}=-L^{+}}^{M^{+}}G_{ij,k}^{+}e_{j,n+k-k^{\prime}}e_{i,m-k+k^{\prime}}$

for each $m,n\in\mathbb{Z}$, under the expansion (1.2.4). This involves finite
summations only, and we can apply the ordinary diamond lemma [Be78]. On the
other hand, the first equality in (1.2.11) yields

$\displaystyle\psi_{i,m}^{+}e_{j,n}=\sum_{k=-N^{+}}^{\infty}g_{ij,k}q^{-kc/2}e_{j,n-k}\psi^{+}_{i,m+k}$
(1.2.19)

for each $m,n\in\mathbb{Z}$. Thus, unless $\psi^{+}_{i}(z)$ is bounded above
or $e_{j}(z)$ is bounded on both sides, we have an infinite summation in the
defining relation, and cannot obtain a genuine algebra. Largely speaking,
there are two solutions of this problem.

  1. 1

Require $\psi^{+}_{i}(z)$ to be bounded above, and $\psi^{-}_{i}(z)$ to be
bounded below, so that the genuine relations contain finite summations only.
This is the case of the Drinfeld realization of quantum affine algebras [D89],
recalled in Example 1.2.5.

  2. 2

Equip the structure functions $g_{ij}(z)$ with a parameter, say $p$, and
require $g_{ij}(z)=g_{ij}(z;p)$ to have a series expansion
$g_{ij}(z;p)=\sum_{k=-N}^{\infty}g_{ij,k}(z)p^{k}\in\mathbb{C}[z^{\pm
1}](\\!(p)\\!)$. Now we equip $U$ with the $p$-adic topology. Then, since
$g_{ij,k}(z)\in\mathbb{C}[z^{\pm 1}]$, we see that each $p$-adic component of
the relations like (1.2.19) contains finite summation only, Thus we can regard
$U$ as a complete algebra with respect to the $p$-adic topology. We only
consider this situation in the following text.

In a general situation, where no parameters are available, we are not sure how
to resolve the problem. One possible solution may be to use the diamond lemma
adapted to the formal power series [H02], which requires an appropriate choice
of the norm on the formal power series ring.

######  Remark 1.2.3.

There are some typos in the quadratic relations of [DI97b, Definitions
2.1,2.2]. We should define the relations between the generating currents
$\psi^{-}_{i}(z)$ and $e_{j}(w),f_{j}(w)$ as (1.2.12), which are consistent
with those of the Drinfeld realization of quantum affine algebras [D89, §2,
Remark] and of the quantum toroidal algebras of simply-laced type [GKV95, §3].

######  Fact 1.2.4 ([DI97b, Theorem 2.2]).

The topological algebra $U=U_{q}(g,A_{l})$ has the following topological Hopf
algebra structure.

  * •

Formal comultiplication $\Delta$:

$\displaystyle\Delta(q^{\pm c/2})\coloneqq q^{c/2}\otimes q^{\pm c/2},\quad$
(1.2.20) $\displaystyle\Delta(e_{i}(z))\coloneqq e_{i}(z)\otimes
1+\psi_{i}^{+}(q^{c_{1}/2}z)\otimes
e_{i}(q^{c_{1}}z),\quad\Delta(f_{i}(z))\coloneqq 1\otimes
f_{i}(z)+f_{i}(q^{c_{2}}z)\otimes\psi^{-}_{i}(q^{c_{2}/2}z),$ (1.2.21)
$\displaystyle\Delta(\psi^{+}_{i}(z))\coloneqq\psi^{+}_{i}(q^{-c_{2}/2}z)\otimes\psi^{+}_{i}(q^{c_{1}/2}z),\quad\Delta(\psi^{-}_{i}(z))\coloneqq\psi^{-}_{i}(q^{c_{2}/2}z)\otimes\psi^{-}_{i}(q^{-c_{1}/2}z),$
(1.2.22)

where $q^{rc_{1}/2}\coloneqq q^{rc/2}\otimes 1$ and $q^{rc_{2}/2}\coloneqq
1\otimes q^{rc/2}$ for $r\in\mathbb{Z}$. In each equality, we expand both
sides to get the value of $\Delta$ on each generator. See [DI97a, DI97b] for
the detail of such a calculation. Since infinite summations appear in this
calculation, the obtained comultiplication $\Delta$ is called formal. The
infinite sums make sense if we take into account the topology given by a good
gradation on the generators (1.2.5).

  * •

Counit $\varepsilon$:

$\displaystyle\varepsilon(q^{c})\coloneqq
1,\quad\varepsilon(\psi^{\pm}_{i}(z))\coloneqq
1,\quad\varepsilon(e_{i}(z))=\varepsilon(f_{i}(z))=0.$ (1.2.23)

Again, we expand the left hand sides to get the value of $\varepsilon$ on each
generator.

  * •

Formal antipode $S$:

$\displaystyle S(q^{c})\coloneqq q^{c}\otimes q^{c},\quad$ $\displaystyle
S(\psi^{\pm}_{i}(z))\coloneqq\psi^{\pm}_{i}(z)^{-1}$ (1.2.24) $\displaystyle
S(e_{i}(z))\coloneqq-\psi^{+}_{i}(q^{-c/2}z)^{-1}e_{i}(q^{-c}z),\quad$
$\displaystyle S(f_{i}(z))\coloneqq-
f_{i}(q^{-c}z)\psi_{i}^{-}(q^{-c/2}z)^{-1}.$ (1.2.25)

We call this topological Hopf algebra $U_{q}(g,A_{l})$ _the Ding-Iohara
quantum algebra of type $A_{l}$_, and _the Ding-Iohara algebra_ for short.

Let us give an example of the Ding-Iohara algebra.

######  Example 1.2.5.

The motivational example of the algebra $U=U_{q}(g,A_{l})$ is given in [DI97b,
Example 2.1, Case I] by setting the structure functions $g=\\{g_{ij}(z)\mid
i,j\in I\\}$ to be

$\displaystyle
g_{ij}(z)=\frac{G^{+}_{ij}(z)}{G^{-}_{ij}(z)}\coloneqq\frac{q^{-a_{ij}}(1-q^{a_{ij}}z)}{1-q^{-a_{ij}}z}.$

If we quote this $U$ by the relations $\psi^{\pm}_{i,n}=0$ for $n>0$ and
$\psi^{+}_{i,0}\psi^{-}_{i,0}=1$, then the resulting object is isomorphic as a
topological Hopf algebra to the quantum affine algebra
$U_{q}(\widehat{\mathfrak{sl}}_{l+1})$ equipped with Drinfeld
comultiplication. See also [DI97a] for the related discussion.

Let us mention that the quantum toroidal algebra
$U_{q,t}(\mathfrak{gl}_{1,\textup{tor}})$, also called the Ding-Iohara-Miki
algebra, is a version of the Ding-Iohara quantum algebra of type $A_{1}$ in
the above meaning. This quantum algebra is nowadays under extensive studies.
See [F+11, F+09, FT11, M07] for example. The structure function is set to be

$\displaystyle
g(z)\coloneqq\frac{(1-qz)(1-z/t)(1-tz/q)}{(1-z/q)(1-tz)(1-qz/t)}.$ (1.2.26)

The additional parameter $t$ corresponds to the (Hecke) parameter of the
Macdonald symmetric polynomials $P_{\lambda}(x;q,t)$.

######  Remark 1.2.6.

As mentioned in the last paragraph of [DI97b], replacing the root system
$A_{l}$ by the simply-raced system $X_{l}=D_{l}$ or $E_{6,7,8}$, we get a
topological Hopf algebra $U_{q}(g,X_{l})$, which we call _the Ding-Iohara
quantum algebra of the simply-laced type $X_{l}$_.

###  1.3. Elliptic algebras $U_{q,p}(\widehat{\mathfrak{g}})$

Here we explain the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$
associated to each untwisted affine Lie algebra $\widehat{\mathfrak{g}}$. As
mentioned in § ‣ 0\. Introduction, there are many objects named “elliptic
quantum groups” or “elliptic algebras” in literature. The algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ was introduced first in [J+99] as a quasi-
Hopf twist of the quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$
associated to $\widehat{\mathfrak{g}}$ with Drinfeld comultiplication. In the
paper [FKO14, §2], the algebra $U_{q,p}(\widehat{\mathfrak{g}})$ was
reformulated as a Hopf algebroid (see § 1.1), and our explanation mainly
follows this Hopf algebroid formulation.

####  1.3.1. Preliminaries on finite root systems

Let us consider the Cartan matrix $A=(a_{ij})_{i,j=1}^{l}$ of finite type
$X_{l}$ in the sense of Kac [Kac90, Theorem 4.3, §4.8, TABLE Fin]. Hence, we
have $X_{l}=A_{l\geq 1}$, $B_{l\geq 3}$, $C_{l\geq 2}$, $D_{l\geq 4}$,
$E_{6,7,8}$, $F_{4}$ or $G_{2}$ as usual. We denote the index set as

$I\coloneqq\\{1,2,\dotsc,l\\},$

so that we can write $A=(a_{ij})_{i,j\in I}$. For definiteness, we write down
some examples of the Cartan matrix $A=A(X_{l})$ of non-simply-laced type
$X_{l}$. Note that, except for type $G_{2}$, they are transpose of those in
Bourbaki [B68].

$A(B_{4})=\left(\begin{smallmatrix}2&{-1}\\\ {-1}&2&{-1}\\\ &{-1}&2&{-1}\\\
&&{-2}&2\end{smallmatrix}\right),\quad
A(C_{4})=\left(\begin{smallmatrix}2&{-1}\\\ {-1}&2&{-1}\\\ &{-1}&2&{-2}\\\
&&{-1}&2\end{smallmatrix}\right),\quad
A(F_{4})=\left(\begin{smallmatrix}2&{-1}\\\ {-1}&2&{-1}\\\ &{-2}&2&{-1}\\\
&&{-1}&2\end{smallmatrix}\right),\quad
A(G_{2})=\left(\begin{smallmatrix}2&{-1}\\\ {-3}&2\end{smallmatrix}\right).$

Let us take a symmetrization $A=DB$ of the Cartan matrix $A$ [Kac90, §2.3]. In
other words, $B$ is symmetric, $D$ is invertible and diagonal, and $B$ and $D$
have rational entries. We denote the entries of $B$ and $D$ as
$B=(b_{ij})_{i,j\in I}$ and $D^{-1}=\operatorname{diag}(d_{i})_{i\in I}$. Note
that $d_{i}$ is defined to be the entry of the _inverse_ of $D$, so that we
have

$\displaystyle b_{ij}=d_{i}a_{ij},\quad b_{ij}=b_{ji}\quad(i,j\in I),$ (1.3.1)

matching the convention in [J+99, A.1] and [FKO14]. Also, recall from [Kac90,
Remark 2.3] that $D$ is uniquely determined by the condition
${}^{t}\\!(D^{-1}A)=D^{-1}A$ up to a constant factor, where ${}^{t}\\!M$
denotes the transpose of a matrix $M$.

Any choice of $D$ will work in the following discussion (see Remark 1.3.3 (5)
for the detail). For definiteness, we make the following choice. Let
$\widehat{A}=(a_{ij})_{i,j\in\widehat{I}}$, $\widehat{I}\coloneqq\\{0\\}\cup
I$ be the generalized Cartan matrix of affine type $X_{l}^{(1)}$ in the sense
of Kac [Kac90, Theorem 4.3, §4.8, TABLE Aff1]. We have the labels
$a_{0},a_{1},\dotsc,a_{l}$ and colabels
$a_{0}^{\vee},a_{1}^{\vee},\dotsc,a_{l}^{\vee}$ [Kac90, §4.8, §6.1], which are
the sequences of positive coprime integers uniquely determined by the
conditions

$\widehat{A}\,{}^{t}\\!(a_{0},a_{1},\dotsc,a_{l})={}^{t}\\!(0,0,\dotsc,0),\quad(a_{0}^{\vee},a_{1}^{\vee},\dotsc,a_{l}^{\vee})\,\widehat{A}=(0,0,\dotsc,0).$

Then, by [Kac90, (6.1.2)], the diagonal matrix
$\widehat{D}^{-1}\coloneqq\operatorname{diag}(a_{0}^{\vee}/a_{0},a_{1}^{\vee}/a_{1},\dotsc,a_{l}^{\vee}/a_{l})$
gives a symmetrization $\widehat{A}=\widehat{D}\widehat{B}$. Since
$A=\left.\widehat{A}\right|_{I\times I}$, we obtain the symmetrization $A=DB$
with

$D^{-1}=\operatorname{diag}(d_{1},\dotsc,d_{n})=\operatorname{diag}(a_{1}^{\vee}/a_{1},a_{2}^{\vee}/a_{2},\dotsc,a_{l}^{\vee}/a_{l}).$

Explicitly, the entries $d_{i}$ are given as:

$\displaystyle\begin{split}A_{l},D_{l},E_{6,7,8}:&\quad(d_{1},\dotsc,d_{l})=(1,\dotsc,1),\\\
B_{l}:&\quad(d_{1},\dotsc,d_{l})=(1,\dotsc,1,\tfrac{1}{2}),\\\
C_{l}:&\quad(d_{1},\dotsc,d_{l})=(1,\dotsc,1,2),\\\
F_{4}:&\quad(d_{1},\dotsc,d_{4})=(1,1,\tfrac{1}{2},\tfrac{1}{2})\\\
G_{2}:&\quad(d_{1},d_{2})=(1,\tfrac{1}{3}).\end{split}$ (1.3.2)

Note that our choices for types $A,D,E$ and $B,C$ coincide with those in
[FKO14].

We denote by $\mathfrak{g}$ the Kac-Moody Lie algebra associated to
$A=(a_{ij})_{i,j\in I}$, and by $(\mathfrak{h},\Pi,\Pi^{\vee})$ a fixed
realization of $A$ [Kac90, §1.1]. Thus, $\mathfrak{g}$ is a simple finite-
dimensional Lie algebra of type $X_{l}$, $\mathfrak{h}\subset\mathfrak{g}$ is
the Cartan subalgebra with $\dim\mathfrak{h}=l$, and $\Pi=\\{\alpha_{i}\mid
i\in I\\}\subset\mathfrak{h}^{*}$ and $\Pi^{\vee}=\\{\alpha^{\vee}_{i}\mid
i\in I\\}\subset\mathfrak{h}$ are linearly independent indexed subsets of the
dual space $\mathfrak{h}^{*}$ and of $\mathfrak{h}$ satisfying

$\langle\alpha_{i},\alpha^{\vee}_{j}\rangle=a_{ji}\quad(i,j\in I).$

Here
$\langle\cdot,\cdot\rangle\colon\mathfrak{h}^{*}\times\mathfrak{h}\to\mathbb{C}$
denotes the canonical pairing of the linear space $\mathfrak{h}$ and its
linear dual $\mathfrak{h}^{*}$. The elements $\alpha_{i}$’s are called _simple
roots_ , and $\alpha^{\vee}_{i}$’s are called _simple coroots_.

We denote by
$\left(\cdot\,|\,\cdot\right)\colon\mathfrak{h}\times\mathfrak{h}\to\mathbb{C}$
the invariant form of $\mathfrak{g}$ with the normalization

$\displaystyle\left(\alpha^{\vee}_{i}\,|\,\alpha^{\vee}_{j}\right)=b_{ij}\quad(i,j\in
I).$ (1.3.3)

Since $\left(\cdot\,|\,\cdot\right)$ is non-degenerate, it induces a linear
isomorphism

$\displaystyle\nu\colon\mathfrak{h}\xrightarrow{\,\sim\,}\mathfrak{h}^{*}\quad\text{s.t.}\quad\langle\nu(h),h^{\prime}\rangle=\left(h\,|\,h^{\prime}\right)\quad(h,h^{\prime}\in\mathfrak{h}).$
(1.3.4)

The induced bilinear form on $\mathfrak{h}$ is denoted by the same symbol
$\left(\cdot\,|\,\cdot\right)$. We have

$\displaystyle\alpha_{i}=\nu(\alpha^{\vee}_{i}),\quad\left(\alpha_{i}\,|\,\alpha_{j}\right)=b_{ij}\quad(i,j\in
I).$ (1.3.5)

####  1.3.2. Dynamical parameters

We keep the notation in the last § 1.3.1. In particular,
$I=\\{1,\dotsc,l\\}$, and $A=(a_{ij})_{i,j\in I}$ denotes the Cartan matrix of
finite type $X_{l}$. We also fix a realization $(\mathfrak{h},\Pi,\Pi^{\vee})$
of $A$, and denote by $\mathfrak{g}$ the corresponding simple Lie algebra.

Let $R\subset\mathfrak{h}^{*}$ be the root system of $\mathfrak{g}$, which is
a reduced irreducible root system of type $X_{l}$ in the sense of Bourbaki
[B68]. The set $\Pi=\\{\alpha_{i}\mid i\in I\\}\subset R$ gives the simple
roots of $R$, which spans _the root lattice_ $\mathcal{Q}\coloneqq\sum_{i\in
I}\mathbb{Z}\alpha_{i}\subset\mathfrak{h}^{*}$. Similarly, the set
$\Pi^{\vee}=\\{\alpha^{\vee}_{i}\mid i\in I\\}\subset\mathfrak{h}$ gives the
simple coroots of $R$, spanning _the coroot lattice_
$\mathcal{Q}^{\vee}\coloneqq\sum_{i\in
I}\mathbb{Z}\alpha^{\vee}_{i}\subset\mathfrak{h}$. Then _the weight lattice_
$\mathcal{P}\subset\mathfrak{h}^{*}$ is defined to be
$\mathcal{P}\coloneqq\\{\lambda\in\mathfrak{h}^{*}\mid\forall i\in
I,\,\langle\lambda,\alpha_{i}^{\vee}\rangle\in\mathbb{Z}\\}$, where
$\langle\cdot,\cdot\rangle\colon\mathfrak{h}^{*}\times\mathfrak{h}\to\mathbb{C}$
is the canonical pairing as before. We have $\mathcal{P}=\sum_{i\in
I}\mathbb{Z}\varpi_{i}\subset\mathfrak{h}^{*}$, where
$\varpi_{i}\in\mathfrak{h}^{*}$ is _the fundamental weight_ determined
uniquely by the condition
$\langle\varpi_{i},\alpha_{j}^{\vee}\rangle=\delta_{i,j}$ for any $j\in I$.
Recall the well-known inclusions

$\mathcal{Q}\subset\mathcal{P}\subset\mathcal{Q}\otimes_{\mathbb{Z}}\mathbb{Q}\subset\mathfrak{h}^{*}.$

Let us consider a copy of the $\mathbb{C}$-linear space $\mathfrak{h}^{*}$,
regarded as $\mathfrak{h}^{*}=\mathcal{P}\otimes_{\mathbb{Z}}\mathbb{C}$, and
denote it by

$\displaystyle H\coloneqq\\{P_{\mu}\mid\mu\in\mathfrak{h}^{*}\\},$ (1.3.6)

following the notation in [FKO14, Ko20]. Thus we have
$P_{c_{1}\mu_{1}+c_{2}\mu_{2}}=c_{1}P_{\mu_{1}}+c_{2}P_{\mu_{2}}$ for
$c_{1},c_{2}\in\mathbb{C}$ and $\mu_{1},\mu_{2}\in\mathfrak{h}^{*}$. A basis
of $H$ is given by

$\displaystyle P_{\alpha_{1}},\dotsc,P_{\alpha_{l}}.$ (1.3.7)

Next, we introduce the $\mathbb{C}$-linear space $\widetilde{H}$ of dimension
$2l$ spanned by the expressions $P_{i}+h_{i}$ and $P_{i}$ ($i\in I$):

$\displaystyle\widetilde{H}\coloneqq\sum_{i\in
I}\mathbb{C}\bigl{(}P_{i}+h_{i}\bigr{)}+\sum_{i\in I}\mathbb{C}P_{i}.$ (1.3.8)

Following the notation in [FKO14, Ko20], for $h=\sum_{i\in
I}c_{i}\alpha^{\vee}_{i}\in\mathfrak{h}$, we denote $P+h\coloneqq\sum_{i\in
I}c_{i}(P_{i}+h_{i})$. Identifying $(P_{i}+h_{i})-P_{i}=h_{i}$ with
$\alpha^{\vee}_{i}$, we have an isomorphism

$\displaystyle\widetilde{H}\cong H\oplus\mathfrak{h}.$ (1.3.9)

Under this identification, the linear space $\widetilde{H}$ has the basis

$\displaystyle\\{P_{i},h_{i}\mid i\in I\\},\quad P_{i}=P_{\alpha_{i}},\quad
h_{i}\cong\alpha_{i}^{\vee},$ (1.3.10)

which extends the basis (1.3.7) of $H$. The elements $P_{i}$ are called _the
dynamical parameters_.

Hereafter we regard $\widetilde{H}$ as a commutative Lie algebra over
$\mathbb{C}$, and consider the field
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ of meromorphic functions on
the dual space of $\widetilde{H}$ as in § 1.1. We can express an element
$F\in\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ as a function

$\displaystyle F=F(P+h,P)$ (1.3.11)

of variables $P+h=(P_{i}+h_{i})_{i\in I}$ and $P=(P_{i})_{i\in I}$. Then the
value of $F$ at $\mu\in\widetilde{H}^{*}$ can be written as
$F(\mu)=F\bigl{(}\langle\mu,P+h\rangle,\langle\mu,P\rangle\bigr{)}$, where
$\langle\cdot,\cdot\rangle\colon\widetilde{H}^{*}\times\widetilde{H}\to\mathbb{C}$
denotes the canonical pairing. In particular, we regard the elements of
$\widetilde{H}$ as meromorphic functions on $\widetilde{H}^{*}$. We also
denote by

$\displaystyle f(P+h),\,f(P)\ \in\
\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ (1.3.12)

a meromorphic function containing the variables $P+h=(P_{i}+h_{i})_{i\in I}$
only, and one containing the variables $P=(P_{i})_{i\in I}$ only.

Let us also introduce the dual spaces $H^{*}$ and $\widetilde{H}^{*}$ as

$\displaystyle\begin{split}&H^{*}=\\{Q_{\alpha}\mid\alpha\in\mathfrak{h}^{*}\\}=\sum_{i\in
I}\mathbb{C}Q_{i},\quad Q_{i}\coloneqq Q_{\alpha_{i}},\\\
&\widetilde{H}^{*}=\sum_{i\in I}\mathbb{C}(Q_{i}+\alpha_{i})+\sum_{i\in
I}\mathbb{C}Q_{i}.\end{split}$ (1.3.13)

Here we consider $H^{*}$ as a copy of $\mathfrak{h}^{*}$ regarded as
$\mathfrak{h}^{*}=\mathcal{Q}\otimes_{\mathbb{Z}}\mathbb{C}$. Thus $H^{*}$ has
a basis

$\displaystyle\\{Q_{i},\alpha_{i}\mid i\in I\\},\quad Q_{i}\coloneqq
Q_{\alpha_{i}}.$ (1.3.14)

We call $Q_{i}$ _the dual_ (_dynamical_) _parameters_. The choice of $Q_{i}$
and $\alpha_{i}$ are as follows: For this basis of $H^{*}$ and the basis
(1.3.10) of $H$, the canonical pairing $\langle\cdot,\cdot\rangle\colon
H^{*}\times H\to\mathbb{C}$ has the values

$\displaystyle\langle
Q_{i},P_{j}\rangle=\left(\alpha_{i}\,|\,\alpha_{j}\right)=b_{ij},\quad\langle\alpha_{i},h_{j}\rangle=a_{ji},$
(1.3.15)

and the other pairings are $0$.

We close this part with

######  Remark 1.3.1.

  1. (1)

We modified the notations in [FKO14] in several places. In particular, the
linear space $\widetilde{H}$ is denoted as $H$ in loc. cit., and defined to be
an extension of ours by $\mathbb{C}c$.

  2. (2)

We use the symbol $P+h$ in two meanings: the one is an element
$P+h\in\widetilde{H}$ associated to $h=\sum_{i\in
I}c_{i}\alpha^{\vee}_{i}\in\mathfrak{h}$, and the other is the set of
variables $P+h=\bigl{(}P_{i}+\nu(h_{i})\bigr{)}_{i\in I}$.

  3. (3)

Let us give an example of the calculation of the pairing (1.3.15): $\langle
Q_{i}+\alpha_{i},P_{j}+h_{j}\rangle=\langle
Q_{i},P_{j}\rangle+\langle\alpha_{i},h_{j}\rangle=\langle
Q_{i},P_{j}\rangle+\langle\alpha_{i},h_{j}\rangle=b_{ij}+a_{ji}$.

####  1.3.3. Definition

Keep the notations in the previous § 1.3.1 and § 1.3.2. We borrow from
[FKO14, p.106] the notations on $q$-numbers. Let $q$ be a complex number with
$0<\left|q\right|<1$, and $i\in I=\\{1,2,\dotsc,l\\}$. For $m,n\in\mathbb{N}$
with $m\leq n$, we define

$\displaystyle q_{i}\coloneqq
q^{d_{i}},\quad[n]_{i}\coloneqq\frac{q_{i}^{n}-q_{i}^{-n}}{q_{i}-q_{i}^{-1}},\quad[n)_{i}\coloneqq\frac{q^{n}-q^{-n}}{q_{i}-q_{i}^{-1}},$
(1.3.16)
$\displaystyle[n]_{i}!\coloneqq[n]_{i}[n-1]_{i}\dotsm[1]_{i},\quad\genfrac{[}{]}{0.0pt}{}{m}{n}_{i}\coloneqq\frac{[m]_{i}!}{[n]_{i}!\,[m-n]_{i}!}.$
(1.3.17)

Let us also fix a formal parameter $p$. Below a _topological algebra over
$\mathbb{C}\llbracket p\rrbracket$_ means the one with respect to the $p$-adic
topology.

The following definition is due to [J+99, Appendix A] and [FKO14, Definition
2.1] for general $\widehat{\mathfrak{g}}$, and due to [Ko98], [Ko09, §2.2] and
[Ko20, Definition 2.3.1] for
$\widehat{\mathfrak{g}}=\widehat{\mathfrak{sl}}_{2}$.

######  Definition 1.3.2.

Let $A=A(X_{l})=(a_{ij})_{i,j\in I}$ be the Cartan matrix of type $X_{l}$. The
elliptic algebra

$U_{q,p}(\widehat{\mathfrak{g}})=U_{q,p}(\widehat{\mathfrak{g}}(X_{l}))$

associated to $A$ is a topological algebra over $\mathbb{C}\llbracket
p\rrbracket$ generated by

$\displaystyle\operatorname{\mathcal{M}}(\widetilde{H}^{*}),\ q^{\pm c/2},\
d,\ K^{\pm}_{i},\ e_{i,m},\ f_{i,m},\ \alpha^{\vee}_{i,n}\quad(i\in
I,\,m\in\mathbb{Z},\,n\in\mathbb{Z}\setminus\\{0\\})$ (1.3.18)

satisfying the following relations.

  * •

$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ is a subalgebra.

  * •

$q^{\pm c/2}$ are central and $q^{c/2}q^{-c/2}=1$.

  * •

$K^{\pm}_{i}$ are invertible.

  * •

It is useful to introduce the generating currents, regarded as formal Laurent
series in $z$, by

$e_{i}(z)\coloneqq\sum_{m\in\mathbb{Z}}e_{i,m}z^{-m},\quad
f_{i}(z)\coloneqq\sum_{m\in\mathbb{Z}}f_{i,m}z^{-m}\quad(i\in I).$

Using the functions $F(P),F(P+h)\in\operatorname{\mathcal{M}}(H^{*})$ (see
(1.3.12)) and the basis elements $Q_{i}=Q_{\alpha_{i}}$ of $H^{*}$ in
(1.3.14), the relations containing the elements in $\mathbb{F}$ are given as
follows.

$\displaystyle F(P+h)\,d=d\,F(P+h),$ $\displaystyle F(P)\,d=d\,F(P),$ (1.3.19)
$\displaystyle F(P+h)\,K_{i}^{\pm}=K^{\pm}_{i}\,F\bigl{(}P+h\mp\langle
Q_{i},P+h\rangle\bigr{)},$ $\displaystyle
F(P)\,K_{i}^{\pm}=K^{\pm}_{i}\,F\bigl{(}P\mp\langle Q_{i},P\rangle\bigr{)},$
(1.3.20) $\displaystyle F(P+h)\,e_{i}(z)=e_{i}(z)\,F(P+h),$ $\displaystyle
F(P)\,e_{i}(z)=e_{i}(z)\,F(P-\langle Q_{i},P\rangle),$ (1.3.21) $\displaystyle
F(P+h)\,f_{i}(z)=f_{i}(z)\,F(P+h-\langle Q_{i},P+h\rangle),$ $\displaystyle
F(P)\,f_{i}(z)=f_{i}(z)\,F(P),$ (1.3.22) $\displaystyle
F(P+h)\,\alpha^{\vee}_{i,n}=\alpha^{\vee}_{i,n}\,F(P+h).$ $\displaystyle
F(P)\,\alpha^{\vee}_{i,n}=\alpha^{\vee}_{i,n}\,F(P),$ (1.3.23)

Similarly as the Ding-Iohara algebra in § 1.2, the equalities containing the
generating currents are understood as those of formal Laurent series in $z$,
and the genuine relations are obtained by comparing the coefficients of
$z^{k}$ ($k\in\mathbb{Z}$) in both sides.

  * •

The remaining relations containing $d$ are given by

$\displaystyle[d,K^{\pm}_{i}]=0,\quad[d,e_{i,m}]=me_{i,m},\quad[d,f_{i,m}]=mf_{i,m},\quad[d,\alpha^{\vee}_{j,n}]=n\alpha^{\vee}_{i,n}.$
(1.3.24)

  * •

The remaining relations containing $K^{\pm}_{i}$ are given by

$\displaystyle K^{\pm}_{i}e_{j}(z)=q_{i}^{\mp a_{ij}}e_{j}(z)K^{\pm}_{i},\quad
K^{\pm}_{i}f_{j}(z)=q_{i}^{\pm
a_{ij}}f_{j}(z)K^{\pm}_{i},\quad[K^{\pm}_{i},\alpha^{\vee}_{j,n}]=0.$ (1.3.25)

  * •

Using (1.3.16), the remaining relations containing $\alpha^{\vee}_{i,n}$ are
given by

$\displaystyle[\alpha^{\vee}_{i,m},\alpha^{\vee}_{j,n}]=\delta_{m+n,0}\frac{[a_{ij}m]_{i}[cm)_{j}}{m}\frac{1-p^{m}}{1-p^{*m}}q^{-cm},$
(1.3.26)
$\displaystyle[\alpha^{\vee}_{i,m},e_{j}(z)]=\frac{[a_{ij}m]_{i}}{m}\frac{1-p^{m}}{1-p^{*m}}q^{-cm}z^{m}e_{j}(z),$
(1.3.27)
$\displaystyle[\alpha^{\vee}_{i,m},f_{j}(z)]=-\frac{[a_{ij}m]_{i}}{m}z^{m}f_{j}(z).$
(1.3.28)

Here and hereafter, we use

$\displaystyle p^{*}\coloneqq pq^{-2c},$ (1.3.29)

and the term $\frac{1}{1-p^{*m}}$ is understood as the formal series
$\sum_{n\geq 0}p^{*mn}$.

  * •

For the other relations, it is useful to introduce additional generating
currents

$\displaystyle\psi^{+}_{i}(q^{-c/2}z)\coloneqq
K^{+}_{i}\exp\Bigl{(}-(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{\alpha^{\vee}_{i,-n}}{1-p^{n}}z^{n}\Bigr{)}\exp\Bigl{(}(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{p^{n}\alpha^{\vee}_{i,n}}{1-p^{n}}z^{-n}\Bigr{)},$
$\displaystyle\psi^{-}_{i}(q^{c/2}z)\coloneqq
K^{-}_{i}\exp\Bigl{(}-(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{p^{n}\alpha^{\vee}_{i,-n}}{1-p^{n}}z^{n}\Bigr{)}\exp\Bigl{(}(q_{i}-q_{i}^{-1})\sum_{n>0}\frac{\alpha^{\vee}_{i,n}}{1-p^{n}}z^{-n}\Bigr{)},$

where $\exp(x)\coloneqq\sum_{n\in\mathbb{N}}\frac{1}{n!}x^{n}$ denotes the
formal exponential, and the expression $\frac{1}{1-p^{n}}$ is understood as
$\sum_{k\geq 0}p^{kn}$. One can check that these generating currents have
well-defined coefficients in the $p$-adic topology (c.f. Remark 1.2.2 2). Then
the remaining relations are given as follows.

    * –

For any $i,j\in I$, the quadratic relations of $e_{i,m}$’s and $f_{i,m}$’s are
given by

$\displaystyle z\,\gamma(\tfrac{w}{z};q^{b_{ij}},p^{*})e_{i}(z)e_{j}(w)$
$\displaystyle=-w\,\gamma(\tfrac{z}{w};q^{b_{ij}},p^{*})e_{j}(w)e_{i}(z),$
(1.3.30) $\displaystyle z\,\gamma(\tfrac{w}{z};q^{-b_{ij}},p)f_{i}(z)f_{j}(w)$
$\displaystyle=-w\,\gamma(\tfrac{z}{w};q^{-b_{ij}},p)f_{j}(w)f_{i}(z),$
(1.3.31)

where the structure function $G$ is given by

$\displaystyle\gamma(x;q,p)\coloneqq\frac{(qx;p)_{\infty}}{(pq^{-1}x;p)_{\infty}}.$

See (0.2.2) for the shifted factorial $(x;p)_{\infty}$. We regard
$G(x;q,p)\in\mathbb{C}[x^{\pm 1}]\llbracket p\rrbracket$ by expanding the
denominator using $\frac{1}{1-xp^{k}}=\sum_{n\geq 0}(xp^{k})^{n}$. The
quantity $b_{ij}$ is the entry of the symmetrization $A=DB$ of the Cartan
matrix $A$ with the diagonal matrix $D^{-1}=(d_{1},\dotsc,d_{l})$ given by
(1.3.2). As before, the genuine relations are obtained by expanding both sides
into formal Laurent series of $z,w$ and comparing the coefficients of
$z^{m}w^{n}$ ($m,n\in\mathbb{Z}$) in both sides.

    * –

For $i,j\in I$ with $i\neq j$, assume $a\coloneqq 1-a_{ij}\geq 2$. We define

$\displaystyle\widetilde{\gamma}(x;q,p)\coloneqq\frac{(pqx;p)_{\infty}}{(pq^{-1}x;p)_{\infty}},$
(1.3.32)

regarded as a formal series of $p$. Then, using (1.3.17), the Serre-type
relation of $e_{i}(z),e_{j}(z)$ is

$\displaystyle\begin{split}\sum_{\sigma\in\mathfrak{S}_{a}}&\prod_{1\leq
m<n\leq
a}\widetilde{\gamma}(\tfrac{z_{\sigma(n)}}{z_{\sigma(m)}};q^{2},p^{*})\\\
&\cdot\sum_{s=0}^{a}(-1)^{s}\genfrac{[}{]}{0.0pt}{}{a}{s}_{i}\prod_{m=1}^{s}\widetilde{\gamma}(\tfrac{w}{z_{\sigma(m)}};q^{b_{ij}},p^{*})\prod_{m=s+1}^{a}\widetilde{\gamma}(\tfrac{z_{\sigma(m)}}{w};q^{b_{ij}},p^{*})\\\
&\cdot e_{i}(z_{\sigma(1)})\dotsm
e_{i}(z_{\sigma(s)})e_{j}(w)e_{i}(z_{\sigma(s+1)})\dotsm
e_{i}(z_{\sigma(a)})=0,\end{split}$ (1.3.33)

and that of $f_{i}(z),f_{j}(z)$ is

$\displaystyle\begin{split}\sum_{\sigma\in\mathfrak{S}_{a}}&\prod_{1\leq
m<n\leq a}\widetilde{\gamma}(\tfrac{z_{\sigma(n)}}{z_{\sigma(m)}};q^{-2},p)\\\
&\cdot\sum_{s=0}^{a}(-1)^{s}\genfrac{[}{]}{0.0pt}{}{a}{s}_{i}\prod_{m=1}^{s}\widetilde{\gamma}(\tfrac{w}{z_{\sigma(m)}};q^{-b_{ij}},p)\prod_{m=s+1}^{a}\widetilde{\gamma}(\tfrac{z_{\sigma(m)}}{w};q^{-b_{ij}},p)\\\
&\cdot f_{i}(z_{\sigma(1)})\dotsm
f_{i}(z_{\sigma(s)})f_{j}(w)f_{i}(z_{\sigma(s+1)})\dotsm
f_{i}(z_{\sigma(a)})=0.\end{split}$ (1.3.34)

Here $\mathfrak{S}_{a}$ denotes the symmetric group of degree $a$.

    * –

Finally, the relations of $e_{i,m}$ and $f_{i,m}$ are given by

$\displaystyle[e_{i}(z),f_{j}(w)]=\frac{\delta_{i,j}}{q-q^{-1}}\bigl{(}\delta(q^{-c}\tfrac{z}{w})\psi^{-}_{i}(q^{c/2}w)-\delta(q^{c}\tfrac{z}{w})\psi^{+}_{i}(q^{-c/2}z)\bigr{)},$
(1.3.35)

where $\delta$ denotes the formal delta function
$\delta(x)\coloneqq\sum_{n\in\mathbb{Z}}x^{n}$ in (1.2.15).

######  Remark 1.3.3.

Some comments on Definition 1.3.2 are in order.

  1. (1)

If we take $p$ to be a complex number such that $\left|p\right|<1$, then
$U_{p,q}(\widehat{g})$ is a genuine algebra over $\mathbb{C}$.

  2. (2)

The definition in [FKO14, Definition 2.1] and [Ko20, Definition 2.3.1] has a
typo. There $U_{q,p}(\widehat{\mathfrak{g}})$ is defined to be a topological
algebra over $\mathbb{F}\llbracket p\rrbracket$ with
$\mathbb{F}\coloneqq\operatorname{\mathcal{M}}(P(\mathfrak{h})^{*})$, but then
the functions $F(P+h),F(P)\in\mathbb{F}$ would commute with the other
generators. A correct definition of the $\widehat{\mathfrak{sl}}_{2}$ case is
given in [Ko09, §2.2].

  3. (3)

The relations $[d,K^{\pm}_{i}]=0$ are missed in [FKO14, Definition 2.1].

  4. (4)

The relations (1.3.30) and (1.3.31) can be rewritten as

$\displaystyle z\,\theta^{*}(q^{b_{ij}}\tfrac{w}{z})e_{i}(z)e_{j}(w)$
$\displaystyle=-w\,\theta^{*}(q^{b_{ij}}\tfrac{z}{w})e_{j}(w)e_{i}(z),$
(1.3.36) $\displaystyle z\,\theta(q^{-b_{ij}}\tfrac{w}{z})f_{i}(z)f_{j}(w)$
$\displaystyle=-w\,\theta(q^{-b_{ij}}\tfrac{z}{w})f_{j}(w)f_{i}(z).$ (1.3.37)

where we used the theta series

$\displaystyle\theta(x)\coloneqq\theta(x;p),\quad\theta^{*}(x)\coloneqq\theta(x;p^{*}),\quad\theta(x;p)\coloneqq(x,px^{-1};p)_{\infty}\in\mathbb{C}[x^{\pm
1}]\llbracket p\rrbracket.$ (1.3.38)

See (0.2.2) for the symbol of infinite product. These relations are given in
[J+99, (A.17), (A.18)].

  5. (5)

The definition of $U=U_{q,p}$ is independent of the choice of symmetrization
$A=DB$, $D^{-1}=(d_{1},\dotsc,d_{l})$, of the Cartan matrix $A$ up to
isomorphism. Indeed, if we take another symmetrization
$A=D^{\prime}B^{\prime}$ with
$D^{\prime-1}=(d^{\prime}_{1},\dotsc,d^{\prime}_{l})$, then by [Kac90, Remark
2.4], there is an $r\in\mathbb{Q}\setminus\\{0\\}$ such that
$d^{\prime}_{i}=rd_{i}$ for any $i\in I$. Thus, replacing

$q\longmapsto q^{\prime}\coloneqq q^{1/r},\quad c\longmapsto
c^{\prime}\coloneqq rc,$

we have $q_{i}=q^{d_{i}}=(q^{\prime})^{d^{\prime}_{i}}=q^{\prime}_{i}$,
$q^{-2c}=(q^{\prime})^{-2c^{\prime}}$ and
$q^{b_{ij}}=q^{d_{i}a_{ij}}=(q^{\prime})^{d^{\prime}_{i}a_{ij}}=(q^{\prime})^{b^{\prime}_{ij}}$,
so that the defining relations (1.3.25)–(1.3.28), (1.3.30)–(1.3.35) are
preserved. Thus, we have an isomorphism $U_{q,p}\cong U_{q^{\prime},p}$ of
topological algebras.

For the later purpose, let us rewrite some of the defining relations.

######  Lemma 1.3.4.

The relations (1.3.27) and (1.3.28) are equivalent (under (1.3.25)) to the
following current relations.

$\displaystyle\psi_{i}^{+}(z)e_{j}(w)$
$\displaystyle=g_{ij}^{*}(q^{-c/2}\tfrac{z}{w})e_{j}(w)\psi_{i}^{+}(z),$
$\displaystyle\psi_{i}^{+}(z)f_{j}(w)$
$\displaystyle=g_{ij}(q^{c/2}\tfrac{z}{w})^{-1}f_{j}(w)\psi_{i}^{+}(z),$
(1.3.39) $\displaystyle\psi_{i}^{-}(z)e_{j}(w)$
$\displaystyle=g_{ij}^{*}(q^{c/2}\tfrac{z}{w})e_{j}(w)\psi_{i}^{-}(z),$
$\displaystyle\psi_{i}^{-}(z)f_{j}(w)$
$\displaystyle=g_{ij}(q^{-c/2}\tfrac{z}{w})^{-1}f_{j}(w)\psi_{i}^{-}(z).$
(1.3.40)

Here the formal series $g_{ij}(x)=g_{ij}(x;p)$,
$g_{ij}^{*}(x)=g_{ij}(x;p^{*})$ are the expansion of

$\displaystyle
g_{ij}(x;p)=\frac{G^{+}_{ij}(x;p)}{G^{-}_{ij}(x;p)}\coloneqq\frac{q^{-b_{ij}}\theta(q^{b_{ij}}x;p)}{\theta(q^{-b_{ij}}x;p)}$
(1.3.41)

with respect to $p$, where $\theta(x;p)$ is given in (1.3.38).

###### Proof.

We only derive
$\psi_{i}^{+}(z)e_{j}(w)=g_{ij}^{*}(q^{-c/2}\tfrac{z}{w})e_{j}(w)\psi_{i}^{+}(z)$
from (1.3.27) and (1.3.25). Using the relation
$\exp(A)\exp(B)=\exp([A,B])\exp(B)\exp(A)$ in case $[A,B]$ commutes with $A$
and $B$, we have
$\psi^{+}_{i}(z)e_{j}(w)=C(\frac{z}{w})e_{j}(w)\psi^{+}_{i}(z)$ with

$\displaystyle C(x)\coloneqq
q_{i}^{-a_{ij}}\exp\Bigl{[}-\sum_{n>0}\frac{1}{n}\frac{q_{i}^{a_{ij}n}-q_{i}^{-a_{ij}n}}{1-p^{*n}}(q^{-c/2}x)^{n}\Bigr{]}\exp\Bigl{[}\sum_{n>0}\frac{1}{n}\frac{q_{i}^{a_{ij}n}-q_{i}^{-a_{ij}n}}{1-p^{*n}}(p^{*}q^{c/2}x^{-1})^{n}\Bigr{]}.$

Then, using $\exp(-\sum_{n>0}\frac{1}{n}x^{n})=1-x$ and
$q_{i}^{a_{ij}}=q^{d_{i}a_{ij}}=q^{b_{ij}}$, we have

$\displaystyle
C(x)=q^{-b_{ij}}\frac{(q^{b_{ij}}q^{-c/2}x;p^{*})_{\infty}}{(q^{-b_{ij}}q^{-c/2}x;p^{*})_{\infty}}\frac{(p^{*}q^{-b_{ij}}q^{c/2}x^{-1};p^{*})_{\infty}}{(p^{*}q^{b_{ij}}q^{c/2}x^{-1};p^{*})_{\infty}}=q^{-b_{ij}}\frac{\theta(q^{b_{ij}}q^{-c/2}x;p^{*})}{\theta(q^{-b_{ij}}q^{-c/2}x;p^{*})}=g^{*}_{ij}(q^{-c/2}x).$

Thus we have the desired equality. ∎

######  Lemma 1.3.5.

The relations (1.3.26) are equivalent (under (1.3.25)) to the current
relations

$\displaystyle\psi_{i}^{\pm}(z)\psi_{j}^{\pm}(w)=\frac{g^{*}_{ij}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\psi_{j}^{\pm}(w)\psi_{i}^{\pm}(z),\quad\psi_{i}^{+}(z)\psi_{j}^{-}(w)=\frac{g^{*}_{ij}(q^{-c}\frac{z}{w})}{g_{ij}(q^{c}\frac{z}{w})}\psi_{j}^{-}(w)\psi_{i}^{+}(z)$
(1.3.42)

with $g_{ij}(x)=g_{ij}(x;p)$ and $g_{ij}^{*}(x)=g_{ij}(x;p^{*})$ given in
(1.3.41).

###### Proof.

Let us derive
$\psi_{i}^{\pm}(z)\psi_{j}^{\pm}(w)=\frac{g^{*}_{ij}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\psi_{j}^{\pm}(w)\psi_{i}^{\pm}(z)$.
The same argument as in Lemma 1.3.4 yields

$\psi_{i}^{\pm}(z)\psi_{j}^{\pm}(w)=C_{ji}(\tfrac{z}{w})C_{ij}(\tfrac{w}{z})^{-1}\psi_{j}^{\pm}(w)\psi_{i}^{\pm}(z),$

where, using
$(x;p,p^{*})_{\infty}\coloneqq\prod_{n\in\mathbb{N}}(xp^{*n};p)_{\infty}=\prod_{m,n\in\mathbb{N}}(1-xp^{m}p^{*n})$,
we set

$C_{ji}(x)\coloneqq\exp\Bigl{[}\sum_{n>0}\frac{1}{n}\frac{q_{j}^{a_{ji}n}-q_{j}^{-a_{ji}n}}{1-p^{n}}\frac{1-q^{-2cn}}{1-p^{*n}}(px)^{n}\Bigr{]}.$

By $\exp(-\sum_{n>0}\frac{1}{n}x^{n})=1-x$ and
$q_{j}^{a_{ji}}=q^{d_{j}a_{ji}}=q^{b_{ji}}$, we have

$C_{ji}(x)=\frac{(q_{j}^{-a_{ji}}xp,q_{i}^{a_{ji}}xp^{*};p,p^{*})_{\infty}}{(q_{j}^{-a_{ji}}xp^{*},q_{j}^{a_{ji}}xp;p,p^{*})_{\infty}}=\frac{(q^{-b_{ji}}xp,q^{b_{ji}}xp^{*};p,p^{*})_{\infty}}{(q^{-b_{ji}}xp^{*},q^{b_{ji}}xp;p,p^{*})_{\infty}}.$

Then, using

$\frac{(xp;p,p^{*})_{\infty}}{(xq;p,p^{*})_{\infty}}=\frac{\prod_{m,n\in\mathbb{N}}(1-xp^{m+1}p^{*n})}{\prod_{m,n\in\mathbb{N}}(1-xp^{m}(p^{*})^{n+1})}=\frac{\prod_{m\in\mathbb{N}}(1-xp^{m+1})}{\prod_{n\in\mathbb{N}}(1-x(p^{*})^{n+1})}=\frac{(xp;p)_{\infty}}{(xp^{*};p^{*})_{\infty}}$

and $b_{ij}=b_{ji}$, we have

$\displaystyle C_{ji}(x)C_{ij}(x^{-1})^{-1}$
$\displaystyle=\frac{(q^{-b_{ji}}xp;p)_{\infty}}{(q^{-b_{ji}}xp^{*},p^{*})_{\infty}}\frac{(q^{b_{ji}}xp^{*};p^{*})_{\infty}}{(q^{b_{ji}}xp;p)_{\infty}}\cdot\frac{(q^{b_{ij}}x^{-1}p^{*};p^{*})_{\infty}}{(q^{b_{ij}}x^{-1}p;p)_{\infty}}\frac{(q^{b_{ij}}x^{-1}p;p)_{\infty}}{(q^{b_{ij}}x^{-1}p^{*};p^{*})_{\infty}}$
$\displaystyle=\frac{(q^{-b_{ij}}x,pq^{b_{ij}}/x;p)_{\infty}}{(q^{b_{ij}}x,pq^{-b_{ij}}/x;p)_{\infty}}\frac{1-q^{b_{ij}}x}{1-q^{-b_{ij}}x}\cdot\frac{(q^{b_{ij}}x,p^{*}q^{-b_{ij}}/x;p^{*})_{\infty}}{(q^{-b_{ij}}x,p^{*}q^{b_{ij}}/x;p^{*})_{\infty}}\frac{1-q^{-b_{ij}}x}{1-q^{b_{ij}}x}$
$\displaystyle=\frac{\theta(q^{b_{ij}}x;p^{*})}{\theta(q^{-b_{ij}}x;p^{*})}\frac{\theta(q^{-b_{ij}}x;p)}{\theta(q^{b_{ij}}x;p)}=\frac{g_{ij}^{*}(x)}{g_{ij}(x)}.$

∎

######  Lemma 1.3.6.

The relations (1.3.30) and (1.3.31) can be rewritten as

$\displaystyle
G_{ij}^{-*}(\tfrac{z}{w})e_{i}(z)e_{j}(w)=G_{ij}^{+*}(\tfrac{z}{w})e_{j}(w)e_{i}(z),\quad
G_{ij}^{+}(\tfrac{z}{w})f_{i}(z)f_{j}(w)=G_{ij}^{-}(\tfrac{z}{w})f_{j}(w)f_{i}(z),$
(1.3.43)

respectively, where we used $G_{ij}^{\pm}(x)\coloneqq G_{ij}^{\pm}(x;p)$ and
$G_{ij}^{\pm*}(x)\coloneqq G_{ij}^{\pm}(x;p^{*})$ with
$G_{ij}^{+}(x;p)\coloneqq q^{-b_{ij}}\theta(q^{b_{ij}}x;p)$,
$G_{ij}^{-}(x;p)\coloneqq\theta(q^{-b_{ij}}x;p)$ as introduced in (1.3.41).

###### Proof.

Recall the rewritten form (1.3.36) of (1.3.30). Then, (1.3.36) can be
rewritten as

$z\,G_{ij}^{+*}(\tfrac{w}{z})e_{i}(z)e_{j}(w)=-w\,G_{ij}^{+*}(\tfrac{z}{w})e_{j}(w)e_{i}(z),$

which is equivalent to the first half of (1.3.43) by the equality
$x^{-1}G_{ij}^{+*}(x^{-1})=G_{ij}^{-*}(x)$. We can similarly show the
equivalence between (1.3.31) and the second half of (1.3.43). ∎

######  Remark 1.3.7.

Note the similarity between the fundamental relations of the Ding-Iohara
quantum algebra $U_{q}(g,A_{l})$ and those of the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ of type $A_{l}$: (1.2.11) and (1.3.39),
(1.2.12) and (1.3.40), (1.2.13) and (1.3.43).

####  1.3.4. Hopf algebroid structure of $U_{q,p}(\widehat{\mathfrak{g}})$

Now we explain a Hopf algebroid structure on the elliptic algebra
$U_{q,p}(\widehat{\mathfrak{g}})$ of Definition 1.3.2.

Let us first remark that the definitions on Hopf algebroids in § 1.1 are
given for algebras over $\mathbb{C}$, and since
$U_{q,p}(\widehat{\mathfrak{g}})$ is not an algebra over $\mathbb{C}$ but a
topological algebra over $\mathbb{C}\llbracket p\rrbracket$, we should modify
the definitions in § 1.1 appropriately. The obtained notions will be called
topological $H$-algebras, topological $H$-bialgebroids, and topological
$H$-Hopf algebroids. We leave it for the reader to write down the precise
definitions.

Let us keep the notation in §§1.3.1–1.3.3, and fix the Cartan matrix
$A=(a_{ij})_{i,j\in I}$ of finite type $X_{l}$. For simplicity, we often
denote the elliptic algebra associated to $A$ by

$U\coloneqq U_{q,p}(\widehat{\mathfrak{g}}).$

Recall the linear space $H$ in (1.3.6) and the basis $\\{P_{i}\mid i\in I\\}$
in (1.3.10). We denote the field of meromorphic functions on the dual space
$H^{*}$ by

$\mathbb{F}=\operatorname{\mathcal{M}}(H^{*}).$

Let us consider the extension $\mathbb{F}\llbracket
p\rrbracket\coloneqq\mathbb{F}\otimes_{\mathbb{C}}\llbracket p\rrbracket$
since we will work over $\mathbb{C}\llbracket p\rrbracket$. Following the
notation (1.3.11), we express an element $F\in\mathbb{F}\llbracket
p\rrbracket$ as

$\displaystyle F=F(P,p)=\sum_{n\in\mathbb{N}}p^{n}F_{n}(P),$ (1.3.44)

where $F_{n}(P)$ is an element of $\mathbb{F}$ for each $n\in\mathbb{N}$.

Let us also recall from (1.3.6) that the linear space $\widetilde{H}$ spanned
by $P_{i}+h_{i}$, $P_{i}$ ($i\in I$). By Definition 1.3.2, the extension
$\operatorname{\mathcal{M}}(H^{*})\llbracket p\rrbracket$ of the meromorphic
function field on $H^{*}$ is a subalgebra of $U$ over $\mathbb{C}\llbracket
p\rrbracket$.

Now, let us regard $H$ as a finite-dimensional commutative Lie algebra, so
that we have the notion of $H$-algebra (see § 1.1). In order to define the
$H^{*}$-bigrading on $U$, we consider the actions of
$q^{P+h},q^{P}\in\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ on $U$, where
we regard $q^{P+h}=\exp((P+h)\log q)$, $q^{P}=\exp(P\log q)$ and apply
(1.3.19)–(1.3.23).

######  Lemma 1.3.8.

The elliptic algebra $U$ has the following topological $H$-algebra structure.

  1. 1

The $H^{*}$-bigrading $U=\bigoplus_{\alpha,\beta\in H^{*}}U_{\alpha,\beta}$ is
given by

$\displaystyle U_{\alpha,\beta}\coloneqq\\{a\in U\mid
q^{P}aq^{-P}=q^{\langle\alpha,P\rangle}a,\
q^{P+h}aq^{-(P+h)}=q^{\langle\beta,P+h\rangle}a\quad(\forall
P,P+h\in\widetilde{H})\\},$

Here $\langle\cdot,\cdot\rangle$ denotes the the canonical pairing on
$\widetilde{H}^{*}\times\widetilde{H}$ (see (1.3.15)) restricted to
$H^{*}\times\widetilde{H}$ with respect to the embedding
$H=\sum_{i}\mathbb{C}Q_{i}\hookrightarrow\widetilde{H}=\sum_{i}\mathbb{C}(Q_{i}+\alpha_{i})+\sum_{i}\mathbb{C}Q_{i}$,
$Q_{i}\mapsto Q_{i}$.

  2. 2

The moment maps $\mu_{l},\mu_{r}\colon\mathbb{F}\llbracket p\rrbracket\to
U_{0,0}$ are defined by

$\displaystyle\mu_{l}(F)\coloneqq F(P,p^{*}),\quad\mu_{r}(F)\coloneqq
F(P+h,p)$ (1.3.45)

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$.

Moreover, the following statements hold.

  1. (1)

The images $\mu_{l}(\mathbb{F})$ and $\mu_{r}(\mathbb{F})$ generate
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$.

  2. (2)

The generators have the following bigrading:

$\displaystyle e_{i}(z)\in U_{-Q_{i},0},\quad f_{i}(z)\in
U_{0,-Q_{i}},\quad\psi^{\pm}_{i}(z)\in U_{-Q_{i},-Q_{i}},\quad K^{\pm}_{i}\in
U_{-Q_{i},-Q_{i}},$

and the others belong to $U_{0,0}$, where, for a current
$x(z)=\sum_{m\in\mathbb{Z}}x_{m}z^{-m}$, the expression $x(z)\in
U^{\prime}_{\alpha,\beta}$ means $x_{m}\in U_{\alpha,\beta}$ for each
$m\in\mathbb{Z}$. In particular, the bigrading values in the lattice
$\mathcal{Q}$:

$U=\bigoplus_{\alpha,\beta\in\mathcal{Q}}U_{\alpha,\beta},\quad\mathcal{Q}\coloneqq\bigoplus_{i\in
I}\mathbb{Z}Q_{i}\subsetneq H^{*}.$

###### Proof.

This is a special case of Lemma 2.2.1. ∎

Note that $\mathcal{Q}$ is isomorphic to the root lattice of the Lie algebra
$\mathfrak{g}$. We also have
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})\llbracket p\rrbracket\subset
U_{0,0}$, so that the values (1.3.45) of the moment maps do belong to
$U_{0,0}$.

######  Proposition 1.3.9.

The topological $H$-algebra $U$ in Lemma 1.3.8 has the following topological
$H$-Hopf algebroid structure $(\Delta,\varepsilon,S)$.

  1. (1)

The comultiplication $\Delta\colon U\to U\mathbin{\widetilde{\otimes}}U$ is
given by the Drinfeld-type formulas. For $F=F(P,p)\in\mathbb{F}\llbracket
p\rrbracket$, we have

$\displaystyle\Delta(\mu_{l}(F))\coloneqq\mu_{l}(F)\mathbin{\widetilde{\otimes}}1,\quad\Delta(\mu_{r}(F))\coloneqq
1\mathbin{\widetilde{\otimes}}\mu_{r}(F).$

By Lemma 1.3.8 (1), these define $\Delta$ on the whose
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$. The formulas for the
generators $q^{\pm c/2}$, $d$ and $K^{\pm}_{i}$ are

$\Delta(q^{\pm c/2})\coloneqq q^{\pm c/2}\mathbin{\widetilde{\otimes}}q^{\pm
c/2},\quad\Delta(d)\coloneqq
d\mathbin{\widetilde{\otimes}}1+1\mathbin{\widetilde{\otimes}}d,\quad\Delta(K^{\pm}_{i})\coloneqq
K^{\pm}_{i}\mathbin{\widetilde{\otimes}}K^{\pm}_{i}.$

The remaining formulas are given in terms of the generating currents:

$\displaystyle\Delta(e_{i}(z))\coloneqq
e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z),$
(1.3.46) $\displaystyle\Delta(f_{i}(z))\coloneqq
1\mathbin{\widetilde{\otimes}}f_{i}(z)+f_{i}(q^{c_{2}}z)\mathbin{\widetilde{\otimes}}\psi^{-}_{i}(q^{c_{2}/2}z),$
(1.3.47)
$\displaystyle\Delta(\psi_{i}^{+}(z))\coloneqq\psi_{i}^{+}(q^{c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{-c_{1}/2}z),$
(1.3.48)
$\displaystyle\Delta(\psi_{i}^{-}(z))\coloneqq\psi_{i}^{-}(q^{-c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{-}(q^{c_{1}/2}z).$
(1.3.49)

For the symbols $q^{\pm c_{1}/2}$ and $q^{\pm c_{2}/2}$, see the comment after
(1.2.22).

  2. (2)

The counit $\varepsilon\colon U\to D_{H}\llbracket p\rrbracket$ is given by

$\displaystyle\varepsilon(\mu_{l}(F))=\varepsilon(\mu_{r}(F))\coloneqq
FT_{0},\quad\varepsilon(q^{\pm c/2})\coloneqq 1,\quad\varepsilon(d)\coloneqq
0,\quad\varepsilon(K^{\pm}_{i})\coloneqq T_{Q_{i}},$
$\displaystyle\varepsilon(e_{i}(z))=\varepsilon(f_{i}(z))\coloneqq
0,\quad\varepsilon(\psi_{i}^{\pm}(z))\coloneqq T_{Q_{i}}.$

Here $D_{H}$ denotes the $H$-algebra of difference operators on $H^{*}$ (see
the paragraph of (1.1.5)), and $D_{H}\llbracket p\rrbracket\coloneqq
D_{H}\otimes_{\mathbb{C}}\mathbb{C}\llbracket p\rrbracket$ is its
$\mathbb{C}\llbracket p\rrbracket$-extension with the naturally induced
topological $H$-algebra structure.

  3. (3)

The antipode $S\colon U\to U$ is given by

$\displaystyle S(\mu_{l}(F))=\mu_{r}(F),\quad
S(\mu_{r}(F))\coloneqq\mu_{l}(F),\quad S(d)\coloneqq d,\quad S(q^{\pm
c/2})\coloneqq q^{\mp c/2},$ $\displaystyle
S(e_{i}(z))\coloneqq-\psi^{+}_{i}(q^{-c/2}z)^{-1}e_{i}(q^{-c}z),\quad
S(f_{i}(z))\coloneqq-f_{i}(q^{-c}z)\psi^{-}_{i}(q^{-c/2}z),$ $\displaystyle
S(K^{\pm}_{i})\coloneqq(K^{\pm}_{i})^{-1},\quad
S(\psi_{i}^{\pm}(z))\coloneqq\psi_{i}^{\pm}(z)^{-1}.$

###### Proof.

The simply-laced types will be proved in § 2, Proposition 2.1.5 and Corollary
2.2.9. The non-simply types will be treated in § 3. The $BCF$ types will be
proved in Proposition 3.2.3, and the type $G_{2}$ will be proved in
Proposition 3.3.3. ∎

The operations in Proposition 1.3.9 are collected and inspected from those in
[J+99, Appendix B], and the topological $H$-algebra structure in Lemma 1.3.8
is designed to fit with these operations.

####  1.3.5. Opposite Hopf algebroid structure

In this part, we will explain another $H$-algebra structure on the elliptic
algebra $U_{q,p}(\widehat{\mathfrak{g}})$ appearing in the literature [Ko09,
§3], [FKO14, Proposition 2.4]111There are some typos in [FKO14], Proposition
2.4 and after, on the choice of the finite-dimensional commutative Lie algebra
$H$. The right choice should be our $H$, denoted by
$P_{\overline{\mathfrak{h}}}$ in loc. cit. and [Ko20, Proposition 3.1.1].

We start with a general argument on $\mathfrak{h}$-algebras. As in § 1.1, let
$\mathfrak{h}$ be a finite-dimensional commutative Lie algebra over
$\mathbb{C}$,
$\mathbb{F}\coloneqq\operatorname{\mathcal{M}}(\mathfrak{h}^{*})$, and
$A=\bigl{(}A,\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}A_{\alpha,\beta},\mu_{l},\mu_{r}\bigr{)}$
be an $\mathfrak{h}$-algebra with moment maps
$\mu_{l},\mu_{r}\colon\mathbb{F}\to A_{0,0}$. Then, exchanging the bigrading
and the moment maps, we have another $\mathfrak{h}$-algebra
$A^{\dagger}=\bigl{(}A^{\dagger},\bigoplus_{\alpha,\beta\in\mathfrak{h}^{*}}A^{\dagger}_{\alpha,\beta},\mu^{\dagger}_{l},\mu^{\dagger}_{r}\bigr{)}$
with

$\displaystyle A^{\dagger}\coloneqq A,\quad
A^{\dagger}_{\alpha,\beta}\coloneqq
A_{\beta,\alpha},\quad\mu^{\dagger}_{l}\coloneqq\mu_{r},\quad\mu^{\dagger}_{r}\coloneqq\mu_{l}.$
(1.3.50)

We now want to introduce an $\mathfrak{h}$-Hopf algebroid structure on the
$\mathfrak{h}$-algebra $A^{\dagger}$. Recall the modified tensor product
$\mathbin{\widetilde{\otimes}}$ of $\mathfrak{h}$-algebras (1.1.1).

######  Lemma 1.3.10.

We have an $\mathfrak{h}$-algebra isomorphism

$\displaystyle\widetilde{\tau}\colon
A\mathbin{\widetilde{\otimes}}A\longrightarrow(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger},\quad
a\mathbin{\widetilde{\otimes}}b\longmapsto b\mathbin{\widetilde{\otimes}}a.$
(1.3.51)

###### Proof.

We prove that $\widetilde{\tau}$ is well-defined as an $\mathfrak{h}$-algebra
homomorphism. Note that, by definition,
$(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger}=A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger}$
as an algebra, and the bigrading and the moment maps are given by

$\displaystyle(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger}_{\alpha,\beta}\coloneqq(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})_{\beta,\alpha}=\bigoplus_{\gamma\in\mathfrak{h}^{*}}A^{\dagger}_{\beta,\gamma}\otimes_{\mathbb{F}}A^{\dagger}_{\gamma,\alpha}=\bigoplus_{\gamma\in\mathfrak{h}^{*}}A_{\gamma,\beta}\otimes_{\mathbb{F}}A_{\alpha,\gamma},$
$\displaystyle\mu_{l}^{\mathbin{\widetilde{\otimes}}^{\dagger}}\coloneqq
1\mathbin{\widetilde{\otimes}}\mu^{\dagger}_{r}=1\mathbin{\widetilde{\otimes}}\mu_{l},\quad\mu_{r}^{\mathbin{\widetilde{\otimes}}^{\dagger}}\coloneqq\mu^{\dagger}_{l}\mathbin{\widetilde{\otimes}}1=\mu_{r}\mathbin{\widetilde{\otimes}}1.$

Now, let $a,a^{\prime}\in A_{\alpha,\gamma}$ and $b,b^{\prime}\in
A_{\gamma,\beta}$, and assume
$a\mathbin{\widetilde{\otimes}}b=a^{\prime}\mathbin{\widetilde{\otimes}}b^{\prime}$
in $A\mathbin{\widetilde{\otimes}}A$. Then, by the definition (1.1.3) of
$\mathbin{\widetilde{\otimes}}$, we have

$\displaystyle a\otimes b-a^{\prime}\otimes
b^{\prime}=\sum_{i}(\mu_{r}(f_{i})a_{i}\otimes
b_{i}-a_{i}\otimes\mu_{l}(f_{i})b_{i})$

for some $f_{i}\in\mathbb{F}$ and $a_{i},b_{i}\in A$, where $\otimes$ denotes
the ordinary tensor product over $\mathbb{C}$ of $A_{\alpha,\gamma}$ and
$A_{\gamma,\beta}$. The definition of the moment maps of $A^{\prime}$ implies

$\displaystyle b\otimes a-b^{\prime}\otimes
a^{\prime}=\sum_{i}(b_{i}\otimes\mu_{r}(f_{i})a_{i}-\mu_{l}(f_{i})b_{i}\otimes
a_{i})=\sum_{i}(b_{i}\otimes\mu^{\prime}_{l}(f_{i})a_{i}-\mu^{\prime}_{r}(f_{i})b_{i}\otimes
a_{i}),$

which yields
$\widetilde{\tau}(a\mathbin{\widetilde{\otimes}}b)=b\mathbin{\widetilde{\otimes}}a=b^{\prime}\mathbin{\widetilde{\otimes}}a^{\prime}=\widetilde{\tau}(a^{\prime}\mathbin{\widetilde{\otimes}}b^{\prime})$
in $(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger}$. Thus
$\widetilde{\tau}$ is well-defined as an algebra homomorphism.

Next, if $a\in A_{\alpha,\gamma}$ and $b\in A_{\gamma,\beta}$, then
$a\mathbin{\widetilde{\otimes}}b\in(A\mathbin{\widetilde{\otimes}}A)_{\alpha,\beta}$
and
$b\mathbin{\widetilde{\otimes}}a\in(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger}_{\alpha,\beta}$.
Also, for $f\in\mathbb{F}$, we have

$\displaystyle\widetilde{\tau}(\mu^{\mathbin{\widetilde{\otimes}}}_{l}(f))=\widetilde{\tau}(\mu_{l}(f)\mathbin{\widetilde{\otimes}}1)=1\mathbin{\widetilde{\otimes}}\mu_{l}(f)=\mu^{\mathbin{\widetilde{\otimes}}^{\prime}}_{l}(f),$
$\displaystyle\widetilde{\tau}(\mu^{\mathbin{\widetilde{\otimes}}}_{r}(f))=\widetilde{\tau}(1\mathbin{\widetilde{\otimes}}\mu_{r}(f))=\mu_{r}(f)\mathbin{\widetilde{\otimes}}1=\mu^{\mathbin{\widetilde{\otimes}}^{\prime}}_{r}(f).$

Thus $\widetilde{\tau}$ is an $\mathfrak{h}$-algebra homomorphism. The
bijectivity is clear. ∎

Recall (see [Kas95, Corollary III.3.5] for example) that, for a Hopf algebra
$H=(H,m,\Delta,S)$ with multiplication $m$, comultiplication $\Delta$ and
antipode $S$, we can associate the following new Hopf algebras.

$H^{\textup{op}}\coloneqq(H,m^{\textup{op}},\Delta,S^{-1}),\quad
H^{\textup{cop}}\coloneqq(H,m,\Delta^{\textup{op}},S^{-1}),\quad
H^{\textup{op},\textup{cop}}\coloneqq(H,m^{\textup{op}},\Delta^{\textup{op}},S),$

where the opposite operations are defined as $m^{\textup{op}}\coloneqq
m\circ\tau$ and $\Delta^{\textup{op}}\coloneqq\tau\circ\Delta$, using the flip
$\tau\colon H\otimes H\to H\otimes H$, $a\otimes b\mapsto b\otimes a$.
Moreover, the antipode $S$ induces Hopf algebra isomorphisms
$H\xrightarrow{\sim}H^{\textup{op},\textup{cop}}$ and
$H^{\textup{op}}\xrightarrow{\sim}H^{\textup{cop}}$.

Using the $\mathfrak{h}$-algebra isomorphism $\widetilde{\tau}$ in Lemma
1.3.10, we have a natural analogue of the above opposite theory for
$\mathfrak{h}$-Hopf algebroids. Besides $A^{\dagger}$ in (1.3.50), we
introduce another $\mathfrak{h}$-algebra
$\dot{A}=(A,\bigoplus_{\alpha,\beta}\dot{A}_{\alpha,\beta},\mu_{l},\mu_{r})$
with

$\dot{A}_{\alpha,\beta}\coloneqq A_{-\alpha,-\beta}.$

######  Proposition 1.3.11.

Let $A$ be an $\mathfrak{h}$-algebra, and $(m,\Delta,S)$ be an
$\mathfrak{h}$-Hopf algebroid structure on $A$ with multiplication $m\colon
A\otimes A\to A$, comultiplication $\Delta\colon A\to
A\mathbin{\widetilde{\otimes}}A$ and antipode $S\colon A\to A$. Then we have
the following new $\mathfrak{h}$-Hopf algebroids.

$\displaystyle
A^{\textup{op}}\coloneqq(\dot{A},m^{\textup{op}},\Delta,S^{-1}),\quad
A^{\textup{cop}}\coloneqq(A^{\dagger},m,\Delta^{\textup{op}},S^{-1}),\quad
A^{\textup{op},\textup{cop}}\coloneqq(\dot{A}^{\dagger},m^{\textup{op}},\Delta^{\textup{op}},S).$
(1.3.52)

Here we defined the opposite operations as $m^{\textup{op}}\coloneqq
m\circ\tau$ and $\Delta^{\textup{op}}\coloneqq\widetilde{\tau}\circ\Delta$,
using the flip $\tau\colon A\otimes A\to A\otimes A$, $a\otimes b\mapsto
b\otimes a$ and the map $\widetilde{\tau}\colon
A\mathbin{\widetilde{\otimes}}A\to(A^{\dagger}\mathbin{\widetilde{\otimes}}A^{\dagger})^{\dagger}$
in Lemma 1.3.10. Moreover, the antipode $S$ induces $\mathfrak{h}$-Hopf
algebroid isomorphisms $A\xrightarrow{\sim}A^{\textup{op},\textup{cop}}$ and
$A^{\textup{op}}\xrightarrow{\sim}A^{\textup{cop}}$.

The proof is straightforward, and we omit it. Note that in the case
$\mathfrak{h}=0$, we have $\mathbin{\widetilde{\otimes}}=\otimes$ and
$\widetilde{\tau}=\tau$, and recover the opposite Hopf algebras in (1.3.52).

Now we go back to the elliptic algebra $U=U_{q,p}(\widehat{\mathfrak{g}})$. We
equip it with the $H$-algebra structure in Lemma 1.3.8. Then the $H$-algebra
$U^{\dagger}$ in (1.3.50) has the $H^{*}$-bigrading
$U^{\dagger}=\bigoplus_{\alpha,\beta\in H^{*}}U^{\dagger}_{\alpha,\beta}$ with

$U^{\dagger}_{\alpha,\beta}\coloneqq\\{a\in U\mid
q^{P+h}aq^{-(P+h)}=q^{\langle\alpha,P+h\rangle}a,\,q^{P}aq^{-P}=q^{\langle\beta,P\rangle}a\
(\forall\,P+h,P\in\widetilde{H})\\},$

and the moment maps are $\mu^{\dagger}_{l}(F)\coloneqq F(P+h,p^{*})$ and
$\mu^{\dagger}_{r}(F)\coloneqq F(P,p)$ for $F=F(P,p)\in\mathbb{F}\llbracket
p\rrbracket$.

######  Remark 1.3.12.

The $H$-algebra structure on $U^{\dagger}$ is used in [Ko09, §3.1] and [Ko20,
§3.2] to study the comultiplication given in terms of the $L$-operators. Also,
it is used in [FKO14] to analyze the quantum $Z$-algebra. The tensor product
$(U^{\dagger}\mathbin{\widetilde{\otimes}}U^{\dagger})^{\dagger}$ is nothing
but the opposite tensor product of $U$ used in [Ko09, Ko20]. Let us denote

$U\mathbin{\widetilde{\otimes}}^{\textup{op}}U\coloneqq
U^{\dagger}\mathbin{\widetilde{\otimes}}U^{\dagger}.$

Then, the opposite comultiplication $\Delta^{\textup{op}}$ in Proposition
1.3.11 can be written as

$\displaystyle\Delta^{\textup{op}}(e_{i}(z))=1\mathbin{\widetilde{\otimes}}^{\textup{op}}e_{i}(z)+e_{i}(q^{c_{1}}z)\mathbin{\widetilde{\otimes}}^{\textup{op}}\psi^{+}_{i}(q^{c_{1}/2}z),$
$\displaystyle\Delta^{\textup{op}}(f_{i}(z))=f_{i}(z)\mathbin{\widetilde{\otimes}}^{\textup{op}}1+\psi^{-}_{i}(q^{c_{2}/2}z)\mathbin{\widetilde{\otimes}}^{\textup{op}}f_{i}(q^{c_{2}}z),$
$\displaystyle\Delta^{\textup{op}}(\psi_{i}^{+}(z))=\psi_{i}^{+}(q^{-c_{1}/2}z)\mathbin{\widetilde{\otimes}}^{\textup{op}}\psi_{i}^{+}(q^{c_{2}/2}z),$
$\displaystyle\Delta^{\textup{op}}(\psi_{i}^{-}(z))=\psi_{i}^{-}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}^{\textup{op}}\psi_{i}^{-}(q^{-c_{2}/2}z).$

These formulas coincide with the Drinfeld-type comultiplication in [J+99,
Appendix B] up to $q$-factors. See also [Fa16, KO] where similar
comultiplication appears.

##  2\. Dynamical Ding-Iohara algebroids of simply-laced root systems

In this section, we introduce an family of Hopf algebroids which includes the
elliptic algebras $U_{q,p}(\widehat{g})$ (see § 1.3) of simply-laced types,
and whose Hopf algebra degeneration is the Ding-Iohara quantum algebra (see §
1.2). As in the previous § 1.3, we use the word “a topological algebra” to
mean the algebra over $\mathbb{C}\llbracket p\rrbracket$ with $p$-adic
topology.

###  2.1. Algebra structure

We use the notation of finite root systems in § 1.3.1. Thus, let

$A=A(X_{l})=(a_{ij})_{i,j\in I},\quad I=\\{1,\dotsc,l\\}$

be the Cartan matrix of finite type $X_{l}$ in the sense of Kac [Kac90,
Theorem 4.3, §4.8, TABLE Fin]. At this moment, we do not restrict $X_{l}$ to
be simply-laced. We choose the symmetrization (1.3.1):

$A=DB,\quad B=(b_{ij})_{i,j\in I},\quad
D^{-1}=\operatorname{diag}(d_{i})_{i\in I}$

with $(d_{1},\dotsc,d_{l})$ in (1.3.2), and fix a realization
$(\mathfrak{h},\Pi,\Pi^{\vee})$ of $A$, consisting of the Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{g}$, the simple roots $\Pi=\\{\alpha_{i}\mid
i\in I\\}\subset\mathfrak{h}^{*}$ and the simple coroots
$\Pi^{\vee}=\\{\alpha^{\vee}_{i}\mid i\in I\\}\subset\mathfrak{h}$.

We also use the notation of dynamical parameters in § 1.3.2. Thus, we have
the linear spaces (1.3.6), (1.3.8), (1.3.13)

$\displaystyle\begin{split}H=\sum_{i\in I}\mathbb{C}P_{i}\ \subset\
&\widetilde{H}=\sum_{i\in I}\mathbb{C}(P_{i}+h_{i})+\sum_{i\in
I}\mathbb{C}P_{i},\\\ H^{*}=\sum_{i\in I}\mathbb{C}Q_{i}\ \subset\
&\widetilde{H}^{*}=\sum_{i\in I}\mathbb{C}(Q_{i}+\alpha_{i})+\sum_{i\in
I}\mathbb{C}Q_{i}\end{split}$ (2.1.1)

equipped with the bilinear form (1.3.15):

$\langle
Q_{i},P_{j}\rangle=b_{ij},\quad\langle\alpha_{i},h_{j}\rangle=a_{ji},\quad\langle\text{others}\rangle=0.$

We will regard $H$ as a commutative Lie algebra, and apply the formalism of
$H$-Hopf algebroids in § 1.1.

Finally, we introduce the structure functions satisfying a modified Ding-
Iohara condition.

######  Definition 2.1.1.

Let $p$ be a formal parameter, and $\\{G^{\pm}_{ij}(z;p)\mid i,j\in I\\}$ be a
set of functions satisfying either of the following _Ding-Iohara condition_.

  1. 1

$G^{\pm}_{ij}(z;p)\in\mathbb{C}[z^{\pm 1}]\llbracket p\rrbracket$, and is
invertible in $\mathbb{C}(\\!(z)\\!)\llbracket p\rrbracket$.

  2. 2

The series

$g_{ij}(z;p)\coloneqq\frac{G^{+}_{ij}(z;p)}{G^{-}_{ij}(z;p)}\in\mathbb{C}(\\!(z)\\!)\llbracket
p\rrbracket$

satisfies

$g_{ij}(z^{-1};p)=g_{ji}(z;p)^{-1}.$

We call $g\coloneqq\\{g_{ij}(z;p)\mid i,j\in I\\}$ _the structure functions_.

Note that we have definite limits

$\displaystyle\overline{G}^{\pm}_{ij}(z)\coloneqq\lim_{p\to
0}G^{\pm}_{ij}(z;p)$ (2.1.2)

which belong to $\mathbb{C}[z^{\pm 1}]$, and these limits are invertible in
the field $\mathbb{C}(\\!(z)\\!)$. In particular, we have a well-defined
function

$\displaystyle\overline{g}_{ij}(z)\coloneqq\frac{\overline{G}^{+}_{ij}(z)}{\overline{G}^{-}_{ij}(z)}\
\in\mathbb{C}(\\!(z)\\!),$ (2.1.3)

which satisfy the condition (1.2.3):

$\displaystyle\overline{g}_{ij}(z^{-1})=\overline{g}_{ji}(z)^{-1}.$ (2.1.4)

Recall that we are considering the $p$-adic topology. Now let us introduce:

######  Definition 2.1.2.

Let $A=A(X_{l})$ be the Cartan matrix of type $X_{l}=A_{l},D_{l}$ or
$E_{6,7,8}$, and take the symmetrization $A=DB$ with
$D^{-1}=(d_{1},\dotsc,d_{l})$ in (1.3.2), a complex parameter $q$ with
$0<\left|q\right|<1$, a formal parameter $p$, and the structure functions
$g=\\{g_{i,j}\mid i,j\in I\\}$ as in Definition 2.1.1. We define

$U_{q,p}(g,X_{l})$

to be the topological algebra over $\mathbb{C}\llbracket p\rrbracket$
generated by

$\displaystyle\operatorname{\mathcal{M}}(\widetilde{H}^{*}),\ q^{\pm c/2},\
d,\ K^{\pm}_{i},\ e_{i,m},\ f_{i,m},\ \psi^{\pm}_{i,m}\quad(i\in
I,\,m\in\mathbb{Z})$ (2.1.5)

with the following relations. Below we use the generating currents

$e_{i}(z)\coloneqq\sum_{m\in\mathbb{Z}}e_{i,m}z^{-m},\quad
f_{i}(z)\coloneqq\sum_{m\in\mathbb{Z}}f_{i,m}z^{-m},\quad\psi^{\pm}_{i}(z)\coloneqq\sum_{m\in\mathbb{Z}}\psi^{\pm}_{i,m}z^{-m},$

and the genuine relations are obtained by expansion with respect to $p$, as in
Definition 1.3.2 of the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$.

  * •

The generators $\operatorname{\mathcal{M}}(\widetilde{H}^{*}),q^{\pm
c/2},\psi_{i}^{\pm}(z)$ and $K^{\pm}_{i}$ should satisfy:

$\displaystyle\begin{split}&\text{The field
$\operatorname{\mathcal{M}}(\widetilde{H}^{*})$ is a subalgebra.}\\\
&\text{$q^{\pm c/2}$ are central and $q^{c/2}q^{-c/2}=1$.}\\\
&\text{$\psi_{i}^{\pm}(z)$ and $K^{\pm}_{i}$ are invertible.}\end{split}$
(2.1.6)

  * •

For $F(P),F(P+h)\in\operatorname{\mathcal{M}}(\widetilde{H}^{*})$, we have

$\displaystyle F(P+h)d=dF(P+h),$ $\displaystyle F(P)d=dF(P),$ (2.1.7)
$\displaystyle F(P+h)K^{\pm}_{i}=K^{\pm}_{i}F(P+h\mp\langle
Q_{i},P+h\rangle),$ $\displaystyle F(P)K^{\pm}_{i}=K^{\pm}_{i}F(P\mp\langle
Q_{i},P\rangle),$ (2.1.8) $\displaystyle F(P+h)e_{i}(z)=e_{j}(z)F(P+h),$
$\displaystyle F(P)e_{i}(z)=e_{i}(z)F(P-\langle Q_{i},P\rangle),$ (2.1.9)
$\displaystyle F(P+h)f_{i}(z)=f_{i}(z)F(P+h-\langle Q_{i},P+h\rangle),$
$\displaystyle F(P)f_{i}(z)=f_{i}(z)F(P),$ (2.1.10) $\displaystyle
F(P+h)\psi^{\pm}_{i}(z)=\psi^{\pm}_{i}(z)F(P+h-\langle Q_{i},P+h\rangle),$
$\displaystyle F(P)\psi^{\pm}_{i}(z)=\psi^{\pm}_{i}(z)F(P-\langle
Q_{i},P\rangle).$ (2.1.11)

  * •

The remaining relations with $d$ are

$\displaystyle[d,K^{\pm}_{i}]=0,\quad[d,e_{i,m}]=me_{i,m},\quad[d,f_{i,m}]=mf_{i,m},\quad[d,\psi^{\pm}_{i}]=m\psi^{\pm}_{i,m}.$
(2.1.12)

  * •

The remaining relations containing $K^{\pm}_{i}$ are given by

$\displaystyle K^{\pm}_{i}e_{j}(z)=q_{i}^{\mp a_{ij}}e_{j}(z)K^{\pm}_{i},\quad
K^{\pm}_{i}f_{j}(z)=q_{i}^{\pm
a_{ij}}f_{j}(z)K^{\pm}_{i},\quad[K^{\pm}_{i},\psi^{\pm}_{i,m}]=0.$ (2.1.13)

  * •

Hereafter we use

$g_{ij}(z)\coloneqq g_{ij}(z;p),\quad g^{*}_{ij}(z)\coloneqq g_{ij}(z;p^{*}).$

Then the remaining relations containing $\psi^{\pm}_{i}$ are given by

$\displaystyle\psi^{\pm}_{i}(z)\psi^{\pm}_{j}(w)=\frac{g^{*}_{ij}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\psi^{\pm}_{j}(w)\psi^{\pm}_{i}(z),$
$\displaystyle\psi^{+}_{i}(z)\psi^{-}_{j}(w)=\frac{g^{*}_{ij}(q^{-c}\frac{z}{w})}{g_{ij}(q^{c}\frac{z}{w})}\psi^{-}_{j}(w)\psi^{+}_{i}(z),$
(2.1.14)
$\displaystyle\psi^{+}_{i}(z)e_{j}(w)=g_{ij}^{*}(q^{-c/2}\tfrac{z}{w})e_{j}(w)\psi^{+}_{i}(z),$
$\displaystyle\psi^{+}_{i}(z)f_{j}(w)=g_{ij}(q^{c/2}\tfrac{z}{w})^{-1}f_{j}(w)\psi^{+}_{i}(z),$
(2.1.15)
$\displaystyle\psi^{-}_{i}(z)e_{j}(w)=g_{ij}^{*}(q^{c/2}\tfrac{z}{w})e_{j}(w)\psi^{-}_{i}(z),$
$\displaystyle\psi^{-}_{i}(z)f_{j}(w)=g_{ij}(q^{-c/2}\tfrac{z}{w})^{-1}f_{j}(w)\psi^{-}_{i}(z).$
(2.1.16)

  * •

Hereafter we use

$G^{\pm}_{ij}(z)\coloneqq G^{\pm}_{ij}(z;p),\quad G^{\pm*}_{ij}(z)\coloneqq
G^{\pm}_{ij}(z;p^{*}).$

Then the remaining relations consist of the quadratic relations

$\displaystyle
G^{-*}_{ij}(\tfrac{z}{w})e_{i}(z)e_{j}(w)=G^{+*}_{ij}(\tfrac{z}{w})e_{j}(w)e_{i}(z),\quad
G^{+}_{ij}(\tfrac{z}{w})f_{i}(z)f_{j}(w)=G^{-}_{ij}(\tfrac{z}{w})f_{j}(w)f_{i}(z),$
(2.1.17)
$\displaystyle[e_{i}(z),f_{j}(w)]=\frac{\delta_{i,j}}{q-q^{-1}}\bigl{(}\delta(q^{-c}\tfrac{z}{w})\psi^{-}_{i}(q^{c/2}w)-\delta(q^{c}\tfrac{z}{w})\psi^{+}_{i}(q^{c/2}z)\bigr{)}$
(2.1.18)

for any $i,j\in I$, and the following Serre-type relations (2.1.20) and
(2.1.21) for $i,j\in I$ such that $a_{ij}=-1$. Using
$\widetilde{g}^{*}_{ij}(z)\coloneqq\overline{g}_{ij}(z)/g^{*}_{ij}(z)$ with
$\overline{g}_{ij}(z)$ given in (2.1.3) and

$\displaystyle
h_{ij}\coloneqq\frac{(\overline{g}_{ii}(\frac{z_{1}}{z_{2}})+1)(\overline{g}_{ij}(\frac{z_{1}}{z})\overline{g}_{ij}(\frac{z_{2}}{z})+1)}{\overline{g}_{ij}(\frac{z_{2}}{z})+\overline{g}_{ii}(\frac{z_{1}}{z_{2}})\overline{g}_{ij}(\frac{z_{1}}{z})},$
(2.1.19)

the Serre-type relation for $e_{i}(z),e_{j}(z)$ is given by

$\displaystyle\begin{split}&\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{1})e_{i}(z_{2})e_{j}(z)-h_{ij}\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})e_{i}(z_{1})e_{j}(z)e_{i}(z_{2})\\\
&+\widetilde{g}^{*}_{ii}(\tfrac{z_{1}}{z_{2}})e_{j}(z)e_{i}(z_{1})e_{i}(z_{2})+\widetilde{g}^{*}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{2})e_{i}(z_{1})e_{j}(z)\\\
&-h_{ij}\widetilde{g}^{*}_{ij}(\tfrac{z_{2}}{z})e_{i}(z_{2})e_{j}(z)e_{i}(z_{1})+e_{j}(z)e_{i}(z_{2})e_{i}(z_{1})=0.\end{split}$
(2.1.20)

Similarly, using $\widetilde{g}_{ij}(z)\coloneqq
g_{ij}(z)/\overline{g}_{ij}(z)$, the Serre-type relation for
$f_{i}(z),f_{j}(z)$ is given by

$\displaystyle\begin{split}&\widetilde{g}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}_{ij}(\tfrac{z_{2}}{z})f_{i}(z_{1})f_{i}(z_{2})f_{j}(z)-h_{ij}\widetilde{g}_{ii}(\tfrac{z_{1}}{z_{2}})\widetilde{g}_{ij}(\tfrac{z_{1}}{z})f_{i}(z_{1})f_{j}(z)f_{i}(z_{2})\\\
&+\widetilde{g}_{ii}(\tfrac{z_{1}}{z_{2}})f_{j}(z)f_{i}(z_{1})f_{i}(z_{2})+\widetilde{g}_{ij}(\tfrac{z_{1}}{z})\widetilde{g}_{ij}(\tfrac{z_{2}}{z})f_{i}(z_{2})f_{i}(z_{1})f_{j}(z)\\\
&-h_{ij}\widetilde{g}_{ij}(\tfrac{z_{2}}{z})f_{i}(z_{2})f_{j}(z)f_{i}(z_{1})+f_{j}(z)f_{i}(z_{2})f_{i}(z_{1})=0.\end{split}$
(2.1.21)

Note that the function $h_{ij}$ of (2.1.19) coincides with (1.2.18) appearing
in the Serre-type relations for the Ding-Iohara quantum algebra with
$g_{ij}(z)$ replaced by $\overline{g}_{ij}(z)$.

######  Remark 2.1.3.

By the same discussion of Remark 1.3.3 (5), the topological algebra
$U_{q,p}(g,X_{l})$ is independent, up to isomorphism, of the choice of a
symmetrization of the Cartan matrix $A$.

######  Remark 2.1.4.

Instead of Definition 2.1.1, we can use broader class of the structure
functions $g=\\{g_{ij}(z;p)\mid i,j\in I\\}$ with some care. Let us explain
two such classes.

  1. (1)

Let $p$ and $p^{\prime}$ be formal parameters, and $\\{G^{\pm}_{ij}(z;p)\mid
i,j\in I\\}$ be a set of series satisfying the following two conditions.

    1. 1

$G^{\pm}_{ij}(z;p)\in\mathbb{C}[z^{\pm 1}]\llbracket
p^{\prime}\rrbracket\llbracket p\rrbracket$, and is invertible in
$\mathbb{C}(\\!(z)\\!)\llbracket p^{\prime}\rrbracket\llbracket p\rrbracket$.

    2. 2

The series $g_{ij}(z;p)\coloneqq
G^{+}_{ij}(z;p)/G^{-}_{ij}(z;p)\in\mathbb{C}(\\!(z)\\!)\llbracket
p^{\prime}\rrbracket\llbracket p\rrbracket$ satisfies
$g_{ij}(z^{-1};p)=g_{ji}(z;p)^{-1}$.

Then we have the definite limits $\overline{g}_{ij}(z)\in\mathbb{C}\llbracket
z\rrbracket\llbracket p^{\prime}\rrbracket$, and considering the $p$\- and
$p^{\prime}$-adic topology, we obtain the topological algebra $U$.

  2. (2)

Let $\\{G^{\pm}_{ij}(z;p)\mid i,j\in I\\}$ satisfy the following two
conditions.

    1. 1

$G^{\pm}_{ij}(z;p)\in\mathcal{A}\llbracket p\rrbracket$, and is invertible in
$(\operatorname{Frac}\mathcal{A})\llbracket p\rrbracket$. Here we used the
function ring $\mathcal{A}$ in (1.2.1).

    2. 2

The series $g_{ij}(z;p)\coloneqq
G^{+}_{ij}(z;p)/G^{-}_{ij}(z;p)\in(\operatorname{Frac}\mathcal{A})\llbracket
p\rrbracket$ satisfies $g_{ij}(z^{-1};p)=g_{ji}(z;p)^{-1}$.

Then we have the definite limits
$\overline{g}_{ij}(z)\in\operatorname{Frac}\mathcal{A}$, and obtain the
genuine relations of $U$ by expanding generating currents and series first in
terms of $p$, and second in terms of the current/series variable. However,
this process yields relations containing infinite summations in general, and
we should take some solution such as 1 in Remark 1.2.2.

The Serre-type relations (2.1.20) and (2.1.21) look rather complicated, but
the other defining relations are the same as those of the elliptic algebra
$U_{q,p}(\widehat{g})$ of type $X_{l}=A_{l},D_{l}$ or $E_{6,7,8}$ in
Definition 1.3.2. Indeed, recalling (1.3.41), we have:

######  Proposition 2.1.5.

In Definition 2.1.2 of $U_{q,p}(g,X_{l})$ for type
$X_{l}=A_{l},D_{l},E_{6,7,8}$, set the structure functions
$g=\\{g_{ij}(z;p)\mid i,j\in I\\}$ as

$\displaystyle
g_{ij}(x;p)=\frac{G_{ij}^{+}(x;p)}{G_{ij}^{-}(x;p)}\coloneqq\frac{q^{-b_{ij}}\theta(q^{b_{ij}}x;p)}{\theta(q^{-b_{ij}}x;p)},$
(2.1.22)

where $B=(b_{ij})_{i,j\in I}$ is the symmetrization $A=DB$ of the Cartan
matrix $A=A(X_{l})$. Note that these satisfy the Ding-Iohara condition
(Definition 2.1.1). Then we have an isomorphism of topological algebras

$U_{q,p}(g,X_{l})\cong U_{q,p}(\widehat{g}).$

###### Proof.

The generators of both topological algebra coincide. Also, using Lemmas 1.3.4,
1.3.5 and 1.3.6 and comparing the fundamental relations of
$U_{q,p}(\widehat{g})$ in Definition 1.3.2 and those of $U_{q,p}(g,X_{l})$ in
Definition 2.1.2, we immediately find that they coincide except the Serre-type
relations (1.3.33) and (1.3.34). So it remains to check that the
specialization (2.1.22) turns the relations (2.1.20) and (2.1.21) into
(1.3.33) and (1.3.34), respectively. First, since we are considering the
simply-laced types, the structure functions $g^{*}_{ij}(z)=g_{ij}(z;p^{*})$
and their limits $\overline{g}_{ij}(z)$ are given by

$\displaystyle
g^{*}_{ii}(z)=q^{-2}\frac{\theta^{*}(q^{2}z)}{\theta^{*}(q^{-2}z)},\quad
g^{*}_{ij}(z)=q\frac{\theta^{*}(q^{-1}z)}{\theta^{*}(qz)},\quad\overline{g}_{ii}(z)=q^{-2}\frac{1-q^{2}z}{1-q^{-2}z},\quad\overline{g}_{ij}(z)=q\frac{1-q^{-1}z}{1-qz}$

with $\theta^{*}(x)\coloneqq\theta(x;p^{*})=(x,p^{*}/x;p^{*})_{\infty}$. Then,
using (1.3.17), the function $h_{ij}$ in (2.1.19) is equal to

$\displaystyle h_{ij}=q+q^{-1}=\genfrac{[}{]}{0.0pt}{}{2}{1}_{i}$ (2.1.23)

for any $i\neq j$. (See § 3.1 for a generalization of this equality.) Also,
denoting $z_{a0}\coloneqq z_{a}/z_{0}$ and $z_{0a}\coloneqq z_{0}/z_{a}$ for
$a=1,2$, we have

$\displaystyle\frac{\overline{g}_{ij}(z_{a0})}{g^{*}_{ij}(z_{a0})}=\frac{(p^{*}qz_{a0},p^{*}/(qz_{a0});p^{*})}{(p^{*}q^{-1}z_{a0},p^{*}/(q^{-1}z_{a0});p^{*})}=\frac{\widetilde{\gamma}^{*}(z_{0a};q^{-1})}{\widetilde{\gamma}^{*}(z_{a0};q^{-1})},$
(2.1.24)

where
$\widetilde{\gamma}^{*}(z;q)\coloneqq\widetilde{\gamma}(z;q,p^{*})=(p^{*}qx;p^{*})_{\infty}/(p^{*}q^{-1}z;p^{*})_{\infty}$
using the function $\widetilde{\gamma}$ in (1.3.32). Now, we can rewrite the
last second term of (2.1.20) as

$\displaystyle-
h_{ij}\frac{\overline{g}_{ij}(z_{20})}{g^{*}_{ij}(z_{20})}e_{i}(z_{2})e_{j}(z)e_{i}(z_{1})=-\genfrac{[}{]}{0.0pt}{}{2}{1}_{i}\frac{\widetilde{\gamma}^{*}(z_{12};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}\frac{\widetilde{\gamma}^{*}(z_{02};q^{-1})}{\widetilde{\gamma}^{*}(z_{20};q^{-1})}\frac{\widetilde{\gamma}^{*}(z_{10};q^{-1})}{\widetilde{\gamma}^{*}(z_{10};q^{-1})}e_{i}(z_{2})e_{j}(z)e_{i}(z_{1}),$
(2.1.25)

where we denoted $z_{12}\coloneqq z_{1}/z_{2}$. Note that we inserted two
trivial fractions in the right hand side. We can similarly rewrite the fourth
term of (2.1.20) as

$\displaystyle\frac{\overline{g}_{ij}(z_{10})}{g^{*}_{ij}(z_{10})}\frac{\overline{g}_{ij}(z_{20})}{g^{*}_{ij}(z_{20})}e_{i}(z_{2})e_{i}(z_{1})e_{j}(z)=\frac{\widetilde{\gamma}^{*}(z_{12};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}\frac{\widetilde{\gamma}^{*}(z_{01};q^{-1})}{\widetilde{\gamma}^{*}(z_{10};q^{-1})}\frac{\widetilde{\gamma}^{*}(z_{02};q^{-1})}{\widetilde{\gamma}^{*}(z_{20};q^{-1})}e_{i}(z_{2})e_{i}(z_{1})e_{j}(z).$
(2.1.26)

Next, using

$\displaystyle\frac{\overline{g}_{ii}(z_{12})}{g^{*}_{ii}(z_{12})}=\frac{(p^{*}q^{-2}z_{12},p^{*}/(q^{-2}z_{12});p^{*})}{(p^{*}q^{2}z_{12},p^{*}/(q^{2}z_{12});p^{*})}=\frac{\widetilde{\gamma}^{*}(z_{21};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}$
(2.1.27)

with $z_{21}\coloneqq z_{2}/z_{1}$, we can rewrite the third term of (2.1.20)
as

$\displaystyle\frac{\overline{g}_{ii}(z_{12})}{g^{*}_{ii}(z_{12})}e_{j}(z)e_{i}(z_{1})e_{i}(z_{2})=\frac{\widetilde{\gamma}^{*}(z_{21};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}\frac{\widetilde{\gamma}^{*}(z_{10};q^{-1})}{\widetilde{\gamma}^{*}(z_{10};q^{-1})}\frac{\widetilde{\gamma}^{*}(z_{20};q^{-1})}{\widetilde{\gamma}^{*}(z_{20};q^{-1})}e_{j}(z)e_{i}(z_{1})e_{i}(z_{2}),$
(2.1.28)

By a similar calculation, we can rewrite the second term as

$\displaystyle\begin{split}&-h_{ij}\frac{\overline{g}_{ii}(z_{12})}{g^{*}_{ii}(z_{12})}\frac{\overline{g}_{ij}(z_{10})}{g^{*}_{ij}(z_{10})}e_{i}(z_{1})e_{j}(z)e_{i}(z_{2})\\\
&=-\genfrac{[}{]}{0.0pt}{}{2}{1}_{i}\frac{\widetilde{\gamma}^{*}(z_{21};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}\frac{\widetilde{\gamma}^{*}(z_{01};q^{-1})}{\widetilde{\gamma}^{*}(z_{10};q^{-1})}\frac{\widetilde{\gamma}^{*}(z_{20};q^{-1})}{\widetilde{\gamma}^{*}(z_{20};q^{-1})}e_{i}(z_{1})e_{j}(z)e_{i}(z_{2}),\end{split}$
(2.1.29)

and the first terms as

$\displaystyle\begin{split}&\frac{\overline{g}_{ii}(z_{12})}{g^{*}_{ii}(z_{12})}\frac{\overline{g}_{ij}(z_{10})}{g^{*}_{ij}(z_{10})}\frac{\overline{g}_{ij}(z_{20})}{g^{*}_{ij}(z_{20})}e_{i}(z_{1})e_{i}(z_{2})e_{j}(z)\\\
&=\frac{\widetilde{\gamma}^{*}(z_{21};q^{2})}{\widetilde{\gamma}^{*}(z_{12};q^{2})}\frac{\widetilde{\gamma}^{*}(z_{01};q^{-1})}{\widetilde{\gamma}^{*}(z_{10};q^{-1})}\frac{\widetilde{\gamma}^{*}(z_{02};q^{-1})}{\widetilde{\gamma}^{*}(z_{20};q^{-1})}e_{i}(z_{1})e_{i}(z_{2})e_{j}(z).\end{split}$
(2.1.30)

Finally, combining (2.1.25)–(2.1.30), reordering, and multiplying by
$\widetilde{\gamma}^{*}(z_{21};q^{2})\widetilde{\gamma}^{*}(z_{20};q^{-1})\widetilde{\gamma}^{*}(z_{10};q^{-1})$,
we can rewrite (2.1.20) as

$\displaystyle 0=$
$\displaystyle\widetilde{\gamma}^{*}(z_{21};q^{2})\Bigl{(}\widetilde{\gamma}^{*}(z_{10};q^{-1})\widetilde{\gamma}^{*}(z_{20};q^{-1})e_{j}(z)e_{i}(z_{1})e_{i}(z_{2})-\genfrac{[}{]}{0.0pt}{}{2}{1}_{i}\widetilde{\gamma}^{*}(z_{01};q^{-1})\widetilde{\gamma}^{*}(z_{20};q^{-1})e_{i}(z_{1})e_{j}(z)e_{i}(z_{2})$
$\displaystyle\hskip
40.00006pt+\widetilde{\gamma}^{*}(z_{01};q^{-1})\widetilde{\gamma}^{*}(z_{02};q^{-1})e_{i}(z_{1})e_{i}(z_{2})e_{j}(z)\Bigr{)}$
$\displaystyle+$
$\displaystyle\widetilde{\gamma}^{*}(z_{12};q^{2})\Bigl{(}\widetilde{\gamma}^{*}(z_{20};q^{-1})\widetilde{\gamma}^{*}(z_{10};q^{-1})e_{j}(z)e_{i}(z_{2})e_{i}(z_{1})-\genfrac{[}{]}{0.0pt}{}{2}{1}_{i}\widetilde{\gamma}^{*}(z_{02};q^{-1})\widetilde{\gamma}^{*}(z_{10};q^{-1})e_{i}(z_{2})e_{j}(z)e_{i}(z_{1})$
$\displaystyle\hskip
40.00006pt+\widetilde{\gamma}^{*}(z_{02};q^{-1})\widetilde{\gamma}^{*}(z_{01};q^{-1})e_{i}(z_{2})e_{i}(z_{1})e_{j}(z)\Bigr{)},$

which is nothing but (1.3.33) for simply-laced types. In the same way, we can
check that (2.1.21) is equivalent to (1.3.34) under the assumption (2.1.22).
Hence we have the statement. ∎

######  Remark 2.1.6.

We discovered the combination
$\overline{g}_{ij}(z_{a}/z_{b})/g_{ij}(z_{a}/z_{b})$ and
$\overline{g}_{ij}(z_{a}/z_{b})/g^{*}_{ij}(z_{a}/z_{b})$ in the Serre-type
relations (2.1.20) and (2.1.21) to establish the equalities (2.1.24) and
(2.1.27) with the function $\widetilde{\gamma}$. The function $h_{ij}$ is
discovered by the well-definedness of the Drinfeld-type comultiplication,
which will be commented in Remark 2.2.3.

###  2.2. $H$-Hopf algebroid structure

Hereafter in this § 2.2, we denote

$U\coloneqq
U_{q,p}(g,X_{l}),\quad\mathbb{F}\coloneqq\operatorname{\mathcal{M}}(H^{*})$

for simplicity. Let us introduce an $H$-Hopf algebroid structure on $U$ with
$H=\sum_{i\in I}\mathbb{C}P_{i}$ in (2.1.1). We begin with:

######  Lemma 2.2.1.

$U=U_{q,p}(g,X_{l})$ is a topological $H$-algebra with the $H^{*}$-bigrading
$U=\bigoplus_{\alpha,\beta\in H^{*}}U_{\alpha,\beta}$ given by

$\displaystyle U_{\alpha,\beta}\coloneqq\\{a\in U\mid
q^{P}aq^{-P}=q^{\langle\alpha,P\rangle}a,\
q^{P+h}aq^{-P-h}=q^{\langle\beta,P+h\rangle}a\quad(P,P+h\in\widetilde{H})\\},$

and the moment maps $\mu_{l},\mu_{r}\colon\mathbb{F}\llbracket p\rrbracket\to
U_{0,0}$ defined by

$\displaystyle\mu_{l}(F)\coloneqq F(P,p^{*}),\quad\mu_{r}(F)\coloneqq
F(P+h,p)$

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$. Moreover, the bigrading
values in the lattice $\mathcal{Q}\coloneqq\bigoplus_{i\in
I}\mathbb{Z}Q_{i}\subsetneq H^{*}$.

###### Proof.

The conditions of the moment maps can be checked directly. For instance,

$\displaystyle\mu_{l}(F)(e_{i}(z))=F(P,p^{*})e_{i}(z)=e_{i}(z)F(P-\langle
Q_{i},P\rangle,p)=e_{i}(z)\mu_{l}\bigl{(}T_{-Q_{i}}(F)\bigr{)},$
$\displaystyle\mu_{r}(F)(e_{i}(z))=F(P+h,p^{*})e_{i}(z)=e_{i}(z)F(P+h,p)=e_{i}(z)\mu_{r}\bigl{(}T_{0}(F)\bigr{)}.$

The last statement on the bigrading holds since

$\displaystyle e_{i}(z)\in U_{-Q_{i},0},\quad f_{i}(z)\in
U_{0,-Q_{i}},\quad\psi^{\pm}_{i}(z)\in U_{-Q_{i},-Q_{i}},\quad K^{\pm}_{i}\in
U_{-Q_{i},-Q_{i}}$

(see Lemma 1.3.8 (1) for the meaning) and the other generators belong to
$U_{0,0}$ ∎

For later use, let us write down the isomorphisms
$D_{H}\mathbin{\widetilde{\otimes}}U\cong U\cong
U\mathbin{\widetilde{\otimes}}D_{H}$ in (1.1.6). By (1.1.2) and (1.1.5), we
have

$\displaystyle(D_{H}\mathbin{\widetilde{\otimes}}U)_{\alpha,\beta}=(D_{H})_{\alpha,\alpha}\otimes_{\mathbb{F}}U_{\alpha,\beta}=\mathbb{F}T_{-\alpha}\otimes_{\mathbb{F}}U_{\alpha,\beta},$
$\displaystyle(U\mathbin{\widetilde{\otimes}}D_{H})_{\alpha,\beta}=U_{\alpha,\beta}\otimes_{\mathbb{F}}(D_{H})_{\beta,\beta}=U_{\alpha,\beta}\otimes_{\mathbb{F}}\mathbb{F}T_{-\beta}$

for $\alpha,\beta\in\mathcal{Q}\subset H^{*}$. Then, by (1.1.3), we find that
the isomorphisms are given by

$\displaystyle(D_{H}\mathbin{\widetilde{\otimes}}U)_{\alpha,\beta}=\mathbb{F}T_{-\alpha}\otimes_{\mathbb{F}}U_{\alpha,\beta}\xrightarrow{\,\sim\,}U_{\alpha,\beta},\quad
FT_{-\alpha}\mathbin{\widetilde{\otimes}}a=T_{-\alpha}\mathbin{\widetilde{\otimes}}\mu_{l}(F)a\longmapsto\mu_{l}(F)a$
(2.2.1)

and

$\displaystyle(U\mathbin{\widetilde{\otimes}}D_{H})_{\alpha,\beta}=U_{\alpha,\beta}\otimes_{\mathbb{F}}\mathbb{F}T_{-\beta}\xrightarrow{\,\sim\,}U_{\alpha,\beta},\quad
a\mathbin{\widetilde{\otimes}}FT_{-\beta}=\mu_{r}(F)a\otimes
T_{-\beta}\longmapsto a\mu_{r}(F)$ (2.2.2)

for $a\in U_{\alpha,\beta}$ and $F\in\mathbb{F}\llbracket p\rrbracket$.

Next, we introduce a Drinfeld-type comultiplication on $U$.

######  Lemma 2.2.2.

We define a homomorphism $\Delta\colon U\to U\mathbin{\widetilde{\otimes}}U$
of topological algebras by

$\displaystyle\Delta(\mu_{l}(F))\coloneqq\mu_{l}(F)\mathbin{\widetilde{\otimes}}1,\quad\Delta(\mu_{r}(F))\coloneqq
1\mathbin{\widetilde{\otimes}}\mu_{r}(F)$ (2.2.3)

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$,

$\displaystyle\Delta(q^{\pm c/2})\coloneqq q^{\pm
c/2}\mathbin{\widetilde{\otimes}}q^{\pm c/2},\quad\Delta(d)\coloneqq
d\mathbin{\widetilde{\otimes}}1+1\mathbin{\widetilde{\otimes}}d,\quad\Delta(K^{\pm}_{i})\coloneqq
K^{\pm}_{i}\mathbin{\widetilde{\otimes}}K^{\pm}_{i},$ (2.2.4)

and

$\displaystyle\Delta(e_{i}(z))\coloneqq
e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z),\qquad$
$\displaystyle\Delta(f_{i}(z))\coloneqq
1\mathbin{\widetilde{\otimes}}f_{i}(z)+f_{i}(q^{c_{2}}z)\mathbin{\widetilde{\otimes}}\psi^{-}_{i}(q^{c_{2}/2}z),$
(2.2.5)
$\displaystyle\Delta(\psi_{i}^{+}(z))\coloneqq\psi_{i}^{+}(q^{-c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{1}/2}z),\qquad$
$\displaystyle\Delta(\psi_{i}^{-}(z))\coloneqq\psi_{i}^{-}(q^{c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{-}(q^{-c_{1}/2}z)$
(2.2.6)

for the generating currents. Then $\Delta$ is a topological $H$-algebra
homomorphism.

###### Proof.

First, we show that $\Delta$ is well-defined as a homomorphism of topological
algebras. We will only write down the check for a part of the defining
relations of $U$. We can check the block (2.1.8)–(2.1.11) as

$\displaystyle\Delta(F(P))\Delta(e_{i}(z))$
$\displaystyle=\bigl{(}1\mathbin{\widetilde{\otimes}}F(P)\bigr{)}\bigl{(}1\mathbin{\widetilde{\otimes}}e_{i}(z)+e_{i}(q^{c_{2}}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{2}/2}z)\bigr{)}$
$\displaystyle=1\mathbin{\widetilde{\otimes}}F(P)e_{i}(z)+e_{i}(q^{c_{2}}z)\mathbin{\widetilde{\otimes}}F(P)\psi_{i}^{+}(q^{c_{2}/2}z)$
$\displaystyle=1\mathbin{\widetilde{\otimes}}e_{i}(z)F(P-\langle
Q_{i},P\rangle)+e_{i}(q^{c_{2}}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{2}/2}z)F(P-\langle
Q_{i},P\rangle)$ $\displaystyle=\Delta(e_{i}(z))\Delta(F(P-\langle
Q_{i},P\rangle)).$

The relations (2.1.13) can be checked as

$\displaystyle\Delta(K^{\pm}_{i})\Delta(\psi^{+}_{j}(w))$
$\displaystyle=\bigl{(}K^{\pm}_{i}\mathbin{\widetilde{\otimes}}K^{\pm}_{i}\bigr{)}\bigl{(}\psi^{+}_{j}(q^{c_{2}/2}w)\mathbin{\widetilde{\otimes}}\psi^{+}_{j}(q^{-c_{1}/2}w)\bigr{)}=K^{\pm}_{i}\psi^{+}_{j}(q^{c_{2}/2}w)\mathbin{\widetilde{\otimes}}K^{\pm}_{i}\psi^{+}_{j}(q^{-c_{1}/2}w)$
$\displaystyle=\psi_{j}^{+}(q^{c_{2}/2}w)K^{\pm}_{i}\mathbin{\widetilde{\otimes}}\psi^{+}_{j}(q^{-c_{1}/2}w)K^{\pm}_{i}=\Delta(\psi^{+}_{j}(w))\Delta(K^{\pm}_{i}).$

The block (2.1.14)–(2.1.16) can be checked as

$\displaystyle\Delta(\psi_{i}^{+}(z))\Delta(\psi_{j}^{+}(w))$
$\displaystyle=\psi_{i}^{+}(q^{c_{2}/2}z)\psi_{j}^{+}(q^{c_{2}/2}w)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{-c_{1}/2}z)\psi_{j}^{+}(q^{-c_{1}/2}w)$
$\displaystyle=\frac{g_{ij}^{*}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\psi_{j}^{+}(q^{c_{2}/2}w)\psi_{i}^{+}(q^{c_{2}/2}z)\mathbin{\widetilde{\otimes}}\frac{g_{ij}^{*}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\psi_{j}^{+}(q^{-c_{1}/2}w)\psi_{i}^{+}(q^{-c_{1}/2}z)$
$\displaystyle=\Delta\left(\frac{g_{ij}^{*}(\frac{z}{w})}{g_{ij}(\frac{z}{w})}\right)\Delta(\psi_{j}^{+}(w))\Delta(\psi_{i}^{+}(z)),$
$\displaystyle\Delta(\psi_{i}^{+}(z))\Delta(\psi_{j}^{-}(w))$
$\displaystyle=\psi_{i}^{+}(q^{c_{2}/2}z)\psi_{j}^{-}(q^{-c_{2}/2}w)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{-c_{1}/2}z)\psi_{j}^{-}(q^{c_{1}/2}w)$
$\displaystyle=\frac{g_{ij}^{*}(q^{-c_{1}+c_{2}}\frac{z}{w})}{g_{ij}(q^{c_{1}+c_{2}}\frac{z}{w})}\psi_{j}^{-}(q^{-c_{2}/2}w)\psi_{i}^{+}(q^{c_{2}/2}z)\mathbin{\widetilde{\otimes}}\frac{g_{ij}^{*}(q^{-c_{1}-c_{2}}\frac{z}{w})}{g_{ij}(q^{-c_{1}+c_{2}}\frac{z}{w})}\psi_{j}^{-}(q^{-c_{2}/2}w)\psi_{i}^{+}(q^{c_{2}/2}z)$
$\displaystyle=\Delta\left(\frac{g_{ij}^{*}(q^{-c}\frac{z}{w})}{g_{ij}(q^{c}\frac{z}{w})}\right)\Delta(\psi_{j}^{-}(w))\Delta(\psi_{i}^{+}(z)),$
$\displaystyle\Delta(\psi_{i}^{+}(z))\Delta(e_{j}(w))$
$\displaystyle=\bigl{(}\psi_{i}^{+}(q^{-c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{1}/2}z)\bigr{)}\bigl{(}e_{j}(w)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{j}(q^{c_{1}/2}w)\mathbin{\widetilde{\otimes}}e_{j}(q^{c_{1}}w)\bigr{)}$
$\displaystyle=\psi_{i}^{+}(q^{-c_{2}/2}z)e_{j}(w)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{1}/2}z)+\psi_{i}^{+}(q^{-c_{2}/2}z)\psi^{+}_{j}(q^{c_{1}/2}w)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{1}/2}z)e_{j}(q^{c_{1}}w)$
$\displaystyle=g^{*}_{ij}(q^{\frac{-c_{1}-c_{2}}{2}}\tfrac{z}{w})e_{j}(w)\psi_{i}^{+}(q^{-c_{2}/2}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{1}/2}z)$
$\displaystyle\quad+\frac{g_{ij}^{*}(q^{\frac{-c_{1}-c_{2}}{2}}\frac{z}{w})}{g_{ij}(q^{\frac{-c_{1}-c_{2}}{2}}\frac{z}{w})}\psi^{+}_{j}(q^{c_{1}/2}w)\psi^{+}_{i}(q^{-c_{2}/2}z)\mathbin{\widetilde{\otimes}}g_{ij}^{*}(q^{\frac{-c_{1}-c_{2}}{2}}\tfrac{z}{w})e_{j}(q^{c_{1}}w)\psi^{+}_{i}(q^{c_{1}/2}z)$
$\displaystyle=\Delta\bigl{(}g_{ij}(q^{-c/2}\tfrac{z}{w})\bigr{)}\Delta(e_{j}(w))\Delta(\psi_{i}^{+}(z)).$

As for the quadratic relation (2.1.17), we have

$\displaystyle[\Delta(e_{i}(z)),\Delta(f_{i}(w))]$
$\displaystyle=\bigl{(}e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\bigr{)}\bigl{(}1\mathbin{\widetilde{\otimes}}f_{i}(w)+f_{i}(q^{c_{2}}w)\mathbin{\widetilde{\otimes}}\psi^{-}_{i}(q^{\frac{c_{2}}{2}}w)\bigr{)}$
$\displaystyle\quad-\bigl{(}1\mathbin{\widetilde{\otimes}}f_{i}(w)+f_{i}(q^{c_{2}}w)\mathbin{\widetilde{\otimes}}\psi^{-}_{i}(q^{\frac{c_{2}}{2}}w)\bigr{)}\bigl{(}e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\bigr{)}$
$\displaystyle=\psi_{i}^{+}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}[e_{i}(q^{c_{1}}z),f_{i}(w)]+[e_{i}(z),f_{i}(q^{c_{2}}w)]\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{2}/2}w)$
$\displaystyle\quad+\bigl{(}\psi^{+}_{i}(q^{c_{1}/2}z)f_{i}(q^{c_{2}}w)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\psi^{-}_{i}(q^{c_{2}/2}w)-f_{i}(q^{c_{2}}w)\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}\psi^{-}_{i}(q^{c_{2}/2}w)e_{i}(q^{c_{1}}z)\bigr{)}.$

The defining relations imply that the last term vanishes. The first and second
terms are

$\displaystyle\psi_{i}^{+}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}[e_{i}(q^{c_{1}}z),f_{i}(w)]+[e_{i}(z),f_{i}(q^{c_{2}}w)]\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{2}/2}w)$
$\displaystyle=\frac{1}{q-q^{-1}}\Bigl{(}\psi_{i}^{+}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}\bigl{(}\delta(q^{c_{1}-c_{2}}\tfrac{z}{w})\psi_{i}^{-}(q^{c_{2}/2}w)-\delta(q^{c_{1}+c_{2}}\tfrac{z}{w})\psi_{i}^{+}(q^{c_{1}+c_{2}/2}z)\bigr{)}$
$\displaystyle\hskip
50.00008pt+\bigl{(}\delta(q^{-c_{1}-c_{2}}\tfrac{z}{w})\psi_{i}^{-}(q^{c_{1}/2+c_{2}}w)-\delta(q^{c_{1}-c_{2}}\tfrac{z}{w})\psi_{i}^{+}(q^{c_{1}/2}z)\bigr{)}\mathbin{\widetilde{\otimes}}\psi_{i}^{-}(q^{c_{2}/2}w)\Bigr{)}$
$\displaystyle=\frac{1}{q-q^{-1}}\bigl{(}\delta(q^{-c_{1}-c_{2}}\tfrac{z}{w})\psi_{i}^{-}(q^{c_{1}/2+c_{2}}w)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{c_{2}/2}w)\mathbin{\widetilde{\otimes}}-\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}\delta(q^{c_{1}+c_{2}}\tfrac{z}{w})\psi_{i}^{+}(q^{c_{1}+c_{2}/2}z)$
$\displaystyle=\frac{1}{q-q^{-1}}\Bigl{(}\Delta\bigl{(}\delta(q^{-c}\tfrac{z}{w})\bigr{)}\Delta\bigl{(}\psi_{i}^{-}(q^{c/2}w)\bigr{)}-\Delta\bigl{(}\delta(q^{c}\tfrac{z}{w})\bigr{)}\Delta\bigl{(}\psi_{i}^{+}(q^{c/2}z)\bigr{)}\Bigr{)},$

by which we have checked (2.1.17). As for the Serre-type relation (2.1.20), we
have

$\displaystyle\begin{split}&\Delta\Bigl{(}\frac{\overline{g}_{ii}(\frac{z_{1}}{z_{2}})}{g^{*}_{ii}(\frac{z_{1}}{z_{2}})}\frac{\overline{g}_{ij}(\frac{z_{1}}{z})}{g^{*}_{ij}(\frac{z_{1}}{z})}\frac{\overline{g}_{ij}(\frac{z_{2}}{z})}{g^{*}_{ij}(\frac{z_{2}}{z})}\Bigr{)}\Delta(e_{i}(z_{1}))\Delta(e_{i}(z_{2}))\Delta(e_{j}(z))\\\
&-\Delta\Bigl{(}h_{ij}\frac{\overline{g}_{ii}(\frac{z_{1}}{z_{2}})}{g^{*}_{ii}(\frac{z_{1}}{z_{2}})}\frac{\overline{g}_{ij}(\frac{z_{1}}{z})}{g^{*}_{ij}(\frac{z_{1}}{z})}\Bigr{)}\Delta(e_{i}(z_{1}))\Delta(e_{j}(z))\Delta(e_{i}(z_{2}))\\\
&+\Delta\Bigl{(}\frac{\overline{g}_{ii}(\frac{z_{1}}{z_{2}})}{g^{*}_{ii}(\frac{z_{1}}{z_{2}})}\Bigr{)}\Delta(e_{j}(z))\Delta(e_{i}(z_{1}))\Delta(e_{i}(z_{2}))+\Delta\Bigl{(}\frac{\overline{g}_{ij}(\frac{z_{1}}{z})}{g^{*}_{ij}(\frac{z_{1}}{z})}\frac{\overline{g}_{ij}(\frac{z_{2}}{z})}{g^{*}_{ij}(\frac{z_{2}}{z})}\Bigr{)}\Delta(e_{i}(z_{2}))\Delta(e_{i}(z_{1}))\Delta(e_{j}(z))\\\
&-\Delta\Bigl{(}h_{ij}\frac{\overline{g}_{ij}(\frac{z_{2}}{z})}{g^{*}_{ij}(\frac{z_{2}}{z})}\Bigr{)}\Delta(e_{i}(z_{2}))\Delta(e_{j}(z))\Delta(e_{i}(z_{1}))+\Delta(e_{j}(z))\Delta(e_{i}(z_{2}))\Delta(e_{i}(z_{1}))=0.\end{split}$
(2.2.7)

The remaining relations can be checked similarly.

Next we prove that $\Delta$ is a homomorphism of topological $H$-algebras. It
is clear that $\Delta$ preserves the $H^{*}$-bigradings. The compatibility
with the moment maps can be checked as

$\displaystyle\Delta(\mu_{l}(F))=\mu_{l}(F)\mathbin{\widetilde{\otimes}}1=\mu_{l}^{\mathbin{\widetilde{\otimes}}}(F),\quad\Delta(\mu_{r}(F))=1\mathbin{\widetilde{\otimes}}\mu_{r}(F)=\mu_{r}^{\mathbin{\widetilde{\otimes}}}(F)$

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$, where
$\mu_{l}^{\mathbin{\widetilde{\otimes}}}$ and
$\mu_{r}^{\mathbin{\widetilde{\otimes}}}$ denote the moment maps of the tensor
product $U\mathbin{\widetilde{\otimes}}U$ (see (1.1.4)). ∎

######  Remark 2.2.3.

The condition that (2.2.7) vanishes uniquely determines the function $h_{ij}$
in (2.1.19).

Next, we introduce the counit of $U$. Recall the topological $H$-algebra
$D_{H}\llbracket p\rrbracket$ explained in Proposition 1.3.9 (2).

######  Lemma 2.2.4.

Let $\varepsilon\colon U\to D_{H}\llbracket p\rrbracket$ be the homomorphism
of topological algebra with

$\displaystyle\varepsilon(\mu_{l}(F))=\varepsilon(\mu_{r}(F))\coloneqq
FT_{0},\quad\varepsilon(q^{\pm c/2})\coloneqq 1,\quad\varepsilon(d)\coloneqq
0$ (2.2.8)

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$ and

$\displaystyle\varepsilon(K^{\pm}_{i})\coloneqq
T_{Q_{i}},\quad\varepsilon(e_{i}(z))=\varepsilon(f_{i}(z))\coloneqq
0,\quad\varepsilon(\psi_{i}^{\pm}(z))\coloneqq T_{Q_{i}}.$ (2.2.9)

Then $\varepsilon$ is a topological $H$-algebra homomorphism.

###### Proof.

We can check that $\varepsilon$ is well-defined as a topological algebra
morphism by direct calculations as in Lemma 2.2.2 (the calculation is much
simpler). On the check of $\varepsilon$ being a topological $H$-algebra
homomorphism, it is clear that $\varepsilon$ preserves the $H^{*}$-bigradings.
The compatibility with the moment maps can be checked as

$\displaystyle\varepsilon(\mu_{l}(F))=\varepsilon(F(P,p^{*}))=F(P,p)=F(P,p)T_{0}=\mu_{l}^{D_{H}}(F),$
$\displaystyle\varepsilon(\mu_{r}(F))=\varepsilon(F(P+h,p))=F(P,p)=F(P,p)T_{0}=\mu_{r}^{D_{H}}(F)$

for $F=F(P,p)\in\mathbb{F}\llbracket p\rrbracket$, where $\mu_{l}^{D_{H}}$ and
$\mu_{r}^{D_{H}}$ are the moment maps of $D_{H}$. ∎

To state the $H$-bialgebroid structure, let us write down the natural
isomorphisms $D_{H}\mathbin{\widetilde{\otimes}}U\cong U\cong
U\mathbin{\widetilde{\otimes}}D_{H}$ in (1.1.6). By the argument of (2.2.1)
and (2.2.2), we have

$\displaystyle(D_{H}\mathbin{\widetilde{\otimes}}U)_{\alpha,\beta}=\mathbb{F}T_{-\alpha}\otimes_{\mathbb{F}}U_{\alpha,\beta}\xrightarrow{\,\sim\,}U_{\alpha,\beta},\quad
FT_{-\alpha}\mathbin{\widetilde{\otimes}}a=T_{-\alpha}\mathbin{\widetilde{\otimes}}\mu_{l}(F)a\longmapsto\mu_{l}(F)a$
(2.2.10)

and

$\displaystyle(U\mathbin{\widetilde{\otimes}}D_{H})_{\alpha,\beta}=U_{\alpha,\beta}\otimes_{\mathbb{F}}\mathbb{F}T_{-\beta}\xrightarrow{\,\sim\,}U_{\alpha,\beta},\quad
a\mathbin{\widetilde{\otimes}}FT_{-\beta}=\mu_{r}(F)a\mathbin{\widetilde{\otimes}}T_{-\beta}\longmapsto
a\mu_{r}(F)$ (2.2.11)

for $a\in U_{\alpha,\beta}$ and $F\in\mathbb{F}\llbracket p\rrbracket$. Now we
can show:

######  Proposition 2.2.5.

$(U,\Delta,\varepsilon)$ is a topological $H$-bialgebroid. That is, the
following relations hold under the isomorphisms
$(U\mathbin{\widetilde{\otimes}}U)\mathbin{\widetilde{\otimes}}U\cong
U\mathbin{\widetilde{\otimes}}(U\mathbin{\widetilde{\otimes}}U)$ and
$D_{H}\mathbin{\widetilde{\otimes}}U\cong U\cong
U\mathbin{\widetilde{\otimes}}D_{H}$.

$\displaystyle(\Delta\mathbin{\widetilde{\otimes}}\operatorname{id})\Delta=(\operatorname{id}\mathbin{\widetilde{\otimes}}\Delta)\Delta,\quad(\varepsilon\mathbin{\widetilde{\otimes}}\operatorname{id})\Delta=\operatorname{id}=(\operatorname{id}\mathbin{\widetilde{\otimes}}\varepsilon)\Delta.$

###### Proof.

We check the relations for each generator. By direct calculation, we have

$\displaystyle(\Delta\mathbin{\widetilde{\otimes}}1)\Delta(e_{i}(z))=(\Delta\mathbin{\widetilde{\otimes}}1)\bigl{(}e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\bigr{)}$
$\displaystyle=e_{i}(z)\mathbin{\widetilde{\otimes}}1\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}+\frac{c_{2}}{2}-\frac{c_{2}}{2}}z)\mathbin{\widetilde{\otimes}}\psi^{+}_{i}(q^{\frac{c_{1}}{2}+\frac{c_{2}}{2}+\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}+c_{2}}z)$
$\displaystyle=e_{i}(z)\mathbin{\widetilde{\otimes}}1\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}\psi^{+}_{i}(q^{c_{1}+\frac{c_{2}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}+c_{2}}z),$
$\displaystyle(1\mathbin{\widetilde{\otimes}}\Delta)\Delta(e_{i}(z))=(1\mathbin{\widetilde{\otimes}}\Delta)\bigl{(}e_{i}(z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{c_{1}/2}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\bigr{)}$
$\displaystyle=e_{i}(z)\mathbin{\widetilde{\otimes}}1\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}}z)\mathbin{\widetilde{\otimes}}1+\psi^{+}_{i}(q^{\frac{c_{1}}{2}}z)\mathbin{\widetilde{\otimes}}\psi^{+}_{i}(q^{c_{1}+\frac{c_{2}}{2}}z)\mathbin{\widetilde{\otimes}}e_{i}(q^{c_{1}+c_{2}}z),$

which yields
$(\Delta\mathbin{\widetilde{\otimes}}1)\Delta(e_{i}(z))=(1\mathbin{\widetilde{\otimes}}\Delta)\Delta(e_{i}(z))$.
We also have

$\displaystyle(\Delta\mathbin{\widetilde{\otimes}}1)\Delta(\psi_{i}^{+}(z))$
$\displaystyle=(\Delta\mathbin{\widetilde{\otimes}}1)(\psi_{i}^{+}(zq^{\frac{c_{2}}{2}})\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(zq^{-\frac{c_{1}}{2}}))$
$\displaystyle=\psi_{i}^{+}(q^{\frac{c_{2}}{2}+\frac{c_{3}}{2}}z)\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(\psi_{i}^{+}(q^{-\frac{c_{1}}{2}+\frac{c_{3}}{2}}z))\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(q^{-\frac{c_{1}}{2}-\frac{c_{2}}{2}}z),$
$\displaystyle(1\mathbin{\widetilde{\otimes}}\Delta)\Delta(\psi_{i}^{+}(z))$
$\displaystyle=(1\mathbin{\widetilde{\otimes}}\Delta)(\psi_{i}^{+}(zq^{\frac{c_{2}}{2}})\mathbin{\widetilde{\otimes}}\psi_{i}^{+}(zq^{-\frac{c_{1}}{2}}))$