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\newtheorem{prop}[thm]{Proposition}
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\newtheorem{conj}[thm]{Conjecture}
\newtheorem{example}[thm]{Example}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{rem}[thm]{Remark}
\newtheorem{convention}[thm]{Convention}
\newtheorem{hy}[thm]{Hypothesis}
\newtheorem*{acks}{Acknowledgements}
\theoremstyle{definition}
\newtheorem{de}[thm]{Definition}
\newtheorem{bfproof}[thm]{{\bf Proof}}

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\title{On quantum toroidal algebra of type $A_1$}

\author{Fulin Chen$^1$}
\address{School of Mathematical Sciences, Xiamen University,
 Xiamen, China 361005} \email{chenf@xmu.edu.cn}\thanks{$^1$Partially supported by the Fundamental
Research Funds for the Central Universities (No.20720190069) and NSF of China (No.11971397).}

\author{Naihuan Jing$^2$}
\address{Department of Mathematics, North Carolina State University, Raleigh, NC 27695,
USA}
\email{jing@math.ncsu.edu}
\thanks{$^2$Partially supported by NSF of China (No.11531004, No.11726016) and Simons Foundation (No.523868).}

\author{Fei Kong$^3$}
\address{Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, China 410081} \email{kongmath@hunnu.edu.cn}
\thanks{$^3$Partially supported by NSF of China (No.11701183).}

 \author{Shaobin Tan$^4$}
 \address{School of Mathematical Sciences, Xiamen University,
 Xiamen, China 361005} \email{tans@xmu.edu.cn}
 \thanks{$^4$Partially supported by NSF of China  (Nos.11531004, 11971397).}

\subjclass[2010]{17B37, 17B67} \keywords{quantum toroidal algebra, triangular decomposition, Hopf algebra}

\begin{abstract} In this paper we introduce a new quantum algebra which  specializes to the $2$-toroidal Lie algebra of type $A_1$.
 We  prove that this quantum toroidal algebra
has a natural triangular decomposition,  a (topological) Hopf algebra structure and a vertex operator realization.
\end{abstract}
\maketitle

\section{Introduction}
Let $\dot\g$  be a finite dimensional simple Lie algebra over $\C$.
The  universal central extension $\mathfrak t(\dot\g)$ of the $2$-loop algebra
$\dot\g\ot \C[t_1^{\pm 1}, t_2^{\pm 1}]$, called the toroidal Lie algebra, has a celebrated presentation given
by Moody-Rao-Yokonuma \cite{MRY} for constructing

the vertex representation for $\mathfrak t(\dot\g)$.

In understanding the Langlands reciprocity for algebraic surfaces,
Ginzburg-Kapranov-Vasserot \cite{GKV} introduced a notion of quantum toroidal algebra $\U_\hbar(\dot{\g}_{tor})$ associated to
 $\dot\g$.
The algebra $\U_\hbar(\dot{\g}_{tor})$ specializes to the Moody-Rao-Yokonuma presentation of $\mathfrak t(\dot\g)$ in general,
except for $\dot\g$ in type $A_1$ when $\U_\hbar(\dot{\g}_{tor})$ specializes to a {\it proper} quotient of the latter
\cite{E-PBW-qaff}.
The theory of quantum toroidal algebras has been extensively studied, especially with a rich representation theory developed by
Hernandez \cite{He-representation-coprod-proof,He-drinfeld-coproduct} and others, see

\cite{He-total} for a survey.

One notices that two major structural properties of $\U_\hbar(\dot{\g}_{tor})$ have played a fundamental role in Hernandez's work:
the triangular decomposition and the (deformed) Drinfeld coproduct.

Let $A$ be the generalized Cartan matrix associated to the affine Lie algebra $\g$ of $\dot\g$.
When $A$ is symmetric, by using the vertex operators calculus, Jing introduced in \cite{J-KM} a quanutm affinization algebra $\U_\hbar(\hat\g)$ associated to $\g$.
Meanwhile, it is remarkable that finite dimensional representations of $\U_\hbar(\hat\g)$ were studied by Nakajima in \cite{Naka-quiver} using powerful geometric approach of quiver varieties.
If $A$ is of simply-laced type, then $\U_\hbar(\hat\g)$ is nothing but the quantum toroidal algebra $\U_\hbar(\dot{\g}_{tor})$.
However, for the case that $A$ is not of simply-laced type, the definition of $\U_\hbar(\hat\g)$
is slightly different from that of $\U_\hbar(\dot{\g}_{tor})$. Explicitly, one notices that $A_1^{(1)}$ is the unique symmetric but non-simply-laced affine generalized Cartan matrix.
In this case, the defining currents $x_0^\pm(z), x_1^\pm(z)$  in $\U_\hbar(\hat\g)$ satisfy the relation
\begin{align}\label{intro-re1}
(z-q^{\mp 2}w)(z-w) x_0^\pm(z)x_1^\pm(w)=(q^{\mp 2}z-w)(z-w) x_1^\pm(w)x_0^\pm(z),
\end{align}
which appeared naturally in calculations of  quantum vertex operators \cite{J-KM} and equivariant K-homology of quiver varieties \cite{Naka-quiver}.
In particular,   $\U_\hbar(\hat\g)$ specializes to the toroidal Lie algebra $\mathfrak t(\dot\g)$  of type $A_1$ as
the classical limit of \eqref{intro-re1} holds in $\mathfrak t(\dot\g)$.
On the other hand, in $\U_\hbar(\dot{\g}_{tor})$ these two currents satisfy the relation
\begin{align}\label{intro-re2}
(z-q^{\mp 2}w) x_0^\pm(z)x_1^\pm(w)=(q^{\mp 2}z-w) x_1^\pm(w)x_0^\pm(z).
\end{align}
This stronger relation was needed in verifying the compatibility with affine quantum Serre relations in $\U_\hbar(\dot{\g}_{tor})$
so that it processes a canonical triangular decomposition \cite{He-representation-coprod-proof}.
For the algebra $\U_\hbar(\hat\g)$, we only know that it has a weak form of triangular decomposition \cite{Naka-quiver}.

From now on, we assume that $\dot\g$ is of type $A_1$.
The main goal of this paper is to define a ``middle" quantum algebra
\[\U_\hbar(\hat\g)\twoheadrightarrow\U\twoheadrightarrow \U_\hbar(\dot\g_{tor})\] of $\U_\hbar(\hat\g)$  and $\U_\hbar(\dot\g_{tor})$,
and prove that this new quantum toroidal algebra $\U$ processes the ``good'' properties
enjoyed by
both of  $\U_\hbar(\hat\g)$  and $\U_\hbar(\dot\g_{tor})$.
Explicitly, we first introduce in Section 2 a quantum algebra $\U$ which specializes to the toroidal Lie algebra $\mathfrak t(\dot\g)$.
By definition, $\U$ is the quotient algebra of  $\U_\hbar(\hat\g)$ obtained by modulo the relation
\begin{align}\label{intro-re3}
[x_0^\pm(z_1),(z_2-q^{\mp 2}w) x_0^\pm(z_2)x_1^\pm(w)-(q^{\mp 2}z_2-w) x_1^\pm(w)x_0^\pm(z_2)]=0.
\end{align}
One notices that $\U_\hbar(\dot\g_{tor})$ is a quotient algebra of $\U$ as the relation \eqref{intro-re2} implies the relations \eqref{intro-re1} and \eqref{intro-re3}.
In Section 3, we prove that $\U$ admits a triangular decomposition (see Theorem \ref{thm:tri-decomp}).
In Section 4, we prove that $\U$ has  a deformed Drinfeld coproduct (see Theorem \ref{thm:bialg-struct}).
As in \cite{Gr-qshuff-qaff}, this allows us to define a (topological) Hopf algebra structure on $\U$ (see Theorem \ref{thm:hopf}).
As usual, the crucial step in establishing Theorems \ref{thm:tri-decomp} and \ref{thm:bialg-struct} is to check the compatibility with affine quantum Serre relations,
in where the new relation \eqref{intro-re3} appeared naturally (see \eqref{newrela1} and \eqref{newrela2}).
Finally, in Section 5 we point out that the quantum vertex operators constructed in \cite{J-KM} satisfy the relation \eqref{intro-re3}, and so
 we obtain a  vertex representation for $\U$.

Throughout this paper, we denote by $\C[[\hbar]]$ the ring of complex formal series in one variable $\hbar$.
By a $\C[[\hbar]]$-algebra, we mean a  topological algebra over $\C[[\hbar]]$ with respect to the $\hbar$-adic topology.
For $n,k,s\in\Z$ with $0\le k\le s$, we denote the usual quantum numbers as follows
\begin{align*}
  [n]_q=\frac{q^n-q^{-n}}{q-q\inv},\quad[s]_q!=[s]_q[s-1]_q\cdots[1]_q,\quad
  \binom{s}{k}_q=\frac{[s]_q!}{[k]_q![s-k]_q!},
  \end{align*}
  where
\[q=\exp(\hbar)\in\C[[\hbar]].\]

\section{Quantum toroidal algebra of type $A_1$.}
In this section we introduce a new quantum algebra which specializes to toroidal Lie algebra of type $A_1$.

Let
\[A=(a_{ij})_{i,j=0,1}=\left(
                         \begin{array}{cc}
                           2 & -2 \\
                          -2 & 2 \\
                         \end{array}
                       \right)\]
 be the generalized Cartan matrix  of type $A_1^{(1)}$.
For $i,j=0,1$, let
\begin{align}
  &g_{ij}(z)=\frac{q^{a_{ij}}-z}{1-q^{a_{ij}}z}
\end{align}
be the formal Taylor series at $z=0$.
The following is  the main object of this paper:

\begin{de}\label{de:q-aff}
The quantum toroidal algebra $\U$  is the $\C[[\hbar]]$-algebra topologically generated by the elements
\begin{align}\label{eq:gen-set}
  h_{i,n},\ x_{i,n}^\pm,\ c\quad i=0,1,\ n\in\Z,
\end{align}
and subject to the relations in terms of generating functions in $z$:
\begin{align*}
  \phi_i^\pm(z)=q^{\pm h_{i,0}}\exp\(\pm(q-q\inv)\sum_{\pm n>0}h_{i,n}z^{-n}\),\quad
  x_i^\pm(z)=\sum_{n\in\Z}x_{i,n}^\pm z^{-n}.
\end{align*}
The relations are:
\begin{align*}
  &\te{(Q1)}\quad c\ \te{is central},\quad [\phi_i^\pm(z),\phi_j^\pm(w)]=0,\\
  &\te{(Q2)}\quad \phi_i^+(z)\phi_j^-(w)=\phi_j^-(w)\phi_i^+(z)g_{ij}(q^cw/z)\inv g_{ij}(q^{-c}w/z),\\
 &\te{(Q3)}\quad \phi_i^+(z)x_j^\pm(w)=x_j^\pm(w)\phi_i^+(z)g_{ij}(q^{\mp\half c}w/z)^{\pm 1},\\
  &\te{(Q4)}\quad \phi_i^-(z)x_j^\pm(w)=x_j^\pm(w)\phi_i^-(z)g_{ji}(q^{\mp\half c}z/w)^{\mp 1},\\
  &\te{(Q5)}\quad [x_i^+(z),x_j^-(w)]=\frac{\delta_{ij}}{q-q\inv}
  \(\phi_i^+(zq^{-\half c})\delta\(\frac{q^cw}{z}\)
  -\phi_i^-(zq^{\half c})\delta\(\frac{q^{-c}w}{z}\)\),\\
  &\te{(Q6)}\quad (z-q^{\pm 2}w)x_i^\pm(z)x_i^\pm(w)=
  (q^{\pm 2}z-w)x_i^\pm(w)x_i^\pm(z),\\
  &\te{(Q7)}\quad (z-q^{\mp 2}w)(z-w)x_i^\pm(z)x_j^\pm(w)
  =(q^{\mp 2}z-w)(z-w)x_j^\pm(w)x_i^\pm(z),\\
  &\te{(Q8)}\quad [x_i^\pm(z_1),\((z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z-w)x_j^\pm(w)x_i^\pm(z_2)\)]=0,\\
  &\te{(Q9)}\, \sum_{\sigma\in S_3}\sum_{r=0}^3(-1)^r\binom{3}{r}_q
   x_i^\pm(z_{\sigma(1)})\cdots x_i^\pm(z_{\sigma(r)})x_j^\pm(w)
   x_i^\pm(z_{\sigma(r+1)})\cdots x_i^\pm(z_{\sigma(3)})=0,
\end{align*}
where $i,j=0,1$ with $i\ne j$ in (Q7), (Q8), (Q9) and $\delta(z)=\sum_{n\in\Z}z^n$ is the usual $\delta$-function.
\end{de}

\begin{rem}{\em As indicated in Introduction,
in literature there are two other definitions of quantum toroidal algebra of type $A_1$:
the algebra $\U_\hbar(\hat\g)$ introduced in \cite{J-KM,Naka-quiver} and the algebra $\U_\hbar(\dot\g_{tor})$ introduced in \cite{GKV,He-representation-coprod-proof}.
By definition, the algebra $\U_\hbar(\hat\g)$ is
the $\C[[\hbar]]$-algebra topologically generated by the elements as in \eqref{eq:gen-set} with relations (Q1)-(Q7) and (Q9),
while the algebra $\U_\hbar(\dot\g_{tor})$ is
the $\C[[\hbar]]$-algebra topologically generated by the elements as in \eqref{eq:gen-set} with relations (Q1)-(Q6), (Q9) and the following relation
\begin{align}\label{strongerrela}
(z-q^{\mp 2}w)x_i^\pm(z)x_j^\pm(w)
  =(q^{\mp 2}z-w)x_j^\pm(w)x_i^\pm(z),\quad i\ne j\in \{0,1\}.
\end{align}
By definition we have that  $\U$  is a  quotient algebra of $\U_\hbar(\hat\g)$, while $\U_\hbar(\dot\g_{tor})$ is a quotient algebra of $\U$.
}\end{rem}

Now we recall the definition of the toroidal Lie algebra of type $A_1$.
Let $\mathcal K$ be the $\C$-vector space spanned by the symbols
\begin{align*}
  &t_1^{m_1}t_2^{m_2}\rk_i,\quad i=1,2,\ m_1,m_2\in\Z
\end{align*}
subject to the relations
\begin{align*}
  &m_1t_1^{m_1}t_2^{m_2}\rk_1+m_2t_1^{m_1}t_2^{m_2}\rk_2=0.
\end{align*}

Let $\dot\g=\ssl_2(\C)$ be the simple Lie algebra of type $A_1$ and $\<\cdot,\cdot\>$ the Killing form on $\dot\g$.
The toroidal Lie algebra (see \cite{MRY})
\begin{align*}
\mathfrak t=\mathfrak t(\dot\g)=\(\dot\g\ot\C[t_1^{\pm 1},t_2^{\pm 1}]\)\oplus \mathcal K
\end{align*}
is the universal central extension of the double loop algebra $\dot\g\ot\C[t_1^{\pm 1},t_2^{\pm 1}]$, where $\mathcal K$ is the center space and
\begin{align*}
  [x\ot t_1^{m_1}t_2^{m_2},y\ot t_1^{n_1}t_2^{n_2}]
  =[x,y]\ot t_1^{m_1+n_1}t_2^{m_2+n_2}+\<x,y\>(\sum_{i=1}^2 m_it_1^{m_1+n_1}t_2^{m_2+n_2}\rk_i),
\end{align*}
for $x,y\in\dot\g$ and $m_1,m_2,n_1,n_2\in\Z$.

Let $\{e^+,\al,e^-\}$ be a standard $\ssl_2$-triple in $\dot\g$, that is,
\begin{align*}
  &[e^+,e^-]=\al,\quad [\al,e^\pm]=\pm 2e^\pm.
\end{align*}
For $i=0,1$ and $m\in \Z$, set
\begin{align*}
  \al_{1,m}=\al\ot t_2^m,\ \al_{0,m}=t_2^mk_1-\al\ot t_2^m,\
  e_{1,m}^\pm=e^\pm\ot t_2^m,\ e_{0,m}^\pm=e^\mp\ot t_1^{\pm 1}t_2^m.
\end{align*}
Note that these elements generate the algebra $\mathfrak t$.

Following \cite{MRY}, we have:
\begin{prop}\label{MRY-pre} The toroidal Lie algebra $\mathfrak t$ is abstractly generated by
the elements $\al_{i,m}, e_{i,m}^\pm, \rk_1$ for $i=0,1, m\in \Z$ with relations
\begin{align*}
  &\te{(L1)}\quad [\rk_1,\mathfrak t]=0,\ [\al_{i,m},\al_{j,n}]=a_{ij}\delta_{m+n,0}m \rk_1,\\
  &\te{(L2)}\quad [\al_{i,m},e_{j,n}^\pm]=\pm a_{ij} e_{j,m+n},\\
  &\te{(L3)}\quad [e_{i,m}^+,e_{j,n}^-]=\delta_{ij}\(\al_{j,m+n}+m\delta_{m+n,0}\rk_1\),\\
  &\te{(L4)}\quad (z-w)[e_i^\pm(z),e_i^\pm(w)]=0,\\
  &\te{(L5)}\quad
  (z-w)^2[e_i^\pm(z),e_j^\pm(w)]=0,\quad i\ne j,\\
  &\te{(L6)}\quad (z_2-w)[e_i^\pm(z_1),[e_i^\pm(z_2),e_j^\pm(w)]]=0,\quad i\ne j,\\
  &\te{(L7)}\quad [e_i^\pm(z_1),[e_i^\pm(z_2),[e_i^\pm(z_3),e_j^\pm(w)]]]=0,\quad i\ne j,
\end{align*}
where $i,j=0,1$, $m,n\in \Z$ and $e_i^\pm(z)=\sum_{n\in\Z}e_{i,n}^\pm z^{-n}$.
\end{prop}
\begin{proof} Denote by $\mathcal L$ the Lie algebra abstractly generated by
the elements $\al_{i,m}, e_{i,m}^\pm, \rk_1$ for $i=0,1, m\in \Z$ with relations
(L1)-(L7). One easily checks that the relations (L1)-(L7) hold in $\mathfrak t$ and so we have a
surjective Lie homomorphism $\psi$ from $\mathcal L$ to $\mathfrak t$.  On the other hand,
denote by $\mathcal L'$ the Lie algebra abstractly generated by
the elements $\al_{i,m}, e_{i,m}^\pm, \rk_1$ for $i=0,1, m\in \Z$ with relations
(L1)-(L4) and (L7). Then there is a quotient map, say $\varphi$, from $\mathcal L'$ to $\mathcal L$.
 It was proved in \cite{MRY}  that
the surjective homomorphism $\psi\circ \varphi:\mathcal L'\rightarrow \mathcal L\rightarrow \mathfrak t$ is an isomorphism,
noting that the relation (L4) is equivalent to the relation $[e_i^\pm(z),e_i^\pm(w)]=0$.
This in turn implies that the map $\psi$ is an isomorphism, as required.
\end{proof}

By combing Definition \ref{de:q-aff} with Proposition \ref{MRY-pre}, one immediate gets the following result.

\begin{thm}
The classical limit $\U/\hbar\U$ of $\U$ is isomorphic to the universal enveloping algebra $\U(\mathfrak t)$ of the torodial Lie algebra $\mathfrak t$.
\end{thm}

\begin{rem}{\em From the proof of Proposition \ref{MRY-pre}, one knows that the algebra $\U_\hbar(\hat\g)$ also specializes $\mathfrak t$.
On the other hand, it is straightforward to see that the current
\[(z-w)[e_0^\pm(z),e_1^\pm(w)]\]  is nonzero in $\mathfrak t$  and its components  lie in the space $\bar{\mathcal K}=\sum_{m_1\in \Z}(\C t_1^{m_1}t_2\rk_1+\C t_1^{m_1}t_2^{-1}\rk_1)$.
 Thus, the algebra $\U(\dot{\g}_{tor})$ specializes to the
quotient algebra
$\mathfrak t/\bar{\mathcal K}$ of $\mathfrak t$ (cf. \cite{E-PBW-qaff}).}
\end{rem}

\section{Triangular decomposition of $\U$}\label{sec:tri-decomp}
In this section, we prove a triangular decomposition of $\U$.
By  a triangular decomposition of a $\C[[\hbar]]$-algebra $A$, we mean  a data of three closed $\C[[\hbar]]$-subalgebras $(A^-,H,A^+)$
of $A$ such that the multiplication $x^-\ot h\ot x^+\mapsto x^-hx^+$
induces an $\C[[\hbar]]$-module isomorphism from $A^-\wh\ot H\wh\ot A^+$ to $A$.
Here and henceforth, for two $\C[[\hbar]]$-modules $U,V$, the notion $U\wh\ot V$ stands for the $\hbar$-adically completed tensor product of $U$ and $V$.

Let $\U^+$ (resp. $\U^-$; resp. $\mathcal H$) be the closed subalgebra of $\U$ generated by $x_{i,m}^+$ (resp. $x_{i,m}^-$; resp. $h_{i,m}$, $c$).
The following is the main result of this section:

\begin{thm}\label{thm:tri-decomp}
$(\U^-,\mathcal H,\U^+)$ is a triangular decomposition of $\U$.
Moreover, $\U^+$ (resp. $\U^-$; resp $\mathcal H$) is isomorphic to the $\C[[\hbar]]$-algebra topologically generated by $x_{i,m}^+$ (resp. $x_{i,m}^-$; resp. $h_{i,m}$, $c$), and subject to the relations (Q6-Q9) with ``$+$'' (resp. (Q6-Q9) with ``$-$''; resp. (Q1-Q2)).
\end{thm}

The rest of this section is devoted to a proof of Theorem \ref{thm:tri-decomp}.
We first introduce some algebras related to $\U$ that will be used later on.

\begin{de}\label{de:algebras}
Let $\wt\U$ be the $\C[[\hbar]]$-algebra topologically generated by the elements in \eqref{eq:gen-set} with defining relations (Q1-Q5),
$\wh\U$ the quotient algebra of $\wt\U$ modulo the relations (Q6-Q7),
and $\bar\U$ the quotient algebra of $\wh\U$ modulo the relation (Q8).
\end{de}

Denote by $\wt \U^+$ (resp. $\wt \U^-$; resp. $\wt{\mathcal H}$) the closed subalgebra of $\wt\U$ generated by $x_{i,m}^+$ (resp. $x_{i,m}^-$; resp. $h_{i,m}$, $c$).
The following result is standard.

\begin{lem}\label{lem:tri-tech0}
$(\wt\U^-, \wt{\mathcal H},\wt\U^+)$ is a triangular decomposition of $\wt\U$. Moreover, $\wt\U^+$ (resp. $\wt\U^-$)
isomorphic to the $\C[[\hbar]]$-algebra topologically free
generated by $x_{i,m}^+$ (resp. $x_{i,m}^-$) and $\wt{\mathcal H}$ is isomorphic to the $\C[[\hbar]]$-algebra topologically generated by  $h_{i,m}$, $c$ with relations (Q1-Q2).
\end{lem}

The following result was proved in (the proof of) \cite[Lemma 8]{He-representation-coprod-proof}.

\begin{lem}\label{lem:tri-tech1}
For $i,j,k=0,1$, the following  hold in $\wt\U$:
\begin{align}\label{eq:tri-tech1}
  [(z-q^{\pm a_{ij}}w)x_i^\pm(z)x_j^\pm(w)-(q^{\pm a_{ij}}z-w)x_j^\pm(w)x_i^\pm(z),x_k^\mp(w_0)]=0.
\end{align}
\end{lem}

Similarly, we have:

\begin{lem}\label{lem:tri-tech2}
For $i,j,k=0,1$ with $i\ne j$, the following  hold in $\wh\U$:
\begin{align}\label{eq:tri-tech2-temp1}
  &[[x_i^\pm(z_1),(z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)],x_k^\mp(w_0)]=0.
\end{align}
\end{lem}

\begin{proof}Let $i,j$ be as in lemma. We first prove that for $\eta=\pm$
\begin{align}\label{eq:tri-tech2-temp2}[\phi_i^\eta(q^{\mp\frac{\eta}{2}c}z_1),\((z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)\)]=0.
\end{align}
Indeed, it follows from (Q3) and (Q7) that
\begin{align*}
  &[\phi_i^+(q^{\mp\frac{1}{2}c}z_1),\((z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)\)]\\
  =&\qquad\qquad\(\frac{q^{\pm 2}z_1-z_2}{z_1-q^{\pm 2}z_2}
    \frac{q^{\mp 2}z_1-w}{z_2-q^{\mp 2}w}-1\)\\
    &\cdot \((z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)\)
    \phi_i^+(q^{\mp\frac{1}{2}c}z_1)\\
  =&\qquad\qquad\frac{(q^{\pm 2}-q^{\mp 2})z_1(z_2-w)}
    {(z_1-q^{\pm 2}z_2)(z_1-q^{\mp 2}w)}\\
  &\cdot\((z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)\)
    \phi_i^+(q^{\mp\frac{1}{2}c}z_1)\\
  =&0.
\end{align*}
Similarly, for the case $\eta=-$, \eqref{eq:tri-tech2-temp2}   follows from (Q4) and (Q7).
Now, in view of \eqref{eq:tri-tech2-temp2} and (Q5), we have
\begin{align}\label{eq:tri-tech2-temp3}
  &[[x_i^\pm(z_1),x_k^\mp(w_0)],(z_2-q^{\mp 2}w)x_i^\pm(z_2)x_j^\pm(w)
    -(q^{\mp 2}z_2-w)x_j^\pm(w)x_i^\pm(z_2)]=0.
\end{align}
This together with \eqref{eq:tri-tech1} (with $i\ne j$) gives \eqref{eq:tri-tech2-temp1}, which proves the lemma.
\end{proof}

\begin{lem}\label{lem:tri-tech3}
For $i,j,k=0,1$ with $i\ne j$, the following equations hold in $\bar\U$:
\begin{equation}\begin{split}\label{eq:tri-decomp-tech-1}
&\sum_{\sigma\in S_{3}}\sum_{r=0}^3
(-1)^r\binom{3}{r}_{q}
x_i^\pm(z_{\sigma(1)})\cdots x_i^\pm(z_{\sigma(r)})\phi^\eta_j( q^{\mp \eta\frac{1}{2}c}w)\\
&\quad\times
x_i^\pm(z_{\sigma(r+1)})\cdots x_i^\pm(z_{\sigma(3)})=0,
\end{split}
\end{equation}
\begin{equation}\begin{split}\label{eq:tri-decomp-tech-2}
&\sum_{\sigma\in S_3}\sum_{r=0}^3
(-1)^r\binom{3}{r}_{q}
\xi_i(z_{\sigma(1)})\cdots \xi_i(z_{\sigma(r)})x_j^\pm(w)\\
&\quad\times
\xi_i(z_{\sigma(r+1)})\cdots \xi_i(z_{\sigma(3)})=0,
\end{split}\end{equation}
where $\eta=\pm$, $\xi_i(z_p)=x_i^\pm(z_p)$ if $p\ne 1$
and $\xi(z_1)=\phi_i^\eta(q^{\mp \eta \frac{1}{2}c}z_1)$.
\end{lem}

\begin{proof} Equation \eqref{eq:tri-decomp-tech-1} can be  proved as that of
\cite[Eq. (20)]{He-representation-coprod-proof} and we omit the details.
For \eqref{eq:tri-decomp-tech-2}, we will prove the case of  $\eta=+$, the other case $\eta=-$ is similar.
Denote by $R^\pm$ the LHS of \eqref{eq:tri-decomp-tech-2}.
Then it follows from the relations (Q3-Q4) that
\begin{align*}
R^\pm=D^\pm\sum_{\pi\in S_{2}}
\sum_{r=1}^{3}&
    P_{r}(z_{1},z_{\pi(2)},z_{\pi(3)},w,q^{\pm 1})
    x_i^\pm(z_{\pi(2)})\cdots x_i^\pm(z_{\pi(r)})x_j^\pm(w)\\
    &\times x_i^\pm(z_{\pi(r+1)})\cdots x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1}),
\end{align*}
where   $S_{2}$ acts on the set $\{2,3\}$ and for $1\le r\le 3$,
\begin{align*}
D^\pm&=\frac{1}{z_1-q^{\mp 2}w}
    \prod_{2\le a\le 3}
    \frac{1}{z_1-q^{\pm 2 }z_a},
\end{align*}
and
\begin{align*}
&P_r(z_1,z_2,z_3,w,q)\\
=&\binom{3}{r}_q(-1)^r
    \sum_{p=1}^r \prod_{2\le a\le p}\(z_1-q^2z_a\)
    \prod_{p<a\le 3}\(q^{2}z_1-z_a\)
    \( q^{a_{ij}}z_1-w\)\\
    &\,+\binom{3}{r-1}_q(-1)^{r-1}
    \sum_{p=r}^{3} \prod_{2\le a\le p}\(z_1-q^2z_a\)
    \prod_{p<a\le 3}\(q^2z_1-z_a\)
    \( z_1-q^{a_{ij}}w\).
\end{align*}

It was proved in \cite[Lemma 6]{He-representation-coprod-proof} that
\begin{align}
P_1(z_1,&z_2,z_3,w,q)=(z_2-q^2w)
f^{(1)}_{3}(z_1,z_3,w,q)+(z_{3}-q^{-2}z_{2})
f^{(1)}_2(z_1,w,q),\label{eq:tri-decomp-temp-P1}\\

P_2(z_1,&z_2,z_3,w,q)=(w-q^2z_2)
f^{(2)}_{2}(z_1,z_3,w,q)
+(z_{3}-q^{2}w)
f^{(2)}_{3}(z_1,z_2,w,q)\label{eq:tri-decomp-temp-P2},\\

P_3(z_1,&z_2,z_3,w,q)=(w-q^2z_3)
f^{(3)}_{3}(z_1,z_{2},w,q)+
(z_{3}-q^{-2}z_{2})
f^{(3)}_2(z_1,w,q),\label{eq:tri-decomp-temp-P3}
\end{align}
where
$f_2^{(r)}$ and $f_3^{(r)}$ are some polynomials
of degree at most $1$ in each variable.

In view of \eqref{eq:tri-decomp-temp-P1}, \eqref{eq:tri-decomp-temp-P2}, \eqref{eq:tri-decomp-temp-P3} and
(Q6), we have
\begin{align*}
&\sum_{\pi\in S_{2}}
P_r(z_1,z_{\pi(2)},z_{\pi(3)},w,q^{\pm 1})
 x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)}) x_j^\pm(w)\phi_i^+(q^{\mp\frac{1}{2}c}z_{1})=0,\\
&\sum_{\pi\in S_{2}}
P_r(z_1,z_{\pi(2)},z_{\pi(3)},w,q^{\pm 1})
 x_j^\pm(w)x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})=0.
\end{align*}
This implies that all the terms in $R^\pm$ which contain the polynomials $f^{(r)}_a$ with $a\ne r,3$ can be erased.
Thus, we  obtain
\begin{align*}
R^\pm=&D^\pm\sum_{\pi\in S_2}(z_{\pi(2)}-q^{\pm 2}w)
    f_3^{(1)}(z_1,z_{\pi(3)},w,q^{\pm 1})
    x_j^\pm(w)x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\\
    -&D^\pm\sum_{\pi\in S_2}(q^{\pm 2}z_{\pi(2)}-w)
    f_2^{(2)}(z_1,z_{\pi(3)},w,q^{\pm 1})
    x_i^\pm(z_{\pi(2)})x_j^\pm(w)x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\\
    +&D^\pm\sum_{\pi\in S_2}(z_{\pi(3)}-q^{\pm 2}w)
    f_3^{(2)}(z_1,z_{\pi(2)},w,q^{\pm 1})
    x_i^\pm(z_{\pi(2)})x_j^\pm(w)x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\\
    -&D^\pm\sum_{\pi\in S_2}(q^{\pm 2}z_{\pi(3)}-w)
    f_3^{(3)}(z_1,z_{\pi(2)},w,q^{\pm 1})
    x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)})x_j^\pm(w)
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1}).
\end{align*}
A straightforward calculation shows that
\begin{align*}
  &f_3^{(1)}(z_1,z_3,w,q)=
  f_2^{(2)}(z_1,z_3,w,q)=Q(z_1-z_3),\\
  &f_3^{(2)}(z_1,z_2,w,q)
  =f_3^{(3)}(z_1,z_2,w,q)=-Q(z_1-z_2),
\end{align*}
where
\begin{align*}
  Q=(q^{-4}-q^{4}+q^{-2}-q^{2})z_1.
\end{align*}
Then we have
\begin{align}
  R^\pm=\pm& Q D^\pm\sum_{\pi\in S_2}(z_{\pi(2)}-q^2w)
    (z_1-z_{\pi(3)})
    x_j^\pm(w)x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\notag\\
    \mp&Q D^\pm\sum_{\pi\in S_2}(q^2z_{\pi(2)}-w)
    (z_1-z_{\pi(3)})
    x_i^\pm(z_{\pi(2)})x_j^\pm(w)x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\notag\\
    \pm&Q D^\pm\sum_{\pi\in S_2}(z_{\pi(3)}-q^2w)
    (z_1-z_{\pi(2)})
    x_i^\pm(z_{\pi(2)})x_j^\pm(w)x_i^\pm(z_{\pi(3)})
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\notag\\
    \mp&Q D^\pm\sum_{\pi\in S_2}(q^2z_{\pi(3)}-w)
    (z_1-z_{\pi(2)})
    x_i^\pm(z_{\pi(2)})x_i^\pm(z_{\pi(3)})x_j^\pm(w)
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\notag\\
    =\mp& Q D^\pm q^{\pm 2}\sum_{\pi\in S_2}(z_1-z_{\pi(2)})\big[x_i^\pm(z_{\pi(2)}),
   (z_{\pi(3)}-q^{\mp 2}w)x_i^\pm(z_{\pi(3)})x_j^\pm(w)\label{newrela1}\\
    &\quad\quad
    -(q^{\mp 2}z_{\pi(3)}-w)x_j^\pm(w)x_i^\pm(z_{\pi(3)})\big]
    \phi_i^+(q^{\mp\frac{1}{2}c}z_{1})\notag\\
    =0&,\notag
\end{align}
where the last equation follows from (Q8).
\end{proof}

As in the proof of \cite[Lemma 10]{He-representation-coprod-proof}, it is obvious that Lemma \ref{lem:tri-tech3} implies the following result.

\begin{lem}\label{lem:tri-tech4}
For $i,j,k=0,1$ with $i\ne j$, the following equations hold in $\bar\U$:
\begin{align*}
[\sum_{\sigma\in S_3}\sum_{r=0}^3(-1)^r\binom{3}{r}_q
   x_i^\pm(z_{\sigma(1)})\cdots x_i^\pm(z_{\sigma(r)})x_j^\pm(w)
   x_i^\pm(z_{\sigma(r+1)})\cdots x_i^\pm(z_{\sigma(3)}),x_k^\mp(w_0)]=0.
   \end{align*}
\end{lem}

\textbf{Proof of Theorem \ref{thm:tri-decomp}:} We first recall a general result of triangular decompositions (cf. \cite[Lemma 3.5]{He-representation-coprod-proof}).
Let $A$ be a completed and separated   $\C[[\hbar]]$-algebra and
$(A^-,H,A^+)$ a triangular decomposition of $A$. Let $B^+$ and $B^-$ be respectively a closed two-sided ideal of
$A^+$ and $A^-$, and let $B$ be the closed ideal of $A$ generated by $B^++B^-$.
Set $C=A/B$ and denote by $C^\pm$ the image of
$B^\pm$ in $C$.
Assume that $A B^+\subset B^+ A$ and $B^- A\subset A B^-$. Then  $(C^+,H,C^-)$ is a triangular decomposition
of $C$ and $C^\pm$ are isomorphic to $A^\pm/B^\pm$.
In view of this criterion, Theorem \ref{thm:tri-decomp}
follows from Lemmas \ref{lem:tri-tech0}, \ref{lem:tri-tech1},
\ref{lem:tri-tech2} and \ref{lem:tri-tech4}.

\begin{rem}{\em It was proved in \cite{He-representation-coprod-proof} that the algebra $\U_\hbar(\dot\g_{tor})$ has a  triangular decomposition
as in Theorem \ref{thm:tri-decomp}.
That is, $\U$ and $\U_\hbar(\dot\g_{tor})$ are two different choices of the quotient algebras of $\U_\hbar(\hat\g)$ with
a triangular decomposition.
}
\end{rem}

\section{Hopf algebra structure}

In this section, we discuss a
Hopf algebra structure on $\U$.
For a $\C[[\hbar]]$-module $M$ and $n\in \mathbb N$, we denote that
\begin{align}
  M^{\wh\ot^n}=
  \belowit{\underbrace{M\wh\ot M \wh\ot\cdots\wh\ot M}}
  {n\te{-copies}}.
\end{align} Let $u,v$ be formal variables.  Motivated by the deformed Drinfeld coproduct given in \cite{He-drinfeld-coproduct}, we have:
\begin{thm}\label{thm:bialg-struct}
There exists a unique $\C[[\hbar]]$-algebra homomorphism
$\Delta_u:\U\rightarrow\big(\U^{\wh\ot^2}\big) ((u))$ defined as follows $(i=0,1)$
\begin{eqnarray*}
&&\te{(Co1) }\qquad\qquad\Delta_u(c)=c\ot 1+1\ot c,\qquad\qquad\\
&&\te{(Co2) }\qquad\qquad\Delta_u\left(\phi_i^\pm(z)\right)
    =\phi_i^\pm(zq^{\pm\frac{c_2}{2}})\otimes \phi_i^\pm(zu\inverse q^{\mp\frac{c_1}{2}}),\qquad\qquad\\
&&\te{(Co3) }\qquad\qquad\Delta_u\left(x_i^+(z)\right)
    =x_i^+(z)\otimes 1+\phi_i^-(zq^{\frac{c_1}{2}})\otimes x_i^+(zu\inverse q^{c_1}),\qquad\qquad\\
&&\te{(Co4) }\qquad\qquad\Delta_u\left(x_i^-(z)\right)
    =1\otimes x_i^-(zu\inverse)+x_i^-(zq^{c_2})\otimes \phi_i^+(zu\inverse q^{\frac{c_2}{2}}),\qquad\qquad
\end{eqnarray*}
where $c_1=c\ot 1$ and $c_2=1\otimes c$.
Moreover, as $\C[[\hbar]]$-algebra homomorphisms
$\U\rightarrow\big(\U^{\wh\ot^3}\big)((u,v))$,
we have
\begin{align*}
  &\(\mathrm{Id}\otimes \Delta_v\)\circ \Delta_u
  =\(\Delta_u\otimes \mathrm{Id}\)\circ \Delta_{uv}.
\end{align*}
and, as $\C[[\hbar]]$-algebra homomorphisms
$\U\rightarrow\big(\U^{\wh\ot^2}\big)((u))$,
we have
\begin{align*}
  &\(\mathrm{Id}\otimes \epsilon\)\circ\Delta_u=\mathrm{Id},\quad
  \(\epsilon\otimes \mathrm{Id}\)\circ\Delta_u=\mathrm{Id}_u,
\end{align*}
where $\mathrm{Id}_u:\U\rightarrow \U\otimes \C((u))$ is the $\C[[\hbar]]$-algebra
homomorphism determined by
\begin{eqnarray*}
&\te{(Id1) }\qquad\qquad&\mathrm{Id}_u(c)=c,\quad \mathrm{Id}_u(\phi_i^\pm(z))=\phi_i^\pm(zu\inverse),\qquad\qquad\\
&\te{(Id2) }\qquad\qquad&\mathrm{Id}_u(x_i^\pm(z))=x_i^\pm(zu\inverse),\qquad\qquad
\end{eqnarray*}
and $\epsilon:\U\rightarrow \C[[\hbar]]$ is the $\C[[\hbar]]$-algebra homomorphism  determined by
\begin{eqnarray*}
&\te{(CoU) }\qquad\qquad \epsilon\(\phi_i^\pm(z)\)=1,\quad
\epsilon\(x_i^\pm(z)\)=0=\epsilon(c).\qquad\qquad
\end{eqnarray*}
\end{thm}

Before proving Theorem \ref{thm:bialg-struct}, we remark that the above
gives a (topological) Hopf algebra structure on $\U$ (by $\Delta_1$, the ``limit" of $\Delta_u$ at $u=1$).
However, one notices that
$\Delta_1$ is not a well-defined $\C[[\hbar]]$-algebra homomorphism from $\U$ to $\U^{\wh\ot^2}$, so
we need to introduce certain topological completions of $\U$ and $\U^{\wh\ot^2}$ as in \cite{Gr-qshuff-qaff}.
Explicitly,
let $\mathcal F$ be the free $\C[[\hbar]]$-algebra topologically generated by the set \eqref{eq:gen-set}.
Now give $h_{i,\pm n}$ degree $n$ for $n>0$, and
all other elements degree $0$. We extend the degree to all the elements of the algebra by summation on the monomials.
For $k\ge0$, let $\mathcal F_k$ be the $\hbar$-adically closed ideal of $\mathcal F$ generated by elements of degree greater than $k$.
Then we obtain an inverse system of $\C[[\hbar]]$-algebras $(\mathcal F/\mathcal F_k, p_k)$,
where $p_k$ is the natural projection $\mathcal F/\mathcal F_k\to \mathcal F/\mathcal F_{k-1}$.
Denote by $\mathcal F_c$ the $\hbar$-adic completion of the inverse limit
$\varprojlim\mathcal F/\mathcal F_k$.
Note that $\mathcal F_c$ is a complete and separated algebra over $\C[[\hbar]]$ with inverse limit topology.
Let $K$ be the closed ideal of $\mathcal F_c$ generated by the relations (Q1-Q10).
Set
\begin{align*}
  \U_c=\mathcal F_c/K,
\end{align*}
 a completion of $\U$.
Note that there is a canonical injection from $\U$ to $\U_c$.

Now, we consider the space $\U^{\wh\ot^2}$. In this case, we view $\mathcal F$ as a $\Z$-graded algebra by giving $x_{i,\pm n}^+$, $x_{i,\pm n}^-$, $h_{i,\pm n}$ degree $n$ for $n\ge 0$, and give other generators degree $0$.
Denote by $\mathcal F_k'$ the closed two sided ideal of $\mathcal F$ of elements of degree at least $k$.
One notices that $\mathcal F_k'$ is a strict subset of $\mathcal F_k$.
Let $\mathcal F\bar\ot\mathcal F$ be the topological completion of the inverse limit
\begin{align*}
  \mathcal F\wh\ot \mathcal F/\mathcal F_k'\wh\ot\mathcal F_k'.
\end{align*}
Then  $\mathcal F\bar\ot\mathcal F$ is also a complete and separated algebra over $\C[[\hbar]]$.
Define $\U_c\wt\ot\U_c$ to be the quotient algebra of $\mathcal F\bar\ot\mathcal F$ modulo the closure of $K\wh\ot\mathcal F+\mathcal F\wh\ot K$.
It is easy to see that there is a canonical injection from $\U\wh\ot \U$ to $\U_c\wt\ot\U_c$.
Using these completions, we deduce from Theorem \ref{thm:bialg-struct} that
$(\U_c,\Delta_1,\epsilon)$ carries a $\C[[\hbar]]$-bialgebra structure.
Furthermore, by the same argument as in the proof of
\cite[Theorem 2.1]{DI-generalization-qaff}, we have the following result.

\begin{thm}\label{thm:hopf}
$\U_c$ is a Hopf algebra with coproduct $\Delta_1$,
counit $\epsilon$
and the antipode $S$ defined by $(i=0,1)$
\begin{align*}
  &S(c)=-c,\qquad
  S\(x_i^+(z)\)=-\phi_i^-(zq^{-\frac{c}{2}})\inverse x_i^+(zq^{-c}),\\
  &S\(x_i^-(z)\)=-x_i^-(zq^{-c})\phi_i^+(zq^{-\frac{c}{2}}),\quad  S\(\phi_i^\pm(z)\)=\phi_i^\pm(z)\inverse.
\end{align*}
\end{thm}

The rest of this section is devoted to proving Theorem \ref{thm:bialg-struct}.
Recall the algebras $\wh\U$ and $\bar\U$ introduced in Definition \ref{de:algebras}.
Firstly, we have the following straightforward result.

\begin{lem}\label{lem:coprod-Q1-Q8}
(Co1-Co4) defines a unique $\C[[\hbar]]$-algebra homomorphism $\wh\Delta_u:\wh\U\to\big(\wh\U^{\wh\ot^2}\big)((u))$.
\end{lem}

Furthermore,  we have

\begin{lem}\label{lem:coprod-Q9}
$\wh\Delta_u$ induces a $\C[[\hbar]]$-algebra homomorphism from $\bar\Delta_u:\bar\U\to\(\bar\U^{\wh\ot^2}\)((u))$.
\end{lem}

\begin{proof}
Fix any $i\ne j\in\{0,1\}$ and denote by
 $I_{ij}^\pm$ the LHS of the relation (Q8).
We need to  show that $\wh\Delta_u(I_{ij}^\eta)=0$ with $\eta=\pm$. We will show
the case $\eta=+$, as the case $\eta=-$ is similar and thus omitted.
Set
\begin{align*}
  &x_{ij}^+(z,w)=(z-q^{-2}w)x_i^+(z)x_j^+(w)-(q^{-2}z-w)x_j^+(w)x_i^+(z),\\
  &A_{ij}^+(z,w)=(z-q^{-2}w)\wh\Delta_u(x_i^+(z))\wh\Delta_u(x_j^+(w))
  -(q^{-2}z-w)\wh\Delta_u(x_j^+(w))\wh\Delta_u(x_i^+(z)).
\end{align*}
A straightforward calculation shows that
\begin{align*}
  &A_{ij}^+(z,w)
  =x_{ij}^+(z,w)\ot 1
  +q^{-c_1}u\phi_i^-(zq^{\frac{c_1}{2}})\phi_j^-(wq^{\frac{c_1}{2}})
  \ot x_{ij}^+(zu\inv q^{c_1},wu\inv q^{c_1}).
\end{align*}
Using this, we obtain
\begin{align}
  &[\wh\Delta_u(x_i^+(z_1)),A_{ij}^+(z_2,w)]
  =[x_i^+(z_1),x_{ij}^+(z_2,w)]\ot 1\label{eq:coprod-Q9-temp1}\\
  +&[\phi_i^-(z_1q^{\frac{c_1}{2}}),x_{ij}^+(z_2,w)]\ot x_i^+(z_1u\inv q^{c_1})
  +q^{-c_1}u\phi_i^-(z_1q^{\frac{c_1}{2}})\phi_i^-(z_2q^{\frac{c_1}{2}})
  \phi_j^-(wq^{\frac{c_1}{2}})\nonumber\\
  &\ot[x_i^+(z_1u\inv q^{c_1}),x_{ij}^+(z_2u\inv q^{c_1},wu\inv q^{c_1})]\nonumber\\
  &+q^{-c_1}u[x_i^+(z_1),\phi_i^-(z_2q^{\frac{c_1}{2}})
    \phi_j^-(wq^{\frac{c_1}{2}})]
  \ot x_{ij}^+(z_2u\inv q^{c_1},wu\inv q^{c_1})\nonumber
\end{align}
By applying (Q4) one gets that
\begin{align}
  &[\phi_i^-(z_1q^{\frac{c}{2}}),x_{ij}^+(z_2,w)]\nonumber\\
  =&\phi_i^-(z_1q^{\frac{c}{2}})x_{ij}^+(z_2,w)
  \(1- g_{ji}(z_1/w)g_{ii}(z_1/z_2) \)\nonumber\\
  =&\frac{(q^2-q^{-2})z_1(z_2-w)}{(w-q^{-2}z_1)(z_2-q^2z_1)}
  \phi_i^-(z_1q^{\frac{c}{2}})x_{ij}^+(z_2,w),\label{eq:coprod-Q9-temp2}\\
  &[x_i^+(z_1),\phi_i^-(z_2q^{\frac{c}{2}})
    \phi_j^-(wq^{\frac{c}{2}})]\nonumber\\
  =&\phi_i^-(z_2q^{\frac{c}{2}})
    \phi_j^-(wq^{\frac{c}{2}})x_i^+(z_1)
    \(g_{ji}(z_1/w)g_{ii}(z_1/z_2)-1\)\nonumber\\
  =&-\frac{(q^2-q^{-2})z_1(z_2-w)}{(w-q^{-2}z_1)(z_2-q^2z_1)}
  \phi_i^-(z_2q^{\frac{c}{2}})
    \phi_j^-(wq^{\frac{c}{2}})x_i^+(z_1).\label{eq:coprod-Q9-temp3}
\end{align}
Recall from (Q7) that
\begin{align}\label{eq:coprod-Q9-temp4}
  (z-w)x_{ij}^+(z,w)=0=(z-w)x_{ij}^+(zu\inv q^{c_1},wu\inv q^{c_1}).
\end{align}
Combining  \eqref{eq:coprod-Q9-temp1}, \eqref{eq:coprod-Q9-temp2}, \eqref{eq:coprod-Q9-temp3} and \eqref{eq:coprod-Q9-temp4}, we deduce from (Q8) that
\begin{align*}
  [\wh\Delta_u(x_i^+(z_1)),A_{ij}^+(z_2,w)]=0.
\end{align*}
This implies that $\wh\Delta_u(I_{ij}^+)=0$, as required.
\end{proof}

To continue the discussion, we need to introduce some notations.
For  $0\le s\le 3$, we denote by $S_{3,s}$ the set of $(s,3-s)$-shuffles in $S_3$, that is
\begin{align*}
  S_{3,s}=\set{\sigma\in S_3}{\sigma(a)<\sigma(b),\,\te{for }a<b\le s\,\,\te{or}\,\,s<a<b}.
\end{align*}
As a convention, we let $\sigma(a)=a$ for any $a<1$ or $a>3$.
For $0\le s\le 3$ and $\sigma\in S_{3,s}$, we define two partitions
\begin{align*}
P_{0,\sigma}^1\cup \cdots \cup P_{3-s,\sigma}^1\quad\te{and}\quad
P_{0,\sigma}^1\cup \cdots \cup P_{s+1,\sigma}^1
\end{align*}
 of the set $\{0,1,2,3\}$,
where
\begin{align*}
  &P_{0,\sigma}^1=\set{p\in\Z}{0\le p<\sigma(s+1)},\\
  &P_{k,\sigma}^1=\set{p\in\Z}{\sigma(s+k)\le p<\sigma(s+k+1)}\quad\te{for } 0<k<3-s,\\
  &P_{3-s,\sigma}^1=\set{p\in\Z}{\sigma(3)\le p\le 3}\quad\te{if }s<3,\\
  &P_{0,\sigma}^2=\set{p\in\Z}{0\le p<\sigma(1)}\quad\te{if }s>0,\\
  &P_{k,\sigma}^2=\set{p\in\Z}{\sigma(k)\le p<\sigma(k+1)}\quad\te{for } 0<k<s,\\
  &P_{s,\sigma}^2=\set{p\in\Z}{\sigma(s)\le p\le 3}.
\end{align*}
Furthermore, for $0\le s\le 3$ and $0\le k\le 3-s$, set
\begin{align*} &T_{s,k}^1(z_1,z_2,z_3,w)=
  \sum_{\sigma\in S_{3,s}}\sum_{r\in P_{k,\sigma}^1}\binom{3}{r}_q(-1)^{r}
  \prod_{\substack{a\le s<b, \sigma(a)<\sigma(b)}}(q^2z_{a}-z_{b})\\
  &\cdot \prod_{\substack{a\le s<b, \sigma(a)>\sigma(b)}}(z_{a}-q^2z_{b})
  \prod_{\substack{a\le s, \sigma(a)\le r}}(q^{-2}z_{a}-w)
  \prod_{\substack{a\le s, \sigma(a)> r}}(z_{a}-q^{-2}w),
\end{align*}
and for $0\le s\le 3$ and $0\le k\le s$, set
\begin{align*}
  &T_{s,k}^2(z_1,z_2,z_3,w)=
  \sum_{\sigma\in S_{3,s}}\sum_{r\in P_{k,\sigma}^2}\binom{3}{r}_q(-1)^{r}
  \prod_{\substack{a\le s<b, \sigma(a)<\sigma(b)}}(q^2z_{a}-z_{b})\\
  &\cdot \prod_{\substack{a\le s<b, \sigma(a)>\sigma(b)}}(z_{a}-q^2z_{b})
\prod_{\substack{a> s, \sigma(a)>r}}(q^{-2}w-z_{a})
  \prod_{\substack{a> s, \sigma(a)\le r}}(w-q^{-2}z_{a}).
\end{align*}
We have:
\begin{lem}\label{lem:coprod-Q10-2}
\textrm{(1)} For  $0\le k\le 3$, one has that
  $$T_{0,k}^1(z_1,z_2,z_3,w)=\binom{3}{k}_q(-1)^k.$$

\textrm{(2)}
There exist  polynomials $f_{1,0}(z_1,z_2),f_{1,1}(z_1,z_2),f_{1,2}(z_1,z_2)\in\C[[\hbar]][z_1,z_2]$  such that
\begin{align*}
  &T_{1,0}^1(z_1,z_2,z_3,w)
  =f_{1,0}(z_1,w)(z_2-q^2 z_3)-f_{1,1}(z_1,z_3)(q^{-2}z_2-w),\\
  &T_{1,1}^1(z_1,z_2,z_3,w)
  =f_{1,1}(z_1,z_3)(z_2-q^{-2}w)+f_{1,1}(z_1,z_2)(q^{-2}z_3-w),\\
  &T_{1,2}^1(z_1,z_2,z_3,w)
  =-f_{1,1}(z_1,z_2)(z_3-q^{-2}w)+f_{1,2}(z_1,w)(z_2-q^2z_3).
\end{align*}
.

\textrm{(3)} There exist polynomials  $$f_{2,0}(z_1,z_2,z_3,w),f_{2,1}(z_1,z_2),f_{2,2}(z_1,z_2,z_3,w)\in\C[[\hbar]][z_1,z_2,z_3,w]$$
such that  $$f_{2,k}(z_1,z_2,z_3,w)=f_{2,k}(z_2,z_1,z_3,w),\quad k=0,2,$$
and
\begin{align*}
  &T_{2,0}^1(z_1,z_2,z_3,w)=f_{2,0}(z_1,z_2,z_3,w)(z_1-q^2 z_2)
  +f_{2,1}(z_1,z_2)(z_3-z_2)(q^{-2}z_3-w),\\
  &T_{2,1}^1(z_1,z_2,z_3,w)=f_{2,2}(z_1,z_2,z_3,w)(z_1-q^2z_2)
  -f_{2,1}(z_1,z_2)(z_3-z_2)(z_3-q^{-2}w).
\end{align*}

\textrm{(4)} There exist polynomials $f_{3,0}(z_1,z_2),f_{3,1}(z_1,z_2)\in\C[[\hbar]][z_1,z_2]$ such that
\begin{align*}
  &T_{3,0}^1(z_1,z_2,z_3,w)=f_{3,0}(z_3,w)(z_1-q^2z_2)+f_{3,1}(z_1,w)(z_2-q^2w).
\end{align*}

\textrm{(5)} $T_{s,k}^2(z_1,z_2,z_3,w)=(-1)^{s}T_{3-s,k}^1(z_3,z_1,z_2,w)$ if $s<3$.

\textrm{(6)}
$T_{3,0}^1(z_1,z_2,z_3,w)=-T_{0,0}^2(z_1,z_2,z_3,w)$.
\end{lem}

\begin{proof}
The lemma follows from the following straightforward facts:
\begin{align*}
  &T_{0,k}^1(z_1,z_2,z_3,w)=T_{3,k}^2(z_1,z_2,z_3,w)=\binom{3}{k}_q(-1)^k,\\
  &T_{1,0}^1(z_1,z_2,z_3,w)=-T_{2,0}^2(z_2,z_3,z_1,w)\\
  &\quad=(q^{4}-1)z_1\big((q^{-4}z_1-w)(z_2-q^2z_3)
    -[3]_{q^2}(z_1-z_3)(q^{-2}z_2-w)\big),\\
  &T_{1,1}^1(z_1,z_2,z_3,w)=-T_{2,1}^2(z_2,z_3,z_1,w)\\
  &\quad=(q^4-1)z_1[3]_{q^2}\big((z_1-z_3)(z_2-q^{-2}w)
  +(z_1-z_2)(q^{-2}z_3-w)\big),\\
  &T_{1,2}^1(z_1,z_2,z_3,w)=-T_{2,2}^2(z_2,z_3,z_1,w)\\
  &\quad=(q^4-1)z_1\big(
    -[3]_{q^2}(z_1-z_2)(z_3-q^{-2}w)
    +(z_1-q^{-4}w)(z_2-q^2z_3)
  \big),\\
  &T_{2,0}^1(z_1,z_2,z_3,w)=T_{1,0}^2(z_2,z_3,z_1,w)\\
  &\quad=z_1z_2(1-q^{-2})(q^4-1)[3]_{q^2}(z_3-w)(q^{-2}z_3-w)
  +(1-q^{-4})(z_1-q^2z_2)\\
  &\quad\times\big(q^{2}[3]_{q^2}(z_3-w)(q^4z_1z_2-z_3w)
  -z_3(1+q^{-2})(q^2z_1-z_3)(q^2z_2-z_3)\big),\\
  &T_{2,1}^1(z_1,z_2,z_3,w)=T_{1,1}^2(z_2,z_3,z_1,w)\\
  &\quad=-z_1z_2(1-q^{-2})(q^4-1)[3]_{q^2}(z_3-w)(z_3-q^{-2}w)
  -(1-q^{-4})(z_1-q^2z_2)\\
  &\quad\times\big(q^{2}[3]_{q^2}(z_3-w)(z_1z_2-q^4z_3w)
  -z_3(1+q^{-2})(z_1-q^2z_3)(z_2-q^2z_3)\big),\\
  &T_{3,0}^1(z_1,z_2,z_3,w)=-T_{0,0}^2(z_1,z_2,z_3,w)\\
  &\quad=w(1-q^{-4})\big(-(z_3-q^{-4}w)
 (z_1-q^2 z_2)+(q^{-4}z_1-w)(z_2-q^2z_3)\big).
\end{align*}
\end{proof}

 For $i\ne j\in \{0,1\}$, denote by $J_{ij}^\pm$  the
LHS of (Q9).
\begin{lem}\label{lem:coprod-Q10-1}
 For $i\ne j\in \{0,1\}$, we have
\begin{align*}
  \bar\Delta_u(J_{ij}^+)
  =&\sum_{\tau\in S_3}\sum_{s=0}^3\sum_{k=0}^{3-s}
  \frac{T_{s,k}^1(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)}
  {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
  \prod_{a\le s}(z_{\tau(a)}-q^{-2}w)}\\
  \cdot&
  \wt\phi_i^-(z_{\tau(s+1)})\cdots \wt\phi_i^-(z_{\tau(3)})\wt\phi_j^-(w)
  x_i^+(z_{\tau(1)})\cdots x_i^+(z_{\tau(s)})\\
  &\ot
  \wt x_i^+(z_{\tau(s+1)})\cdots \wt x_i^+(z_{\tau(k)})\wt x_j^+(w)
  \wt x_i^+(z_{\tau(k+1)})\cdots \wt x_i^+(z_{\tau(3)})\\
  +\sum_{\tau\in S_3}&\sum_{s=0}^3\sum_{k=0}^{s}
  \frac{T_{s,k}^2(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)}
    {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
    \prod_{a> s}(w-q^{-2}z_{\tau(a)})}\\
  \cdot&
  \wt\phi_i^-(z_{\tau(s+1)})\cdots \wt\phi_i^-(z_{\tau(3)})
  x_i^+(z_{\tau(1)})\cdots x_i^+(z_{\tau(k)})x_j^+(w)\\
  \cdot&
  x_i^+(z_{\tau(k+1)})
  \cdots x_i^+(z_{\tau(s)})
  \ot \wt x_i^+(z_{\tau(s+1)})\cdots \wt x_i^+(z_{\tau(3)}),
\end{align*}
where $\wt\phi_k^-(z)=\phi_k^-(zq^{\frac{c_1}{2}})$ and $\wt x_k^+(z)=x_k^+(zu\inv q^{c_1})$ for $k=i,j$.
\end{lem}

\begin{proof}
For $\sigma\in S_{3,s}$, we define
\begin{align*}
  \xi_\sigma(z_a)=\begin{cases}
           x_i^+(z_a),&\te{if }\sigma\inv(a)\le s,\\
           \wt\phi_i^-(z_a),&\te{if }\sigma\inv(a)>s,
         \end{cases},
  \quad
  \wt\xi_\sigma(z_a)=\begin{cases}
                \wt x_i^+(z_a),&\te{if }\sigma\inv(a)>s,\\
                1,&\te{if }\sigma\inv(a)\le s.
              \end{cases}
\end{align*}
For $0\le r,s\le 3$ and $\sigma\in S_{3,s}$, set
\begin{align*}
  \xi_{r,\sigma}(z_1,z_2,z_3,w)=&\xi_\sigma(z_1)\cdots \xi_\sigma(z_r)\phi_j^-(wq^{\frac{c}{2}})\xi_\sigma(z_{r+1})
  \cdots \xi(z_3),\\
  \eta_{r,\sigma}(z_1,z_2,z_3,w)=&\xi_\sigma(z_1)\cdots \xi_\sigma(z_r)x_j^+(w)\xi_\sigma(z_{r+1})
  \cdots \xi_\sigma(z_3),\\
  \wt \xi_\sigma(z_1,z_2,z_3)=&\wt\xi_\sigma(z_1)\wt\xi_\sigma(z_2)\wt\xi_\sigma(z_3),\\
  \wt \eta_{r,\sigma}(z_1,z_2,z_3,w)=&\wt\xi_\sigma(z_1)\cdots \wt\xi_\sigma(z_r)x_j^+(wu\inv q^{c_1})\wt\xi_\sigma(z_{r+1})
  \cdots \wt\xi_\sigma(z_3).
\end{align*}
And for $i_1,i_2,\dots,i_n\in \{0,1\}$, $\zeta_{i_r}(z_r)=x_{i_r}^+(z_r)$ or $\phi_{i_r}^-(z_rq^{\frac{c}{2}})$, we define an ordered product
\begin{align*}
  \:\zeta_{i_1}(z_1)\cdots \zeta_{i_n}(z_n)\;
\end{align*}
by moving $\phi_{i_r}^-(z_rq^{\frac{c}{2}})$ to the left.
Then it follows from (Q4) that
\begin{align*}
  &\xi_{r,\sigma}(z_1,z_2,z_3,w)=\prod_{\substack{ a\le s<b}}
  g_{ii}(z_{\sigma(b)}/z_{\sigma(a)})
  \prod_{\substack{a\le s\\ \sigma(a)\le r}}g_{ij}(w/z_{\sigma(a)})
  \:\xi_{r,\sigma}(z_1,z_2,z_3,w)\;,\\
  &\eta_{r,\sigma}(z_1,z_2,z_3,w)=\prod_{\substack{ a\le s<b}}
  g_{ii}(z_{\sigma(b)}/z_{\sigma(a)})
  \prod_{\substack{a\le s\\ \sigma(a)> r}}g_{ji}(z_{\sigma(a)}/w)
  \:\eta_{r,\sigma}(z_1,z_2,z_3,w)\;.
\end{align*}
Moreover, it is straightforward to see that
\begin{align*}
  &\Delta_u(x_i^+(z_1))\cdots\Delta_u(x_i^+(z_r))\Delta_u(x_j^+(w))
  \Delta_u(x_i^+(z_{r+1}))\cdots\Delta_u(x_i^+(z_3))\\
  =&\sum_{s=0}^3\sum_{\sigma\in S_{3,s}}
  \prod_{\substack{ a\le s<b}}
  g_{ii}(z_{\sigma(b)}/z_{\sigma(a)})
  \prod_{\substack{a\le s\\ \sigma(a)\le r}}g_{ij}(w/z_{\sigma(a)})
  \:\xi_{r,\sigma}(z_1,z_2,z_3,w)\;\ot\wt\eta_{r,\sigma}(z_1,z_2,z_3,w)\\
  &+\sum_{s=0}^3\sum_{\sigma\in S_{3,s}}
  \prod_{\substack{ a\le s<b}}
  g_{ii}(z_{\sigma(b)}/z_{\sigma(a)})
  \prod_{\substack{a\le s\\ \sigma(a)> r}}g_{ji}(z_{\sigma(a)}/w)
  \:\eta_{r,\sigma}(z_1,z_2,z_3,w)\;\ot\wt\xi_\sigma(z_1,z_2,z_3).
\end{align*}

Now the lemma follows from a direct calculation and the following facts:
\begin{align*}
  &\:\xi_{r_1,\sigma}(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)\;\ot \wt\eta_{r_1,\sigma}(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)\\
  =&\:\xi_{0,1}(z_{\tau\sigma(1)},z_{\tau\sigma(2)},z_{\tau\sigma(3)},w)\;\ot \wt\eta_{p_{k,\sigma}^1,1}(z_{\tau\sigma(1)},z_{\tau\sigma(2)},z_{\tau\sigma(3)},w),\\
  &\:\eta_{r_2,\sigma}(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)\;\ot \wt\xi_{\sigma}(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)\\
  =&\:\eta_{p_{k,\sigma}^2,1}(z_{\tau\sigma(1)},z_{\tau\sigma(2)},z_{\tau\sigma(3)},w)\;\ot \wt\xi_{1}(z_{\tau\sigma(1)},z_{\tau\sigma(2)},z_{\tau\sigma(3)},w),
\end{align*}
where $\tau\in S_3$, $\sigma\in S_{3,s}$ ($0\le s\le 3$), $r_a\in P_{k,\sigma}^a$ ($a=1,2$) and
$p_{k,\sigma}^a$ the minimal element in $P_{k,\sigma}^a$ ($a=1,2$).

\end{proof}

\textbf{Proof of Theorem \ref{thm:bialg-struct}}:
In view of Lemmas \ref{lem:coprod-Q1-Q8} and \ref{lem:coprod-Q9}, it suffices to show that
\begin{align*}
  \bar\Delta_u(J_{ij}^\pm)=0,\quad \te{for }i\ne j\in\{0,1\}.
\end{align*}
Let  $i\ne j\in\{0,1\}$ be fixed.
From Lemma \ref{lem:coprod-Q10-1} and (1), (5) of Lemma \ref{lem:coprod-Q10-2}, it follows  that
\begin{align*}
  \bar\Delta_u(J_{ij}^+)
  =&\sum_{\tau\in S_3}\sum_{s=1}^3\sum_{k=0}^{3-s}
  \frac{T_{s,k}^1(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)}
  {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
  \prod_{a\le s}(z_{\tau(a)}-q^{-2}w)}\\
  \cdot&
  \wt\phi_i^-(z_{\tau(s+1)})\cdots \wt\phi_i^-(z_{\tau(3)})\wt\phi_j^-(w)
  x_i^+(z_{\tau(1)})\cdots x_i^+(z_{\tau(s)})\\
  &\ot
  \wt x_i^+(z_{\tau(s+1)})\cdots \wt x_i^+(z_{\tau(k)})\wt x_j^+(w)
  \wt x_i^+(z_{\tau(k+1)})\cdots \wt x_i^+(z_{\tau(3)})\\
  +\sum_{\tau\in S_3}&\sum_{s=0}^2\sum_{k=0}^{s}
  \frac{T_{s,k}^2(z_{\tau(1)},z_{\tau(2)},z_{\tau(3)},w)}
    {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
    \prod_{a> s}(w-q^{-2}z_{\tau(a)})}\\
  \cdot&
  \wt\phi_i^-(z_{\tau(s+1)})\cdots \wt\phi_i^-(z_{\tau(3)})
  x_i^+(z_{\tau(1)})\cdots x_i^+(z_{\tau(k)})x_j^+(w)\\
  \cdot&
  x_i^+(z_{\tau(k+1)})
  \cdots x_i^+(z_{\tau(s)})
  \ot \wt x_i^+(z_{\tau(s+1)})\cdots \wt x_i^+(z_{\tau(3)}).
\end{align*}
Combining this with Lemma \ref{lem:coprod-Q10-2} and (Q6-Q7), a similar argument of Lemma \ref{lem:tri-tech3} shows that
\begin{equation}\begin{split}\label{newrela2}
  \bar\Delta_u(J_{ij}^+)
  &=\sum_{\tau\in S_3}
  \frac{f_{1,1}(z_{\tau(1)},z_{\tau(3)})}
  {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
  \prod_{a\le s}(z_{\tau(a)}-q^{-2}w)}\wt\phi_i^-(z_{\tau(2)})\wt\phi_i^-(z_{\tau(3)})\wt\phi_j^-(w)\\
  \cdot&
  x_i^+(z_{\tau(1)})
  \ot\big[(z_{\tau(2)}-q^{-2}w)\wt x_i^+(z_{\tau(2)})\wt x_j^+(w)
    -(q^{-2}z_{\tau(2)}-w)\wt x_j^+(w)
  \wt x_i^+(z_{\tau(2)}),\wt x_i^+(z_{\tau(3)})\big]\\
  +\sum_{\tau\in S_3}&
  \frac{f_{1,1}(z_{\tau(3)},z_{\tau(2)})}
    {\prod_{a\le s<b}(z_{\tau(a)}-q^2z_{\tau(b)})
    \prod_{a> s}(w-q^{-2}z_{\tau(a)})}\wt\phi_i^-(z_{\tau(3)})\\
  \cdot&
  \big[(z_{\tau(1)}-q^{-2}w)x_i^+(z_{\tau(1)})x_j^+(w)
  -(q^{-2}z_{\tau(1)}-w)
  x_j^+(w)x_i^+(z_{\tau(1)})
  , x_i^+(z_{\tau(2)})\big]
  \ot \wt x_i^+(z_{\tau(3)}).
\end{split}\end{equation}
Then it follows from (Q8) that $\bar\Delta_u(J_{ij}^+)=0$.
The proof of $\bar\Delta_u(J_{ij}^-)=0$ is similar.
Therefore, we complete the proof of Theorem \ref{thm:bialg-struct}.

\begin{rem}{\em Let $\wt{\U}_\hbar(\dot\g_{tor})$ be the quotient algebra of $\wt{\U}$ modulo the relation \eqref{strongerrela}.
It was shown in \cite[Proposition 29]{He-representation-coprod-proof} that the action (Co1-Co4) defines a unique
algebra homomorphism $\wt\Delta_u:\wt\U_\hbar(\dot\g_{tor})\rightarrow \big(\wt\U_\hbar(\dot\g_{tor})^{\wh\ot^2}\big) ((u))$.
Recall that $\U_\hbar(\dot\g_{tor})$ is the quotient algebra of $\wt{\U}_\hbar(\dot\g_{tor})$ modulo the relation (Q9).
 As the relations (Q6-Q8) also hold in  $\U_\hbar(\dot\g_{tor})$,
it follows from (the proof of) Theorem \ref{thm:bialg-struct}  that $\wt\Delta_u$ is compatible with the affine quantum Serre relation (Q9).
This implies that $\wt\Delta_u$ induces an algebra homomorphism from
$\U_\hbar(\dot\g_{tor})$ to $\big(\U_\hbar(\dot\g_{tor})^{\wh\ot^2}\big) ((u))$, as expected in \cite[Remark 6]{He-representation-coprod-proof}.
}\end{rem}

\section{Vertex representation of $\U$}

In this section we show the  vertex representation for $\U_\hbar(\hat\g)$ given in \cite{J-KM} induces a representation for $\U$.

We first recall the vertex representation given in \cite{J-KM}.
Let
\begin{align}
  \symalg=\C[h_{i,n}\mid i=0,1,n<0][[\hbar]]
\end{align}
be the symmetric $\C[[\hbar]]$-algebra in the variable $h_{i,n}$, which is topologically free as a $\C[[\hbar]]$-module.
It is known that there is an $\mathcal H$-action on $\symalg$ such that  $c=1$, $h_{i,0}=0$, $h_{i,-n}=$ multiplication operator and
$h_{i,n}=$ annihilation operator subject to the relation
\begin{eqnarray*}
[h_{i,m},h_{j,-n}]=\frac{\delta_{m+n,0}}{n}[n a_{ij}]_q\frac{q^{nc}-q^{-nc}}{q-q\inv},
\end{eqnarray*}
where $i=0,1$ and $n>0$.
From now on, we take $c$ and $h_{i,n}$ ( $i=0,1$, $n\in\Z$)
as operators on $\symalg$.
For $i=0,1$, we introduce the following fields on $\symalg$
\begin{align*}
&h_i^\pm(z)=\sum\limits_{\pm m>0}h_{i,m}z^{-m},\quad
E_i^\pm(z)=\te{exp}\left(\mp\sum\limits_{m>0}
    \frac{h_{i,\pm m}}{[m]_q}z^{\mp m}\right).
\end{align*}

Let $Q=\Z\al_0\oplus\Z\al_1$ be a rank $2$  lattice equipped with a semi-positive form $\<\cdot,\cdot\>$ determined by
$\<\al_i,\al_j\>=a_{ij}$ for $i,j=0,1$.
We fix a $2$-cocycle $\varepsilon:Q\times Q\to \{\pm 1\}$ such that
\begin{align*}
  &\varepsilon(\al,\be)\varepsilon(\be,\al)\inv=C(\al,\be),\quad
  \varepsilon(\al,0)=1=\varepsilon(0,\al),
\end{align*}
where
\begin{eqnarray*}\label{eq:group-commutator}
C: Q\times Q\rightarrow \{\pm 1\},\quad (\al,\be)\mapsto (-1)^{\<\al|\be\>}.
\end{eqnarray*}
Denote by $\C_\varepsilon[Q]$ the $\varepsilon$-twisted group $\C$-algebra of $Q$,
which by definition has a designated basis $\{e_\al\mid \al\in Q\}$
such that $e_\al\cdot e_\be=\varepsilon(\al,\be)e_{\al+\be}$ for $\al,\be\in Q$.
Then $\C_\hbar\{Q\}=\C_\varepsilon [Q][[\hbar]]$ becomes an $\mathcal H$-module under the action that
 $c$ and $h_{i,n}$ act trivially for $i=0,1$, $n\ne 0$, and $h_{i,0}$ ($i=0,1$) act by
\begin{align*}
  h_{i,0}\cdot e_\be=\<\be,\al_i\> e_\be\quad\te{for }\be\in Q.
\end{align*}
For $i=0,1$, define a linear operator $z^{h_{i,0}}:\C_\hbar\{Q\}\to\C_\hbar\{Q\}[[z,z\inv]]$ by
\begin{align*}
  z^{h_{i,0}}\cdot e_\be=z^{\<\beta,\al_i\>}e_\be\quad\te{for }\be\in Q.
\end{align*}

We define the Fock space
\begin{align}
  V=\symalg\wh\ot\C_\hbar\{Q\}
\end{align}
to be the tensor product  $\mathcal H$-module.
Recall the vertex operators define in \cite{J-KM}:
\begin{align}
  &X_i^\pm(z)=\sum_{n\in\Z}X_{i,n}z^{-n-1}=E_i^-(q^{\mp\half}z)^{\pm 1}E_i^+(q^{\pm\half}z)^{\pm 1}e_{\pm \al_i}z^{\pm h_{i}},
\end{align}
As usual, the normal ordered product $\:X_i^\pm(z)X_j^\pm(w)\;$ is defined by moving annihilation operators
$h_{i,n}$ ($i=0,1$, $n\ge 0$) to the right.
It was proved in \cite[(3.6)]{J-KM} that
\begin{align}\label{eq:locality}
  &X_i^\pm(z)X_j^\pm(w)=\:X_i^\pm(z)X_j^\pm(w)\;(z-q^{\mp 1}w)_{q^2}^{a_{ij}},\\
  &\:X_i^\pm(z)X_j^\pm(w)\;=(-1)^{a_{ij}}\:X_j^\pm(w)X_i^\pm(z)\;
  =\:X_j^\pm(w)X_i^\pm(z)\;,
\end{align}
where for $n\in\N$,
\begin{align*}
  &(1-x)_{q^2}^n=(1-q^{1-n}x)(1-q^{3-n}x)\cdots(1-q^{n-1}x),\\
  &(1-x)_{q^2}^{-n}=1/(1-x)_{q^2}^n,\quad
  (z-w)_{q^2}^{\pm n}=(1-w/z)_{q^2}^{\pm n}z^{\pm n},
\end{align*}
and $(1-x)_{q^2}^{-n}$ is understood as power series in $x$.
We have:

\begin{thm}\label{thm:vr}
There is a $\U$-module structure on the Fock space $V$  with the action given by
\[c\mapsto 1,\quad h_{i,n}\mapsto h_{i,n}\quad\te{and}\quad x_{i,n}^\pm\mapsto X_{i,n}^\pm\]
where  $i=0,1$ and $n\in\Z$.
\end{thm}

\begin{proof}
The relations (Q1-Q7) and (Q9) have been checked in  \cite[Theorem 3.1]{J-KM} and it remains to prove
 the relation (Q8).
For $i\ne j\in \{0,1\}$,
it follows from  \eqref{eq:locality} that
\begin{align}
  X_{ij}^\pm(z,w)=
  &(z-q^{\mp 2}w)X_i^\pm(z)X_j^\pm(w)-(q^{\mp 2}z-w)X_j^\pm(w)X_i^\pm(z)
  \nonumber\\
  =&\:X_i^\pm(z)X_j^\pm(w)\;z\inv\delta\(\frac{w}{z}\)
  =\:X_i^\pm(w)X_j^\pm(w)\;z\inv\delta\(\frac{w}{z}\).\label{eq:Q9-temp1}
\end{align}
It is straightforward to check that
\begin{align}
  &X_i^\pm(z)\:X_i^\pm(w)X_j^\pm(w)\;
  =\:X_i^\pm(z)\:X_i^\pm(w)X_j^\pm(w)\;\;\nonumber\\
  =&\:\:X_i^\pm(w)X_j^\pm(w)\;X_i^\pm(z)\;
  =\:X_i^\pm(w)X_j^\pm(w)\;X_i^\pm(z).\label{eq:Q9-temp2}
\end{align}
Then it follows from \eqref{eq:Q9-temp1} and \eqref{eq:Q9-temp2} that (Q8) holds on $V$, as required.
\end{proof}

\begin{rem}{\em From \eqref{eq:locality}, it follows that for $i\ne j\in \{0,1\}$,
\[(z-q^{\mp 2}w)X_i^\pm(z)X_j^\pm(w)\ne (q^{\mp 2}z-w)X_j^\pm(w)X_i^\pm(z).\]
In particular,  the action given in Theorem \ref{thm:vr} can not induce a $\U_\hbar(\dot\g_{tor})$-module structure on $V$.
}\end{rem}

\bigskip


TITLE: Does the 'arrow' internal category forgetful functor have a right adjoint?
QUESTION [3 upvotes]: Let $\mathcal{E}$ be a category with finite limits, and let $Arr : Cat(\mathcal{E}) \to \mathcal{E}$ be the forgetful functor that sends an internal category to its object of arrows/morphisms. Does this functor always have a right adjoint? In particular, does $Arr : Cat \to Set$ have a right adjoint?
One conjecture I have is that if $\mathcal{E}$ also has binary coproducts and $X$ is an object of $\mathcal{E}$, then the right adjoint to $Arr$ would send $X$ to the indiscrete internal category on the $\mathcal{E}$-object $X + X$ of objects.
I am particularly interested in knowing whether $Arr : Cat(\mathcal{E}) \to \mathcal{E}$ preserves (binary) coproducts in general, which is true in the case $\mathcal{E} = Set$.

REPLY [4 votes]: No, $Arr$ does not even have a right adjoint in the case $\mathcal{E}=\mathsf{Set}$. The reason is that it fails to preserve coequalizers.
Any monoid $M$ can be viewed as a category with one object (so the monoid is the set of arrows under composition). The coequalizer of a pair of monoid homomorphisms in the category $\mathsf{Mon}$ of monoids agrees with the coequalizer of the corresponding pair of functors in $\mathsf{Cat}$. But the forgetful functor from $\mathsf{Mon}$ to $\mathsf{Set}$ does not preserve coequalizers, so neither does $Arr$.


text:
\begin{document}

\title{Exact Solution of Schrodinger Equation in (Anti-)deSitter Spaces for Hydrogen Atom}
\author{{Mokhtar FALEK}$^{1}$
\and {Noureddine BELGHAR}$^{2}$
\and {Mustafa MOUMNI}$^{1}$\\m.moumni@univ-biskra.dz\\$^{1}$Laboratory of Photonic Physics and Nano-Materials (LPPNNM)\\Department of Matter Sciences, University of Biskra, ALGERIA \\$^{2}$Laboratory of Energy Engineering and Materials (LGEM)\\Department of Mechanics, University of Biskra, ALGERIA}
\maketitle

\begin{abstract}
We write Schrödinger equation for the Coulomb potential in both deSitter and Anti-deSitter spaces using the Extended Uncertainty Principle formulation. We use the Nikiforov-Uvarov method to solve the equations. The energy eigenvalues for both systems are given in their exact forms and the corresponding radial wave functions are expressed in associated Jacobi polynomials for deSitter space, while those of Anti-deSitter space are given in terms of Romanovski polynomials. We have also studied the effect of the spatial deformation parameter on the bound states in the two cases.

\textbf{PACS:} 03.65.Ge, 03.65.Pm.

\end{abstract}
\tableofcontents

\section{Introduction}

\label{Int}

The extension of the quantum field theory to curved space-time, which can be
considered as a first approximation of quantum gravity has attracted
considerable interest as there are strong motivations for absorption of
infinities lying in standard field theories. In such situation of curved
space-time, we deal with a structure perturbed by the gravitational field.
Such modifications can also be found in Snyder model where the measurements in
noncommutative quantum mechanics can be governed by a Generalized Uncertainty
Principle (GUP) \cite{Snyder}. This model admits a fundamental length scale
supposed to be of the order of the Planck length and this is equivalent to a
nonzero minimal uncertainty in the measurement of the position \cite{Kempf}
\cite{Mendes}. Because there are many arguments showing that quantum gravity
implies also a minimal measurable length in the order of the Planck length, a
large amount of efforts have been devoted to extend the study of the quantum
mechanics to a curved space--time via the Extended Uncertainty Principle (EUP)
\cite{Chung}. A significant consequence deduced from this extension is that
the minimal length uncertainty in quantum gravity can be related also to a
modification of the standard Heisenberg algebra by adding small corrections to
the canonical commutation relations \cite{Mignemi}\cite{Ghosh}\cite{Nozari}.
This was motivated by Doubly Special Relativity (DSR) \cite{Amelino}, string
theory \cite{Capozziello}, non-commutative geometry \cite{Douglas} and also
black hole physics \cite{Scardigli}.

In the context of deformed quantum theory with EUP, there are only a few
available exact solutions. At the level of relativistic quantum mechanics the
list of the exactly solved problems is very restricted, e.g. the case of
one-dimensional Dirac and Klein-Gordon oscillators on anti-deSitter (AdS)
space was recently considered in \cite{Hamil1}, the three and two-dimensional
Dirac oscillator in the presence of minimal uncertainty in momentum was
studied in \cite{Hamil2} and the exact solution of (1+1)-dimensional bosonic
oscillator subject to the influence of an uniform electric field in AdS space
too \cite{Merad}. On the other hand, the non-relativistic case is also of
great interest and remains unexplored within this framework. Despite the fact that, in conventional field theory approach in static de Sitter and anti de Sitter space-time models, we cannot derive any nonrelativistic covariant Schr\"{o}dinger-like equation from covariant Klein-Fock-Gordon equation, we can use the EUP formulation to write the dS and AdS versions of the Schr\"{o}dinger equation. Indeed Hamil et al treat the exact solution of the D-dimensional Schrödinger equation for the free-particle and the harmonic-oscillator in AdS space \cite{Hamil4}. In \cite{Chung}, Chung study analytically the one dimensional box problem and the harmonic oscillator problem. Also in \cite{Ghosh}, Ghosh and Mignemi use perturbative methods to study both harmonic oscillator and Hydrogen atom.

Regarding the hydrogen atom and because of the physics that comes from
studying and understanding such system, there has been a growing interest in
the study of exact solutions of this kind of problem in the ordinary case
\cite{Reyes}\cite{Ran}\cite{Gordeyev}\cite{Tsutsui}\cite{Yepez} as well as
in\ the context of deformed quantum mechanics based on GUP and we cite here
the study of Schr\"{o}dinger equation for the Coulomb potential with minimal
length in one dimension \cite{Pedram}\cite{Nouicer}\cite{Fityo} and in three
dimensions \cite{Brau}\cite{Akhoury}\cite{Benczik}.

In this paper, we are looking for the analytical treatment of the hydrogen
atom when subject to gravitational effects governed by EUP because it has been
studied only perturbatively \cite{Ghosh}. For this purpose we solve the
non-relativistic Coulomb problem to get the exact form of the energy
eigenvalues and eigenfunctions. The paper is organized as follows: In Sec. II
\ref{Def}, we give a review dS and AdS models while In Sec. III \ref{N-U}, we
introduce Nikiforov-Uvarov (NU) method that we use to solve equation of our
system. We expose in Sec. IV \ref{Sch}, the explicit computations for the
hydrogen atom of the deformed Schr\"{o}dinger equation with EUP and in both dS
and AdS cases of the algebra. The energy eigenvalues are given in their exact
form and the corresponding radial wave functions are expressed in associated
Jacobi polynomials for dS space \ref{dS}\ and in terms of Romanovski
polynomials for the AdS space \ref{AdS}. In the end of this section, we
investigate numerically the spectroscopic implications of the EUP deformation.
Finally, the concluding remarks come in Sec. V \ref{cc}.

\section{Review on the Deformed Quantum Mechanics Relation}

\label{Def}

In three-dimensional space, the deformed Heisenberg algebra leading to EUP is
defined by the following commutation relations \cite{Mignemi1}\cite{Stetsko}
\begin{equation}
\left[  X_{i},X_{j}\right]  =0\text{ , }\left[  P_{i},P_{j}\right]
=i\hbar\tau\lambda\epsilon_{ijk}L_{k}\text{ , }\left[  X_{i},P_{j}\right]
=i\hbar\left(  \delta_{ij}-\tau\lambda X_{i}X_{j}\right)  \text{ with }
\tau=-1,+1 \label{eqt1}
\end{equation}
where $\lambda$ is the parameter of the deformation and it is very small
because, in the context of quantum gravity, this EUP parameter is determined
as a fundamental constant associated to the scale factor of the expanding
universe and it is proportional to the cosmological constant $\Gamma
=3\tau\lambda=3\tau/a^{2}$ where $a$ is the deSitter radius \cite{Bolen}.
$L_{k}$ is the component of the angular momentum expressed by:
\begin{equation}
L_{k}=\epsilon_{ijk}X_{i}P_{j} \label{eqt2}
\end{equation}
and satisfying the usual algebra:
\begin{equation}
\left[  L_{i},P_{j}\right]  =i\hbar\varepsilon_{ijk}P_{k}\text{ , }\left[
L_{i},X_{j}\right]  =i\hbar\varepsilon_{ijk}X_{k}\text{ , }\left[  L_{i}
,L_{j}\right]  =i\hbar\varepsilon_{ijk}L_{k} \label{eqt3}
\end{equation}

As in ordinary quantum mechanics, the commutation relation \ref{eqt1} gives
rise to a Heisenberg uncertainty relation:
\begin{equation}
\Delta X_{i}\Delta P_{i}\geq\frac{\hbar}{2}\left(  1-\tau\lambda\left(  \Delta
X_{i}\right)  ^{2}\right)  \label{eqt4}
\end{equation}
where we choose the states for which $\left\langle X_{i}\right\rangle =0$.

According to the value of $\tau$ we distinguish two kinds of subalgebra. For
$\tau=-1$, the deformed algebra is characterized by the presence of a nonzero
minimum uncertainty in momentum and it is called Anti-deSitter model. For
simplicity, we assume isotropic uncertainties $X_{i}=X$ and this allows us to
write the minimal uncertainty for the momentum in AdS model:
\begin{equation}
\left(  \Delta P_{i}\right)  _{\min}=\hbar\sqrt{\tau\lambda} \label{eqt5}
\end{equation}
For de Sitter model where $\tau=+1$, the relation \ref{eqt4} does not imply an
non-zero minimal value for momentum uncertainties.

\begin{figure}[ptb]
\centering
\includegraphics[
height=2.3255in,
width=3.659in
]
{fig1.eps}
\caption{Graphic of HUP and EUP in both dS and AdS Cases }
\label{fig1}
\end{figure}

This is shown in figure1 \ref{fig1}, where the uncertainty relations are
plotted according to the modified relation found in \ref{eqt4}. The colored
region in \ref{fig1} is the forbidden area for position and momentum
measurements in AdS space.

The noncommutative operators $X_{i}$ and $P_{i}$ satisfy the modified algebra
\ref{eqt1} which gives rise to rescaled uncertainty relation \ref{eqt4} in
momentum space. In order to study the exact solutions of the deformed
Schr\"{o}dinger equation, we represent these operators as functions of the
operators $x_{i}$and $p_{i}$ that satisfy the ordinary canonical commutation
relations; This is done thanks to the following transformations:
\begin{subequations}
\begin{align}
X_{i}  &  =\frac{x_{i}}{\sqrt{1+\tau\lambda r^{2}}}\label{eqt6a}\\
P_{i}  &  =-i\hbar\sqrt{1+\tau\lambda r^{2}}\partial_{x_{i}} \label{eqt6b}
\end{align}
If $\tau=-1$, the variable $r$ varies in the domain $\left]  -1/\sqrt{\lambda
},1/\sqrt{\lambda}\right[  $.

\section{Nikiforov--Uvarov Method}

The Nikiforov-Uvarov (NU) method was developed basically on the hypergeometric
differential equation. The formulas used in NU method reduce the second order
differential equations to the hypergeometric type with an appropriate
coordinate transformation $s=s\left(  x\right)  $:
\end{subequations}
\begin{equation}
\psi^{\backprime\backprime}\left(  s\right)  +\frac{\tilde{\tau}\left(
s\right)  }{\sigma\left(  s\right)  }\psi^{\backprime}\left(  s\right)
+\frac{\tilde{\sigma}\left(  s\right)  }{\sigma^{2}\left(  s\right)  }
\psi\left(  s\right)  =0 \label{eqt7}
\end{equation}
where $\sigma\left(  s\right)  $ and $\widetilde{\sigma}\left(  s\right)  $
are polynomials of the second degree at most and the degree of the polynomial
$\widetilde{\tau}\left(  s\right)  $ is strictly less than 2 \cite{Egrifes}
\cite{Nikiforov}. If we take the following factorization:
\begin{equation}
\psi\left(  s\right)  =\phi\left(  s\right)  y\left(  s\right)  \label{eqt8}
\end{equation}
\ref{eqt7} becomes \cite{Nikiforov}:
\begin{equation}
\sigma\left(  s\right)  y^{\backprime\backprime}\left(  s\right)  +\tau\left(
s\right)  y^{\backprime}\left(  s\right)  +\Lambda y\left(  s\right)  =0
\label{eqt9}
\end{equation}
where:
\begin{equation}
\pi\left(  s\right)  =\sigma\left(  s\right)  \frac{d}{ds}\left(  \ln
\phi\left(  s\right)  \right)  \text{ \ and \ }\tau\left(  s\right)
=\tilde{\tau}\left(  s\right)  +2\pi\left(  s\right)  \label{eqt10}
\end{equation}
$\Lambda$ is defined as:
\begin{equation}
\Lambda_{n}+n\tau^{\backprime}+\frac{n\left(  n-1\right)  \sigma
^{\backprime\backprime}}{2}=0,\text{ \ \ }n=0,1,2,... \label{eqt11}
\end{equation}
And the energy eigenvalues are calculated from the above equation. We first
have to determine $\pi\left(  s\right)  $ and $\Lambda$ by defining:
\begin{equation}
k=\Lambda-\pi^{\backprime}\left(  s\right)  \label{eqt12}
\end{equation}
Solving the quadratic equation for $\pi\left(  s\right)  $ with \ref{eqt12},
we get
\begin{equation}
\pi\left(  s\right)  =\left(  \frac{\sigma^{\backprime}-\widetilde{\tau}}
{2}\right)  \pm\sqrt{\left(  \frac{\sigma^{\backprime}-\tilde{\tau}}
{2}\right)  ^{2}-\tilde{\sigma}+\sigma k} \label{eqt13}
\end{equation}
Here, $\pi\left(  s\right)  $ is a polynomial of the parameter $s$ and the
prime denotes the first derivative.

One has to note that the determination of $k$ is the essential point in the
calculation of $\pi\left(  s\right)  $ and It is simply defined by stating
that the expression under the square root in \ref{eqt13} must be a square of a
polynomial; This gives us a general quadratic equation for $k$.

To determine the polynomial solutions $y_{n}\left(  s\right)  $, we use
\ref{eqt10} and the Rodrigues relation:
\begin{equation}
y_{n}\left(  s\right)  =\frac{C_{n}}{\rho\left(  s\right)  }\frac{d^{n}
}{ds^{n}}\left[  \sigma^{n}\left(  s\right)  \rho\left(  s\right)  \right]
\label{eqt14}
\end{equation}
where $C_{n}$ is normalizable constant and the weight function $\rho\left(
s\right)  $ satisfies the following relation:
\begin{equation}
\frac{d}{ds}\left[  \sigma\left(  s\right)  \rho\left(  s\right)  \right]
=\tau\left(  s\right)  \rho\left(  s\right)  \label{eqt15}
\end{equation}
This last equation refers to the classical orthogonal polynomials that have
many important properties and especially orthogonality defined by:
\begin{equation}
\int_{a}^{b}y_{n}\left(  s\right)  y_{m}\left(  s\right)  \rho\left(
s\right)  ds=0\text{ if }m\neq n \label{eqt16}
\end{equation}

\section{Schr\"{o}dinger Equation for the Hydrogen Atom in (Anti-)deSitter
Space}

\label{Sch}

In this section, we study the effects of deformed space on the energy
eigenvalues and eigenfunctions of a hydrogen atom in the context of the
non-relativistic quantum mechanics. In the case of a three-dimensional
space,we consider the following stationary Schr\"{o}dinger equation with a
Coulomb-type interaction:
\begin{equation}
\left[  \frac{\mathbf{p}^{2}}{2m}-\frac{e^{2}}{r}\right]  \psi\left(
\mathbf{r}\right)  =E\psi\left(  \mathbf{r}\right)  \, \label{eqt17}
\end{equation}

In order to include the effect of EUP on the above Schr\"{o}dinger equation,
we use the transformations \ref{eqt16a} and \ref{eqt6b} to obtain:
\begin{equation}
\left[  -\frac{\hbar^{2}}{2m}\left[  \left(  1+\tau\lambda r^{2}\right)
\left(  \frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial
}{\partial r}-\frac{L^{2}}{\hbar^{2}r^{2}}\right)  +\tau\lambda r\frac
{\partial}{\partial r}\right]  -\frac{e^{2}\sqrt{1+\tau\lambda r^{2}}}
{r}\right]  \psi=E\psi\label{eqt18}
\end{equation}
$\,$

In order to separate the variables, we write the solution as $\psi
(r,\theta)=r^{-1/2}R(r)Y_{l}^{m_{l}}(\theta,\varphi)$ and this enables us to
split the equation into two parts, one angular and the other radial (where
$\chi\equiv\sqrt{1+\tau\lambda r^{2}}$):
\begin{equation}
L^{2}Y_{l}^{m_{l}}(\theta,\varphi)=\hbar^{2}l\left(  l+1\right)  Y_{l}^{m_{l}
}(\theta,\varphi)\, \label{eqt19}
\end{equation}
\begin{equation}
\left[  \left(  \chi\frac{d}{dr}\right)  ^{2}+\frac{\chi^{2}}{r}\frac{d}
{dr}-\frac{\left(  l\left(  l+1\right)  +\frac{1}{4}\right)  \chi^{2}}{r^{2}
}+\frac{2me^{2}\chi}{\hbar^{2}r}\right]  R(r)=-\left(  \frac{2mE}{\hbar^{2}
}+\frac{\tau\lambda}{2}\right)  R(r) \label{eqt20}
\end{equation}
The angular equation of the system is just the usual one for spherical
harmonics, so we are interested in the resolution of the radial one. In order
to do this, we use the following transformations:
\begin{equation}
s=\frac{\sqrt{1+\tau\lambda r^{2}}}{\sqrt{\lambda}r} \label{eqt21}
\end{equation}
Then, the new form of \ref{eqt20} becomes:
\begin{equation}
\left[  \left(  1-\tau s^{2}\right)  ^{2}\frac{d^{2}}{ds^{2}}-\tau s\left(
1-\tau s^{2}\right)  \frac{d}{ds}-\left(  l+\frac{1}{2}\right)  ^{2}s^{2}+\eta
s+\varepsilon\right]  R_{1,2}(s)=0\, \label{eqt22}
\end{equation}
where
\begin{equation}
\eta=\frac{2me^{2}}{\hbar^{2}\sqrt{\lambda}}\text{ and\ }\varepsilon
=\frac{2mE}{\lambda\hbar^{2}}+\frac{\tau}{2} \label{eqt23}
\end{equation}

\subsection{Solutions for deSitter Space ($\tau=+1$)}

\label{dS}

Comparison between \ref{eqt22} and \ref{eqt7} allows us to use the NU method
where the expressions of the polynomials appearing in \ref{eqt7} are given by:
\begin{equation}
\sigma\left(  s\right)  =\left(  1-s^{2}\right)  \text{ , }\tilde{\tau}\left(
s\right)  =-s\text{ \ and }\tilde{\sigma}\left(  s\right)  =-\left(
l+\frac{1}{2}\right)  ^{2}s^{2}+\eta s+\varepsilon\label{eqt24}
\end{equation}
Substituting them into \ref{eqt13} we obtain:
\begin{equation}
\pi\left(  s\right)  =\frac{-s}{2}\pm\sqrt{\left(  \frac{1}{4}+\left(
l+\frac{1}{2}\right)  ^{2}-k\right)  s^{2}-\eta s+k-\varepsilon} \label{eqt25}
\end{equation}
Where the parameter $k$ is to be determined by the condition mentioned in the
section III \ref{N-U}. One then obtains the following possible solutions for
each $k$:
\begin{equation}
\pi\left(  s\right)  =\left\{
\begin{array}
[c]{c}
\pi_{1,2}=\left(  \frac{-1}{2}\pm\delta_{1}\right)  s\mp\frac{\eta}
{2\delta_{1}}\text{\ for\ }k_{1}=\frac{1}{2}\left[  \varepsilon+\frac{1}
{4}+\left(  l+\frac{1}{2}\right)  ^{2}+\sqrt{\Delta}\right] \\
\pi_{3,4}=\left(  \frac{-1}{2}\pm\delta_{2}\right)  s\mp\frac{\eta}
{2\delta_{2}}\text{\ for\ }k_{2}=\frac{1}{2}\left[  \varepsilon+\frac{1}
{4}+\left(  l+\frac{1}{2}\right)  ^{2}-\sqrt{\Delta}\right]
\end{array}
\right.  \label{eqt26}
\end{equation}
with:

\begin{equation}
\delta_{1,2}=\sqrt{\frac{1}{4}+\left(  l+\frac{1}{2}\right)  ^{2}-k_{1,2}
}\text{ and }\Delta=\left(  \varepsilon-\frac{1}{4}-\left(  l+\frac{1}
{2}\right)  ^{2}\right)  ^{2}-\eta^{2} \label{eqt27}
\end{equation}
Here, we choose the proper value $\pi_{1}$, so that:
\begin{equation}
\tau\left(  s\right)  =2\left(  \delta_{1}-1\right)  s-\frac{\eta}{\delta_{1}}
\label{eqt28}
\end{equation}
From \ref{eqt11} we obtain:
\begin{equation}
\Lambda=k_{1}-\frac{1}{2}+\delta_{1}=n_{r}\left(  n_{r}+1-2\delta_{1}\right)
\text{ , }n_{r}=0,1,2,... \label{eqt29}
\end{equation}
Hence, the energy eigenvalues are found as:
\begin{equation}
E_{n,l}=-\frac{me^{4}}{2\hbar^{2}n^{2}}-\frac{\lambda\hbar^{2}}{2m}\left(
n^{2}-l\left(  l+1\right)  -1\right)  \label{eqt30}
\end{equation}
where $n=n_{r}+l+1$ is the principal quantum number.

We remark that the above expression of energies contains the usual Hydrogen
term and an additional correction term proportional to the deformation
parameter $\lambda$, so we recover the Bohr energies when the deformation
disappears. It should be noted here that the first term of the correction is
proportional to $n

{{}^2}

$ and so it is equivalent to the energy of a non-relativistic quantum particle
moving in a square well potential; In our case, the boundaries of the well are
placed at $\pm\pi/2\sqrt{\lambda}$. The second term in the correction contains
the azimuthal quantum number $l$ and it removes the $\left(  2l+1\right)  $
degeneracy of the energy levels. We also notice that the correction
deformation affects all energy levels except the ground level ($n=1$) which
remains not affected by the deformation even for large values of $\lambda$.

Now let us find the corresponding eigenfunctions. Taking the expression of
$\pi_{1}\left(  s\right)  $ from \ref{eqt26}, the $\phi\left(  s\right)  $
part is defined from the relation \ref{eqt10} as below:
\begin{equation}
\phi\left(  s\right)  =\left(  1+s\right)  ^{\frac{1}{4}\left(  1-2\delta
_{1}-\frac{\eta}{\delta_{1}}\right)  }\left(  1-s\right)  ^{\frac{1}{4}\left(
1-2\delta_{1}+\frac{\eta}{\delta_{1}}\right)  } \label{eqt31}
\end{equation}
and according to the form of $\sigma\left(  s\right)  $ \ref{eqt24}, the
$y\left(  s\right)  $ part is given by Rodrigues relation:
\begin{equation}
y_{n}\left(  s\right)  =\frac{C_{n}}{\rho\left(  s\right)  }\frac{d^{n}
}{ds^{n}}\left[  \left(  1-s^{2}\right)  ^{n}\rho\left(  s\right)  \right]
\label{eqt32}
\end{equation}
where $\rho\left(  s\right)  =\left(  1+s\right)  ^{\left(  -\delta_{1}
-\frac{\eta}{2\delta_{1}}\right)  }\left(  1-s\right)  ^{-\left(  \delta
_{1}-\frac{\eta}{2\delta_{1}}\right)  }$. The expression \ref{eqt32} stands
for the Jacobi polynomials as:
\begin{equation}
y_{n}\left(  s\right)  \equiv P_{n_{r}}^{\left(  -\delta_{1}-\frac{\eta
}{2\delta_{1}},-\delta_{1}+\frac{\eta}{2\delta_{1}}\right)  }\left(  s\right)
\label{eqt33}
\end{equation}
Hence, $R_{dS}(s)$ can be written in the following form:
\begin{equation}
R_{dS}(s)=C_{n}\left(  1-s\right)  ^{\frac{1}{4}\left(  1-2\delta_{1}
+\frac{\eta}{\delta_{1}}\right)  }\left(  1+s\right)  ^{\frac{1}{4}\left(
1-2\delta_{1}-\frac{\eta}{\delta_{1}}\right)  }P_{n_{r}}^{\left(  -\delta
_{1}-\frac{\eta}{2\delta_{1}},-\delta+\frac{\eta}{2\delta_{1}}\right)
}\left(  s\right)  \label{eqt34}
\end{equation}
In terms of the variables $r$, $\theta$ and $\varphi$, we can now write the
general form of the wave function $\Psi$ as follows:
\begin{align}
\Psi_{n_{r}}\left(  r,\theta,\varphi\right)   &  =C_{n}\left(  1-\frac
{\sqrt{1+\lambda r^{2}}}{\sqrt{\lambda}r}\right)  ^{\frac{1}{4}\left(
1-2\delta_{1}+\frac{\eta}{\delta_{1}}\right)  }\left(  1+\frac{\sqrt{1+\lambda
r^{2}}}{\sqrt{\lambda}r}\right)  ^{\frac{1}{4}\left(  1-2\delta_{1}-\frac
{\eta}{\delta_{1}}\right)  }\times\nonumber\\
&  P_{n_{r}}^{\left(  -\delta_{1}-\frac{\eta}{2\delta_{1}},-\delta_{1}
+\frac{\eta}{2\delta_{1}}\right)  }\left(  \frac{\sqrt{1+\lambda r^{2}}}
{\sqrt{\lambda}r}\right)  Y_{l}^{m_{l}}(\theta,\varphi)\, \label{eqt35}
\end{align}
where $C_{n}$ is a normalization constant.

\subsection{Solutions for Anti-deSitter Space ($\tau=-1$)}

\label{AdS}

By comparing \ref{eqt22} with \ref{eqt7}, we determine NU polynomials as
follows:
\begin{equation}
\sigma\left(  s\right)  =\left(  1+s^{2}\right)  \text{ ,\ }\tilde{\tau
}\left(  s\right)  =s\text{ and }\tilde{\sigma}\left(  s\right)  =-\left(
l+\frac{1}{2}\right)  ^{2}s^{2}+\eta s+\varepsilon\label{eqt36}
\end{equation}
Substituting them into \ref{eqt10}, we obtain:
\begin{equation}
\pi\left(  s\right)  =\frac{s}{2}\pm\sqrt{\left(  k+\frac{1}{4}+\left(
l+\frac{1}{2}\right)  ^{2}\right)  s^{2}-\eta s+k-\varepsilon} \label{eqt37}
\end{equation}
The constant $k$ is determined in the same way as in dS case. Therefore, we
get:
\begin{equation}
\pi\left(  s\right)  =\left\{
\begin{array}
[c]{c}
\pi_{1,2}=\left(  \frac{1}{2}\pm\delta_{1}^{\backprime}\right)  s\mp\frac
{\eta}{2\delta_{1}^{\backprime}}\text{\ for\ }k_{1}^{\backprime}=\frac{1}
{2}\left[  \varepsilon-\frac{1}{4}-\left(  l+\frac{1}{2}\right)  ^{2}
-\sqrt{\vartriangle^{\backprime}}\right] \\
\pi_{3,4}=\left(  \frac{1}{2}\pm\delta_{2}^{\backprime}\right)  s\mp\frac
{\eta}{2\delta_{2}^{\backprime}}\text{\ for\ }k_{2}^{\backprime}=\frac{1}
{2}\left[  \varepsilon-\frac{1}{4}-\left(  l+\frac{1}{2}\right)  ^{2}
+\sqrt{\vartriangle^{\backprime}}\right]
\end{array}
\right.  \label{eqt38}
\end{equation}
where:
\begin{equation}
\delta_{1,2}^{\backprime}=\sqrt{\frac{1}{4}+\left(  l+\frac{1}{2}\right)
^{2}+k_{1,2}^{\backprime}}\text{ and }\Delta^{\backprime}=\left(
\varepsilon+\frac{1}{4}+\left(  l+\frac{1}{2}\right)  ^{2}\right)  ^{2}
+\eta^{2} \label{eqt39}
\end{equation}
Here, we choose the proper value $\pi_{2}$, so that we have:
\begin{equation}
\tau\left(  s\right)  =2\left(  1-\delta_{1}^{\backprime}\right)  s-\frac
{\eta}{\delta_{1}^{\backprime}} \label{eqt40}
\end{equation}
From \ref{eqt11}, we calculate:
\begin{equation}
\Lambda=k_{1}^{\backprime}+\frac{1}{2}-\sqrt{\frac{1}{4}+\left(  l+\frac{1}
{2}\right)  ^{2}+k_{1}^{\backprime}}=-n_{r}\left(  n_{r}+1-2\sqrt{\frac{1}
{4}+\left(  l+\frac{1}{2}\right)  ^{2}+k_{1}^{\backprime}}\right)
\label{eqt41}
\end{equation}
Hence, the energy eigenvalues are found as:
\begin{equation}
E_{n,l}=-\frac{me^{4}}{2\hbar^{2}n^{2}}+\frac{\lambda\hbar^{2}}{2m}\left(
n^{2}-l\left(  l+1\right)  -1\right)  \label{eqt42}
\end{equation}

The same remarks made in the case of dS space apply here except that in this
case the correcting terms are inversely proportional to the deformation
parameter $\lambda$, so that the energies increases with increasing values
{}of $\lambda$ and the bound states in AdS space become less bounded than
those of the dS case for the same value of $\lambda$ \ref{eqt30}\ref{eqt42}.

In both cases, the spectral corrections due to EUP are qualitatively different
to those associated to GUP \cite{Brau}.

Now, to deduce the complete expression of the wave functions $\Psi_{n}\left(
x\right)  $, we use the expression \ref{eqt38} of $\pi_{2}\left(  s\right)  $
as follows:
\begin{equation}
\phi\left(  s\right)  =\left(  1+s^{2}\right)  ^{\frac{1}{2}\left(  \frac
{1}{2}-\delta_{1}^{\backprime}\right)  }e^{\frac{-\eta}{2\delta_{1}
^{\backprime}}\tan^{-1}\left(  s\right)  } \label{eqt43}
\end{equation}
and using Rodrigues formula \ref{eqt14}, we find
\begin{equation}
y_{n}\left(  s\right)  =\frac{C_{n}^{\backprime}}{\rho\left(  s\right)  }
\frac{d^{n}}{ds^{n}}\left[  \left(  1+s^{2}\right)  ^{n}\rho\left(  s\right)
\right]  \label{eqt44}
\end{equation}
where $\rho\left(  s\right)  =\left(  1+s^{2}\right)  ^{-\delta_{1}
^{\backprime}}e^{\frac{\eta}{\delta_{1}^{\backprime}}\tan^{-1}\left(
s\right)  }$.

The relation \ref{eqt44} stands for the Romanovski polynomials \cite{Alvaro}
as:
\begin{equation}
y_{n}\left(  s\right)  \equiv R_{n}^{\left(  -\delta_{1}^{\backprime}
,\frac{-\eta}{\delta_{1}^{\backprime}}\right)  }\left(  s\right)  =\frac
{C_{n}^{\backprime}}{\left(  1+s^{2}\right)  ^{-\delta_{1}^{\backprime}
}e^{\frac{-\eta}{\delta_{1}^{\backprime}}\tan^{-1}\left(  s\right)  }}
\frac{d^{n}}{ds^{n}}\left[  \left(  1+s^{2}\right)  ^{n-\delta_{1}
^{\backprime}}e^{\frac{-\eta}{\delta_{1}^{\backprime}}\tan^{-1}\left(
s\right)  }\right]  \label{eqt45}
\end{equation}
Consequently, the expression of $R_{AdS}(s)$ is written as:
\begin{equation}
R_{AdS}(s)=C_{n}\left(  1+s^{2}\right)  ^{\frac{1}{2}\left(  \frac{1}
{2}-\delta_{1}\right)  }e^{\frac{-\eta}{2\delta_{1}}\tan^{-1}\left(  s\right)
}R_{n}^{\left(  -\delta_{1},\frac{-\eta}{\delta_{1}}\right)  }\left(
s\right)  \label{eqt46}
\end{equation}
and the expression of the wave function $\Psi$ with the former variables $r$,
$\theta$ and $\varphi$ is given by:
\begin{equation}
\Psi_{n}\left(  r,\theta,\varphi\right)  =C_{n}^{\backprime}\left(
\sqrt{\lambda}r\right)  ^{\left(  \delta_{1}^{\backprime}-\frac{1}{2}\right)
}e^{\frac{-\eta}{2\delta_{1}^{\backprime}}\tan^{-1}\left(  \frac
{\sqrt{1-\lambda r^{2}}}{\sqrt{\lambda}r}\right)  }R_{n}^{\left(  -\delta
_{1}^{\backprime},\frac{-\eta}{\delta_{1}^{\backprime}}\right)  }\left(
\frac{\sqrt{1-\lambda r^{2}}}{\sqrt{\lambda}r}\right)  Y_{l}^{m_{l}}
(\theta,\varphi)\, \label{eqt47}
\end{equation}
with $C_{n}^{\backprime}$ is a normalization constant.

In order to show the effects of the deformed Heisenberg algebra leading to EUP
on the bound-states of the\ Coulomb potential in non-relativistic quantum
mechanics systems, we plot, as an example, the energies levels of the
$s-$states $E_{n,0}$ versus the deformation parameters $\lambda$ for different
values of $n$ (we use the Hartree atomic units $m=\hbar=e=4\pi\varepsilon_{0}$
$=1$).

\begin{figure}[ptb]
\centering
\includegraphics[
height=2.3255in,
width=3.659in
]
{fig2.eps}
\caption{$E_{n,0}(\lambda)$ for $n=1,2$ \& $3$ in dS and AdS cases}
\label{fig2}
\end{figure}

According to the results shown in figure2 \ref{fig2} and to the expression of
the energies \ref{eqt42}, it is clear that the deformation increases the
energies in AdS case and thus decreases the binding energies of the states. We
thus arrive at a critical point where the value of the deformation parameter
cancels the bound state or $E_{n,l}=0$:
\begin{equation}
\lambda_{c}(n,l)=\frac{1}{n^{2}\left(  n^{2}-l\left(  l+1\right)  -1\right)  }
\label{eqt48}
\end{equation}
This critical values of the spatial deformation parameter can be interpreted
as a resonance point because the corresponding state of the atomic system
ionizes. We give in table1 \ref{tab1} some critical values $\lambda_{c}(n,l)$
corresponding to the first levels. Note from \ref{eqt30} that this is not the
case for dS space because the deformation increases the bonding of atomic
states and so no ionization effect occurs here.
\begin{equation}
\begin{tabular}
[c]{|c|c|c|c|c|c|}\hline
$\lambda_{c}(n,l)$ & $l=0$ & $l=1$ & $l=2$ & $l=3$ & $l=4$\\\hline
$n=2$ & $0.0833$ & $0.2500$ & $-$ & $-$ & $-$\\\hline
$n=3$ & $0.1389$ & $0.0185$ & $0.0556$ & $-$ & $-$\\\hline
$n=4$ & $0.0042$ & $0.0048$ & $0.0069$ & $0.0208$ & $-$\\\hline
$n=5$ & $0.0017$ & $0.0018$ & $0.0022$ & $0.0033$ & $0.0100$\\\hline
\end{tabular}
\ \tag{Table1}\label{tab1}
\end{equation}

Figure2 \ref{fig2} and to the expression of the dS energies \ref{eqt30} show
that the deformation can reverse the order of energy levels since the
correction depends on the main quantum number. If we take the level $n=3$ as
an example, we see that it decreases faster than the $2^{nd}$ level and
therefore it becomes lower. Then it continues to decrease until it becomes
lower than the $1^{st}$ level, which will no longer be the fundamental one.
The value of $\lambda$ that causes this inversion between the upper levels and
the fundamental one is calculated from \ref{eqt30}:
\begin{equation}
\lambda_{f}(n,l)=\frac{n^{2}-1}{n^{2}\left(  n^{2}-l\left(  l+1\right)
-1\right)  } \label{eqt49}
\end{equation}

In table2 \ref{tab2}, we give some numerical values of $\lambda_{f}(n,l)$.
\begin{equation}
\begin{tabular}
[c]{|c|c|c|c|c|c|}\hline
$\lambda_{f}(n,l)$ & $l=0$ & $l=1$ & $l=2$ & $l=3$ & $l=4$\\\hline
$n=2$ & $0.250$ & $0.750$ & $-$ & $-$ & $-$\\\hline
$n=3$ & $0.111$ & $0.148$ & $0.444$ & $-$ & $-$\\\hline
$n=4$ & $0.063$ & $0.072$ & $0.104$ & $0.313$ & $-$\\\hline
$n=5$ & $0.040$ & $0.044$ & $0.053$ & $0.080$ & $0.240$\\\hline
\end{tabular}
\ \tag{Table2}\label{tab2}
\end{equation}

Using \ref{eqt5}, we see that the main feature of the hydrogen spectrum in AdS
model \ref{eqt42} is the presence of an additional positive correction
proportional to the nonzero minimal uncertainty:
\begin{equation}
E_{n,l}=-\frac{mk^{2}e^{4}}{2\hbar^{2}n^{2}}+\frac{\Delta P_{\min}^{2}}
{2m}\left(  n^{2}-l\left(  l+1\right)  -1\right)  \label{eqt50}
\end{equation}
One can use this relation to obtain an upper bound on the EUP deformed
parameter $\lambda\ $from spectroscopic considerations and we choose the
$2s-1s$\ transition line:
\begin{equation}
\frac{E_{2s}-E_{1s}}{E_{1s}}=-\frac{3}{4}-\frac{3}{2}\frac{\hbar^{2}}
{m^{2}k^{2}e^{4}}\Delta P_{\min}^{2} \label{eqt51}
\end{equation}
Taking the experimental results for this transition in the hydrogen atom where
the precision is of the order of $\varepsilon\approx10^{-15}$ \cite{Matveev}
and if we attribute this error entirely to the EUP correction \ref{eqt50}, we
can write:
\begin{equation}
\varepsilon=\frac{3}{2}\frac{\hbar^{2}}{m^{2}k^{2}e^{4}}\Delta P_{\min}^{2}
\label{eqt52}
\end{equation}
Therefore, the upper bound of the minimal uncertainty in momentum is given by
$\Delta P_{\min}\sim10^{-32}Jsm^{-1}$; This value is much smaller than the one
obtained in \cite{Hamil2}.

\section{Conclusion}

\label{cc}

In this work, we have analytically studied the deformed Schr\"{o}dinger equation in three dimensions for a Hydrogen atom in deSitter and Anti-deSitter spaces by using the position representation of the Extended Uncertainty Principle formulation and the Nikiforov–Uvarov method. For both cases, we obtained
the exact eigen-energies and eigen-functions. The radial wave functions were
expressed as associated Jacobi polynomials for deSitter space and in terms of
Romanovski polynomials for Anti-deSitter space.

The deformed energy spectrum was written as the usual Coulomb term with an
additional correction term that removes the $l$ degeneracy of Bohr energies.
The main effect of the deformation parameter $\lambda$ is an increase of the
energies for AdS spaces and a decrease of these energies for dS spaces. It should be noted here that the two spectra are similar to those arising when considering the non-relativistic model for Hydrogen atom in space of constant negative and positive curvature: hyperbolic Lobachevsky and spherical Riemann models \cite{Redkov} (and the references therein).

In the AdS case, we showed that, due to the decrease of the binding energies
with increasing $\lambda$ values, we reach a critical point where the
corresponding state is no longer bound and thus becomes ionized or diffusive.
The critical values $\lambda_{c}(n,l)$ are inversely proportional to the
quantum numbers $n$ and $l$; so higher levels ionize one after the other as
$\lambda$ increases, until the stage where the atomic system contains only the
fundamental level. This is explained by the fact that the higher states are
more easily ionized even in the ordinary case.

On the other hand, in the case of dS space, all the energies levels are more
bounded proportionally to the values of the EUP parameter. Because bound
energies are increased according to $\lambda$ in this case, the deformation
can cause a reversal of the order of the levels where the energy of the higher
levels are diminished until becoming smaller than that of the fundamental
level. The corresponding values to this phenomenon $\lambda_{f}(n,l)$\ are
also inversely proportional to the quantum numbers $n$ and $l$.

These two effects of ionization and inversion are comparable to an extension
of the higher levels in the case AdS and a contraction of these same levels in
the case of dS.

Finally, in order to see the effect of the deformation on the physical
systems, we compared them with the experimental results of the
non-relativistic hydrogen atom and we have determined a satisfactory value of
the upper bound of minimal momentum uncertainty for AdS space.

\section*{Acknowledgment}

The authors would like to thank the referee for the remarks made; these have greatly improved the manuscript and thus contribute to a better understanding of the work.

config: arxiv


TITLE: Can the real line be partitioned into two sets, each with positive Lebesgue measure and no interior points?
QUESTION [0 upvotes]: Is there such example that $A\subset \mathbb{R}$, such that both $A$ and $\mathbb{R}\backslash A$ are sets without interior points and with positive Lebesgue measure?

REPLY [3 votes]: You cannot write $\Bbb R$ as a union of two nowhere dense sets, because the union of nowhere dense sets is still nowhere dense.
And without interior points (after the question change): take $A$ negative rationals plus positive irrationals), and the other set its complement.


TITLE: Functional equations for spectral zeta function
QUESTION [4 upvotes]: Functional equation in the theory of  zeta functions is one of the important components of this theory.
I am interested to know whether the similar property, having functional equation, for the spectral zeta functions has been studied?
This might be a search for such a property of the spectral zeta function (for the Laplacian) of spaces with specific symmetries, e.g. symmetric spaces, or homogeneous ones.

REPLY [2 votes]: The Selberg zeta function for, say compact Riemann surfaces (other results are known but I forget the exact generality), satisfies a functional equation.  This is built out of eigenfunctions of the Laplacian, but perhaps is not what you mean by a "spectral zeta function."
If you mean something like the Minakshisundaram-Pleijel zeta function, then McKean, in his 1972 paper on Selberg's trace formula, that Minakshisundaram-Pleijel and Weil hoped this may have a functional equation, but there were no positive results thus far.  A quick search reveals this paper from 1995 which obtains functional equations for the MP zeta function of spheres and real or complex projective spaces, but I didn't find any other cases where a functional equation has been established.


TITLE: Function continuous substitution
QUESTION [1 upvotes]: Let $$f(x) = \frac{1}{1+x}.$$
Evaluate $f(f(f(f(f(f(1))))))$.
Shortcut or easy way for this. Really need to know that concept. Or do we have to do it with brute force?

REPLY [0 votes]: Hint: Using the Fibonacci Numbers:
$$
\frac1{1+\frac{F_n+xF_{n-1}}{F_{n+1}+xF_n}}=\frac1{\frac{F_{n+2}+xF_{n+1}}{F_{n+1}+xF_n}}=\frac{F_{n+1}+xF_n}{F_{n+2}+xF_{n+1}}
$$
Note that with $n=0$, $\frac{F_n+xF_{n-1}}{F_{n+1}+xF_n}=x$.
Thus, the composition of $n$ copies of $f$ would be
$$
\frac{F_n+xF_{n-1}}{F_{n+1}+xF_n}
$$


TITLE: total variation distance between two solutions of SDE
QUESTION [7 upvotes]: Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a standard one-dimensional Brownian motion, the functions $b_1$ and $b_2$ are uniformly bounded and Lipschitz.
Using Gronwall's inequality one can easily derive bounds on $|X_t^1-X_t^2|$ and hence it is possible to estimate the Wasserstein distance between $X_t^1$ and $X_t^2$.
My question is, how one can estimate the total variation distance between $X_t^1$ and $X_t^2$? Intuitively, because the processess $X_t^1$ and $X_t^2$ start from the same point, the total variation distance between them should be small (at least for small $t$). So, is it possible to obtain the bounds of the form
$$d_{TV} (X_t^1,X_t^2)\le \phi(t) $$
for some function $\phi$? (here $d_{TV}$ means the total variation distance between two measures)
Thanks!

REPLY [6 votes]: Such a bound can be derived with Girsanov's theorem and Pinsker's inequality. Let $X_t = x_0 + W_t$. Supposing $b$ has linear growth in $x$, we may define measures $P_i$, $i=1,2$ by
$\frac{dP_i}{dP} = \exp\left(\int_0^Tb_i(t,X_t)dW_t - \frac{1}{2}\int_0^T|b_i(t,X_t)|^2dt\right)$.
Then, under $P_i$, $W^i_t := W_t - \int_0^tb_i(s,X_s)ds$ is a Brownian motion, and so $X$ is a weak solution of SDE (i). Let $P^i \circ X^{-1}$ denote the $P^i$-law of the entire process, a measure on the space of continuous functions. Then
$\frac{dP_2}{dP_1} = \exp\left(\int_0^T(b_2(t,X_t) - b_1(t,X_t))dW^1_t - \frac{1}{2}\int_0^T|b_2(t,X_t) - b_1(t,X_t)|^2dt\right)$.
Let $\mathcal{H}(\cdot | \cdot)$ denote the relative entropy. By Pinsker's inequality,
$
\begin{align}
d_{TV}^2(P_1 \circ X^{-1}, P_2 \circ X^{-1}) &= d_{TV}^2(P_1, P_2) \le 2\mathcal{H}(P_1 | P_2) \newline
 &= -2\mathbb{E}^{P_1}\left[\log \frac{dP^2}{dP^1}\right] \newline
 &= \mathbb{E}^{P_1}\left[\int_0^T|b_1(s,X_s) - b_2(s,X_s)|^2ds\right] \newline
 &= \mathbb{E}^{P}\left[\frac{dP_1}{dP}\int_0^T|b_1(s,X_s) - b_2(s,X_s)|^2ds\right]
\end{align}
$
since the stochastic integral term is a true martingale. This is actually a much stronger control than you requested, since it is easy to see that $d_{TV}(P_1 \circ X_t^{-1}, P_2 \circ X_t^{-1}) \le d_{TV}(P_1 \circ X^{-1}, P_2 \circ X^{-1})$ for any $t$. If you have a uniform bound on $|b_1 - b_2|$, you have a good bound on the $TV$ distance between your processes. You can probably get away with an $L^2$ bound, but you'll then need to fuss with the $dP_1/dP$ term a bit. Hope this helps!
EDIT 1: It's interesting to note that this approach breaks down if you have two different volatility coefficients in your SDE, because the laws of the processes are then singular. But you could still bound the TV distance between the time-$t$ laws, as requested, probably via Malliavin calculus.
EDIT 2: I should elaborate on the first of the string of equalities above. Since $X$ and $W$ generate the same $\sigma$-fields, $dP^i/dP$ is $X$-measurable, and so
$\frac{dP^i \circ X^{-1}}{dP \circ X^{-1}} (X) = \mathbb{E}\left[\frac{dP^i}{dP}|X\right] = \frac{dP^i}{dP}$.
From this it is clear that the TV-distances above are the same, from the formula $d_{TV}(\mu,\nu) = \int d\lambda|d\mu/d\lambda - d\nu/d\lambda|$ for $\mu,\nu \ll \lambda$.


TITLE: Best strategy to determine the most efficient cargo capacity of a ship?
QUESTION [0 upvotes]: I'm using the example of cargo on a ship to demonstrate one variable reliant on another and how to find the peak value of each variable that gets the most out of both.
So lets say you want to ship cargo from one country to another. You want to build the ship but you want to find out what is the most efficient cargo weight that provides the most efficient speed, meaning, would carrying 100 tons at 60mph work out better than 150 tons at 45mph?
Focusing solely on the relationship between maximum speed and cargo weight, how would you propose working out the most efficient cargo weight that also obtains the highest speed.
I thought about multiplying the cargo weight by the speed and using a points system?
I'm curious to know this forums outlook to a situation like this or perhaps if there are any implemented methods already.
Thank you in advance.

REPLY [1 votes]: If you multiply the weight by the speed you get a number with units of ton-miles/hour.  If you know how much total cargo there is to move and the distance to move it, you can multiply them and divide by the ship product to get a number with units of hours.  This is the time it will take to move the cargo.  If you want to minimize that, you want to be dividing by the largest number you can, so you want to maximize the product of weight on the ship and ship speed.
What you need to make use of this is the relationship between weight and speed for the ships you can build.  You solve that for one in terms of the other and plug it into the weight-speed product, giving an expression in one variable.  Take the derivative, set to zero.....