text:
\begin{document}

\title[Fixed point theorems of various  nonexpansive actions]{Fixed point theorems of various  nonexpansive actions of semitopological semigroups on weakly/weak* compact convex sets}

\author[Muoi]{Bui Ngoc Muoi}
\address[Bui Ngoc Muoi]{Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, and Department of Mathematics, Hanoi Pedagogical University 2, Vinh Phuc, Vietnam.}
\email{\tt buingocmuoi@hpu2.edu.vn, muoisp2@gmail.com}

\author[Wong]{Ngai-Ching Wong}
\address[Ngai-Ching Wong]{Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan; Department of Healthcare Administration and Medical Information, and Center of
Fundamental Science, Kaohsiung  Medical University, 80708 Kaohsiung, Taiwan.}
\email{\tt wong@math.nsysu.edu.tw}

\thanks{Corresponding author: Ngai-Ching Wong, E-mail: wong@math.nsysu.edu.tw}

\dedicatory{In memory of  Wataru Takahashi (Jan.\ 22, 1944 --- Nov.\ 19, 2020)}

\begin{abstract}
Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of
left uniformly continuous functions on $S$.
Suppose that  $\operatorname{LUC}(S)$ has a left invariant mean.
Let $K$ be a weakly compact convex subset of a Banach space. We show that there always exists a common fixed point for any
jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$.
Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.
\end{abstract}

\keywords{Semitopological semigroups, amenability, reversibility, invariant means, asymptotically nonexpansive actions, Radon-Nikod\'{y}m property, distality, fixed points.}

\subjclass[2010]{Primary 47H10; Secondary 47H20, 47H09.}

\maketitle

\section{Introduction}
Let $K$ be a non-empty convex subset of a Banach space $E$. Let $T: K\to K$ be a \emph{nonexpansive} map, namely $\|Tx-Ty\|\leq\|x-y\|$ for all $x, y$ in $K$.
Schauder \cite{Schauder30} shows that $T$ has a fixed point if $K$ is norm compact. Kirk \cite{Kirk65} shows that
 $T$ has a fixed point if $E$ is
reflexive  and $K$ is weakly compact with normal structure.
Nevertheless, Alspach \cite{Alspach81} gives an example in which $K$ is weakly compact but without normal structure, and $T$ has no fixed point.

After Johnson \cite{Johnson72} introduced the notion of amenability, many authors
 studied the common fixed point problem
  of actions of semitopological semigroups on metric spaces, see e.g., \cite{Lau73,LauZhang08,LauZhang12,Mit68,Taka81}.
  In \cite[Problem 4]{Lau1976}
 Lau  raised a question about whether the left amenability of a semitopological semigroup ensures a common fixed point for
 its nonexpansive action.
 Recently, Salame \cite{Sal2019PAMS} gave a positive answer.

 \begin{Theorem}[see {\cite[Theorem 3.1]{Sal2019PAMS}}]
 	A semitopological semigroup $S$ is left amenable if and only if
 	\begin{quote}
 		$\mathbf{(F_{w^*})}$ every jointly weak* continuous nonexpansive action of $S$ on a weak* compact convex subset
 $K$ of a dual Banach space has a common fixed point.
 	\end{quote}
 \end{Theorem}
\noindent By embedding a Banach space into its double dual space, we have
\begin{Corollary}[see{\cite[Corollary 3.2]{Sal2019PAMS}}]
	If $S$ is left amenable then
	\begin{quote}
		$\mathbf{(F_{w})}$ every jointly weakly continuous nonexpansive action of $S$ on a weakly
 compact convex subset $K$ of a Banach space has a common fixed point.
	\end{quote}
\end{Corollary}

\begin{question}
  What can we say if the action is \emph{not nonexpansive}?
\end{question}

We can construct an action of a commutative (and thus amenable) discrete semigroup on a compact convex set without common fixed point.
Indeed, Boyce \cite{Boyce69} showed that there are two commutative continuous functions $f, g: [0,1]\to[0,1]$ with no common fixed point.
Therefore, in general, the assumption of nonexpansiveness can not be dropped. For a resolve, Holmes and Lau \cite{HolLau71} considered
asymptotically nonexpansive actions (see section 2 for definitions) on norm-compact convex sets, while Amini, Medghalchi and Naderi
\cite{AMN2016JMAA} considered pointwise eventually nonexpansive actions on weakly compact convex set with normal structure. Under some
conditions, these actions have  common fixed points.

\begin{question}
 \emph{Without normal structure}, what can we say for asymptotically nonexpansive like actions on weakly/weak* compact convex sets?
\end{question}

In this paper, we provide a positive answer for the {super asymptotically nonexpansive} actions (see Section
\ref{section2} for definitions).
We  show in Theorem \ref{mainThm}
that if $S$ is right reversible and left amenable then it holds a fixed point property similar to $\mathbf{(F_{w})}$
for super asymptotically nonexpansive actions. When we assume further that the domain $K$ of
 the action is norm-separable, we show in  Theorem \ref{mainthm2} that
 it holds a fixed point property similar to $\mathbf{(F_{w^*})}$. We also provide some results about the existence of a common fixed point
 for an action of any reversible semigroup. In Section \ref{section4}, motivated by recent results of Wi\'{s}nicki \cite{WiS2020}, we
 establish fixed point theorems involving the Radon-Nikod\'{y}m property or the distality.
 In Section \ref{section5}, applying the results in previous sections, we establish  fixed point theorems for
 commutative pointwise eventually
 nonexpansive mappings on weakly/weak* compact convex sets.

\section{Preliminaries}\label{section2}
A \emph{semitopological semigroup} $S$ is a semigroup with a Hausdorff topology such that the product is separately continuous,
i.e., for each fixed $t\in S$, both the maps $s\mapsto ts$ and $s\mapsto st$ from $S$ into $S$ are continuous. Let $\operatorname{CB}(S)$ be the Banach space of bounded and continuous real-valued functions on $S$ equipped with the supremum norm.

For each $s\in S$ and $f\in \operatorname{CB}(S)$, we denote by $l_sf$ the \emph{left translation}
of $f$ by $s$, where $l_sf(t)=f(st)$ for all $t\in S$. Let $\operatorname{LUC}(S)$ be the space of   \textit{left uniformly continuous}
functions on $S$, namely those $f\in\operatorname{CB}(S)$ for which the map $s\mapsto l_sf$ from $S$ into $\operatorname{CB}(S)$ is norm
continuous. A bounded linear functional $m$ on $\operatorname{LUC}(S)$ is called a \emph{mean} if $\|m\|=m(1)=1$.
A mean $m$ is called a \textit{left invariant mean}, or $\operatorname{LIM}$ in short, if $m(l_sf)=m(f)$ for all $s\in S$ and all $f\in\operatorname{LUC}(S)$. We call $S$ \emph{left amenable} if $\operatorname{LUC}(S)$ has a $\operatorname{LIM}$.

An \emph{action} of a semitopological semigroup  $S$ on a Hausdorff topological space $K$ is a mapping of $S\times K$ into $K$, denoted by $(s,x) \mapsto s.x$ (or simply $sx$),
such that $(st).x=s.(t.x)$ for all $s, t\in S$ and $x\in K$.  We call the action  \emph{separately} (resp.\ \emph{jointly}) \emph{continuous}
 if the mapping $(s, x)\mapsto s.x$ is separately (resp.\ jointly) continuous. A point $x_0\in K$ is called a \emph{common fixed point} for $S$ if $s.x_0=x_0$ for all $s\in S$.

Recall that a \emph{left} (resp.\ \emph{right}) \emph{ideal} $I$ of a semigroup $S$ is a subset of $S$ such that $SI\subseteq I$
(resp.\ $IS\subseteq I$).
We say that a left ideal $I$ is \emph{supported by an element $t$} in $S$ if $I\subseteq St$.
In this case, we can set the left ideal $I^t= \{s\in S: st\in I\}$ and write $I=I^t t$.

\begin{Definition}\label{DefSuperAssym}
	An action $S\times K\to K$ of a semitopological semigroup
 $S$ on a subset $K$ of a Banach space is called
 \begin{enumerate}[{\quad 1.}]
	\item \textit{nonexpansive} if
	$\|s.x-s.y\|\leq\|x-y\|$ for all $s\in S$ and $x, y\in K$.
	\item \textit{asymptotically nonexpansive} (see {\cite{HolLau71}}) if for each given $x, y\in K$, there exists a left ideal
	$I_{xy}$ of $S$ such that
$\|s.x-s.y\|\leq \|x-y\|$ for all $s\in I_{xy}$;
	\item \textit{pointwise eventually nonexpansive} (see {\cite{AMN2016JMAA}}) if for each given  $x\in K$, there exists a left ideal
	$I_x$ of $S$ such that
 $\|s.x-s.y\|\leq \|x-y\|$  for all $s\in I_x$ and all $y\in K$;

    \item \emph{super asymptotically nonexpansive} if for each given $x\in K$ and $t\in S$, there exists a left ideal $I_{x,t}$ of $S$ supported by $t$
 such that
$\|s.x-s.y\|\leq \|x-y\|$ for all $s\in I_{x,t}$ \text{and all} $y\in K$.
\end{enumerate}
By taking $I_x^t = \{s\in S: st\in I_{x,t}\}$,
the definition of a super asymptotically nonexpansive   action of $S$ on $K$   can be restated that
for each given $x\in K$ and $t\in S$, there exists a left ideal $I_x^t$ of $S$
 such that
$$
\|st.x-st.y\|\leq \|x-y\|  \quad \text{for all $s\in I_x^t$ and all $y\in K$}.
$$
 \end{Definition}

\begin{Remark}\label{RemarkPointwiseEN}
(a)
Note that   pointwise eventually nonexpansive actions are also called  \emph{semi-asymptotically nonexpansive actions}
 in \cite{AMN2016}, and  \emph{strongly asymptotically nonexpansive actions} in \cite{AAR2018}.

(b)
The notion of pointwise eventually nonexpansive action extends the one introduced by Kirk and Xu in \cite{KirkXu08}, in which

a map $T: K\to K$ is called \emph{pointwise eventually nonexpansive} if for each $x\in K$ there exists $N(x)\in\mathbb{N}$ such that \begin{align}\label{eq:PEN}
\|T^nx-T^ny\|\leq\|x-y\|,\quad \forall y\in K \mbox{ and } n\geq N(x).
\end{align}

(c)	Let $\{T_1,\ldots,T_k\}$ be a commutative family of pointwise eventually nonexpansive maps on $K$. Let $S$ be the
 discrete semigroup generated by this family. Then the action of $S$ on $K$ is indeed
 super asymptotically nonexpansive. Indeed, for each $x\in K$,
 there exist $N_1(x),\ldots,N_k(x)\in\mathbb{N}$ satisfying \eqref{eq:PEN}
 for  $T_1, \ldots, T_k$, respectively.
 For  any $T_0= T_1^{p_1}\cdots T_k^{p_k}\in S$, set $q_i^{T_0}(x)= \max\{p_i, N_i(x)\}$ for $i=1,\ldots, k$.
Consider the left ideal $I_{x, T_0}=\{T_1^{n_1}\cdots T_k^{n_k}: n_i\geq q_i^{T_0}(x), i=1,\ldots k\}$ of $S$ supported by $T_0$.
Then for each $T\in I_{x,T_0}$ and $y\in K$, we have $\|Tx-Ty\|\leq\|x-y\|$.

\end{Remark}

We have the following implications for the above mentioned actions of semitopological semigroups:
\begin{quote}
	nonexpansiveness $\implies$ super asymptotic nonexpansiveness $\implies$ pointwise eventual nonexpansiveness $\implies$ asymptotic nonexpansiveness.
\end{quote}
As seen in Examples \ref{Ex1} and \ref{Ex2} below, these implications can be strict. However,
as seen in  Proposition \ref{remark210}, when the semigroup $S$ is  compact and right reversible, all three asymptotically nonexpansiveness coincide.

\begin{Example}[based on {\cite[Example 3.3(ii)]{AMN2016}}]\label{Ex1}
	Let $K$ be the closed unit ball in $\mathbb{R}^2$ and $f:[-1,1]\to[-1,1]$ be
 given by $f(x)=x^3$. Consider two maps $T_1$ and $T_2$ from $K$ into $K$ defined by
 $$T_1(x_1,x_2)=(f(x_2),0)\quad \mbox{ and }\quad T_2(x_1,x_2)=(0,f(x_1)).$$
	Clearly, $T_1^n=0$ and $T_2^n=0$ for all $n\geq 2$. Consider the non-commutative discrete semigroup $S$ generated by $T_1$ and $T_2$, that is
	$$S=\{0, T_1, T_2, (T_1T_2)^n, (T_2T_1)^n, T_2(T_1T_2)^n, T_1(T_2T_1)^n: n\in\mathbb{N}\}.$$
	Since $f$ is not nonexpansive, so are $T_1$ and $T_2$.  Hence the action of $S$ on $K$,
 defined by $(T,(x_1,x_2))\mapsto T(x_1,x_2)$, is not nonexpansive.

 However,
 for each $T\in S$ we can find an element $T'\in S$ such that $T'T=0$. Thus the left ideal $\{0\}=ST'.T$ is supported by any $T$ in $S$.
 Therefore, the action of $S$ on $K$ is super asymptotically nonexpansive.
 We also see that $(0,0)$ is a common fixed point of the action.
\end{Example}

\begin{Example}[{based on \cite[Example]{HolLau71}}]\label{Ex2}
	Let $K=\left\lbrace (r,\theta): 0\leq r\leq 1, 0\leq \theta<2\pi\right\rbrace$ be the closed unit disk in $\mathbb{R}^2$
	in polar coordinates and the usual Euclidean norm.
Define two continuous mappings $f, g$ from $K$ into $K$ by
	\begin{equation*}
	f(r,\theta)=(r/2,\theta)\quad\text{and}\quad g(r,\theta)=(r,2\theta\ ({\operatorname{mod} 2\pi})).
	\end{equation*}
	Let $S$ be the discrete semigroup generated by $f$ and $g$ under composition.
	Any left ideal $I$ of $S$ must have the form
	$I=\left\lbrace f^ng^m: n\geq n_0, m\geq m_0\right\rbrace, \text{for some $n_0, m_0\in\mathbb{N}_0$}$, where $\mathbb{N}_0 =\{0,1,2,\ldots\}$.

	An action of $S$ on $K$ is given by
	$$
	f^ng^m(r,\theta)=(\frac{r}{2^n},2^m\theta\; (\operatorname{mod} 2\pi)).
	$$
As seen in \cite{HolLau71}, the action is asymptotically nonexpansive. However, it is not super asymptotically nonexpansive.
To see this, choose
	$x =(1,0)\in K$, $t =f\in S,$
	and any left ideal $I_0=\left\lbrace f^ng^m: n\geq n_0, m\geq m_0\right\rbrace$, together with
 the left ideal $I=I_0 t$  supported by $t$.
For each $n\geq n_0, m\geq m_0$, consider $s=f^ng^m\in I$ and $y_m=(1,\frac{\pi}{2^m})\in K$.  We have
	$$\|f^{n+1}g^m(x)-f^{n+1}g^m(y_m)\|=\|(\frac{1}{2^{n+1}},0)-(\frac{1}{2^{n+1}},\pi)\|=\frac{1}{2^n},$$ and
	$\|x-y_m\|=\left[2(1-\cos\frac{\pi}{2^m})\right]^{1/2}\rightarrow 0$ as $m\rightarrow \infty$.
	Thus, the inequality
$$\|s.t_0.x-s.t_0.y_m\|\leq \|x-y_m\|$$
fails to hold for all $m\geq m_0$ when $n$ is fixed.
\end{Example}

\begin{Remark} (a) The semigroup $S$ in Example \ref{Ex1} is left amenable because the point evaluation at $0$,
defined by $\delta_0(f)=f(0)$, is a $\operatorname{LIM}$ of $\operatorname{CB}(S)$.

	(b) Theorem 3.1 in \cite{HolLau71} dose not apply to  Example \ref{Ex1}, since the action in Example \ref{Ex1}
 does not satisfy the property
$\mathbf{(B)}$ in its assumptions. The property $\mathbf{(B)}$ says that for each $x\in K$ whenever a net $\{s_\alpha x\}$ converges
to $x$, the net $\{s_\alpha sx\}$ converges to $sx$ for any $s\in S$. However, consider $x=(1,0)\in K$, $s=T_2$ and the sequence
$s_n=(T_1T_2)^n$ in $S$. We see that $(T_1T_2)^n(1,0)=(1,0)$ and $T_2(1,0)=(0,1)$ while $(T_1T_2)^nT_2(1,0)=(0,0)$ for all $n\in\mathbb{N}$.
\end{Remark}

A semitopological semigroup $S$ is called \emph{right} (resp.\ \emph{left}) \emph{reversible} if any two closed left (resp.\ right) ideals of $S$ always intersect. We call $S$ \textit{reversible} if it is both left and right reversible. For example, the semigroup $S$ in Example \ref{Ex1} is reversible.

\begin{Lemma}[see {\cite[Lemma 3.4]{LauZhang12}}]\label{lemmaLeftReversible}
	For every separately continuous action of a left reversible topological
	semigroup $S$ on a compact Hausdorff space $K$, there is a non-empty closed subset $A$ of $K$ such that $A\subset s.A$ for all $s\in S$.
\end{Lemma}

Recall that a Hausdorff topological space $X$ is \emph{countably compact}
if every countable open covering of $X$ has a finite subcovering; in other words,
every countable collection of closed sets in $X$ with the finite intersection property has a nonempty intersection.
A topological space $Y$ is called \emph{C-closed} if every countably compact set in $Y$ is closed (see \cite{IN80}).
For example,  first countable and countably compact spaces are regular, and thus they are C-closed spaces by \cite[Proposition 1.4]{IN80}.

\begin{Proposition}\label{remark210}
	Let $S$ be a  countably compact right reversible semitopological semigroup.
	Consider a   separately  weakly continuous and asymptotically nonexpansive action of $S$ on  a  set  $K$ in a Banach space.
	\begin{enumerate}[(a)]
		\item If  $K$ is norm separable, then the action is pointwise eventually nonexpansive.
		\item If $S$ is   C-closed and $K$ is norm separable, then the action is super asymptotically nonexpansive.
		\item If $S$ is   compact, then the action is  super asymptotically nonexpansive.
	\end{enumerate}
\end{Proposition}
\begin{proof}
	(a)	Let $\left\lbrace y_n\right\rbrace_{n=1}^\infty$ be a norm dense subset of $K$.
	Suppose
	an action of $S$ on $K$ is asymptotically nonexpansive. For each $x\in K$ and $n\in\mathbb{N}$,
	there is a left ideal $I_n$ such that
	$$
	\|a.x- a.y_n\| \leq  \|x-y_n\|, \quad\forall a\in I_n.
	$$
	By the separately weak continuity of the action, we can assume that $I_n$ is closed.

	By the right reversibility and the countable compactness of $S$, there is $r\in \bigcap_{n} I_n$.
	Consider the left ideal $I_x:=Sr\subseteq \bigcap_{n} I_n$.
	It follows
	$$
	\|s.x- s.y_n\| \leq \|x-y_n\|, \quad\forall s\in I_x,\ \forall n=1,2,\ldots.
	$$
	By the norm denseness of $\left\lbrace y_n\right\rbrace_{n=1}^\infty$ in $K$ and the separately weak continuity of the action, we see that
	$$
	\|s.x- s.y\| \leq \|x - y\|,\quad\forall s\in I_x, \forall y\in K.
	$$
	In other words, the action is pointwise eventually nonexpansive.

	(b)
	Continuing with the arguments in (a), we see that the action is already pointwise eventually nonexpansive.
	Thus, for any $n=1,2,\ldots$, there is a  left ideal $J_n$ of $S$ such that
	$$
	\|s.y_n - s.y\|\leq \|x-y\|, \quad\forall s\in J_n, \ \forall y\in K.
	$$
	By the separate weakly continuity of the action, we can assume  that $J_n$ is closed.
	By the right reversibility and the countable compactness of $S$,
	we have a (nonempty) closed left ideal $J:= \bigcap_n J_n$.
	For any $t\in S$, the left ideal $St$ is countably compact, and thus closed since $S$ is C-closed.
	By the right reversibility of $S$, we have $(St)\cap J$ is nonempty.
	Consequently,
	$J^t =\{s\in S: st\in  J\}$
	is  a (nonempty) left ideal of S.
	We have
	$$
	\|s.(t.y_n) - s.(t.y)\| = \|(st).y_n - (st).y\| \leq \|y_n-y\|, \quad\forall s\in J^t, \ \forall y\in K.
	$$
	Since $\{y_n\}_{n=1}^\infty$ is norm dense in $K$ and the action is separately weakly continuous,
	$$
	\|s.(t.x) - s.(t.y)\|  \leq \|x-y\|, \quad\forall s\in J^t, \ \forall y\in K.
	$$
	Therefore, the action is super asymptotically nonexpansive.

	(c) Suppose that $S$ is assumed compact but $K$ not necessarily norm separable.
	Note that the left ideal $St$ is compact and thus closed for every $t$ in $S$.
	We can go through the same arguments as in (a) and (b), but by considering the whole set $K$ rather than the sequence $\{y_n\}$,
	to  get the desired conclusion.
\end{proof}

Let $A$ be bounded subset of a Banach space $E$. A point $x\in A$ is called \emph{diametral} if $\sup_{y\in A}\|x-y\|=\operatorname{diam}A$,
 where $\operatorname{diam}A=\sup_{a,b\in A}\|a-b\|$. A convex set $K$ of $E$ is said to have \emph{normal structure} if each bounded,
 convex subset $A$ of $K$ with $\operatorname{diam}A>0$ contains a non-diametral point. Any
 norm compact convex set has normal structure, see
 \cite[Lemma 1]{DeMarr63}, while a weakly (resp.\ weak*) compact convex set might not be so, see \cite{Alspach81,Lim80}.

A closed convex subset $K$ of a Banach space $E$ is said to have the \emph{Radon-Nikod\'{y}m property} ($\operatorname{RNP}$ for short)
 if for any measure space $(\Omega,\mathcal{F},\mu)$ and any $E-$valued measure $F:\mathcal{F}\to E$ which is absolutely continuous with
 respect to $\mu$ such that $\{F(B)/\mu(B): B\in\mathcal{F},\mu(B)>0\}\subset K$, there is a Bochner integral function

$f\in L^1_X(\Omega,\mathcal{F},\mu)$ such that $F(B)=\int_{B}fd\mu$ for every $B\in\mathcal{F}$
(see, e.g., \cite[Definition 2.1.1]{Bourgin83}).

\begin{Lemma}[see {\cite[Theorem 4.2.13]{Bourgin83}}]\label{LemmaRNP}
	Let $K$ be a weak* compact convex subset of a dual Banach space $E^*$. Then $K$ has  the $\operatorname{RNP}$ if and only if
for any weak* compact subset $Y$ of $K$, the identity map $\operatorname{id}: (Y,\mathrm{wk}^*)\to (Y,\|\cdot\|)$ has a point of continuity.
\end{Lemma}

\begin{Lemma}[based on {\cite[Proposition 3.5]{Megrelishvili98}}]\label{LemmaRNPWeak}
	Let $K$ be a weakly compact subset of a locally convex space $(E,Q)$, where $Q$ is a family of seminorms of $E$
 determining the topology. Let $q\in Q$ and $\varepsilon>0$. Then there is a weakly open subset $U$ of $E$, and a point $x\in K\cap U$ such that $q(x-y)<\varepsilon$ for every $y\in K\cap U$.
\end{Lemma}

An action $S\times K\to K$ of a semitopological semigroup
$S$ on a subset $K$ of the locally convex space $(E,\tau)$ is called \textit{$\tau$-distal} if
$0\notin\overline{\{sx-sy: s\in S\}}^{\,\tau}$ for every pair of distinct points $x,y\in K$.

\begin{Theorem}[Ryll-Nardzewski {\cite[Theorem 3]{RyllNar67}}]\label{ThmAffineDistal}
	Let $S\times A\to A$ be a continuous affine $\tau$-distal action of a semitopological semigroup $S$ on a compact convex subset
$A$ of a locally convex space. Suppose there exists a nonempty compact subset $K$ of $A$ such that $s.K\subset K$ for all $s\in S$. Then there is a common fixed point of $S$.
\end{Theorem}

We also consider other classical  function spaces  on $S$ instead of $\operatorname{LUC}(S)$.
Let $\operatorname{AP}(S)$ (resp.\ $\operatorname{WAP}(S)$) be the subspace of $\operatorname{CB}(S)$ consisting of
\textit{almost periodic} (resp.\ \textit{weakly almost periodic}) functions;
namely those $f$ for which the set $\left\lbrace l_sf: s\in S\right\rbrace$ is relatively compact in the norm (resp.\ weak) topology of $\operatorname{CB}(S)$.
In general, we have
$$
\operatorname{AP}(S)\subseteq \operatorname{LUC}(S)\subseteq \operatorname{CB}(S)\quad \text{and}\quad
\operatorname{AP}(S)\subseteq \operatorname{WAP}(S)\subseteq \operatorname{CB}(S).
$$
For the existence of $\operatorname{LIM}$ on these spaces and the associated fixed point properties, the reader can see \cite{Lau73,LauZhang08,Taka92}.

\section{Fixed point theorems assured by the amenability and the reversibility}\label{section3}

\begin{Theorem}\label{mainThm}
	Let $S$ be a right reversible and left amenable semitopological semigroup. Let $K$ be a weakly compact  convex subset of a Banach space. Then every jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$ has a common fixed point.

\end{Theorem}

The proof of  Theorem \ref{mainThm} needs several lemmas.
The first one arises from the proof of \cite[Theorem 3.1]{HolLau71}.

\begin{Lemma}
\label{lemma1}
	Let $S$ be a right reversible semitopological semigroup.  Assume $S\times K\to K$ is a
separately  continuous action of  $S$
 on a  compact convex subset $K$ of a locally convex space.
Then there exists a subset $L_0$ of $K$ which is minimal with respect to being nonempty,
compact, convex and satisfying the following conditions $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$.
	\begin{enumerate}
 	   \item[$\mathbf{(\star 1)}$] there exists a collection $\Lambda = \left\lbrace\Lambda_i: i\in I\right\rbrace$ of  closed subsets of $K$ such that $L_0 =\bigcap \Lambda$, and
    \item[$\mathbf{(\star 2)}$]  for each $x\in  L_0$  there is a left ideal $J_i\subseteq S$ such that $J_i.x\subseteq\Lambda_i$ for each $i\in I$.
	\end{enumerate}
	Furthermore, $L_0$ contains a subset $Y$ that is minimal with respect to being nonempty, compact, and $S$-invariant,
i.e., $s.Y\subseteq Y$ for all $s\in S$.
\end{Lemma}

The following is a variant of \cite[Lemma 5.1]{LauTaka95}, we sketch a proof here since we need to consult its argument later.
\begin{Lemma}\label{lemma2}
	Let $S\times Y\to Y$ be a jointly weakly continuous action of a semitopological semigroup
 $S$ on a weakly compact subset $Y$ of a normed space. Assume  that $S$ is left amenable.
Assume further that
$Y$ is minimal with respect to being an $S$-invariant, nonempty and weakly compact subset of $Y$. Then $Y$ is norm separable, and $S$-preserving, i.e., $s.Y=Y$ for all $s\in S$.
\end{Lemma}

\begin{proof}
	 For each pair of $y\in Y$ and $f\in \operatorname{C}(Y)$ ($=\operatorname{CB}(Y)$ while $Y$ is the compact space
equipped with the weak topology), define a function $R_yf$ by $R_yf(s)=f(s.y)$.
It can be shown that $R_yf\in \operatorname{LUC}(S)$.

	Let $m$ be a $\operatorname{LIM}$ on  $\operatorname{LUC}(S)$.
Define a left invariant linear functional $\psi$ on $\operatorname{C}(Y)$ by $\psi(f)=m(R_yf)$.
 Let $\mu$ be the Radon probability measure on $Y$ defining $\psi$, and
let $Y_0$ be the support of
$\mu$, i.e.,
$$
Y_0=\operatorname{supp}(\mu)=\bigcap\left\lbrace F\subseteq Y: F \ \text{is weakly closed and}\ \mu(F)=1\right\rbrace.
$$
It can be shown that $Y_0$ is $S$-preserving, hence $Y=Y_0$ by the minimality of $Y$. Since every finite Radon measure on a
weakly compact set in a Banach space has a norm separable support (see, e.g., \cite[Theorem 4.3, page 256]{Joram72}), $Y$ is norm separable.
\end{proof}

\begin{Lemma}\label{lem:3}
	Let Y be a norm separable and weakly compact set in a Banach space $E$.
For a super asymptotically nonexpansive action of a right reversible semitopological semigroup $S$ on $Y$,
suppose that $Y$ is minimal with respect to being weakly compact and $S$-invariant. Let $F$ be any nonempty weakly closed subset of $Y$ such that $F\subset sF$ for all $s\in S$. Then $F$ is norm compact.
In particular, $Y$ is norm compact if $sY=Y$ for all $s\in S$.
\end{Lemma}
\begin{proof} We follow the idea in \cite[Lemma 5.2]{LauTaka95} in which a nonexpansive action of $S$ is considered instead.

	Define $N_\varepsilon=\left\lbrace x\in E:\; \|x\|\leq \varepsilon\right\rbrace$ for any given $\varepsilon>0$. Since $Y$ is norm separable,
there exists $\left\lbrace x_i: i\in \mathbb{N}\right\rbrace\subseteq Y$ such that
$Y\subseteq\bigcup\left\lbrace x_i+N_\varepsilon: i\in \mathbb{N}\right\rbrace$.
By the Baire category theorem, there exist $\bar{x}\in\left\lbrace x_i: i\in \mathbb{N}\right\rbrace$ such that $(\bar{x}+N_\varepsilon)\cap Y$
has nonempty interior in $Y$ in the relative weak topology.
Hence, there exist a $z\in Y$ and a weakly open neighborhood $V$ of $0$
such that $(z+V)\cap Y\subseteq(\bar{x}+N_\varepsilon)\cap Y$. We can choose a weak neighborhood $U$ of $0$ such that $U+U\subseteq V$.
Since $U$ also contains a norm open neighborhood of $0$, there exists $\delta>0$ such that
$N_\delta\subseteq U$. By the norm separability of $Y$ again, we can assume, with a new sequence $\{x_i: i\in \mathbb{N}\}$,

that
\begin{align}\label{eq:Yi}
Y=\bigcup\left\lbrace (x_i+N_\delta)\cap Y: i\in\mathbb{N}\right\rbrace.
\end{align}

	By the definition of the super asymptotic nonexpansiveness, for each given $r_0\in S$, there exists a left ideal
$I_1=I_{x_1}^{r_0}$ of $S$ such that $\|sr_0.x_1-sr_0.y\|\leq \|x_1-y\|$ for all  $s\in I_1, y\in Y$.
Since $I_1r_0.x_1$ is $S$-invariant in $Y$, by the minimality of $Y$, its weak
closure must be exactly $Y$. Thus, there exists an $s_1\in I_1$ such that $s_1r_0.x_1\in (z+U)\cap Y$.
Let $r_1=s_1r_0$, we have $r_1.x_1\in (z+U)\cap Y$ and  $\|r_1.x_1-r_1.y\|\leq \|x_1-y\|$ for all  $y\in Y$.

	Similarly, there exists a left ideal $I_2=I_{x_2}^{r_1}$ of $S$ such that $\|sr_1.x_2-sr_1.y\|\leq \|x_2-y\|$ for
all  $s\in I_2, y\in Y$. There exists an $s_2\in I_2$ such that
$s_2r_1.x_2\in (z+U)\cap Y$. Let $r_2:=s_2r_1=s_2s_1r_0$,
we have $r_2.x_2\in (z+U)\cap Y$ and $\|r_2.x_2-r_2.y\|\leq \|x_2-y\|$ for all  $y\in Y$.
		By induction, we can choose a sequence $\left\lbrace r_i: i\in \mathbb{N}\right\rbrace$ in $S$ such that
\begin{gather*}
r_i.x_i\in (z+U)\cap Y,\quad  r_i=s_is_{i-1}\cdots s_1r_0,
\intertext{and}
\|r_i.x_i-r_i.y\|\leq \|x_i-y\|, \quad\forall  y\in Y,\ i\geq 1.
\end{gather*}

	For each $y\in (x_i+N_\delta)\cap Y$, we can write $r_iy=(r_iy-r_ix_i)+r_ix_i$ where $r_ix_i\in (z+U)\cap Y$ and $\|r_ix_i-r_iy\|\leq\|x_i-y\|<\delta$. Thus, $r_i((x_i+N_\delta)\cap Y)\subseteq (z+U+N_\delta)\cap Y\subseteq (z+V)\cap Y$.
	We rewrite the action $r.x$ in the form of $L_rx$.
Then $(x_i+N_\delta)\cap Y\subseteq L_{r_i}^{-1}((z+V)\cap Y)$, where $L_{r_i}^{-1}((z+V)\cap Y)$ is weakly open by the weak continuity of the action.
By the weak compactness and \eqref{eq:Yi}, we can cover $Y$ by  finitely many such weakly open sets.  Let
$$
Y=\bigcup_{i=1}^{n}L_{r_i}^{-1}((z+V)\cap Y).
$$

	It follows from the super asymptotic nonexpansiveness of the action that
 there exist left ideals $J_i=J_{\bar{x}}^{t_i}$, where $t_i=s_{n+1}s_{n}\cdots s_{i+1}$, $i=1,\ldots, n$, such that
 $\|st_i.\bar{x}-st_i.y\|\leq \|\bar{x}-y\|$ for all  $y\in Y, s\in J_i$. Since $S$ is right reversible, there exists $t_0\in\cap_{i=1}^{n}\overline{J_i} $.

	For each  $i$, there exists a net $\left\lbrace s_\lambda\right\rbrace\subseteq J_i$ converging to $t_0$. Hence $s_\lambda t_i\bar{x}-s_\lambda t_i y$ converges to $t_0 t_i\bar{x}-t_0t_i y$ weakly. Therefore, from the lower continuity of norm in weak topology,
\begin{align}\label{eq:dist-bar}
\|t_0t_i.\bar{x}-t_0t_i.y\|\leq \|\bar{x}-y\| \quad\text{for all}\  y\in Y,\ i=1,\ldots, n.
\end{align}

Since $F\subset sF$ for all $s\in S$, we have
	\begin{equation*}
	\begin{array}{rl}
	F\subset L_{t_0}L_{r_{n+1}}F\subset L_{t_0}L_{r_{n+1}}Y &=L_{t_0}L_{r_{n+1}}\left\lbrace \bigcup_{i=1}^{n}L_{r_i}^{-1}((z+V)\cap Y)\right\rbrace\\
	&\subseteq \bigcup_{i=1}^{n}\left\lbrace L_{t_0}L_{s_{n+1}\cdots s_{i+1}}((\bar{x}+N_\varepsilon)\cap Y)\right\rbrace\\
	&= \bigcup_{i=1}^{n}\left\lbrace L_{t_0t_i}((\bar{x}+N_\varepsilon)\cap Y)\right\rbrace\\
	&\subseteq \bigcup_{i=1}^{n}\left\lbrace (L_{t_0t_i}\bar{x}+N_\varepsilon)\cap Y)\right\rbrace\\
	&\subseteq \bigcup_{i=1}^{n}\left\lbrace L_{t_0t_i}\bar{x}+N_\varepsilon\right\rbrace.
	\end{array}
	\end{equation*}
The second last inclusion above follows from \eqref{eq:dist-bar}. This proves that the norm closed set $F$ can be covered by a finite $\epsilon$-net for
any $\epsilon>0$.  Hence, $F$ is norm compact.
	\end{proof}

Together with above lemmas and motivated by \cite[Theorem 3.1]{HolLau71}  and \cite[Theorem 4.2]{AAR2018}, we are ready to prove Theorem \ref{mainThm}.  Note that, if the subset $Y$ in Lemma \ref{lem:3} is known to be
 convex then we can apply \cite[Lemmas 2.5]{SGF2018} for a shorter proof.
 However, at the current stage, we do not have the convexity of $Y$.

\begin{proof}[Proof of Theorem \ref{mainThm}]
By Lemmas \ref{lemma2} and \ref{lem:3}, the nonempty
 $S$-preserving subset $Y$ given in Lemma \ref{lemma1} is separable and norm compact in the Banach space $E$. Consequently, the norm topology and the weak topology  agree on $Y$.
If  $Y$ contains exactly one point  then we are done. Otherwise, let
$$
r=\mbox{diam} (Y)=\sup\left\lbrace\|x-y\|: x, y\in Y\right\rbrace.
$$
By DeMarr's Lemma \cite[Lemma 1]{DeMarr63}, there is an element $u\in\overline{\operatorname{conv}}(Y)$
such that
$$
r_0=\sup\left\lbrace\|u-y\|: y\in Y\right\rbrace<r.
$$
Let $0<\varepsilon<r-r_0$.
Let $L_0$ and $\left\lbrace\Lambda_i: i\in I\right\rbrace$ be given in Lemma \ref{lemma1}.
For each $\Lambda\in \left\lbrace\Lambda_i: i\in I\right\rbrace$,
set
\begin{align*}
N_{\varepsilon,\Lambda}&= \bigcap_{y\in Y} \{x\in \Lambda : \|x-y\|\leq  r_0+\varepsilon\}
\intertext{and}
N_0&=\bigcap\left\lbrace N_{\varepsilon, \Lambda_i}:  \ i\in I\right\rbrace= L_0\cap \bigcap_{y\in Y}\bar{B}[y, r_0+\varepsilon],
\end{align*}
where $\bar{B}[y, \delta]$ stands for the norm closed ball centered at $y$ of radius $\delta$.

We show that $N_0$  satisfies $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$. Indeed, every $N_{\varepsilon,\Lambda_i}$ is weakly compact.
Thus $N_0$ is a weakly compact subset of $L_0$, and contains $u$.
For each $x\in N_0$ and  $i\in I$, there exists a left ideal $I\subseteq S$ such that $I.x\subseteq \Lambda_i$.
By the super asymptotic nonexpansiveness of the action, for each $t\in S$ there exists a left ideal $I_{x}^t$
such that  $\|st.y-st.x\|\leq \|y-x\|$ for all  $y\in K$ and $s\in I_{x}^t$.
By the right reversibility of $S$, there exists a  $t_0\in\overline{I}\cap\overline{I_{x}^t t}$. Since $\Lambda_i$ is weakly closed, $St_0.x\subseteq \Lambda_i$.
   Consider a net $s_\lambda\in I_{x}^t$ such that $s_\lambda t\to t_0$.
From  $\|ss_\lambda t.y-ss_\lambda t.x\|\leq \|y-x\|\leq r_0+\varepsilon$ for all $\lambda$, $y\in Y$ and $s\in S$,
we have $\|st_0.y-st_0.x\|\leq \|y-x\|\leq r_0+\varepsilon$.
Since $Y\subset st_0.Y$, we have $\|y'- st_0.x\|\leq r_0+\varepsilon$ for all $y'\in Y$.
In other words, there exists a left ideal $J=St_0$ of $S$
such that $J.x\subseteq N_{\varepsilon, \Lambda_i}$.  Consequently, the nonempty  weakly compact
convex subset $N_0$ of $L_0$ also satisfies conditions $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$.

By the minimality of $L_0$, we have
$Y\subseteq L_0= N_0 \subseteq \bigcap_{y\in Y}\bar{B}[y, r_0+\varepsilon]$.
This gives us a contradiction that $\operatorname{diam}(Y) \leq r_0+\epsilon < r$.
Therefore, $Y$ contains a unique point and it is the common fixed point for the action of $S$ on $K$.
\end{proof}

 Lau and Zhang \cite[Theorem 6.2]{LauZhang12} established that  a left amenable semitopological semigroup $S$
  has the following fixed point property.

\begin{quote}
 $\mathbf{(F_{w^*,sep})}$ Let $K$ be a weak* compact convex and norm-separable subset of a dual Banach space. Then every  jointly weak* continuous and nonexpansive  action   of  $S$ on  $K$ has a common fixed point.
\end{quote}
Replacing the assumption of nonexpansiveness with the weaker one of
super asymptotic nonexpansiveness, but together with the  right reversibility of the semigroup, we obtain the following result.

\begin{Theorem}\label{mainthm2}
Let  $S$ be a right reversible and left amenable semitopological semigroup. Let $K$ be a weak* compact convex and norm-separable subset of a dual Banach space. Then every jointly weak* continuous and super asymptotically nonexpansive action of $S$ on $K$ has a common fixed point.

\end{Theorem}
\begin{proof}
It follows similarly as in proving Theorem \ref{mainThm}, but with the norm-separability coming from the assumption, and noticing that
Lemma \ref{lem:3} is valid for the weak* compact case.
	\end{proof}

\begin{Remark} Since the support of a finite Radon measure on a weak* compact set in a dual Banach space \emph{may not}
 be norm separable,  the conclusion of Lemma \ref{lemma2} about the norm separability of $Y$ may not hold for the weak* compact case.
\end{Remark}

The following result supplements {\cite[Theorem 3.4]{LauZhang08}}.

\begin{Theorem}\label{APWAP}
	Let $S$ be a right reversible semitopological semigroup.
	\begin{itemize}
		\item[(i)] Assume $\operatorname{AP}(S)$ has a $\operatorname{LIM}$.  Let $K$ be a weakly compact (resp.\ weak* compact
and norm-separable) convex subset of a Banach (resp.\ dual Banach) space. Then every separately weakly (resp.\ weak*) continuous, equicontinuous and super asymptotically nonexpansive action of $S$ on $K$ has a common fixed point.
		\item[(ii)] Assume $\operatorname{WAP}(S)$ has a $\operatorname{LIM}$.  Let $K$ be a weakly compact (resp.\ weak* compact
and norm-separable) convex subset of a Banach (resp.\ dual Banach) space. Then every separately weakly (resp.\ weak*) continuous, quasi-equicontinuous and super asymptotically nonexpansive action of $S$ on $K$ has a common fixed point.

	\end{itemize}
\end{Theorem}
\begin{proof}
	These are direct consequences of
\cite[Lemma 3.1]{Lau73}, \cite[Lemma 3.2]{LauZhang08}, and the proofs of Theorems \ref{mainThm} and  \ref{mainthm2}.
	\end{proof}

\begin{Corollary}\label{corr.Normal}
	Let $S$ be a  semitopological semigroup as well as a normal topological
space.  Assume that $\operatorname{CB}(S)$ has an invariant mean. Let $K$ be a weakly compact  (resp.\ weak* compact  and
	norm-separable) convex subset of a Banach (resp.\ dual Banach) space.
Then every separately weakly (resp.\ weak*) continuous and super asymptotically nonexpansive action of $S$ on $K$ has a common fixed point.
\end{Corollary}
\begin{proof}
It is  known that if $S$ is normal and $\operatorname{CB}(S)$ has a right invariant mean then $S$ is right reversible. The assertion now follows from Theorems \ref{mainThm} and \ref{mainthm2}.
\end{proof}

\begin{Remark}
	The only place we need the joint continuity of the action in
	Theorems \ref{mainThm} and \ref{mainthm2}  is where we derive that $R_yf$ belongs to $\operatorname{LUC}(S)$ for each $f\in \operatorname{C}(Y)$ and $y\in Y$.  Then we can construct a $\operatorname{LIM}$ $\psi$ of $\operatorname{C}(Y)$.
	For Theorem \ref{APWAP} and Corollary \ref{corr.Normal}, we need only separate continuity since other assumptions there suffice to ensure that
	such a $\operatorname{LIM}$ $\psi$ exists for the stated function spaces on $S$.
\end{Remark}

Without any amenability assumption,
we consider in the following some fixed point properties of reversible semitopological semigroups. Borzdyński and Wi\'{s}nicki
\cite[Theorem 3.5]{BW2014} established that any commutative (and thus left amenable and reversible) semigroup has the fixed point property
$\mathbf{(F_{w^*})}$.
For a discrete semigroup, the left amenability implies
the left reversibility (see \cite[page 2549]{LauZhang08}), while in general it might not be the case.
The following two results supplement Theorems \ref{mainThm} and \ref{mainthm2}.  The key point in their proofs is that we can bypass Lemma \ref{lemma2}.

\begin{Proposition}\label{coroLeftRever}
	Let $S$ be a reversible semitopological semigroup. Let $K$ be a weak* compact convex and norm-separable subset of a dual Banach space.
 Then every separately weak* continuous and super asymptotically nonexpansive action of $S$
		on $K$  has a common fixed point.
\end{Proposition}
\begin{proof}
	By  Lemma \ref{lemma1}, there is a subset $L_0$ of $K$ which is minimal with respect to being nonempty, weak* compact,
	convex, and satisfying conditions $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$
where the weak* topology is involved. Moreover, $L_0$ contains
	a subset $Y$ that is minimal with respect to being nonempty, weak* compact and $S$-invariant.

By Lemma \ref{lemmaLeftReversible}, there is a non-empty weak* closed subset $F$ of $Y$ such that $F\subset sF$ for all $s\in S$.
 Applying Lemma \ref{lem:3}, we see that $F$ is compact.
	The remaining parts now follow similarly as in proving Theorem \ref{mainThm} where the set $Y$ is replaced by its norm compact subset $F$.
\end{proof}

\begin{Proposition}\label{sep+reversibility}
	Let $S$ be a separable and reversible semitopological semigroup. Let $K$ be a weakly compact  convex subset of a Banach space. Then every separately weakly continuous and super asymptotically nonexpansive action of  $S$ on $K$ has a common fixed point.
\end{Proposition}
\begin{proof}
	By Lemma \ref{lemma1}, we establish a subset $Y$ of $L_0\subset K$ that is minimal with respect to being nonempty, weakly compact
	and $S$-invariant.
	We then follow an idea in \cite[Lemma 3.3]{LauZhang08} to show that $Y$ is norm separable. Indeed, for any
	fixed $y\in Y$, we have $Sy=\{s.y: s\in S\}$ is an $S$-invariant subset of $Y$. Thus, its weak closure $\overline{Sy}^{\,\mathrm{wk}}$ must be exactly $Y$. Assume $S$ contains   a countable dense subset $S_c$. Since the action is separately weakly
	continuous, $\overline{Sy}^{\,\mathrm{wk}}=\overline{S_cy}^{\,\mathrm{wk}}$. Moreover,
	$\overline{\operatorname{conv}}^{\,\mathrm{wk}}(Sy)=\overline{\operatorname{conv}}^{\,\mathrm{wk}}(S_cy)
=\overline{\operatorname{conv}}^{\,\|\cdot\|}(S_cy)$ by Mazur's Theorem.
It follows that $Y=\overline{\operatorname{conv}}^{\,\mathrm{wk}}(Sy)$ is norm separable.
From Lemma \ref{lemmaLeftReversible}, the left reversibility of $S$ ensures that there is a  weakly compact subset $F$ of $Y$ satisfying $F\subset sF$ for all $s\in S$.
	The remaining parts follow similarly as in the proof of Theorem \ref{mainThm}.
\end{proof}

\begin{Remark}
 In Example \ref{Ex1}, all assumptions of Theorem \ref{mainThm} and Proposition \ref{sep+reversibility} are satisfied. Thus we can conclude that the action there has a fixed point. Indeed, the action has a common fixed point $(0,0)$.
\end{Remark}
	When the action is nonexpansive, we do not need the assumption on the
 right reversibility to guarantee a common fixed point. The following provides a variance to a result of Lim
  \cite[Theorem 3]{Lim74} in which the subset $K$ is assumed to have normal structure.

	\begin{Proposition}\label{QuesSalame}
		Let $S$ be a separable and left reversible semitopological semigroup. Let $K$ be a weakly compact  convex subset of a Banach space. Then every separately weakly continuous nonexpansive action of $S$ on $K$ has a common fixed point.
	\end{Proposition}
\begin{proof}
	Let $K_1$ be a subset of $K$ which is minimal with respect to being nonempty, weakly compact, convex,
 and $S$-invariant. Let $Y$ be a subset of $K_1$ which is minimal with respect to being nonempty, weakly compact,
  and $S$-invariant. Following the proof of Proposition \ref{sep+reversibility},
  we can show that $Y$ is norm-separable.
  Moreover, by the left reversibility of $S$, there is a weakly compact subset $F$ of $Y$ such that $F\subset sF$ for all $s\in S$.
  Following the arguments in Lemma \ref{lem:3}, but for (the stronger) nonexpansive actions (see also \cite[Lemma 3.3]{LauZhang08}),
   we see that $F$ is norm compact.

	We now make use of an idea in the proof of \cite[Theorem 4.1]{Lau73}, see also \cite[Theorem 3.4]{LauZhang08}, to show that $F$ consists
of a single point. For a contradiction, suppose that
 $r=\operatorname{diam}F>0$. Since $F$ is compact,  the closure $\overline{\operatorname{conv}}  F$ of its convex hull has
 normal structure. Hence, there is $u\in\overline{\operatorname{conv}}  F$ such that $r_0=\sup_{y\in F}\|u-y\|<r$.
	Let $$F_0=\{x\in K_1: \|x-y\|\leq r_0\; \mbox{ for all }\; y\in F\}.$$
Since $u\in F_0$, we see that $F_0$ is a nonempty weakly compact convex proper subset of $K_1$.
 Furthermore, since $F\subset sF$ for all $s\in S$, we can check that $F_0$ is $S$-invariant which contradicts the minimality of $K_1$.
 Hence, $F$ consists of a single point which is the common fixed point of $S$.
	\end{proof}

A semitopological semigroup $S$ is called \emph{strongly left reversible} (resp.\ \emph{strongly reversible}) if any separable subsemigroup of $S$ is left reversible (resp.\ reversible).

\begin{Corollary}
	Let $S$ be a strongly left reversible (resp.\ strongly reversible) semitopological semigroup. Let $K$ be a weakly compact  convex subset of a Banach space. Then every separately weakly continuous and nonexpansive (resp.\ super asymptotically nonexpansive) action of  $S$ on $K$ has a common fixed point.
\end{Corollary}
\begin{proof}
	For each $s\in S$, let $F(s)=\{x\in K: s.x=x\}$ be the nonempty
 set of fixed points for $s$.  By the separately weak continuity of the action, $F(s)$ is
	weakly closed.  Indeed, for any net $\{x_\alpha\}$ in $F(s)$ weakly converging to $x$ in $K$,
	we have $s.x = \lim_\alpha s.x_\alpha = \lim_\alpha x_\alpha = x$.
	By the weak compactness of $K$, the common fixed point set $F(S)=\bigcap_{s\in S} F(s)\neq \emptyset$ if
	every finite subset of $S$ has a common fixed point in $K$.
	In other words, if the assertion would fail then there were $s_1, s_2,\ldots, s_n$ in
 $S$ such that $\bigcap_{i=1}^n F(s_i) = \emptyset$.
	Consider the closed subsemigroup $S'$  of $S$ generated
	by $s_1, s_2,\ldots, s_n$, which is separable and left reversible (resp.\ reversible).
	By Propositions \ref{sep+reversibility} and  \ref{QuesSalame},
there is a common fixed point for the action of $S'$ on $K$. Therefore, every finite subset   of $S$
has a common fixed point, and hence so is $S$.
\end{proof}

\section{Fixed point theorems involving the Radon-Nikod\'{y}m property or the distality}\label{section4}

Recently, Wi\'{s}nicki \cite{WiS2020} provided an extension for the Ryll-Nardzewski's Theorem \ref{ThmAffineDistal}.
It is about the existence of a common fixed point for a nonlinear action of a semigroup on a weakly compact convex set
in a locally convex space. Following his idea, we establish in this section some extensions of the results in
Section \ref{section3} as well as the results in \cite{WiS2020}, for the asymptotically nonexpansive
type actions of right reversible semigroups.

The main idea is to replace the norm-separability assumption in \cite[Theorem 6.2]{LauZhang08} and in Theorem \ref{mainthm2}
with the Radon-Nikod\'{y}m property or the norm-distality, and
 to derive the norm compactness of an $S$-invariant subset of $K$ in the action.

\begin{Theorem}\label{ThmRNP}
	Let  $S$ be a right reversible and left amenable semitopological semigroup. Let $K$ be a weak* compact convex subset of a dual Banach space with the $\operatorname{RNP}$. Then every jointly weak* continuous and pointwise eventually nonexpansive action of $S$ on $K$ has a common fixed point.
\end{Theorem}

\begin{proof}
	 By Lemmas \ref{lemma1} and \ref{lemma2},
there is a Radon probability measure $\mu$ such that the  support $Y=\operatorname{supp}(\mu)$
 is minimal with respect to being a nonempty
 $S$-invariant weak* compact subset of $K$.  Below, we follow
 the approach in \cite[Theorem 3.1]{WiS2020}, see also
 \cite[Theorem 4.2]{WiS2019Arxiv}, in which   nonexpansive actions are considered instead.

 By Lemma \ref{LemmaRNP}, there exists an $x\in Y$ such that for each $\varepsilon>0$, there is a weak* open neighbourhood $U$ of $x$ such
 that $\|x-y\|<\varepsilon$ for all $y\in U\cap Y$. Hence

	\begin{equation}\label{eqSec4}
	\delta:=\mu(\{y\in Y: \|x-y\|<\varepsilon\})\geq\mu(U\cap Y)>0.
	\end{equation}
Since the action is pointwise eventually nonexpansive, there is a left ideal $I$ of $S$ such that
$$\|sx-sy\|\leq\|x-y\|,\quad \forall s\in I, \forall y\in Y.$$
	Therefore, for each $s\in I$, we have
$$
\{y\in Y: \|x-y\|<\varepsilon\}\subset L_s^{-1}\{z\in Y: \|sx-z\|<\varepsilon\},
$$
where $L_s(y):=sy$ for $y\in K$.
	Since $\mu$ is invariant,
	\begin{equation*}
	\begin{array}{rl}
	\mu(\{y\in Y: \|sx-y\|<\varepsilon\})&=\mu(L_s^{-1}\{y\in Y: \|sx-y\|<\varepsilon\})\\
	&\geq \mu(\{y\in Y: \|x-y\|<\varepsilon\})=\delta>0.
	\end{array}
	\end{equation*}

	Since $Ix$ is $S$-invariant, by the minimality of $Y$, we have $Y=\overline{\{Ix\}}^{\,\mathrm{wk}^*}$.
We shall see that there are only finitely many elements $s_1,\ldots,s_k$ in $I$ such that $\|s_ix-s_jx\|\geq 2\varepsilon$ for any $i\neq j$.
	In fact, all $Y_i=\{y\in Y: \|y-s_ix\|<\varepsilon\}$ are pairwise disjoint subsets of $Y$, and $\mu(Y_i)\geq\delta$ for all $i$.
Now the fact $\mu(Y)=1$ ensures that at most finitely many of such elements exist. In other words, $Ix$ is totally bounded in norm,
and hence $\overline{\{Ix\}}^{\,\|\cdot\|}$ is norm-compact.
Since the identity map $\mathrm{id}: (Y,\|\cdot\|)\to (Y,\mathrm{wk}^*)$ is always continuous,
$Y=\overline{\{Ix\}}^{\,\mathrm{wk}^*}=\overline{\{Ix\}}^{\,\|\cdot\|}$ is norm-compact. The remaining   follows similarly as the proof of
Theorem \ref{mainThm}.
\end{proof}

 The following result supplements Theorem \ref{ThmRNP}, and applies to the case when we do not have the left amenability in stock.

\begin{Theorem}\label{ThmRNPDis}
	Let $S$ be a right reversible semitopological semigroup. Let $K$ be a weak* compact convex subset of a dual Banach space with the $\operatorname{RNP}$. Then every separately weak* continuous, pointwise eventually nonexpansive and norm-distal action of $S$ on $K$ has a common fixed point.
\end{Theorem}
\begin{proof}
By  Lemma \ref{lemma1}, there is a subset $L_0$ of $K$ which is minimal with respect to being nonempty, weak* compact,
convex,
 and satisfying conditions $\mathbf{(\star 1)}$ and $\mathbf{(\star 2)}$ where the weak* topology is involved. Moreover, $L_0$ contains
a subset $Y$ that is minimal with respect to being nonempty, weak* compact and $S$-invariant.

As in the proof of \cite[Theorem 3.1]{WiS2020}, we are going to construct an $S$-invariant Radon probability measure $\mu$
on $K$ with respect to
the weak* topology.
Notice that the weak*-distality of the action together with the minimality of $Y$ implies that the action is weak*-distal.

Let ${\operatorname{C}(Y)}$ be the space of continuous functions on $Y$. Let $\mathbf{P}(Y)$ be the weak* compact convex set of all means on $\operatorname{C}(Y)$. Consider an action of $S$ on $\mathbf{P}(Y)$ given by $s.\mu=l_s^*\mu$,
 where $l_s$ is the left translation operator by $s$, and $\left\langle l_s^*\mu,f\right\rangle=\left\langle \mu,l_sf\right\rangle$ for all $s\in S$ and $f\in {\operatorname{C}(Y)}$.
 Let $\phi: Y\to \mathbf{P}(Y)$ be the isometric natural embedding,  $x\mapsto\phi(x)=\hat{x}$, defined by $\hat{x}(f)=f(x)$ for all
 $f\in {\operatorname{C}(Y)}$.
 Then, we can form an action of $S$ on $\phi(Y)$ by defining
  $s.\phi(x)=\phi(s.x)$ for all $s\in S$ and $x\in Y$.

  Since the action of $S$ on $Y$ is
 weak*-distal, so is  the action of $S$ on $\phi(Y)$. By Theorem \ref{ThmAffineDistal}, there is a
 common fixed point $\mu$ of $S$ in $\mathbf{P}(Y)$.
 In other words, $\mu$ is an $S$-invariant Radon probability measure on $Y$ with respect to the weak* topology.
As in the proof of Theorem \ref{ThmRNP}, we see that $S$ has a common fixed point in $K$.
\end{proof}

For the super asymptotically (resp.\ pointwise eventually) nonexpansive actions on a weakly compact convex subset of a locally convex space, we obtain the following fixed point properties  without assuming neither $\operatorname{RNP}$ nor norm-separability.

Let $(E,Q)$ be a locally convex space where $Q$ is a family of seminorms determining the topology. An action $S\times K\to K$ is said to be
\begin{enumerate}
	\item  \emph{$Q$-pointwise eventually nonexpansive} if for each $x\in K$ and $q\in Q$, there exists a left ideal $I=I(x,q)$ of $S$ such that $q(s.x-s.y)\leq q(x-y)$ for all $s\in I$ and all $y\in K$;

	\item \emph{$Q$-supper asymptotically nonexpansive} if for each $x\in K$, $t\in S$ and $q\in Q$, there exists a left ideal $I=I(x,t,q)$ of $S$ supported by $t$ such that $q(s.x-s.y)\leq q(x-y)$ for all $s\in I$ and all $y\in K$.
\end{enumerate}
See \cite{MuoiWong2020Frechet} for more discussions.

The following result is an extension of Theorem \ref{mainThm} for pointwise eventually nonexpansive actions.
\begin{Proposition}\label{Prop4.3}
	Let $S$ be a right reversible and left amenable semitopological semigroup. Let $K$ be a weakly compact  convex subset of a locally convex space $(E,Q)$. Then every jointly weakly continuous and $Q$-pointwise eventually nonexpansive action of $S$ on $K$ has a common fixed point.
\end{Proposition}
\begin{proof}
	Using Lemma \ref{LemmaRNPWeak} and arguing as in proving Theorem \ref{ThmRNP}, we obtain a similar inequality as \eqref{eqSec4}
in which the norm is replaced by a seminorm in $Q$.  We can then
 derive that $Y$ is $Q$-compact. Finally, with an argument similar to the one proving Theorem \ref{mainThm}, see also \cite[Theorem 2.14]{MuoiWong2020Frechet}, we will arrive at the conclusion.
\end{proof}

\begin{Theorem}\label{ThmDistal}
	Let $S$ be a separable and right reversible semitopological semigroup. Let $K$ be a weakly compact convex subset of a locally convex space $(E,Q)$. Then every separately weakly continuous, $Q$-super asymptotically nonexpansive and $Q$-distal action of $S$
		on $K$ has a common fixed point.
\end{Theorem}

\begin{proof}
	By Lemma \ref{lemma1}, there is a subset $Y$ of $K$, that is minimal with respect to being nonempty,
 weakly compact and $S$-invariant.  As in proving Proposition \ref{sep+reversibility}, we can show that $Y$ is $Q$-separable.

	As in proving \cite[Theorem 4.1]{WiS2020}, the $Q$-distality of the action together with the minimality of $Y$ ensures
 that the action is indeed weakly-distal. Following the proof of Theorem \ref{ThmRNPDis}, we can show the existence of a Radon probability measure $\mu$ defined on $Y$ such that $Y=\operatorname{supp}(\mu)$ and $sY=Y$ for all $s\in S$.
  Arguing as in proving Lemma \ref{lem:3}, but with the  seminorms in $Q$ replacing the norm,
  we see that $Y$ is $Q$-compact. The remaining now follows as in the proof of Theorem \ref{mainThm}.
\end{proof}

\section{Fixed point theorems for pointwise eventually nonexpansive mappings}\label{section5}

In the following, applying the result in previous sections, we establish fixed point theorems for a finite commutative
family of continuous maps on weakly/weak* compact convex sets which are pointwise eventually nonexpansive.

\begin{Corollary}\label{CoroKirkXu}
	Let $K$ be a non-empty weakly compact convex subset of a Banach space. Let $\{T_1,\ldots,T_k\}$ be a commutative family of weakly continuous and pointwise eventually nonexpansive maps on $K$. Then they have a common fixed point.
\end{Corollary}
\begin{proof}
	Let $S$ be the discrete semigroup generated by this family.
As seen in Remark \ref{RemarkPointwiseEN}(c), the action of $S$ on $K$ is super asymptotically nonexpansive. Moreover, since $S$ is commutative, it is reversible and left amenable.
	From Theorem \ref{mainThm}, as well as Proposition \ref{sep+reversibility}, $S$ has a common fixed point. Hence $\{T_1,\ldots,T_k\}$ has a common fixed point in $K$.
\end{proof}

\begin{Corollary}
	Let $K$ be a weak* compact convex subset of a dual Banach space with the $\operatorname{RNP}$. Let $\{T_1,\ldots,T_k\}$ be a commutative family of weak* continuous and pointwise eventually nonexpansive maps on $K$. Then they have a common fixed point.
\end{Corollary}
\begin{proof}
Let $S$ be the semigroup   generated by this finite family.
	From Theorem \ref{ThmRNP}, the canonical
 action of $S$ on $K$  has a common fixed point. Therefore, the finite family has a common fixed point in $K$.
\end{proof}
Since every weak* compact convex and norm-separable subset has the $\operatorname{RNP}$, we have the following result.
\begin{Corollary}\label{CoroKirkXuWeakStar}
	Let $K$ be a weak* compact convex and norm-separable subset of a dual Banach space. Let $\{T_1,\ldots,T_k\}$ be a commutative family of weak* continuous and pointwise eventually nonexpansive maps on $K$. Then they have a common fixed point.
\end{Corollary}
\begin{proof}
	This is a consequence of Theorem \ref{mainthm2}, as well as Remark \ref{RemarkPointwiseEN}(c) and Proposition \ref{coroLeftRever}.
\end{proof}
A map $T: K\to K$ on a locally convex space $(E, Q)$ is called \emph{eventually nonexpansive} if for each $x, y\in K$ and each seminorm $q\in Q$, there exists an $n(x,y,q)\in\mathbb{N}$ such that $q(T^nx-T^ny)\leq q(x-y)$ for all $n\geq n(x,y,q)$.

\begin{Corollary}
	Let $K$ be a compact convex subset of a locally convex space $(E,Q)$. Let $\{T_1,\ldots,T_k\}$ be a commutative family of continuous and eventually nonexpansive maps on $K$. Then they have a common fixed point.
\end{Corollary}
\begin{proof}
	This follows Theorem 3.1 in \cite{HolLau71}. Noting that for commutative semigroups, the property \textbf{(B)} is always   satisfied.
\end{proof}

We end this paper with an open problem about possible extensions of our results.
Under some conditions,
we establish that a super asymptotically nonexpansive or pointwise eventually nonexpansive action  of
a semitopological semigroup $S$ on weakly/weak* compact convex sets has a common fixed point.

\begin{question}\label{question:2}
Do we have similar results as   Theorems \ref{mainThm}, \ref{mainthm2}, \ref{ThmRNP}, \ref{ThmRNPDis}, \ref{ThmDistal}, and Propositions \ref{coroLeftRever}, \ref{sep+reversibility} for asymptotically nonexpansive actions?
\end{question}

From Proposition \ref{remark210}, if $S$ is compact and right reversible then the asymptotic nonexpansiveness coincides
with the super asymptotic nonexpansiveness. Hence
the question has an affirmative answer in this case.

\section*{Acknowledgment}
The authors would like to thank K. Salame for valuable suggestions and comments.
B. N. Muoi is supported by Hanoi Pedagogical University 2, Vietnam grant HPU2.UT-2021.03, and N.-C. Wong is supported by Taiwan MOST grant 108-2115-M-110-004-MY2.


TITLE: How to programmatically find the inverse of a 2x2 matrix (mod 26).
QUESTION [1 upvotes]: I'm trying to create a hill cipher utility. One feature I want is to be able to compute the key if you have the plaintext and ciphertext.
$C$ = ciphertext matrix ($2\times 2$), $P$ = plaintext matrix $\left(2\times\frac{N}{2}\right)$, $K$ = key ($2\times 2$).
$$PK = C$$
So if I have $C$ and $P$, then $K = P^{-1}C$. How do I best find $P^{-1}$?
EDIT: It is important to note that this is all mod $26$. Sorry.

REPLY [3 votes]: You could use the direct formula for $2 \times 2$:
$$ \pmatrix{a & b\\
c & d}^{-1} = \dfrac{1}{ad-bc}\pmatrix{d & -b\\-c & a}$$


text:
\begin{document}
\title[Classification of quasi-trigonometric solutions]{Classification of quasi-trigonometric solutions of the classical
Yang--Baxter equation}
\author{Iulia Pop and Alexander Stolin}

\address{Department of Mathematical Sciences, University of Gothenburg, Sweden}
\email{iulia@math.chalmers.se, astolin@math.chalmers.se}

\subjclass[2000]{Primary 17B37, 17B62; Secondary 17B81}
\keywords{Classical Yang--Baxter equation, $r$-matrix, Manin triple,
parabolic subalgebra, generalized Belavin--Drinfeld data}
\begin{abstract}
It was proved by Montaner and Zelmanov that
up to classical twisting Lie bialgebra structures
on $\mathfrak{g}[u]$ fall into four classes.
Here $\mathfrak{g}$ is a simple complex finite-dimensional
Lie algebra.
It turns out that classical twists within one of these
four classes are in a one-to-one correspondence with the
so-called quasi-trigonometric solutions of the
classical Yang-Baxter equation.
In this paper we give a complete list of the
quasi-trigonometric solutions in terms of
sub-diagrams of the certain Dynkin diagrams
related to $\mathfrak{g}$.
We also explain how to quantize the corresponding
Lie bialgebra structures.
\end{abstract}
\maketitle
\section{Introduction}
The present paper constitutes a step towards the classification
of quantum groups. We describe an algorithm for the quantization
of all Lie bialgebra structures on the polynomial Lie algebra
$P=\mathfrak{g}[u]$, where $\mathfrak{g}$ is a simple complex
finite-dimensional Lie algebra.

Lie bialgebra structures
on $P$, up to so-called \textit{classical twisting}, have been classified by
F. Montaner and E. Zelmanov in \cite{MZ}. We recall that given a Lie
co-bracket $\delta$ on $P$, a \textit{classical twist}
is an element $s\in P\wedge P$ such that
\begin{equation}
\mathrm{CYB}(s)+\mathrm{Alt}(\delta\otimes \mathrm{id})(s)=0,
\end{equation}
where $\mathrm{CYB}$ is the l.h.s. of the classical Yang-Baxter equation.

We also note that a classical twist does not change the classical double
$D_{\delta}(P)$
associated to a given Lie bialgebra structure $\delta$.
If $\delta^s$ is the twisting co-bracket
via $s$, then the Lie bialgebras $(P,\delta)$ and $(P,\delta^s)$ are in the same class, i.e. there exists a Lie algebra isomorphism between
$D_{\delta}(P)$ and $D_{\delta^s}(P)$, preserving the canonical forms and compatible with the canonical embeddings of $P$ into the doubles.

According to the results of Montaner and Zelmanov, there are four Lie bialgebra
structures on $P$ up to classical twisting. Let us present them:
\vspace{0.5cm}

\textbf{Case 1.} Consider $\delta_1=0$. Consequently, $D_1(P)=P+\varepsilon P^{*}$, where $\varepsilon^2=0$. The symmetric nondegenerate invariant form $Q$ is
given by the canonical pairing between $P$ and $\varepsilon P^{*}$.

Lie bialgebra structures which fall in this class are the elements $s\in P\wedge P$ satisfying $\mathrm{CYB}(s)=0$. Such elements are in a one-to-one correspondence
with finite-dimensional quasi-Frobenius Lie subalgebras of $P$.
\vspace{0.5cm}

\textbf{Case 2.} Let us consider the co-bracket $\delta_2$ given by
\begin{equation}
\delta_2(p(u))=[r_2(u,v),p(u)\otimes 1+1\otimes p(v)],
\end{equation}
where $r_2(u,v)=\Omega/(u-v)$. Here $\Omega$ denotes the quadratic Casimir element on $\mathfrak{g}$.

It was proved in \cite{S1} that the associated classical double is $D_2(P)=\mathfrak{g}((u^{-1}))$, together with the canonical invariant form
\begin{equation}\label{eq1}
Q(f(u),g(u))=Res_{u=0}K(f,g),
\end{equation}
where $K$ denotes the Killing form of the Lie algebra $\mathfrak{g}((u^{-1}))$ over $\mathbb{C}((u^{-1}))$.

Moreover, the Lie bialgebra structures which are obtained by twisting $\delta_2$ are in a one-to-one correspondence
with so-called \textit{rational solutions} of the CYBE, according to \cite{S1}.
\vspace{0.5cm}

\textbf{Case 3.} In this case, let us consider the Lie bialgebra structure given by
\begin{equation}\label{Delta_3}
\delta_3(p(u))=[r_3(u,v),p(u)\otimes 1+1\otimes p(v)],
\end{equation}
with $r_3(u,v)=v\Omega/(u-v) +\Sigma_{\alpha} e_{\alpha}\otimes f_{\alpha}+{\frac{1}{2}}\Omega_0$,
where ${e_{\alpha},  f_{\alpha}}$ are root vectors of $\mathfrak{g}$ and $\Omega_0$ is the Cartan part
of  $\Omega$.

It was proved in \cite{KPST} that the associated classical double is
$D_3(P)=\mathfrak{g}((u^{-1}))\times\mathfrak{g}$, together with the invariant nondegenerate form $Q$ defined by
\begin{equation}\label{eq2}
Q((f(u),a),(g(u),b))=K(f(u),g(u))_0-K(a,b),
\end{equation}
where the index zero means that one takes the free term in the series expansion.
According to \cite{KPST}, there is a one-to-one correspondence between Lie bialgebra structures which are obtained by twisting
$\delta_3$ and so-called \textit{quasi-trigonometric solutions} of the CYBE.

\vspace{0.5cm}

\textbf{Case 4.} We consider the co-bracket on $P$ given by
\begin{equation}
\delta_4(p(u))=[r_4(u,v),p(u)\otimes 1+1\otimes p(v)],
\end{equation}
with $r_4(u,v)=uv\Omega/(v-u)$.

It was shown in \cite{SY} that the classical double associated to the Lie bialgebra structure $\delta_4$ is $D_4(P)=\mathfrak{g}((u^{-1}))\times(\mathfrak{g}\otimes
\mathbb{C}[\varepsilon])$, where $\varepsilon^2=0$. The form $Q$ is described as follows: if $f(u)=\sum_{-\infty}^{N} a_ku^k$ and
$g(u)=\sum_{-\infty}^{N}b_ku^k$, then
\[
Q(f(u)+A_0+A_1\varepsilon,g(u)+B_0+B_1\varepsilon)=
Res_{u=0}u^{-2}K(f,g)-K(A_0,B_1)-K(A_1,B_0).
\]

Lie bialgebra structures which are in the same class as $\delta_4$ are in a one-to-one correspondence with \textit{quasi-rational r-matrices}, as it was proved in \cite{SY}.

Regarding the quantization of these Lie bialgebra structures on $P$, the following conjecture stated in \cite{KPST} and proved by G. Halbout in \cite{H} plays
a crucial role.
\begin{thm} \label{Hal} Any classical twist can be extended to a quantum twist, i.e.,
if $(L,\delta)$ is any Lie bialgebra, $s$ is a classical twist, and $(A,\Delta,\varepsilon)$ is a quantization of $(L,\delta)$, there exists
$F\in A\otimes A$ such that

(1) $F=1+O(\hbar)$ and $F-F^{21}=\hbar s+O(\hbar^2)$,

(2) $(\Delta\otimes \mathrm{id})(F)F^{12}-(\mathrm{id}\otimes \Delta)(F)F^{23}=0$,

(3) $(\varepsilon\otimes \mathrm{id})(F)=(\mathrm{id} \otimes \varepsilon)(F)=1$.

Moreover gauge equivalence classes of quantum twists for $A$ are in bijection
with gauge equivalence classes of $\hbar$-dependent classical twists $s_\hbar=\hbar s_1+O(\hbar^2)$ for $L$.

\end{thm}

Let us suppose that we have a Lie bialgebra structure $\delta$ on $P$.
Then $\delta$ is obtained by twisting one of the four structures $\delta_i$ from
Cases 1--4. This above theorem implies that in order to find a quantization for $(P,\delta)$,
it is sufficient to determine the quantization of $\delta_i$ and then find the quantum twist whose classical
limit is $s$. Let us note that the quantization of $(P, \delta_3)$ is well-known.
The corresponding quantum algebra was introduced by V. Tolstoy in \cite{tol}
and it is denoted by $U_q  (\mathfrak{g}[u])$.

The quasi-trigonometric solutions of the CYBE were
studied in \cite{KPSST}, where it was proved that they fall into classes, which are in a one-to-one correspondence
with vertices of the extended Dynkin diagram of $\mathfrak{g}$. Let us consider corresponding roots, namely simple roots
$\alpha_1,\alpha_2,\cdots \alpha_r$ and $\alpha_0=-\alpha_{\rm{max}}$.
In \cite{KPSST}
quasi-trigonometric solutions corresponding to the simple roots which have coefficient one in the decomposition of the
maximal root
were classified. It was also proved there that quasi-trigonometric solutions corresponding to
$\alpha_0$ are in a one-to-one correspondence with constant solutions of the modified CYBE
classified in \cite{BD} and the polynomial part of these solutions is constant.
The aim of our paper is to obtain a
complete classification of quasi-trigonometric solutions of the CYBE. In particular,
we describe all the quasi-trigonometric solutions with non-trivial polynomial
part for  $\mathfrak{g}=o(5)$.

\section{Lie bialgebra structures associated with quasi-trigonometric solutions}\begin{defn}
A solution $X$ of the CYBE is called \emph{quasi-trigonometric} if
it is of the form $X(u,v)=v\Omega/(u-v)+p(u,v)$, where $p$ is a
polynomial with coefficients in $\mathfrak{g}\otimes\mathfrak{g}$.
\end{defn}
The class of quasi-trigonometric solutions is closed under
\emph{gauge transformations}. We first need to introduce the following notation:
Let $R$ be a commutative ring and let $L$ be a Lie algebra over $R$. Let us denote by
$\mathrm{Aut}_R (L)$ the group of automorphisms of $L$ over $R$. In other words
we consider such automorphisms of $L$, which satisfy the condition $f(rl)=rf(l)$, where
$r\in R,\ l\in L.$

At this point we note that there exists a natural embedding
$$\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])\hookrightarrow\mathrm{Aut}_{\mathbb{C}((u^{-1}))}(\mathfrak{g}((u^{-1}))),$$
defined by the formula
$$\sigma(u^{-k}x)=u^{-k}\sigma(x)\,,$$
for any $\sigma\in\mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ and $x\in\mathfrak{g}[u]$.

 Now if $X$ is a
quasi-trigonometric solution and
$\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$, one can check that the function $Y(u,v):=(\sigma(u)\otimes\sigma(v))(X(u,v))$ is
again a quasi-trigonometric solution. $X$ and $Y$ are said to be
\emph{gauge equivalent}.

\begin{thm}
There exists a natural one-to-one correspondence between
quasi--trigonometric solutions of CYBE for $\mathfrak{g}$ and
linear subspaces $W$ of $\mathfrak{g}((u^{-1}))\times\mathfrak{g}$
which satisfy the following properties:

(1) $W$ is a Lie subalgebra of
$\mathfrak{g}((u^{-1}))\times\mathfrak{g}$ and $W\supseteq
u^{-N}\mathfrak{g}[[u^{-1}]]$ for some positive integer $N$.

(2)
$W\oplus\mathfrak{g}[u]=\mathfrak{g}((u^{-1}))\times\mathfrak{g}$.

(3) $W$ is a Lagrangian subspace of
$\mathfrak{g}((u^{-1}))\times\mathfrak{g}$ with respect to the
invariant bilinear form $Q$ given by (\ref{eq2}).
\end{thm}

Let $\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$.
Let $\widetilde{\sigma}(u)=\sigma(u)\oplus \sigma(0)$ be the induced automorphism of $\mathfrak{g}((u^{-1}))\times\mathfrak{g}$.

\begin{defn}
We will say that $W_{1}$ and $W_{2}$ are \emph{gauge equivalent} if  there exists
$\sigma(u)\in \mathrm{Aut}_{\mathbb{C}[u]}(\mathfrak{g}[u])$ such that $W_{1}=\widetilde{\sigma}(u)W_{2}$.
\end{defn}
It was checked in \cite{KPST} that two quasi-trigonometric solutions are gauge equivalent if and only if the corresponding subalgebras are gauge equivalent.

Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ with
 the corresponding set of roots $R$ and a choice of simple
 roots $\Gamma$. Denote by $\mathfrak{g}_{\alpha}$ the root space
 corresponding to a root $\alpha$. Let $\mathfrak{h}(\mathbb{R})$
 be the set of all $h\in\mathfrak{h}$ such that
 $\alpha(h)\in\mathbb{R}$ for all $\alpha\in R$.
 Consider the valuation on $\mathbb{C}((u^{-1}))$ defined by
 $v(\sum_{k\geq n}a_{k}u^{-k})=n$. For any root $\alpha$ and any
 $h\in\mathfrak{h}(\mathbb{R})$, set
 $M_{\alpha}(h)$:=$\{f\in\mathbb{C}((u^{-1})):v(f)\geq \alpha(h)\}$.
 Consider
\begin{equation}
 \mathbb{O}_{h}:=\mathfrak{h}[[u^{-1}]]\oplus(\oplus_{\alpha\in R}M_{\alpha}(h)\otimes\mathfrak{g}_{\alpha}).
\end{equation}

As it was shown in \cite{KPSST}, any maximal order $W$ which corresponds to a quasi-trigonometric solution of the CYBE, can be embedded (up to some gauge equivalence) into $\mathbb{O}_{h}\times\mathfrak{g}$. Moreover $h$ may be taken as a vertex of the standard simplex
$\Delta_{st}= \{h\in\mathfrak{h}(\mathbb{R}):$ $\alpha(h)\geq 0$ for
all $\alpha\in\Gamma$ and $\alpha_{\max}\leq 1\}$.

Vertices of the above simplex correspond to vertices of the extended Dynkin diagram of
$\mathfrak{g}$, the correspondence being given by the following rule:
\[0\leftrightarrow\alpha_{\max}\]\[h_{i}\leftrightarrow\alpha_{i},\]
where $\alpha_{i}(h_{j})=\delta_{ij}/k_{j}$ and $k_{j}$ are given by the relation $\sum
k_{j}\alpha_{j}=\alpha_{\max}$. We will write $\mathbb{O}_{\alpha}$ instead of
$\mathbb{O}_{h}$ if $\alpha$ is the root which corresponds to the vertex $h$.

By straightforward computations, one can check the following two results:

\begin{lem}\label{Lemma1}
Let $R$ be the set of all roots and $\alpha$ an arbitrary simple root. Let $k$ be the coefficient of $\alpha$ in the decomposition of $\alpha_{\max}$.

For each $r$, $-k\leq r\leq k$, let $R_{r}$ denote the set of all roots which contain $\alpha$ with coefficient $r$. Let $\mathfrak{g}_{0}=\mathfrak{h}\oplus\sum_{\beta\in R_{0}}\mathfrak{g}_\beta$ and $\mathfrak{g}_r=\sum_{\beta\in R_r}\mathfrak{g}_\beta$. Then
\begin{equation}
\mathbb{O}_\alpha=\sum_{r=1}^k u^{-1}\mathbb{O}\mathfrak{g}_r+
\sum_{r=1-k}^0 \mathbb{O}\mathfrak{g}_r+u\mathbb{O}\mathfrak{g}_{-k},
\end{equation}
where $\mathbb{O}:=\mathbb{C}[[u^{-1}]]$.
\end{lem}

\begin{lem} Let $\alpha$ be a simple root and $k$ its coefficient in the decomposition of $\alpha_{\max}$. Let $\Delta_\alpha$ denote the set of all pairs $(a,b)$, $a\in\mathfrak{g}_0+\mathfrak{g}_{-k}$, $b\in\mathfrak{g}_0+\mathfrak{g}_{-1}+...+\mathfrak{g}_{-k}$, $a=a_0+a_{-k}$, $b=b_0+b_{-1}+...+b_{-k}$ and $ a_0=b_0$. Then

(i) The orthogonal complement of $\mathbb{O}_\alpha\times\mathfrak{g}$ with respect to $Q$ is given by
\begin{equation}
(\mathbb{O}_\alpha\times\mathfrak{g})^\perp=
\sum_{r=-k}^{-1}\mathbb{O}\mathfrak{g}_r+
\sum_{r=0}^{k-1}u^{-1}\mathbb{O}\mathfrak{g}_r+
u^{-2}\mathbb{O}\mathfrak{g}_k.
\end{equation}

(ii) There exists an isomorphism $\sigma$
\begin{equation}
\frac{\mathbb{O}_\alpha\times\mathfrak{g}}{(\mathbb{O}_\alpha\times\mathfrak{g})^\perp}\cong(\mathfrak{g}_k\oplus\mathfrak{g}_0\oplus\mathfrak{g}_{-k})\times\mathfrak{g}
\end{equation}
given by
\[\sigma((f,a)+(\mathbb{O}_\alpha\times\mathfrak{g})^\perp)=(a_0+b_0+c_0,a),\]
where the element $f\in\mathbb{O}_\alpha$ is decomposed according to Lemma \ref{Lemma1}:
\[f=u^{-1}(a_0+a_1u^{-1}+...)+(b_0+b_1u^{-1}+...)+u(c_0+c_1u^{-1}+...)+...,\]
$a_i\in\mathfrak{g}_k$, $b_i\in\mathfrak{g}_0$, $c_i\in\mathfrak{g}_{-k}$
and $a\in\mathfrak{g}$.

(iii) $(\mathbb{O}_{\alpha}\times\mathfrak{g})\cap\mathfrak{g}[u]$ is sent via the isomorphism $\sigma$ to $\Delta_\alpha$.
\end{lem}

Let us make an important remark. The Lie subalgebra $\mathfrak{g}_k+\mathfrak{g}_0+\mathfrak{g}_{-k}$ of $\mathfrak{g}$
coincides with the semisimple Lie algebra whose Dynkin diagram is obtained from the extended Dynkin diagram of $\mathfrak{g}$ by
crossing out $\alpha$. Let us denote this subalgebra by $L_\alpha$.
The Lie algebra $L_\alpha\times\mathfrak{g}$ is endowed with the following invariant bilinear form:
\begin{equation}
Q'((a,b),(c,d))=K(a,c)-K(b,d),
\end{equation}
for any $a,c\in L_\alpha$ and $b,d\in\mathfrak{g}$.

On the other hand, $\mathfrak{g}_0+\mathfrak{g}_{-k}$ is the parabolic subalgebra $P_{-\alpha_{\max}}^{+}$ of $L_\alpha$ which corresponds to $-\alpha_{\max}$.
The Lie subalgebra $\mathfrak{g}_0+\mathfrak{g}_{-1}+...+\mathfrak{g}_{-k}$ is the parabolic subalgebra $P_{\alpha}^{-}$ of $\mathfrak{g}$ which corresponds to the root $\alpha$ and contains the negative Borel subalgebra. Let us also note that $\mathfrak{g}_0$ is precisely the reductive part of $P_{\alpha}^{-}$ and of $P_{-\alpha_{\max}}^{+}$. We can conclude that the set $\Delta_{\alpha}$ consists of all pairs $(a,b)\in P_{-\alpha_{\max}}^{+}\times P_{\alpha}^{-}$ whose reductive parts are equal.

\begin{thm}

Let $\alpha$ be a simple root. There is a one-to-one correspondence between Lagrangian subalgebras $W$ of $\mathfrak{g}((u^{-1}))\times\mathfrak{g}$ which are contained in $\mathbb{O}_\alpha\times\mathfrak{g}$ and transversal to $\mathfrak{g}[u]$, and Lagrangian subalgebras $\mathfrak{l}$ of $L_{\alpha}\times\mathfrak{g}$ transversal to $\Delta_\alpha$ (with respect to the bilinear form $Q'$).
\end{thm}
\begin{proof}
Since $W$ is a subspace of $\mathbb{O}_{\alpha}\times\mathfrak{g}$, let $\mathfrak{l}$ be its image in $L_{\alpha}\times\mathfrak{g}$. Because $W$ is transversal to $\mathfrak{g}[u]$, one can check that $\mathfrak{l}$ is transversal to the image of $(\mathbb{O}_{\alpha}\times\mathfrak{g})\cap\mathfrak{g}[u]$ in $L_{\alpha}\times\mathfrak{g}$, which is exactly $\Delta_{\alpha}$. The fact that $W$ is Lagrangian implies that $\mathfrak{l}$ is also Lagrangian.

Conversely, if $\mathfrak{l}$ is a Lagrangian subalgebra of $L_{\alpha}\times\mathfrak{g}$ transversal to $\Delta_{\alpha}$, then its preimage $W$ in $\mathbb{O}_\alpha\times\mathfrak{g}$ is transversal to $\mathfrak{g}[u]$ and Lagrangian as well.

\end{proof}
The Lagrangian subalgebras $\mathfrak{l}$ of $L_{\alpha}\times\mathfrak{g}$ which are transversal to $\Delta_{\alpha}$, can be determined using results of P. Delorme \cite{Del} on the classification of Manin triples. We are interested in determining Manin triples of the form $(Q',\Delta_{\alpha},\mathfrak{l})$.

 Let us recall Delorme's construction of so-called \textit{generalized Belavin-Drinfeld data}. Let $\mathfrak{r}$ be a finite-dimensional complex, reductive, Lie algebra and $B$ a
symmetric, invariant, nondegenerate bilinear form on $\mathfrak{r}$. The goal in
\cite{Del} is to classify all Manin triples of $\mathfrak{r}$ up to conjugacy under the
action on $\mathfrak{r}$ of the simply connected Lie group $\mathcal{R}$ whose Lie
algebra is $\mathfrak{r}$.

One denotes by $\mathfrak{r}_{+}$ and $\mathfrak{r}_{-}$ respectively the sum of the
simple ideals of $\mathfrak{r}$ for which the restriction of $B$ is equal to a positive
(negative) multiple of the Killing form. Then the derived ideal of $\mathfrak{r}$ is the
sum of $\mathfrak{r}_{+}$ and $\mathfrak{r}_{-}$.

Let $\mathfrak{j}_{0}$ be a Cartan subalgebra of $\mathfrak{r}$, $\mathfrak{b}_{0}$ a
Borel subalgebra containing $\mathfrak{j}_{0}$ and $\mathfrak{b}_{0}'$ be its opposite.
Choose $\mathfrak{b}_{0}\cap\mathfrak{r}_{+}$ as Borel subalgebra of $\mathfrak{r}_{+}$
and $\mathfrak{b}_{0}'\cap\mathfrak{r}_{-}$ as Borel subalgebra of $\mathfrak{r}_{-}$.
Denote by $\Sigma_{+}$ (resp., $\Sigma_{-}$) the set of simple roots of
$\mathfrak{r}_{+}$ (resp., $\mathfrak{r}_{-}$) with respect to the above Borel
subalgebras. Let $\Sigma=\Sigma_{+}\cup\Sigma_{-}$ and denote by $\mathcal{W}=
(H_{\alpha},X_{\alpha},Y_{\alpha})_{\alpha\in\Sigma_+}$ a Weyl system of generators of
$[\mathfrak{r},\mathfrak{r}]$.
\begin{defn}[Delorme, \cite{Del}]\label{BD data}
One calls $(A,A',\mathfrak{i}_{\mathfrak{a}},\mathfrak{i}_{\mathfrak{a}'})$
\emph{generalized Belavin-Drinfeld data} with respect to $B$ when the following five
conditions are satisfied:

(1) $A$ is a bijection from a subset $\Gamma_{+}$ of $\Sigma_{+}$ on a subset
$\Gamma_{-}$ of $\Sigma_{-}$ such that
\[
B(H_{A\alpha},H_{A\beta})=-B(H_{\alpha},H_{\beta}), \alpha, \beta\in\Gamma_{+}.
\]

(2) $A'$ is a bijection from a subset $\Gamma'_{+}$ of $\Sigma_{+}$ on a subset
$\Gamma'_{-}$ of $\Sigma_{-}$ such that
\[
B(H_{A'\alpha},H_{A'\beta})=-B(H_{\alpha},H_{\beta}),\alpha,\beta\in\Gamma'_{+}.
\]

(3) If $C=A^{-1}A'$ is the map defined on $\rm{dom}(C)=\{\alpha\in\Gamma'_{+}:
A'\alpha\in\Gamma_{-}\}$ by $C\alpha=A^{-1}A'\alpha$, then $C$ satisfies:

For all $\alpha\in \rm{dom}(C)$, there exists a positive integer $n$ such that $\alpha$,...,
$C^{n-1}\alpha\in \rm{dom}(C)$ and $C^{n}\alpha\notin \rm{dom}(C)$.

(4) $\mathfrak{i}_{\mathfrak{a}}$ (resp., $\mathfrak{i}_{\mathfrak{a}'}$) is a complex
vector subspace of $\mathfrak{j}_{0}$, included and Lagrangian in the orthogonal
$\mathfrak{a}$ (resp., $\mathfrak{a}'$) to the subspace generated by $H_{\alpha}$,
$\alpha\in\Gamma_{+}\cup\Gamma_{-}$ (resp., $\Gamma'_{+}\cup\Gamma'_{-}$).

(5) If $\mathfrak{f}$ is the subspace of $\mathfrak{j}_{0}$ generated by the family
$H_{\alpha}+H_{A\alpha}$, $\alpha\in\Gamma_{+}$, and $\mathfrak{f}'$ is defined
similarly, then
\[
(\mathfrak{f}\oplus\mathfrak{i}_{\mathfrak{a}})\cap(\mathfrak{f}'\oplus
\mathfrak{i}_{\mathfrak{a}'})={0}.
\]
\end{defn}

Let $R_{+}$ be the set of roots of $\mathfrak{j}_{0}$ in $\mathfrak{r}$ which are linear
combinations of elements of $\Gamma_{+}$. One defines similarly $R_{-}$, $R'_{+}$ and
$R'_{-}$. The bijections $A$ and $A'$ can then be extended by linearity to bijections
from $R_{+}$ to $R_{-}$ (resp., $R'_{+}$ to $R'_{-}$). If $A$ satisfies condition (1),
then there exists a unique isomorphism $\tau$ between the subalgebra $\mathfrak{m}_{+}$
of $\mathfrak{r}$ spanned by $X_{\alpha}$, $H_{\alpha}$ and $Y_{\alpha}$,
$\alpha\in\Gamma_{+}$, and the subalgebra $\mathfrak{m}_{-}$ spanned by $X_{\alpha}$,
$H_{\alpha}$ and $Y_{\alpha}$, $\alpha\in\Gamma_{-}$, such that
$\tau(H_{\alpha})=H_{A\alpha}$, $\tau(X_{\alpha})=X_{A\alpha}$,
$\tau(Y_{\alpha})=Y_{A\alpha}$ for all $\alpha\in\Gamma_{+}$. If $A'$ satisfies (2), then
one defines similarly an isomorphism $\tau'$ between $\mathfrak{m'}_{+}$ and
$\mathfrak{m'}_{-}$.

\begin{thm}[Delorme, \cite{Del}]\label{Manin}
(i) Let
$\mathcal{B}\mathcal{D}=(A,A',\mathfrak{i}_{\mathfrak{a}},\mathfrak{i}_{\mathfrak{a}'})$
be generalized Belavin-Drinfeld data, with respect to $B$. Let $\mathfrak{n}$ be the sum
of the root spaces relative to roots $\alpha$ of $\mathfrak{j}_{0}$ in
$\mathfrak{b}_{0}$, which are not in $R_{+}\cup R_{-}$. Let $\mathfrak{i}:=
\mathfrak{k}\oplus\mathfrak{i}_{\mathfrak{a}}\oplus\mathfrak{n}$, where
$\mathfrak{k}:=\{X+\tau(X):X\in\mathfrak{m}_{+}\}$.

Let $\mathfrak{n'}$ be the sum of the root spaces relative to roots $\alpha$ of
$\mathfrak{j}_{0}$ in $\mathfrak{b}_{0}'$, which are not in $R'_{+}\cup R'_{-}$. Let
$\mathfrak{i'}:=\mathfrak{k'}\oplus \mathfrak{i}_{\mathfrak{a'}}\oplus\mathfrak{n'}$,
where $\mathfrak{k'}:=\{X+\tau'(X):X\in\mathfrak{m'}_{+}\}$.

Then $(B,\mathfrak{i},\mathfrak{i}')$ is a Manin triple.

(ii) Every Manin triple is conjugate by an element of $\mathcal{R}$ to a unique Manin
triple of this type.
\end{thm}
Let us consider the particular case
$\mathfrak{r}=L_{\alpha}\times\mathfrak{g}$.
We set $\Sigma_{+}:=(\Gamma^{ext} \setminus \{\alpha\})\times\{0\}$,
$\Sigma_{-}:=\{0\}\times\Gamma$ and $\Sigma:=\Sigma_{+}\cup\Sigma_{-}$.

Denote by $(X_{\gamma},Y_{\gamma}, H_{\gamma})_{\gamma\in\Gamma}$ a Weyl system of generators for $\mathfrak{g}$ with respect to the root system $\Gamma$.
Denote $-\alpha_{\max}$ by $\alpha_0$.
Let $H_{\alpha_0}$ be the coroot of $\alpha_0$. We choose $X_{\alpha_0}\in\mathfrak{g}_{\alpha_0}$, $Y_{\alpha_0}\in\mathfrak{g}_{-\alpha_0}$ such that $[X_{\alpha_0},Y_{\alpha_0}]=H_{\alpha_0}$.

A Weyl system of generators in $L_{\alpha}\times\mathfrak{g}$ (with respect to the root system $\Sigma$) is the following:
$X_{(\beta,0)}=(X_\beta,0)$, $H_{(\beta,0)}=(H_\beta,0)$, $Y_{(\beta,0)}=(Y_\beta,0)$, for any $\beta\in\Gamma^{ext} \setminus \{\alpha\}$, and
$X_{(0,\gamma)}=(0,X_{\gamma})$, $H_{(0,\gamma)}=(0,H_\gamma)$, $Y_{(0,\gamma)}=(0,Y_\gamma)$, for any $\gamma\in\Gamma$.

By applying the general result of Delorme, one can deduce the description of the Manin triples of the form $(Q',\Delta_{\alpha},\mathfrak{l})$.

First, let us denote by $i$ the embedding
\[\Gamma\setminus \{\alpha\}\hookrightarrow\Gamma^{ext}\setminus\{\alpha\},\]
Recall that $\mathfrak{g}_0$ denotes the reductive part of $P_{\alpha}^{-}$
and has the Dynkin diagram $\Gamma\setminus \{\alpha\}$. Then $i$ induces an inclusion
\[\mathfrak{g}_{0}\hookrightarrow L_\alpha.\]
We will also denote this embedding by $i$.

\begin{cor}\label{special Manin}
Let $S:=\Gamma\setminus \{\alpha\}$
and $\zeta_S:=\{h\in\mathfrak{h}:\beta(h)=0, \forall \beta\in S\}$.
For any Manin triple $(Q',\Delta_{\alpha},\mathfrak{l})$, there exists a unique
generalized Belavin-Drinfeld data
$\mathcal{B}\mathcal{D}=(A,A',\mathfrak{i}_{\mathfrak{a}},\mathfrak{i}_{\mathfrak{a}'})$
where $A:i(S)\times \{0\}\longrightarrow \{0\}\times S$,
$A(i(\gamma),0)=(0,\gamma)$ and
$\mathfrak{i}_{\mathfrak{a}}=\rm{diag}(\zeta_S)$,
such that $(Q',\Delta_{\alpha},\mathfrak{l})$ is conjugate to the Manin triple
$\mathcal{T}_{\mathcal{B}\mathcal{D}}=(Q',\mathfrak{i},\mathfrak{i'})$. Moreover, up to a conjugation which preserves $\Delta_{\alpha}$, one has
$\mathfrak{l}=\mathfrak{i'}$.
\end{cor}
\begin{proof}
Let us suppose that $(Q',\Delta_{\alpha},\mathfrak{l})$ is a Manin triple. Then there exists a unique generalized Belavin-Drinfeld data
$\mathcal{B}\mathcal{D}=(A,A',\mathfrak{i}_{\mathfrak{a}},\mathfrak{i}_{\mathfrak{a}'})$ such that the corresponding $\mathcal{T}_{\mathcal{B}\mathcal{D}}=(Q',\mathfrak{i},\mathfrak{i'})$ is conjugate to $(Q',\Delta_{\alpha},\mathfrak{l})$. Since $\mathfrak{i}$ and $\Delta_{\alpha}$ are conjugate and $\Delta_{\alpha}$ is ``under'' the parabolic subalgebra $P_{\alpha_{0}}^{+}\times P_{\alpha}^{-}$, it follows that
$\mathfrak{i}$ is also ``under'' this parabolic and thus $\mathfrak{a}=
\zeta_S\times\zeta_S$. According to \cite{Del}, p. 136, the map $A$ should be an isometry between $i(S)\times\{0\}$ and $\{0\}\times S$. Let us write
$A(i(\gamma),0)=(0,\tilde{A}(\gamma))$, where $\tilde{A}:S\rightarrow S$ is an
isometry which will be determined below.

Let $\mathfrak{m}$ be the image of $\mathfrak{g}_0$ in $L_\alpha$ via the embedding $i$. Then $\mathfrak{m}$ is spanned by $X_{i(\beta)}$, $H_{i(\beta)}$, $Y_{i(\beta)}$ for all $\beta\in S$.

According to Theorem \ref{Manin}, the Lagrangian subspace $\mathfrak{i}$ contains $\mathfrak{k}:=\{(X,\tau(X)):X\in\mathfrak{m}\}$, where $\tau$ satisfies the following conditions:
$\tau(X_{i(\beta)})=X_{\tilde{A}\beta}$, $\tau(H_{i(\beta)})=H_{\tilde{A}\beta}$, $\tau(Y_{i(\beta)})=Y_{\tilde{A}\beta}$, for all $\beta\in S$.

We obtain
$\tau i(X_{\beta})=X_{\tilde{A}\beta}$,
$\tau i(Y_{\beta})=Y_{\tilde{A}\beta}$,
$\tau i(H_{\beta})=H_{\tilde{A}\beta}$, for all $\beta\in S$.

Since $\mathfrak{i}$ and $\Delta_{\alpha}$ are conjugate, we must have
$\mathfrak{i}_{\mathfrak{a}}=\rm{diag}(\zeta_S)$. Moreover
$\tau i$ has to be an inner automorphism of $\mathfrak{g}_0$.
It follows that the isometry $\tilde{A}:S\rightarrow S$ which corresponds to this inner automorphism must be the identity.
Thus $\tilde{A}=\rm{id}$ and this ends the proof.
\end{proof}

We will consider triples of the form $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$, where $\Gamma'_{1}\subseteq\Gamma^{ext} \setminus \{\alpha\}$,
$\Gamma'_{2}\subseteq\Gamma$ and $\tilde{A'}$ is an isometry between $\Gamma'_{1}$ and $\Gamma'_{2}$.
\begin{defn}\label{types}
We say that a triple $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$
is of \textit{type I} if $\alpha\notin\Gamma'_{2}$ and $(\Gamma'_{1},i(\Gamma'_{2}),i\tilde{A'})$ is an admissible triple in the sense of \cite{BD}.
The triple $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ is of \textit{type II} if $\alpha\in\Gamma'_{2}$ and
$\tilde{A'}(\beta)=\alpha$, for some
$\beta\in\Gamma'_{1}$ and $(\Gamma'_{1}\setminus\{\beta\},
i(\Gamma'_{2}\setminus\{\alpha\}),i\tilde{A'})$ is an
admissible triple in the sense of \cite{BD}.

\end{defn}

Using the definition of generalized Belavin-Drinfeld data, one can easily check the following:
\begin{lem}
Let  $A:i(S)\times \{0\}\longrightarrow \{0\}\times S$,
$A(i(\gamma),0)=(0,\gamma)$ and $\mathfrak{i}_{\mathfrak{a}}=\rm{diag}(\zeta_S)$.
A quadruple $(A,A', \mathfrak{i}_{\mathfrak{a}},\mathfrak{i}_{\mathfrak{a'}})$ is
generalized Belavin-Drinfeld data if and only if the pair
$(A',\mathfrak{i}_{\mathfrak{a'}})$ satisfies the following conditions:

(1) $A':\Gamma'_{1}\times \{0\} \longrightarrow \{0\}\times \Gamma'_{2}$ is given by
$A'(\gamma,0)=(0,\tilde{A'}(\gamma))$ and $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ is of type I or II from above.

(2) Let $\mathfrak{f}$ be the subspace of $\mathfrak{h}\times\mathfrak{h}$
spanned by pairs $(H_{i(\gamma)},H_{\gamma})$ for all $\gamma\in S$ and
$\mathfrak{f'}$ be the subspace of $\mathfrak{h}\times\mathfrak{h}$ spanned by pairs
$(H_{\beta},H_{\tilde{A'}(\beta)})$ for all $\beta\in\Gamma'_{1}$. Let
$\mathfrak{i}_{\mathfrak{a'}}$ be Lagrangian subspace of
$\mathfrak{a'}:=\{(h_{1},h_{2})\in\mathfrak{h}\times\mathfrak{h}:\beta(h_{1})=0,
\gamma(h_{2})=0, \forall\beta\in\Gamma'_{1}, \forall \gamma\in\Gamma'_{2}\}$. Then
\begin{equation}\label{condf}
(\mathfrak{f'}\oplus\mathfrak{i}_{\mathfrak{a'}})\cap(\mathfrak{f}\oplus\mathfrak{i}_{\mathfrak{a}})={0}.
\end{equation}
\end{lem}
\begin{rem}
One can always find $\mathfrak{i}_{\mathfrak{a'}}$ which is a Lagrangian subspace of $\mathfrak{a'}$ and satisfies condition (\ref{condf}). This is a consequence of \cite{Del} Remark 2, p. 142. A proof of this elementary fact can also be found in \cite{S3}, Lemma 5.2.
\end{rem}
Summing up the previous results we conclude the following:

\begin{thm}\label{main}
Let $\alpha$ be a simple root. Suppose that $\mathfrak{l}$ is a Lagrangian subalgebra of $L_{\alpha}\times\mathfrak{g}$ transversal to $\Delta_{\alpha}$. Then, up to a conjugation which preserves $\Delta_{\alpha}$, one has $\mathfrak{l}=\mathfrak{i'}$, where $\mathfrak{i'}$ is constructed from a pair formed by
a triple $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ of type I or II and
a Lagrangian subspace $\mathfrak{i}_{\mathfrak{a'}}$ of
$\mathfrak{a'}$ such that (\ref{condf}) is satisfied.

\end{thm}
\begin{rem}
The above algorithm covers also the case when the simple root $\alpha$
has coefficient $k=1$ in the decomposition of $\alpha_{\rm{max}}$,
which was considered in \cite{KPSST}. In this form we do not need to
use the Cartan involution which appeared there.
\end{rem}
\begin{rem}
In \cite{KPSST} we constructed two examples of quasi-trigonometric solutions for $\mathfrak{g}=\mathfrak{sl}(n+1)$ related to the Cremmer-Gervais $r$-matrix.
Let us apply the present algorithm for the simple root $\alpha_{n}$.
Let $\Gamma=\{\alpha_{1},...,\alpha_{n}\}$ and $\alpha_{0}=-\alpha_{\rm{max}}$.

We consider the triple $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$, where
$\Gamma'_{1}=\{\alpha_{0},\alpha_{1},...\alpha_{n-2}\}$,
$\Gamma'_{2}=\{\alpha_{1},...\alpha_{n-1}\}$ and $\tilde{A'}(\alpha_j)=\alpha_{j+1}$, for $j=0,...,n-2$.
This is a triple of type I, cf. Definition \ref{types}.

Another example is given by the following triple:
$\Gamma'_{1}=\{\alpha_{0},\alpha_{1},...\alpha_{n-1}\}$,
$\Gamma'_{2}=\{\alpha_{1},...\alpha_{n}\}$ and $\tilde{A'}(\alpha_{j})=\alpha_{j+1}$,
for $j=1,...,n-1$. This is a triple of type II, cf. Definition \ref{types}.

The above triples together with the corresponding subspaces $\mathfrak{i}_{\mathfrak{a}'}$ (which we do not compute here) induce two Lagrangian subalgebras of Cremmer-Gervais type.

\end{rem}

Theorem \ref{main} can also be applied to roots which have coefficient
greater than one.
As an example, let us classify solutions in $\mathfrak{g}=o(5)$.
\begin{cor}
Let $\alpha_1$, $\alpha_2$ be the simple roots in $o(5)$ and
$\alpha_0=-2\alpha_{1}-\alpha_{2}$. Up to gauge equivalence, there exist
two quasi-trigonometric solutions with non-trivial polynomial part.
\end{cor}
\begin{proof}
The root $\alpha_{2}$ has coefficient $k=1$ in the decomposition of the maximal root. The only possible choice for a triple
$(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ with
$\Gamma'_{1}\subseteq\{\alpha_{0},\alpha_{1}\}$,
$\Gamma'_{2}\subseteq\{\alpha_{1},\alpha_{2}\}$ to be of type I or II is
$\Gamma'_{1}=\{\alpha_{0},\}$, $\Gamma'_{2}=\{\alpha_{2},\}$,
$\tilde{A'}(\alpha_{0})=\alpha_{2}$. One can check that
$\mathfrak{i}_{\mathfrak{a'}}$ is a 1-dimensional space spanned by
the following pair
$(\rm{diag}(-2,1,0,-1,2),\rm{diag}(0,\sqrt{5},0,-\sqrt{5},0))$.
 The Lagrangian subalgebra $\mathfrak{i}'_{2}$ constructed from this triple is transversal to $\Delta_{\alpha_{2}}$ in $\mathfrak{g}\times\mathfrak{g}$.

The root $\alpha_{1}$ has coefficient $k=2$ in the decomposition of the maximal root. The only possible choice for a triple
$(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ with
$\Gamma'_{1}\subseteq\{\alpha_{0},\alpha_{2}\}$,
$\Gamma'_{2}\subseteq\{\alpha_{1},\alpha_{2}\}$ is again
$\Gamma'_{1}=\{\alpha_{0},\}$, $\Gamma'_{2}=\{\alpha_{2},\}$,
$\tilde{A'}(\alpha_{0})=\alpha_{2}$ and
$\mathfrak{i}_{\mathfrak{a'}}$ is as in the previous case.
The Lagrangian subalgebra $\mathfrak{i}'_{1}$ constructed from this triple is transversal to $\Delta_{\alpha_{1}}$ in
$L_{\alpha_{1}}\times\mathfrak{g}$.

\end{proof}

\section{Quantization of quasi-trigonometric solutions}
Let us consider the Lie bialgebra structure on $\mathfrak{g}[u]$ given by
the simplest quasi-trigonometric solution:
\begin{equation}\label{Delta}
\delta(p(u))=[r(u,v),p(u)\otimes 1+1\otimes p(v)],
\end{equation}
with $r(u,v)=v\Omega/(u-v) +\Sigma_{\alpha} e_{\alpha}\otimes f_{\alpha}+{\frac{1}{2}}\Omega_0$,
where ${e_{\alpha},  f_{\alpha}}$ are root vectors of $\mathfrak{g}$ and $\Omega_0$ is the Cartan part of  $\Omega$. Here $\delta$ and $r$ are exactly the same as $\delta_3$ and $r_3$ from the Introduction.

Quasi-trigonometric solutions correspond to Lie bialgebra
structures on $\mathfrak{g}[u]$ which are obtained by twisting $\delta$.
The quantization of the Lie bialgebra
$(\mathfrak{g}[u],\delta)$ is the quantum algebra $U_q(\mathfrak{g}[u])$
introduced by V. Tolstoy in \cite{tol}. We will recall its construction with a
slight modification.

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra
of rank $l$ with a standard Cartan matrix $A=(a_{ij})_{i,j=1}^l$, with
a system of simple roots $\Gamma= \{\alpha_1,\ldots, \alpha_l\}$, and with a
Chevalley basis $h_{\alpha_i}$, $e_{\pm\alpha_i}$ $(i=1,2,\ldots, l)$.
Let $\theta$ be the maximal (positive) root of $\mathfrak{g}$. The corresponding
non-twisted affine algebra with zero central charge $\hat{\mathfrak{g}}$ is generated by
$\mathfrak{g}$ and the additional affine elements $e_{\delta-\theta}:=ue_{-\theta}$,
$e_{-\delta+\theta}:=u^{-1}e_{\theta}$.

The Lie algebra $\mathfrak{g}[u]$ is generated by $\mathfrak{g}$,
the positive root vector $e_{\delta-\theta}$ and the Cartan element
$h_{\delta-\theta}=[e_{\delta-\theta},e_{-\delta+\theta}]$. The standard
defining relations of the universal
enveloping algebra $U(\mathfrak{g}[u])$ are given by the formulas:
\begin{eqnarray}
[h_{\alpha_{i}}^{},h_{\alpha_{j}}^{}]\!\!&=\!\!&0~,
\label{tolsA1'}
\\[5pt]
[h_{\alpha_{i}}^{},e_{\pm\alpha_{j}}]\!\!&=\!\!&
\pm(\alpha_{i},\alpha_{j})e_{\pm\alpha_{j}}~,
\label{tolsA2}
\\[5pt]
[e_{\alpha_{i}},e_{-\alpha_{j}}]\!\!&=\!\!&\delta_{ij}h_{\alpha_{i}}~,
\label{tolsA2'}
\\[5pt]
(\mathrm{ad} e_{\pm\alpha_{i}}\!)^{n_{ij}} e_{\pm\alpha_{j}}\!\!&=\!\!&0
\qquad{\rm for}\;\;i\neq j,\;\;n_{ij}\!:=\!1\!-\!a_{ij}~,
\label{tolsA3}
\\[5pt]
[h_{\alpha_i}^{},e_{\delta-\theta}]\!\!&=\!\!&-
(\alpha_i,\theta)\,e_{\delta-\theta}'~,
\label{tolsA3'}
\\[5pt]
[e_{-\alpha_i},e_{\delta-\theta}]\!\!&=\!\!&0~,
\label{tolsA4}
\\[5pt]

(\mathrm{ad} e_{\alpha_i})^{n_{i0}}e_{\delta-\theta}\!\!&=\!\!&0
\quad\;{\rm for}\;\,n_{i0}=1\!+\!2(\alpha_i,\theta)/(\alpha_i,\alpha_i),
\label{tolsA5}
\\[5pt]
[[e_{\alpha_i},e_{\delta-\theta}],e_{\delta-\theta}]\!\!&=\!\!&0\quad\;
{\rm for}\;\,\mathfrak{g}\ne\mathfrak{sl}_2 \;\;{\rm and}\;\;(\alpha_i,\theta)\ne 0~,
\label{tolsA6}
\\[5pt]
[[[e_{\alpha},e_{\delta-\alpha}],e_{\delta-\alpha}],e_{\delta-\alpha}]
\!\!&=\!\!&0\quad\;{\rm for}\;\,\mathfrak{g}=\mathfrak{sl}_2
\;\;(\theta=\alpha)~.
\label{tolsA7}
\end{eqnarray}

The quantum algebra $U_q(\mathfrak{g}[u])$ is
a $q$-defor\-mation of $U(\mathfrak{g}[u])$.
The Chevalley generators for  $U_q(\mathfrak{g}[u])$ are
$k_{\alpha_i}^{\pm 1}:=q^{\pm h_{\alpha_i}}$, $e_{\pm\alpha_i}$
$(i=1,2,\ldots, l)$, $e_{\delta-\theta}$
and $k_{\delta-\theta}:=q^{h_{\delta-\theta}}$.  Then the defining relations of
$U_q(\mathfrak{g}[u])$ are the following:
\begin{eqnarray}
k_{\alpha_i}^{\pm 1}k_{\alpha_j}^{\pm 1}\!\!&=\!\!&
k_{\alpha_j}^{\pm 1}k_{\alpha_i}^{\pm 1}~,
\label{tolsA8'}
\\[5pt]
k_{\alpha_i}^{}k^{-1}_{\alpha_i}\!\!&=\!\!&
k^{-1}_{\alpha_i}k_{\alpha_i}^{}=1~,
\label{tolsA9}
\\[5pt]
k_{\alpha_i}^{}e_{\pm\alpha_j}^{}k^{-1}_{\alpha_i}\!\!&=\!\!&
q^{\pm(\alpha_i,\alpha_j)}e_{\pm\alpha_j}^{}~,
\label{tolsA9'}
\\[5pt]
[e_{\alpha_i}^{},e_{-\alpha_i}^{}]\!\!&=\!\!&
\frac{k_{\alpha_i}-k_{\alpha_i}^{-1}}{q-q^{-1}}~,
\label{tolsA10}
\\[5pt]
(\mathrm{ad}_{q}e_{\pm\alpha_{i}}^{}\!)^{n_{ij}}e_{\pm\alpha_{j}^{}}\!\!&=\!\!&0
\quad{\rm for}\;\,i\neq j,\;\;n_{ij}\!:=\!1\!-\!a_{ij}~,
\label{tolsA11}
\\[5pt]
k_{\alpha_i}^{}e_{\delta-\theta}^{}k^{-1}_{\alpha_i}\!\!&=\!\!&
q^{-(\alpha_i,\theta)}e_{\delta-\theta}^{}~,
\label{tolsA11'}
\\[5pt]
[e_{-\alpha_i}^{},e_{\delta-\theta}^{}]\!\!&=\!\!&0~,
\label{tolsA12}
\\[5pt]

(\mathrm{ad}_{q}e_{\alpha_i}^{})^{n_{i0}}e_{\delta-\theta}^{}\!\!&=\!\!&0
\quad{\rm for}\;\;n_{i0}=1\!+\!2(\alpha_i,\theta)/(\alpha_i,\alpha_i),
\label{tolsA13}
\\[5pt]
[[e_{\alpha_i}^{},e_{\delta-\theta}^{}]_{q},
e_{\delta-\theta}^{}]_{q}\!\!&=\!\!&0\quad{\rm for}\;\;
\mathfrak{g}\ne\mathfrak{sl}_2\;\;{\rm and}\;\;(\alpha_i,\theta)\ne0~,
\label{tolsA14}
\\[5pt]
[[[e_{\alpha}^{},e_{\delta-\alpha}^{}]_{q},
e_{\delta-\alpha}^{}]_{q},e_{\delta-\alpha}^{}]_{q}\!\!&=\!\!&0
\quad{\rm for}\;\;\mathfrak{g}=\mathfrak{sl}_2~,
\label{tolsA15}
\end{eqnarray}
where $(\mathrm{ad}_q e_{\beta})e_{\gamma}$ is the q-commutator:
\begin{equation}
\quad\quad\;{}
(\mathrm{ad}_q e_{\beta}^{})e_{\gamma}^{}:= [e_\beta^{},e_\gamma^{}]_q:=
e_\beta^{}e_\gamma^{}-q^{(\beta,\gamma)}e_\gamma^{}e_\beta^{}~.
\label{tolsA16}
\end{equation}
The comultiplication $\Delta_{q}$, the antipode $S_{q}$, and
the co-unit $\varepsilon_{q}$ of $U_{q}(\mathfrak{g}[u])$
are given by
\begin{equation}
\begin{array}{rcl}

\Delta_{q}(k_{\alpha_i}^{\pm 1})\!\!&=\!\!&
k_{\alpha_i}^{\pm 1}\otimes k_{\alpha_i}^{\pm 1}~

\\[5pt]

\Delta_{q}(e_{-\alpha_i}^{})\!\!&=\!\!&
e_{-\alpha_i}^{}\otimes k_{\alpha_i}^{}+1\otimes
e_{-\alpha_i}^{}~,

\\[5pt]
\Delta_{q}(e_{\alpha_i}^{})\!\!&=\!\!&
e_{\alpha_i}^{}\otimes1+k_{\alpha_i}^{-1}\otimes
e_{\alpha_i}^{}~,\qquad

\\[5pt]
\Delta_{q}(e_{\delta-\theta}^{})\!\!&=\!\!&e_{\delta-\theta}^{}
\otimes1+k_{\delta-\theta}^{-1}\otimes
e_{\delta-\theta}^{}~,\qquad

\label{tolsA19}
\end{array}
\end{equation}
\begin{equation}
\begin{array}{rcl}

S_{q}(k_{\alpha_i}^{\pm 1})\!\!&=\!\!&k_{\alpha_i}^{\mp 1}~,

\\[5pt]
S_{q}(e_{-\alpha_i}^{})\!\!&=\!\!&-e_{-\alpha_i}^{}k_{\alpha_i}^{-1}~,

\\[5pt]
S_{q}(e_{\alpha_i}^{})\!\!&=\!\!&-k_{\alpha_i}^{}e_{\alpha_i}^{}~,

\\[5pt]
S_{q}(e_{\delta-\theta}^{})\!\!&=\!\!&
-k_{\delta-\theta}^{}e_{\delta-\theta}^{}~,

\end{array}
\end{equation}
\begin{equation}
\varepsilon_{q}(e_{\pm\alpha_i})~=~\varepsilon_{q}(e_{\delta-\theta})=0~,
\qquad\varepsilon_{q}(k_{\alpha_i}^{\pm 1})=1.
\label{tolsA22'}
\end{equation}
We see that
\begin{equation}
\qquad\;{}k_{\delta-\theta}^{}=\,k_{\alpha_1}^{-n_1}
k_{\alpha_2}^{-n_2}\cdots k_{\alpha_l}^{-n_l}
\label{tolsA23}
\end{equation}
if $\theta=n_1^{}\alpha_1^{}+n_2^{}\alpha_2^{}+\cdots\,+ n_l^{}\alpha_l^{}$.
\par

Let us consider now $U_{\hbar}(\mathfrak{g}[u])$, which is an algebra over
$\mathbb{C}[\hbar]]$ defined by the previous relations in which we set
$q=\mathrm{exp}(\hbar)$.

\begin{thm}
The classical limit of $U_{\hbar}(\mathfrak{g}[u])$ is
$(\mathfrak{g}[u],\delta)$.
\end{thm}
\begin{proof}
Since the image of $e_{\delta-\theta}$ in $U_{\hbar}(\mathfrak{g}[u])/{\hbar}U_{\hbar}(\mathfrak{g}[u])$ is $ue_{\theta}$, we can identify the classical
limit of $U_{\hbar}(\mathfrak{g}[u])$ with $U(\mathfrak{g}[u])$.

It remains to prove that for any $a\in\mathfrak{g}[u]$ and its preimage
$\tilde{a}\in U_{\hbar}(\mathfrak{g}[u])$, we have
\[{\hbar}^{-1}(\Delta(\tilde{a})-\Delta^{op}(\tilde{a}))\;\; \mathrm{mod}\;\; \hbar=
\delta(a).\]

It is clear that $U_{\hbar}(\mathfrak{g}[u])$ is a Hopf subalgebra
of $U_{\hbar}(\hat{\mathfrak{g}})$ with zero central charge.
It follows from \cite{D} that
$\Delta^{op}(\tilde{a})=R\Delta(\tilde{a})R^{-1}$,
where $R=1\otimes 1+\hbar r+...$ is the universal $R$-matrix for  $U_{\hbar}(\hat{\mathfrak{g}})$ (see also \cite{KT} and \cite{J}).

Writing $\tilde{a}=a+\hbar s$, we see that
\[{\hbar}^{-1}(\Delta(\tilde{a})-R\Delta(a)R^{-1})\;\; \mathrm{mod}\;\; \hbar=
[r,a\otimes 1+1\otimes a]=\delta(a).\]
This ends the proof of the theorem.
\end{proof}

The Lie bialgebra structures induced by quasi-trigonometric solutions
are obtained by twisting $\delta$ via a classical twist.
According to Theorem \ref{main}, any such
twist is given by a triple $(\Gamma'_{1},\Gamma'_{2},\tilde{A'})$ of type I or II together with a Lagrangian subspace $\mathfrak{i}_{\mathfrak{a'}}$ of
$\mathfrak{a'}$ such that (\ref{condf}) is satisfied.

On the other hand, Theorem \ref{Hal} states that classical twists can be extended to quantum twists.

We conclude that any data of the form $(\Gamma'_{1},\Gamma'_{2},\tilde{A'},\mathfrak{i}_{\mathfrak{a'}})$ provides a twisted comultiplication and antipode in the quantum algebra $U_{\hbar}(\mathfrak{g}[u])$.
In case $\mathfrak{g}=\mathfrak{sl}(n)$, some exact formulas were obtained in
\cite{KPSST}.

\vspace*{1cm}

\textbf{Acknowledgement.} The authors are thankful to V. Tolstoy
for valuable discussions.

\bibliographystyle{amsalpha}

config: arxiv


TITLE: Language to describe that a multivariate function is convex with respect to only certain variables
QUESTION [1 upvotes]: Consider the following function:
$$f(x, y, z) = x^2 + y^2 - z^2$$
I am writing a paper with such a function in which I stated "$f$ is convex only in variables $x$ and $y$". By this I mean that for any constant $a$, $f(x, y, a)$ is a convex function of $x$ and $y$. I figured my meaning would be clear from context, but my advisor told me that he thinks that this language is potentially ambiguous and asked me to consult the literature to make sure my verbiage is consistent with that which other mathematicians and scientists use for such functions.
The problem is that wherever I have looked I can only find the definition for a function being convex as a whole. I have found no reference that gives any kind of standard notation/name for the situation I described. Can anyone shed some light on this?
Perhaps the answer is right in the title of my question. I could say "$f$ is convex with respect to only $x$ and $y$". But this begs the same: is this standard usage of the language/is the meaning clear and unambiguous?

REPLY [1 votes]: I would write: "For each value of $a$, the function $f(\cdot, \cdot, a)$ is convex"
or
(very similarly)
"For each value of $a$, the function $(x,y) \mapsto f(x,y,a)$ is convex".


TITLE: Solutions of a linear system
QUESTION [0 upvotes]: $$A=
        \begin{pmatrix}
        1 & 1 & 0 \\
        1 & 2 & 1 \\
        1 & 3 & 2 \\
        \end{pmatrix}
$$
If $b = [ 2, 4 ,6 ]^T$ , how many solutions are
there to the system $Ax = b$?
My guess is infinitely many because the echelon form produces last row to 0's but not sure

REPLY [0 votes]: Did you agument your matrix as follows:
\begin{bmatrix}
1 & 1 & 0 & 2\\
1 & 2 & 1 & 4\\
1 & 3 & 2 & 6\\
\end{bmatrix}
For your row reduced echelon form:
\begin{bmatrix}
1 & 0 & -1 & 0\\
0 & 1 & 1 & 2\\
0 & 0 & 0 & 0\\
\end{bmatrix}
The last row shows us we have a free variable.  Call it $z$.  Then our second row shows us that $y+z=2$ and our first row shows us that $x-z=0$.  Or, $y=2-z$ and $x=z$.