%The template file to use preparing contributions to the Marinov Memorial Volume

%If you have figures in your text and using, say, epsf.sty change the next line to
%  \documentstyle[sprocl,twoside,epsf]{article}

\documentstyle[sprocl,twoside,amssymb,amsmath,amsthm,xypic]{article}

% Macroses below are optional, you can introduce your own ones.

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

%Our macros
\newcommand{\g}{\mathfrak g}
\newcommand{\h}{\mathfrak h}
\renewcommand{\l}{\mathfrak l}
\newcommand{\m}{\mathfrak m}
\newcommand{\n}{\mathfrak n}
\renewcommand{\u}{\mathfrak u}
\newcommand{\z}{\mathfrak z}
\newcommand{\Dg}{D(\g)}
\newcommand{\geps}{\g [\varepsilon]}
\newcommand{\Ker}{\mbox{\rm Ker} \, }
\newcommand{\Lie}{\mbox{\rm Lie} \, }
\newcommand{\CYB}{\mbox{\rm CYB}}
\newcommand{\Alt}{\mbox{\rm Alt}}
\newcommand{\Ad}{\mbox{\rm Ad}}
\newcommand{\ad}{\mbox{\rm ad}}
\newcommand{\id}{\mbox{\rm id}}
\newcommand{\Span}{\mbox{\rm Span}}
\newcommand{\gxg}{\g \! \times \! \g}
\newcommand{\gotg}{\g \otimes \g}
\newcommand{\gwg}{\wedge^2 \g}
\newcommand{\gotgu}{(\g \otimes \g)^{\u}}
\newcommand{\mwm}{\wedge^2 \m}
\newcommand{\mwmu}{(\wedge^2 \m)^{\u}}
\newcommand{\scpr}{\langle \cdot \, , \cdot \rangle}
\newcommand{\RRR}{{\bf R}}
\newcommand{\NNN}{{\bf N}}
\newcommand{\UUU}{{\bf U}}
\newcommand{\Pairs}{\mbox{\bf Pairs}(\g)}
\newcommand{\Lagr}{\mbox{\bf Lagr}(\g)}
\newcommand{\Pairsu}{\mbox{\bf Pairs}(\g, \u)}
\newcommand{\Lagru}{\mbox{\bf Lagr}(\g, \u)}
\newcommand{\Dynr}{\mbox{\bf Dynr}({\g, \u}, \Omega)}
\newcommand{\Modr}{{\cal M}({\g, \u}, \Omega)}
\newcommand{\Momega}{{\cal M}_\Omega}
\newcommand{\Maps}{\mbox{\bf Map}(D, G)^{\u}}
\newcommand{\Mapszero}{\mbox{\bf Map}_0(D, G)^{\u}}
\newcommand{\Dom}{\mbox{\rm Domain}}
\newcommand{\halfomega}{\frac{\Omega}{2}}
\newcommand{\modomega}{\frac{1}{4}[\Omega^{12}, \Omega^{23}]}
\newcommand{\nh}{\NNN,\,h}

\newtheorem{proposition}{Proposition}%[section]
\newtheorem{theorem}[proposition]{Theorem}
\newtheorem{corollary}[proposition]{Corollary}
\newtheorem{lemma}[proposition]{Lemma}
\newtheorem{remark}[proposition]{Remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% We ask you to use pagestyle defined below.
% Substitute the authors' names and the title
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagestyle{myheadings}  %%{Authors}{Title}
\markboth{\small \em \quad E.~Karolinsky and A.~Stolin \hfill}{\hfill
\small
\em Classical dynamical r-matrices \quad}
\thispagestyle{plain}
\setcounter{page}{252}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\sloppy

\title{CLASSICAL DYNAMICAL R-MATRICES AND POISSON HOMOGENEOUS SPACES}

\author{EUGENE KAROLINSKY}

\address{Department of Mathematics, Kharkov National University,\\
4 Svobody Sqr., Kharkov, 61077, Ukraine}

\author{ALEXANDER STOLIN}

\address{Department of Mathematics,
University of G\"oteborg,\\ SE-412 96 G\"oteborg, Sweden}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle\abstracts{Lu\cite{Lu CDYBE} showed that any dynamical
r-matrix for the pair $(\g,\u)$ naturally induces a Poisson
homogeneous structure on $G/U$. She also proved that if $\g$ is
complex simple, $\u$ is its Cartan subalgebra and $r$ is
quasitriangular, then this correspondence is in fact 1--1. In the
present paper we find some general conditions under which the Lu
correspondence is 1--1. Then we apply this result to describe all
triangular Poisson homogeneous structures on $G/U$ for a simple
complex group $G$ and its reductive subgroup $U$ containing a
Cartan subgroup.}

\vspace{0.4cm}

\tableofcontents

\newpage

\section*{Foreword by Alexander Stolin} 
\addcontentsline{toc}{section}{\numberline{}Foreword}
I met Misha Marinov at the Max
Born Symposium in Poland in 1996, where Misha presented a group-theoretical
approach to construction of dynamical systems.\cite{Mar1} His construction
was based on a homogeneous space $M=G/K$ equipped with a K\"ahler structure.
We immediately realized that the construction is related to some solution of
the classical Yang-Baxter equation and Misha's quantization of this K\"ahler
structure leads to a solution of the quantum Yang-Baxter equation.

Interplay between physics and geometry interested Misha for a long
time.\cite{Mar2,Mar3} We planned further development of ideas about
connections between geometry of homogeneous spaces and the Yang-Baxter
equation. Unfortunately Misha's unexpected death stopped our plans.


\section{Introduction}

The notion of a Poisson-Lie group was introduced almost 20 years
ago by Drinfeld.\cite{Drinfeld PL}  Its infinitesimal counterpart,
Lie bialgebras, were introduced in the same paper and later it was
explained that these objects are in fact quasiclassical limits of
quantum groups.\cite{Drinfeld QG} Lie bialgebra structures on a
Lie algebra $\g$ are in a natural 1--1 correspondence with Lie
algebra structures on the vector space $\Dg = \g \oplus \g^*$ with
some compatibility conditions. $\Dg$ with this Lie algebra
structure is called the double of the Lie bialgebra $\g$.

The most popular and important class of Lie bialgebras is the
class of quasitriangular Lie bialgebras.\cite{Drinfeld QG} They
can be defined by an element $r \in \g \otimes \g$ (called the
classical r-matrix) such that
%
$$\Omega := r + r^{21}$$
%
is $\g$-invariant, and the classical Yang-Baxter equation (CYBE)
%
$$[r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}] = 0$$
%
is satisfied. If $r$ is skew-symmetric, then one says that the
corresponding Lie bialgebra is triangular. In general, $\Lambda :=
r - \halfomega$ (i.e., the skew-symmetric part of $r$) satisfies
the modified CYBE
%
$$[\Lambda^{12}, \Lambda^{13}] + [\Lambda^{12}, \Lambda^{23}] +
[\Lambda^{13}, \Lambda^{23}] = \modomega.$$
%

It is well known (and can be easily shown) that if $\g$ is a
complex simple finite-dimensional Lie algebra, then any Lie
bialgebra structure on $\g$ is quasitriangular. For the case
$\Omega \neq 0$ (``quasitriangular case in the strict sense") they
were classified by Belavin and Drinfeld.\cite{Bel-Dr FA,Bel-Dr}
The triangular case was studied by
Stolin.\cite{Stolin0,Stolin2,Stolin3}

It was shown\,\cite{Stolin1} that for such $\g$ there are only two
possible structures of the $\Dg$. In the triangular case $\Dg =
\geps = \g \oplus \g\varepsilon$, where $\varepsilon^2 = 0$ and
otherwise, $\Dg = \gxg$ (and $\g$ is embedded diagonally into
$\gxg$). Then it is clear that solutions of the CYBE (resp.\ the
modified CYBE with $\Omega \neq 0$) are in a 1--1 correspondence
with Lagrangian subalgebras $\l$ in $\geps$ (resp.\ in $\gxg$)
such that ${\l \cap \g} = 0$.

Along with the Poisson-Lie groups it is natural to study their
Poisson actions, and in particular their Poisson homogeneous
spaces. Drinfeld\,\cite{Drinfeld Lagr} gave a general approach to
the classification of Poisson homogeneous spaces. Namely, he
showed that if $G$ is a Poisson-Lie group, $\g$ is the
corresponding Lie bialgebra, then Poisson homogeneous $G$-spaces
are essentially in a 1--1 correspondence with $G$-orbits on the
set of all Lagrangian subalgebras in $\Dg$. A classification of
Lagrangian subalgebras in some important cases (including the case
$\g$ is complex simple, $\Dg = \gxg$) was obtained by
Karolinsky.\cite{Kar2,Kar3,Kar1}

At the same time an important generalization of the CYBE, the dynamical
classical Yang-Baxter equation, was introduced in physics and mathematics.
Notice that this equation is defined for a pair $(\g,\u)$, where $\u$ is a
subalgebra of $\g$. From the mathematical point of view it was presented by
Felder.\cite{Felder,Felder1} This equation and its quantum analogue were
studied in many papers.\cite{E-V lectures,E-V,Sch,Xu} First classification
results for the solutions of the classical dynamical Yang-Baxter equation
(dynamical r-matrices) were obtained by Etingof and Varchenko,\cite{E-V} and
Schiffmann.\cite{Sch}

Later Lu\,\cite{Lu CDYBE} found a connection (which is essentially a 1--1
correspondence) between dynamical r-matrices for the pair $(\g, \u)$ (where
$\u $ is a Cartan subalgebra of the complex simple finite-dimensional
algebra $\g$), and Poisson homogeneous $G$-structures on $G/U$. Here $U
\subset G$ are connected Lie groups corresponding to $\u \subset \g$, and
$G$ is equipped with the standard quasitriangular (with $\Omega \neq 0$)
Poisson-Lie structure. Lu also noticed that this connection can be
generalized to the case $\u$ is a subspace in a Cartan subalgebra (with some
``regularity'' condition). The dynamical r-matrices for the latter case were
classified by Schiffmann.\cite{Sch}

Now let $G$ be a complex connected semisimple Lie group, and let $U$ be its
connected subgroup. Suppose $\u \subset \g$ are the corresponding Lie
algebras. In the present paper we consider connections between Poisson
homogeneous structures on $G/U$ related to the triangular Poisson-Lie
structures on $G$ (i.e., with $\Omega=0$), where $U$ is a reductive subgroup
containing a Cartan subgroup of $G$, and triangular dynamical r-matrices for
the pair $(\g,\u)$.

In fact, our results are based on a general result on relations between
dynamical classical r-matrices and Poisson homogeneous structures (see
Theorem \ref{twistdyn1}), which is valid also in the quasitriangular case.
Notice that the results of Sections \ref{sectdyn} and \ref{secttwist} can be
used to describe a 1--1 correspondence between dynamical r-matrices for the
pair $(\g, \u)$, where $\u \subset \g$ is a Cartan subalgebra, and Poisson
homogeneous $G$-structures on $G/U$, where $G$ is equipped with any
quasitriangular (with $\Omega \neq 0$) Poisson-Lie structure (of course the
latter result is due to Lu). Our approach is based on some strong
classification results for dynamical r-matrices given recently by Etingof
and Schiffmann.\cite{E-S}

The paper is organized as follows. In Section \ref{sectdyn} we describe a
correspondence between the (moduli space of) dynamical r-matrices for a pair
$(\g, \u)$ and Poisson homogeneous $G$-structures on $G/U$ proving that
under certain assumptions it is a bijection. In Section \ref{secttwist} we
consider a procedure of twisting for Lie bialgebras and examine its impact
on the double $D(\g)$ and Poisson homogeneous spaces for corresponding
Poisson-Lie groups. Then we use the twisting to weaken some restrictions
needed in Section \ref{sectdyn}. Finally, in Section \ref{sectex} we
consider the basic example of our paper: $\g$ is semisimple, $\u \subset \g$
is a reductive Lie subalgebra that contains some Cartan subalgebra of $\g$,
and the Lie bialgebra structure on $\g$ is triangular (i.e., $\Dg = \geps$).


\section{Classical dynamical r-matrices and Poisson homogeneous spaces}
\label{sectdyn}

In this section we assume $\g$ to be any finite-dimensional Lie algebra over
$\mathbb C$. Let $G$ be a connected Lie group such that $\Lie G = \g$. Let
$\u \subset \g$ be a Lie subalgebra (not necessary abelian). By $U$ denote
the connected subgroup in $G$ such that $\Lie U = \u$. We propose (under
certain conditions) a connection between dynamical r-matrices for the pair
$(\g, \u)$ and Poisson structures on $G/U$ that make $G/U$ a {\it Poisson}
homogeneous $G$-space (for certain Poisson-Lie structures on $G$). Note that
this connection was first introduced by Jiang-Hua Lu\,\cite{Lu CDYBE} for
the case $\g$ is simple, $\u$ is a Cartan subalgebra, and the dynamical
r-matrix has non-zero coupling constant.

In order to recall the definition of the classical dynamical
r-matrix we need some notation. Let $x_1,...,x_r$ be a basis of
$\u$. By $D$ denote the formal neighborhood of $0$ in $\u^*$. By
functions from $D$ to a vector space $V$ we mean the elements of
the space $V[[x_1,...,x_r]]$, where $x_i$ are regarded as
coordinates on $D$. If $\omega \in \Omega^k(D,V)$ is a $k$-form
with values in a vector space $V$, we denote by
$\overline{\omega}: D \to \wedge^k {\u} \otimes V$ the
corresponding function. Finally, for an element $r \in \gotg$ we
define the classical Yang-Baxter operator $$\CYB(r)=
[r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}].$$

Recall that a {\it classical dynamical r-matrix} for the pair
$(\g, \u)$ is an $\u$-equivariant function $r: D \to \gotg$ that
satisfies the classical dynamical Yang-Baxter equation (CDYBE):
\begin{equation}
\Alt(\overline{dr}) + \CYB(r) = 0,
\end{equation}
where for $x \in {\g}^{\otimes 3}$, we let $\Alt(x) = x^{123} + x^{231} +
x^{312}$ (see Etingof and Schiffmann,\cite{E-V lectures,E-S} Etingof and
Varchenko\,\cite{E-V}). Usually one requires also an additional {\it
quasi-unitarity condition}: $$r + r^{21} = \Omega \in (S^2 \g)^{\g}.$$ Note
that if $r$ satisfies the CDYBE and the quasi-unitarity condition then
$\Omega$ is a constant function.

Suppose $\Omega \in (S^2 \g)^{\g}$. Let us denote by $\Dynr$ the
set of all classical dynamical r-matrices $r$ for the pair $(\g,
\u)$ such that $r + r^{21} = \Omega$.

Denote by $\Maps$ the set of all regular $\u$-equivariant maps from $D$ to
$G$. Suppose $g \in \Maps$. For any $\mathfrak u$-equivariant function $r: D
\to \gotg$ set $$r^g = (\Ad_g \otimes \Ad_g)(r - \overline{\eta_g} +
\overline{\eta_g}^{21} + \tau_g),$$ where $\eta_g=g^{-1}dg$, and
$\tau_g(\lambda) = (\lambda \otimes 1 \otimes 1)([\overline{\eta_g}^{12},
\overline{\eta_g}^{13}](\lambda))$. Then $r$ is a classical dynamical
r-matrix iff $r^g$ is (see Etingof and Schiffmann\,\cite{E-S}). The
transformation $r \mapsto r^g$ is called a gauge transformation. It is
indeed an action of the group $\Maps$ on $\Dynr$ (i.e., $(r^{g_1})^{g_2} =
r^{g_2g_1}$). Following Etingof and Schiffmann\,\cite{E-S} denote by $\Modr$
the moduli space $\Mapszero\backslash\Dynr$, where $\Mapszero$ is the
subgroup in $\Maps$ consisting of maps $g$ satisfying $g(0) = e$.

In what follows we need some notation. Suppose $a \in
{\g}^{\otimes k}$. By $\overrightarrow{a}$ (resp.
$\overleftarrow{a}$) denote the left (resp. right) invariant
tensor field on $G$ corresponding to $a$.

Suppose $\rho \in \gotg$ satisfies the classical Yang-Baxter
equation (CYBE), i.e., $\CYB(\rho) = 0$. Assume also that $\rho +
\rho^{21} = \Omega$ (i.e., $\rho = \halfomega + \Lambda$, where
$\Lambda \in \gwg$). Introduce a bivector field $\pi_\rho =
\overrightarrow{\rho} - \overleftarrow{\rho} =
\overrightarrow{\Lambda} - \overleftarrow{\Lambda}$ on $G$. It is
well known that $(G, \pi_\rho)$ is a Poisson-Lie group.

Now let $r \in \Dynr$. We have $r = \halfomega + A$, where $A \in
\gwg$. Set $\tilde\pi_r := \overrightarrow{r(0)} -
\overleftarrow{\rho} = \overrightarrow{A(0)} -
\overleftarrow{\Lambda}$. Consider a bivector field $\pi_r$ on
$G/U$ defined by $\pi_r\left(\underline{g}\right) =
p_*\tilde\pi_r(g)$, where $p: G \to G/U$ is the natural
projection, and $\underline{g} = p(g)$. Note that $\pi_r$ is well
defined since $r(0) \in \gotgu$.

The following proposition belongs to Jiang-Hua Lu\,\cite{Lu CDYBE}
(note that it is stated there for the case $\mathfrak g$ is
simple, $\u$ is a Cartan subalgebra, but the proof fits the
general case).

\medskip

\begin{proposition} \label{Lu}
The bivector field $\pi_r$ is Poisson, and $(G/U, \pi_r)$ is a
Poisson homogeneous $(G, \pi_\rho)$-space. \hfill $\square$
\end{proposition}

\medskip

\begin{proposition}
Suppose $g \in \Mapszero$. Then $\pi_r = \pi_{r^g}$.
\end{proposition}

\proof Since $(G/U, \pi_r)$ is a Poisson homogeneous $(G,
\pi_\rho)$-space, we see that $\pi_r$ depends only on $\pi_r
(\underline{e}) =$ the image of $r(0) - \rho$ in $\wedge^2 (\g /
\u)$. Thus it is enough to note that $r^g(0) - r(0) \in \u \otimes
\g + \g \otimes \u$. \hfill $\square$

\begin{corollary}
The correspondence $r \mapsto \pi_r$ defines a map from $\Modr$ to
the set of all Poisson $(G, \pi_\rho)$-homogeneous structures on
$G/U$. \hfill $\square$
\end{corollary}

Suppose now that the following conditions hold:

\begin{itemize}
\item[(a)] $\u$ has an $\u$-invariant complement
$\m$ in $\g$ (we fix one).
\item[(b)] $\Omega \in (\u \otimes \u) \oplus (\m
\otimes \m)$.
\item[(c)] $\rho \in \halfomega + \mwmu$.
\end{itemize}

\begin{theorem} \label{dyn}
Under the assumptions above the correspondence $r \mapsto \pi_r$
is a bijection between $\Modr$ and the set of all Poisson $(G,
\pi_\rho)$-homogeneous structures on $G/U$.
\end{theorem}

The rest of this section is devoted to the proof of Theorem
\ref{dyn}. First we recall some results belonging to Etingof and
Schiffmann.\cite{E-S} Assume that (a) holds. Set $$\Momega =
\left\{ x \in \halfomega + {\mwmu} \, \Bigm| \, \CYB(x) = 0 \
\mbox{in}\, \wedge^3(\g / \u) \right\}.$$

\begin{theorem}[Etingof, Schiffmann\,\cite{E-S}] \label{ESmain}

1. Any class ${\cal C} \in \Modr$ has a representative $r \in \cal
C$ such that $r(0) \in \Momega$. Moreover, this defines an
embedding $\Modr \to \Momega$.

2. Assume that (b) holds. Then the map $\Modr \to \Momega$ defined
above is a bijection. \hfill $\square$
\end{theorem}

Now suppose $b \in (\wedge^2 ({\g / \u}))^{\u} = (\wedge^2
\m)^{\u}$. Set $\pi\left(\underline{g}\right) = (L_g)_*b +
p_*\pi_\rho(g)$. Since $\rho$ is $\u$-invariant, we see that
$\pi_\rho(g) = 0$ for $g \in U$; therefore $\pi$ is a well-defined
bivector field on $G/U$.

\begin{proposition} \label{poiss}
The bivector field $\pi$ is Poisson iff $\CYB(\rho + b) = 0$ in
$\wedge^3 (\g / \u)$.
\end{proposition}

\proof Set $a = \Lambda + b$. Define a bivector field
$\tilde{\pi}$ on $G$ by the formula $\tilde\pi =
\overrightarrow{a} - \overleftarrow{\Lambda}$. Note that
$\tilde\pi = \overrightarrow{b} + \pi_\rho$, therefore $\pi =
p_*\tilde\pi$. Let us normalize the Schouten bracket of the
bivector fields on $G$ in a way that $[\overrightarrow{x},
\overrightarrow{x}] = \overrightarrow{\CYB(x)}$ for all $x \in
\wedge^2 \g$. Then we have $$[\tilde\pi, \tilde\pi] =
[\overrightarrow{a}, \overrightarrow{a}] - 2[\overrightarrow{a},
\overleftarrow{\Lambda}] + [\overleftarrow{\Lambda},
\overleftarrow{\Lambda}] = \overrightarrow{\CYB(a)} -
\overleftarrow{\CYB(\Lambda)}.$$ Since $\rho = \halfomega +
\Lambda$ satisfies the CYBE, we see that $\CYB(\Lambda) =
\modomega \in (\wedge^3 \g)^{\g}$. Thus $$[\tilde\pi, \tilde\pi] =
\overrightarrow{\CYB(a) - \modomega} =
\overrightarrow{\CYB\left(\halfomega + a\right)} =
\overrightarrow{\CYB(\rho + b)}.$$ To finish the proof it is
enough to note that $[\pi, \pi] = p_*[\tilde\pi, \tilde\pi]$.
\hfill $\square$

\bigskip

\noindent {\it Proof of Theorem \ref{dyn}.} \ Let us construct the
inverse map. Suppose $(G/U, \pi)$ is a Poisson homogeneous $(G,
\pi_\rho)$-space. Set $b = \pi(\underline{e}) \in \wedge^2({\g /
\u}) = \wedge^2\m$. The condition (c) implies that in fact $b \in
(\wedge^2({\g / \u}))^{\u} = \mwmu$. Furthermore, (c) yields that
$\rho + b \in \halfomega + \mwmu$. By Proposition \ref{poiss}, we
have $\CYB(\rho + b) = 0$ in $\wedge^3(\g / \u)$, i.e., $\rho + b
\in \Momega$. Then, by Theorem \ref{ESmain}, there exists $r \in
\Dynr$ such that $r(0) = \rho + b$, and the image of $r$ in
$\Modr$ is uniquely determined. It is now easy to verify that $\pi
= \pi_r$. \hfill $\square$


\section{Twisting of Poisson homogeneous structures}\label{secttwist}

Assume again that $\g$ is an arbitrary finite-dimensional Lie
algebra over $\mathbb C$. Recall that a Lie bialgebra structure on
$\g$ is a 1-cocycle $\delta: {\g} \to \gwg$ which satisfies the
co-Jacobi identity. Denote by ${\mathcal D}(\g, \delta)$ the
classical double of $(\g, \delta)$.

We say that two Lie bialgebra structures $\delta_1$, $\delta_2$ on $\g$ are
in the same class if there exists a Lie algebra isomorphism $f: {\mathcal
D}({\g}, \delta_1)\rightarrow {\mathcal D}({\g}, \delta_2)$ which
intertwines the canonical forms $Q_i$ on ${\mathcal D}({\g}, \delta_i)$, and
such that the following diagram is commutative:
%
$$ \diagram {\g} \rto \drto & {\mathcal D}({\g},\delta_1) \dto^f\\
           & {\mathcal D}({\g}, \delta_2).
\enddiagram
$$
%

\begin{theorem} \label{twistth}
Two Lie bialgebra structures $\delta$, $\delta'$ on $\g$ are in
the same class if and only if $\delta' = \delta + ds$, where $s
\in \gwg$ and
%
\begin{equation} \label{twisteq}
\CYB(s) = \Alt(\delta \otimes \id)(s).
\end{equation}
%
\end{theorem}

\proof Let us consider ${\mathcal D}(\g, \delta) = \g \oplus \g^*$. Notice
that Lie bialgebra structures on $\g$ that are in the same class with
$\delta$ are in a 1--1 correspondence with Lagrangian subalgebras $\l
\subset {\mathcal D}(\g, \delta)$ such that ${\l} \cap {\g} = 0$.

Any $s = \sum_i s_i' \otimes s_i'' \in \gwg$ defines a linear map $S:
\g^\ast \rightarrow \g$,\
%
$S(l)=\sum_i \langle l, s_i' \rangle s_i''$.
%
Let $\l \subset {\mathcal D}(\g, \delta)$ be the graph of $S$. Clearly,
${\l} \cap {\g} = 0$, and $\l$ is a Lagrangian subspace because $s$ is
skew-symmetric.

It is easy to verify that for any $l_1, l_2, l_3 \in \g^\ast$,
%
$$\langle l_1 \otimes l_2 \otimes l_3, \CYB(s) - \Alt(\delta \otimes \id)(s)
\rangle = Q([l_1+S(l_1), l_2+S(l_2)], l_3+S(l_3)),$$
%
where $\scpr$ is the canonical pairing, and $Q$ is the canonical bilinear
form on ${\mathcal D}(\g, \delta)$. Therefore $\l$ is a subalgebra if and
only if (\ref{twisteq}) holds.

Suppose $s \in \gwg$ satisfies (\ref{twisteq}). Let $\delta'$ be the Lie
bialgebra structure on $\g$ defined by $s$. It is straightforward to verify
that $\delta' = \delta + ds$. This completes the proof of the theorem.
\hfill $\square$

\begin{remark}
{\rm If we consider our Lie bialgebra $(\g, \delta)$ as a Lie
quasibialgebra, then $({\g}, \delta + ds)$ is called ``twisting
via $s$''. The notions of Lie quasibialgebra and twisting via $s$
was introduced by Drinfeld.\cite{Drinfeld QH} The theorem above
can be also deduced from the general results on Lie
quasibialgebras (see Drinfeld\,\cite{Drinfeld QH}).}
\end{remark}

Further, we are going to examine the effect of the twisting on
Poisson homogeneous spaces. First we recall some definitions and
rather well-known results.

Let $G$ be a connected complex Poisson-Lie group, $(\g,\delta)$
its Lie bialgebra, and ${\mathcal D}(\g) = {\mathcal
D}(\g,\delta)$ the corresponding classical double of $\g$ with the
canonical invariant form $Q$.

Recall that an action of $G$ on a Poisson manifold $M$ is called
Poisson if the defining map $G\times M\rightarrow M$ is a Poisson
map, where $G\times M$ is equipped with the product Poisson
structure. If the action is transitive, we say that $M$ is a {\it
Poisson homogeneous $G$-space}.

Let $M$ be a homogeneous $G$-space, and let $\pi$ be any bivector
field on $M$. For any $x \in M$ let us consider the map
%
$$\pi_x: T^\ast_xM\rightarrow T_xM,\ \pi_x(l)=(l \otimes
\id)(\pi(x)).$$
%
On the other hand, $M \cong G/H_x$ and $T_xM = {\g/\h}_x$, $T^\ast_xM =
({\g/\h}_x)^\ast = {\h}^\perp_x \subset \g^\ast$, where $H_x$ is the
stabilizer of $x$, and ${\h}_x = \Lie H_x$. Therefore we can consider
$\pi_x$ as a map $\pi_x: {\h}^\perp_x \rightarrow {\g/\h}_x$.

Now let us consider the following set of subspaces in ${\mathcal
D}(\g) = \g \oplus \g^\ast$:
%
\begin{equation} \label{Lagrdef}
{\l}_x = \{ a + l \, | \, a \in {\g}, \, l \in {\h}^\perp_x,\,
\pi_x(l) = \overline{a} \},
\end{equation}
%
where $\overline{a}$ is the image of $a$ in ${\g/\h}_x$. Observe that
${\l}_x$ are Lagrangian subspaces, and ${\l}_x \cap {\g} = {\h}_x$. The
following result was obtained by Drinfeld.\cite{Drinfeld Lagr}

\begin{theorem}[Drinfeld\,\cite{Drinfeld Lagr}] \label{Drinfeld}
$(M, \pi)$ is a Poisson homogeneous $G$-space if and only if for any $x \in
M$ the subspace ${\l}_x$ is a subalgebra of ${\mathcal D}(\g)$, and
${\l}_{gx} = \Ad_g{\l}_x$ for all $g \in G$. \hfill $\square$
\end{theorem}

Now set $\delta' = \delta + ds$, where $s \in \gwg$ satisfies
(\ref{twisteq}). Then we have two Poisson-Lie groups, $(G,
\pi_\delta)$ and $(G, \pi_{\delta'})$, whose Lie bialgebras are
$(\g, \delta)$ and $(\g, \delta')$ respectively. Let $(M, \pi)$ be
a Poisson homogeneous $(G, \pi_\delta)$-space. Consider a bivector
field $\xi$ on $M$ defined by the formula $\xi(x) =$ the image of
$s$ in $\wedge^2 ({\g / \h}_x) = \wedge^2 T_x M$. Set $\pi' = \pi
- \xi$.

\begin{proposition} \label{homtwisted}
$(M, \pi')$ is a Poisson homogeneous $(G, \pi_{\delta'})$-space,
and thus one obtains a bijection between the sets of all Poisson
$(G, \pi_\delta)$- and $(G, \pi_{\delta'})$-homogeneous structures
on $M$.
\end{proposition}

\proof Theorem \ref{twistth} allows one to identify ${\mathcal
D}(\g, \delta)$ and ${\mathcal D}(\g, \delta')$. It is easy to
verify that under this identification the sets of Lagrangian
subspaces that correspond to $(M, \pi)$ and $(M, \pi')$ are the
same. This completes the proof, according to
Theorem~\ref{Drinfeld}. \hfill $\square$

\medskip

Finally, we are going to generalize the main result of the
previous section to the twisted case. Assume that $(\g, \delta)$
is a quasitriangular Lie bialgebra, i.e., $\delta = d\rho$, where
$\rho \in \g \otimes \g$ and $\CYB(\rho) = 0$. It is easy to
verify that the condition (\ref{twisteq}) for an element $s \in
\gwg$ is equivalent to
\begin{equation} \label{twisteqtriang}
\CYB(s) + [\![\rho, s]\!] + [\![s, \rho]\!] = 0,
\end{equation}
where for $a, b \in {\g}^{\otimes 2}$ we set $[\![a, b]\!] = [a^{12},
b^{13}] + [a^{12}, b^{23}] + [a^{13}, b^{23}] \in {\g}^{\otimes 3}$ (i.e.,
$\CYB(a) = [\![a, a]\!]$).

Fix $\Omega \in (S^2\g)^{\g}$ and assume that $\rho \in \halfomega
+ \gwg$. As before, consider the Poisson-Lie group $(G,
\pi_\delta)$, where $\pi_\delta = \pi_\rho = \overrightarrow{\rho}
- \overleftarrow{\rho}$. Suppose $s \in \gwg$ satisfies
(\ref{twisteqtriang}). Set $\delta' = \delta + ds = d(\rho + s)$;
then $\pi_{\delta'} = \pi_{\rho, s} := \overrightarrow{\rho + s} -
\overleftarrow{\rho + s}$, and $(G, \pi_{\rho, s})$ is a
Poisson-Lie group.

Let $U$ be a connected Lie subgroup in $G$, and ${\u} = \Lie U$.
Consider $r \in \Dynr$. As usually, set $\tilde{\pi}_r =
\overrightarrow{r(0)} - \overleftarrow{\rho}$ and denote by
$\pi_r$ the natural projection of $\tilde{\pi}_r$ on $G/U$. By
Proposition \ref{Lu}, $(G/U, \pi_r)$ is a Poisson homogeneous $(G,
\pi_\rho)$-space. Set also $\tilde{\pi}_{r, s} = \tilde{\pi}_r -
\overleftarrow{s} = \overrightarrow{r(0)} - \overleftarrow{\rho +
s}$ and denote by $\pi_{r, s}$ its projection on $G/U$. According
to Proposition \ref{homtwisted}, $(G/U, \pi_{r, s})$ is a Poisson
homogeneous $(G, \pi_{\rho, s})$-space.

Moreover, if we combine Theorem \ref{dyn} and Proposition
\ref{homtwisted}, we get the following

\begin{theorem} \label{twistdyn}
Assume that $\u$, $\Omega$, and $\rho$ satisfy the conditions (a),
(b), and (c) from the previous section. Then the correspondence $r
\mapsto \pi_{r, s}$ is a bijection between $\Modr$ and the set of
all Poisson $(G, \pi_{\rho, s})$-homogeneous structures on $G/U$.
\hfill $\square$
\end{theorem}

Clearly, this can be reformulated as follows:

\begin{theorem} \label{twistdyn1}
Assume that $\u$ and $\Omega$ satisfy the conditions (a) and (b)
from the previous section. Suppose also that there exists $s \in
\gwg$ such that (\ref{twisteqtriang}) holds, and $\rho + s \in
\halfomega + \mwmu$. Then the correspondence $r \mapsto \pi_r$ is
a bijection between $\Modr$ and the set of all Poisson $(G,
\pi_\rho)$-homogeneous structures on $G/U$. ~ \hfill $\square$
\end{theorem}

Let us apply our previous results to the triangular case.

\begin{corollary} \label{twistdyntriang}
Assume that $\u$ satisfies the condition (a) from the previous
section. Set $\Omega = 0$. Consider any $\rho \in \gwg$ that
satisfies the CYBE. Then the correspondence $r \mapsto \pi_r$ is a
bijection between $\Modr$ and the set of all Poisson $(G,
\pi_\rho)$-homogeneous structures on $G/U$.
\end{corollary}

\proof Set $s = -\rho$ and apply Theorem \ref{twistdyn1}. \hfill
$\square$


\section{Poisson homogeneous structures in the triangular case}\label{sectex}

Now assume that $\g$ is semisimple. Fix a Cartan subalgebra $\h \subset \g$
and denote by $\RRR$ the corresponding root system. In this section we apply
the results of the previous sections to the case $\u$ is a reductive Lie
subalgebra in $\g$ containing $\h$, $\Omega = 0$, and $\rho \in \gwg$ such
that $\CYB(\rho) = 0$.

To be more precise, consider $\UUU \subset \RRR$, and suppose $\u
= \h \oplus (\bigoplus_{\alpha \in \UUU} \g_\alpha)$ is a
reductive Lie subalgebra in $\g$. If this is the case, then we say
that a subset $\UUU \subset \RRR$ is {\it reductive} (i.e., $(\UUU
+ \UUU) \cap \RRR \subset \UUU$ and $-\UUU = \UUU$; see
Gorbatsevich, Onishchik, and Vinberg\,\cite{Lie 3}). Condition (a)
is satisfied since $\m = \bigoplus_{\alpha \in \RRR \setminus
\UUU} \g_\alpha$ is an $\u$-invariant complement to $\u$ in $\g$.

Applying Corollary \ref{twistdyntriang} (and results of Etingof
and Schiffmann cited in Section \ref{sectdyn}), we get:

\begin{enumerate}
\item Any structure of a Poisson homogeneous $(G, \pi_\rho)$-space
on $G/U$ is of the form $p_*(\overrightarrow{x} -
\overleftarrow{\rho})$, where $x \in \Momega$.
\item If $x \in \Momega$, then there exists (a unique up to the
$\Mapszero$-action) $r\in \Dynr$ such that $r(0) = x$.
\end{enumerate}

Let us now describe $\Momega$ and thus get an explicit description
of all Poisson $(G, \pi_\rho)$-homogeneous structures on $G/U$.
Recall that in our case by definition $$\Momega =  \left\{ x \in
{\mwmu} \, \Bigm| \, \CYB(x) = 0 \ \mbox{in}\, \wedge^3(\g / \u)
\right\}.$$

We need to fix some notation. Fix a nondegenerate invariant
bilinear form ({\it invariant scalar product}) $\scpr$ on $\g$.
For any $\alpha \in \RRR$ choose $E_\alpha \in \g_\alpha$ such
that $\langle E_\alpha, E_{-\alpha} \rangle = 1$. Further, suppose
$\NNN$ is a reductive subset which contains $\UUU$. We say that $h
\in \h$ is {\it $(\NNN, \UUU)$-regular} if $\alpha (h) = 0$ for
all $\alpha \in \UUU$, and $\alpha (h) \neq 0$ for all $\alpha \in
\NNN \setminus \UUU$.

\begin{proposition} \label{Momega}
$x \in \Momega$ iff
\begin{equation} \label{xNh}
x = x_{\nh} = \sum_{\alpha \in \NNN \setminus \UUU}
\frac{1}{\alpha(h)}E_\alpha \otimes E_{-\alpha},
\end{equation}
where $\NNN$ is a reductive subset in $\RRR$ containing $\UUU$,
and $h \in \h$ is $(\NNN, \UUU)$-regular.
\end{proposition}

\proof One can easily prove the following lemmas:

\begin{lemma} \label{l1}
$x \in \mwmu$ iff $$x = \sum_{\alpha \in \RRR \setminus \UUU} x_\alpha \cdot
E_\alpha \otimes E_{-\alpha},$$ where $x_{-\alpha} = -x_\alpha$, and for all
$\alpha, \beta \in \RRR \setminus \UUU$ such that $\alpha + \beta \in \UUU$,
we have $x_\alpha + x_\beta = 0$. \hfill $\square$
\end{lemma}

\begin{lemma} \label{l2}
Suppose $$x = \sum_{\alpha \in \RRR \setminus \UUU} x_\alpha \cdot E_\alpha
\otimes E_{-\alpha} \in \mwmu.$$ Then $x \in \Momega$ iff the following
condition holds: for all $\alpha, \beta, \gamma \in \RRR \setminus \UUU$,
$\alpha + \beta + \gamma = 0$, we have $x_\alpha x_\beta + x_\beta x_\gamma
+ x_\gamma x_\alpha = 0$. \hfill $\square$
\end{lemma}

\bigskip

Now consider the following properties of the function $\RRR
\setminus \UUU \to \mathbb C$, $\alpha \mapsto x_\alpha$:

\begin{itemize}
\item[(d)] $x_{-\alpha} = -x_\alpha$ for all $\alpha \in \RRR \setminus
\UUU$.
%
\item[(e)] If $\alpha, \beta \in \RRR \setminus \UUU$, $\alpha +
\beta \in \UUU$, then $x_\alpha + x_\beta = 0$.
%
\item[(f)] If $\alpha, \beta, \gamma \in \RRR \setminus \UUU$, $\alpha +
\beta + \gamma = 0$, then $x_\alpha x_\beta + x_\beta x_\gamma + x_\gamma
x_\alpha = 0$.
\end{itemize}

It is also straightforward to prove the following:

\begin{lemma} \label{l3}
$x_\alpha$ satisfies (d)--(f) iff
%
$$x_\alpha = \left\{
\begin{array}{cl}
1/\alpha(h), & \mbox{\rm if } \alpha \in \NNN \setminus \UUU
\\ 0, & \mbox{\rm if } \alpha \in \RRR \setminus \NNN ,
\end{array} \right.$$
%
for a certain reductive subset $\NNN \subset \RRR$ such that $\NNN \supset
\UUU$, and $(\NNN, \UUU)$-regular element $h \in \h$. \hfill $\square$
\end{lemma}

The last lemma proves the proposition. \hfill $\square$

\begin{remark}
{\rm We note that Lemmas \ref{l1}, \ref{l2}, and \ref{l3} are
essentially contained in the paper by Donin, Gurevich, and
Shnider.\cite{DGS}

In that paper, among other results, the {\it symplectic}
$G$-invariant structures on $G/U$ are classified if $U$ is a Levi
subgroup of $G$. Actually, in this case there exists a
$G$-equivariant symplectomorphism from $G/U$ to a semisimple
coadjoint $G$-orbit equipped with the Kirillov-Kostant-Souriau
bracket.

Moreover, it is easy to show that if $G/U$ has a $G$-invariant
symplectic structure, then $U$ is a Levi subgroup. Indeed, let
$p_*\overrightarrow{x_{\nh}}$ (where $x_{\nh}$ is defined by
(\ref{xNh})) be a $G$-invariant Poisson structure on $G/U$.
Obviously, it is symplectic iff $\NNN = \RRR$. Since $h$ is
$(\RRR, \UUU)$-regular, i.e., $\alpha(h) = 0$ for all $\alpha \in
\Span \UUU$ and $\alpha(h) \neq 0$ for all $\alpha \in \RRR
\setminus \UUU$, we see that $(\Span \UUU) \cap \RRR = \UUU$. It
is well known that the latter condition is equivalent to the fact
that $U$ is a Levi subgroup.

Let us also remark that the existence of reductive non-Levi
subgroups is the main difference between the triangular and the
strictly quasitriangular cases. Indeed, suppose $U$ is a Cartan
subgroup. Then in the triangular case the Poisson homogeneous
structures on $G/U$ relate to {\it all}\/ reductive subgroups of
$G$, while in the strictly quasitriangular case they relate to the
Levi subgroups only (see Karolinsky\,\cite{Kar1} and Lu\,\cite{Lu
CDYBE}).}
\end{remark}

Finally we are going to describe the Lagrangian subalgebras corresponding to the
Poisson $(G, \pi_\rho)$-homogeneous structures on $G/U$. Since the Lie
bialgebras corresponding to $(G, \pi_\rho)$ are all in the same class, we
may assume without loss of generality that $\rho = 0$. It is clear that the
corresponding Manin triple is $(\geps, \g, \g\varepsilon)$, where $\geps =
\g \oplus \mathfrak g\varepsilon$, $\varepsilon^2 = 0$, and the canonical
form is given by the formula
%
$$Q(a + b\varepsilon, c + d\varepsilon) = \langle a, d \rangle
+ \langle b, c \rangle.$$
%

Suppose $\rho = 0$. Assume that $\NNN$ and $h$ are as in
Proposition \ref{Momega}. Set $\pi_{\nh} =
p_*\overrightarrow{x_{\nh}}$, where $x_{\nh}$ is defined by
(\ref{xNh}). By ${\l}_{\nh}$ denote the Lagrangian subalgebra
corresponding to $(G/U, \pi_{\nh})$ at the base point
$\underline{e}$.
%Let us calculate $\l_{\nh}$.

\begin{proposition}
${\l}_{\nh} = {\u} \oplus \left( \bigoplus_{\alpha \in \RRR \setminus \NNN}
\varepsilon{\g}_\alpha \right) \oplus \left( \bigoplus_{\alpha \in \NNN
\setminus \UUU} (1 - \alpha (h) \varepsilon){\g}_\alpha \right)$
\end{proposition}

\proof By definition (see (\ref{Lagrdef})), $${\l}_{\nh} = \{ a +
b\varepsilon \, | \, a \in {\g}, b \in {\u}^\perp = {\m}, (b
\otimes 1)(x_{\nh}) = \overline{a} \}, $$ where $\overline{a}$ is
the image of $a$ in $\g/\u = \m$. Suppose $b = E_\alpha$, where
$\alpha \in \RRR \setminus \UUU$. Then $$(b \otimes 1)(x_{\nh}) =
\left\{
\begin{array}{cl}
-\frac{1}{\alpha(h)}E_\alpha, & \mbox{\rm if } \alpha \in \NNN
\setminus \UUU \\ 0, & \mbox{\rm if } \alpha \in \RRR \setminus
\NNN.
\end{array} \right. $$
This completes the proof. \hfill $\square$


%\appendix


\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

The authors are grateful to Pavel Etingof and Olivier Schiffmann
for valuable discussions. Research of E.K. was supported in part
by CRDF grant UM1--2091.


\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

\begin{thebibliography}{99}

  \bibitem{Mar2} D.~Bar-Moshe and M.~Marinov,
                 ``Quantization on homogeneous compact
                 K\"ahler manifolds,'' in {\em Generalized
                 symmetries in physics (Clausthal, 1993)},
                 (World Sci.\ Publishing,
                 River Edge, NJ, 1994) pp.\ 411--424.
  \bibitem{Bel-Dr FA} A.~A.~Belavin and  V.~G.~Drinfeld,
                   ``On classical Yang-Baxter equation for simple Lie
                   algebras,'' {\em Funct.~An.~Appl.} \textbf{16}
                   1-29 (1982).
  \bibitem{Bel-Dr} A.~A.~Belavin, V.~G.~Drinfeld,
                   ``Triangle equations and
                   simple Lie algebras,'' in {\em Soviet Scientific 
Reviews} Section C 4, pp.\ 93--165,
                   2nd edition: {\em Classic Reviews in Mathematics and
                   Mathematical Physics}, Vol.~1 (Harwood, Amsterdam,
                   1998).
  \bibitem{DGS} J.~Donin, D.~Gurevich and S.~Shnider,
                ``Double quantization on some orbits in the coadjoint
                representations of simple Lie groups,'' {\em
                Commun.~Math.~Phys.} \textbf{204}, 39-60 (1999)
                [e-print math.QA/9807159].
  \bibitem{Drinfeld PL} V.~G.~Drinfeld,
                        ``Hamiltonian structures on Lie groups, Lie
                        bialgebras, and the geometric meaning of the
                        classical Yang-Baxter equations,'' {\em
                        Soviet Math.~Dokl.} \textbf{27}, 68-71 (1983).
  \bibitem{Drinfeld QG} V.~G.~Drinfeld, ``Quantum Groups,'' in {\em
                        Proceedings of the International Con\-gress of
                        Mathematicians, Berkeley 1986}, pp. 798--820.
  \bibitem{Drinfeld QH} V.~G.~Drinfeld, ``Quasi-Hopf algebras,''
                        {\em Leningrad Math.~J.} \textbf{1}, 1419-1457 (1990).
  \bibitem{Drinfeld Lagr} V.~G.~Drinfeld,
                          ``On Poisson homogeneous spaces
                          of Poisson-Lie groups,'' {\em Theor.\
                          Math.~Phys.} \textbf{95}, 226-227 (1993).
  \bibitem{E-V lectures} P.~Etingof and O.~Schiffmann, {\it Lectures on the
                         dynamical Yang-Baxter equations}, e-print
                         math.QA/9908064.
  \bibitem{E-S} P.~Etingof and O.~Schiffmann, ``On the moduli space of
                classical dynamical r-matrices,'' {\em  Math.~Res.~Letters},
                \textbf{8}, 157-170 (2001) [e-print math.QA/0005282].
  \bibitem{E-V} P.~Etingof and  A.~Varchenko, ``Geometry and classification
                of solutions of the classical dynamical Yang-Baxter
                equation,'' {\em Commun.~Math.~Phys.} \textbf{196}, 591-640 
(1998) [e-print q-alg/9703040].
  \bibitem{Felder} G.~Felder, ``Conformal field theory and integrable
                   systems associated to elliptic curves,'' in 
{\em Proceedings
                   of the International Congress of Mathematicians,
                   Z\"urich 1994}, pp.\ 1247--1255.
  \bibitem{Felder1} G.~Felder, `` Elliptic quantum groups,'' in {\em
                    Proceedings of the ICMP, Paris 1994}
                    [hep-th/9412207].
  \bibitem{Lie 3} V.~V.~Gorbatsevich, A.~L.~Onishchik and
                  E.~B.~Vinberg,
                  ``Structure of Lie groups and Lie algebras,'' in {\em
                  Encyclopaedia of Math.~Sci.} Vol.~41, (Springer-Verlag,
                  Berlin, 1994).
  \bibitem{Kar2} E.~Karolinsky,
                 ``A classification of Poisson homogeneous
                 spaces of a compact Poisson-Lie group,'' {\em
                 Mathematical Physics, Analysis, and Geometry}, \textbf{3}, 
               274-289 (1996) (in Russian).
  \bibitem{Kar3} E.~Karolinsky,
                 ``A classification of Poisson homogeneous
                 spaces of compact Poisson-Lie groups,'' {\em  Doklady
                 Math.} \textbf{57}, 179-181 (1998).
  \bibitem{Kar1} E.~Karolinsky, ``A classification of Poisson homogeneous
                 spaces of complex reductive Poisson-Lie groups,''
            {\em  Banach Center Publications}, \textbf{51}, 103-108 (2000)
                 [e-print math.QA/9901073].
  \bibitem{Lu CDYBE} J.-H.~Lu,
                     ``Classical dynamical r-matrices and homogeneous
                     Poisson structures on $G/H$ and $K/T$,'' {\em
                     Commun.~Math.~Phys.} \textbf{212}, 337-370 (2000)
                     [e-print math.SG/9909004].
  \bibitem{Mar1} M.~Marinov,
                 ``Homogeneous phase-space manifolds:
                 classical and quantum theory,''
                 in {\em  New symmetries in the theories
                 of fundamental interactions (Max Born
                 Symposium, Karpacz, 1996)}, pp.\ 313--323,
                 (PWN, Warsaw, 1997).
  \bibitem{Mar3} M.~Marinov and E.~Strahov,
                 ``A geometrical approach to non-adiabatic
                 transitions in quantum theory:
                 applications to NMR, over barrier reflection
                 and parametric excitation of
                 quantum oscillator,'' {\em J.~Phys.~A},
                 \textbf{34}, 1741-1752 (2001).
  \bibitem{Sch} O.~Schiffmann,~``On classification of dynamical
                r-matrices,'' {\em  Math.\ Res.\ Letters}, 
              \textbf{5}, 13-30 (1998) [e-print q-alg/9706017].
  \bibitem{Stolin0} A.~Stolin,
                    ``On rational solutions of Yang-Baxter equation for
                    $sl(n)$,'' {\em Math.\ Scand.} \textbf{69}, 57-80 (1991).
  \bibitem{Stolin2} A.~Stolin,
                    ``Constant solutions of Yang-Baxter equation for
                    $sl(2)$, $sl(3)$,'' {\em Math.\ Scand.} \textbf{69}, 
                    81-88 (1991).
  \bibitem{Stolin3} A.~Stolin,
                    ``On rational solutions of Yang-Baxter equation.
                    Maximal orders in loop algebra,'' 
                    {\em  Commun.~Math.~Phys.}
                    \textbf{141}, 533-548 (1991).
  \bibitem{Stolin1} A.~Stolin,
                    ``Some remarks on Lie bialgebra structures on
                    simple complex Lie algebras,'' {\em
                    Comm.~Algebra}, \textbf{27}, 4289-4302 (1999).
  \bibitem{Xu} P.~Xu, {\it Triangular dynamical r-matrices and
               quantization}\/, e-print math.QA/0005006.

\end{thebibliography}

\end{document}
