# Classical dynamical $r$-matrices for the Chern-Simons formulation of
generalised 3d gravity

Juan Carlos Morales Parra$\dagger$ Bernd J Schroers$\ddagger$
$\dagger$ Maxwell Institute of Mathematical Sciences and Department of
Mathematics,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
<EMAIL_ADDRESS>
$\ddagger$ Maxwell Institute of Mathematical Sciences and School of
Mathematics,
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
<EMAIL_ADDRESS>

Abstract

Classical dynamical $r$-matrices arise naturally in the combinatorial
description of the phase space of Chern-Simons theories, either through the
inclusion of dynamical sources or through a gauge-fixing procedure involving
two punctures. Here we consider classical dynamical $r$-matrices for the
family of Lie algebras which arise in the Chern-Simons formulation of 3d
gravity, for any value of the cosmological constant. We derive differential
equations for classical dynamical $r$-matrices in this case, and show that
they can be viewed as generalised complexifications, in a sense which we
define, of the equations governing dynamical $r$-matrices for
$\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. We obtain explicit
families of solutions and relate them, via Weierstrass factorisation, to
solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of
chiral WZWN models.

##  1 Introduction

###  1.1 Motivation

One important reason for studying classical dynamical $r$-matrices, and the
one which motivates this paper, is their role in the gauge-fixed Poisson
structures of character varieties over punctured Riemann surfaces. Such
Poisson spaces appear in particular as phase-spaces of lower dimensional gauge
theories, like the Chern-Simons formulation of gravity in three dimensions
(see e.g. [1], [2], [3], [4], [5]). As we shall explain, the gauge fixing is
interesting and natural classically. In addition, it offers a route to
quantisation of constraint systems which avoids some of the technical
challenges of imposing constraints after quantisation. This is of particular
interest in the Chern-Simons formulation of 3d gravity where the quantised
theory is an interesting toy model for quantum gravity.

The goal of this paper is to study the space of classical dynamical
$r$-matrices up to a naturally defined notion of gauge equivalence for some
particular Lie algebras $\mathfrak{g}_{\lambda}$ that appear in the setting of
3d gravity, namely the Lie algebras of the symmetry groups of maximally
symmetric three dimensional Riemannian and pseudo-Riemannian manifolds. In
this extended introduction we define classical dynamical $r$-matrices,
illustrate their appearance in gauge-fixed character varieties in a simple
example and then prepare the ground for a full classification of the classical
dynamical $r$-matrices which are relevant for 3d gravity in the remainder of
this paper.

###  1.2 Definitions

A classical dynamical $r$-matrix is an equivariant solution of the Classical
Dynamical Yang-Baxter Equation. More precisely, given a finite dimensional Lie
algebra $\mathfrak{g}$, a Lie subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$
and an element $K\in(S^{2}\mathfrak{g})^{\mathfrak{g}}$, a classical dynamical
$r$-matrix associated to the triple $(\mathfrak{g},\mathfrak{h},K)$ is a
$\mathfrak{h}$-equivariant function $K+r\in
C^{1}(\mathfrak{h}^{*})\otimes(\mathfrak{g}\otimes\mathfrak{g})$ that solves
the Classical Dynamical Yang-Baxter Equation (CDYBE)

$\text{CDYB}(r)\equiv[[r,r]]+\text{Alt}(dr)=-[[K,K]].$ (1)

Explicitly,

$[[r,r]]=\text{CYB}(r)\equiv[r_{12},r_{23}]+[r_{13},r_{23}]+[r_{12},r_{13}]$

is the standard Schouten bracket and

$\text{Alt}(dr)\equiv h_{i}^{(1)}\frac{\partial r_{23}}{\partial
h^{i}}-h_{i}^{(2)}\frac{\partial r_{13}}{\partial
h^{i}}+h_{i}^{(3)}\frac{\partial r_{12}}{\partial h^{i}},$

in terms of a basis $\\{h_{i}\\}_{i=1,\cdots,\dim\mathfrak{h}}$ of
$\mathfrak{h}$ and its dual $\\{h^{i}\\}_{i=1,\cdots,\dim\mathfrak{h}}$ for
$\mathfrak{h}^{*}$, where

$h_{i}^{(1)}\frac{\partial r_{23}}{\partial h^{i}}=\frac{\partial
r^{ab}}{\partial h^{i}}h_{i}\otimes T_{a}\otimes T_{b},\quad
h_{i}^{(2)}\frac{\partial r_{13}}{\partial h^{i}}=\frac{\partial
r^{ab}}{\partial h^{i}}T_{a}\otimes h_{i}\otimes T_{b},\quad
h_{i}^{(3)}\frac{\partial r_{12}}{\partial h^{i}}=\frac{\partial
r^{ab}}{\partial h^{i}}T_{a}\otimes T_{b}\otimes h_{i},$

with $r=\sum_{ab}r^{ab}T_{a}\otimes T_{b}$ and
$\\{T_{a}\\}_{a=1,\cdots,\dim\mathfrak{g}}$ a basis of $\mathfrak{g}$. The
Einstein summation convention will be used throughout the paper.

The $\mathfrak{h}$-equivariance condition means

$\frac{d}{ds}\bigg{|}_{s=0}r(\text{Ad}^{*}(e^{sh})x)+[r(x),h\otimes 1+1\otimes
h]=0$ (2)

holds for all $x\in\mathfrak{h}^{*}$ and $h\in\mathfrak{h}$, i.e. the
coadjoint action of $\mathfrak{h}$ on the argument of $r$ equals to the
adjoint action of $\mathfrak{h}$ on the Lie algebra part of $r$.

The CDYBE originally appeared in research related to integrability in
conformal field theories (see e.g. [6], [7]). Since then, this equation, its
solutions and its quantum counterpart have been widely studied, especially its
applications to the theory of integrable systems, (Quasi-) Poisson geometry
and special functions (see e.g. [8], [9], [10]).

The definition of a classical dynamical $r$-matrix varies according to the
reference under consideration, and may include or omit the equivariance
condition. For example, in literature on the origin and application of
solutions of the CDYBE to (Quasi-)Poisson structures, equivariance is part of
the definition (see e.g. [11]), while in literature concerning their
application to quantum integrable systems the equivariance condition is
usually not considered (see e.g. [12]). Here we make the distinction between
solutions to the CDYBE and classical dynamical $r$-matrices, where the later
are obtained from the former after imposing the equivariance condition (2).

Let $G$ be a Lie group and denote by $\mathfrak{g}$ its Lie algebra. Given a
Lie subalgebra $\mathfrak{h}$ and an element
$K\in(S^{2}\mathfrak{g})^{\mathfrak{g}}$, the set of classical dynamical
($\mathfrak{g},\mathfrak{h},K$) $r$-matrices is denoted by
$\text{Dyn}(\mathfrak{g},\mathfrak{h},K)$. By taking the union of these spaces
for all Lie subalgebras $\mathfrak{h}$ of $\mathfrak{g}$, we get the set of
dynamical $r$-matrices associated to the pair $(\mathfrak{g},K)$, denoted by

$\text{Dyn}(\mathfrak{g},K)=\bigcup_{\mathfrak{h}\leq\mathfrak{g}}\text{Dyn}(\mathfrak{g},\mathfrak{h},K),$
(3)

whose geometry and structure are studied e.g. in [13] and [14].

If $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are conjugate-equivalent Lie
subalgebras of $\mathfrak{g}$, say by $g\in G$, then the map

$\begin{split}\text{Dyn}(\mathfrak{g},\mathfrak{h},K)&\to\text{Dyn}(\mathfrak{g},\mathfrak{h}^{\prime},K)\\\
r(x)&\mapsto\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x)\end{split}$
(4)

is well-defined and bijective (see Lemma A in Appendix A), allowing us to
define an action of the Lie group $G$ over the space (3).

In this paper, as is usual in the literature regarding the CDYBE, we focus in
the case where the Lie subalgebras considered are Cartan subalgebras of
$\mathfrak{g}$. Along the rest of the paper $\mathfrak{h}$ will be mainly used
to name a Cartan subalgebra of a Lie algebra.

Let $G$ be a Lie group and denote by $\mathfrak{C}_{\mathfrak{g}}$ the set of
Cartan subalgebras of the Lie algebra $\mathfrak{g}$ and by
$\mathfrak{C}^{\text{Ad}}_{\mathfrak{g}}$ the set of conjugacy classes of
Cartan subalgebras of $\mathfrak{g}$. Analogous to the construction of the set
of classical dynamical $r$-matrices $\text{Dyn}(\mathfrak{g},K)$, we restrict
to Cartan subalgebras and define the set of Cartan classical dynamical
$r$-matrices associated to $(\mathfrak{g},K)$ by

$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g},K)=\bigcup_{\mathfrak{h}\in\mathfrak{C}_{\mathfrak{g}}}\text{Dyn}(\mathfrak{g},\mathfrak{h},K).$
(5)

By picking one representative in each conjugacy class of
$\mathfrak{C}_{\mathfrak{g}}^{\text{Ad}}$ and collecting them in a set denoted
by $[\mathfrak{C}^{\text{Ad}}_{\mathfrak{g}}]$, the set (5) cab be decomposed
as

$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g},K)=\bigcup_{g\in
G}\bigsqcup_{\mathfrak{h}\in[\mathfrak{C}_{\mathfrak{g}}^{\text{Ad}}]}\text{Dyn}^{g}(\mathfrak{g},\mathfrak{h},K),$

where $\text{Dyn}^{g}(\mathfrak{g},\mathfrak{h},K)$ is a short notation for
the image under the map (4) for $g\in G$.

Over the set of classical dynamical $r$-matrices
$\text{Dyn}(\mathfrak{g},\mathfrak{h},K)$, for each
$\mathfrak{h}\in[\mathfrak{C}_{\mathfrak{g}}^{\text{Ad}}]$, the group of
smooth functions from $\mathfrak{h}^{*}$ to the stabilizer of $H$

$\mathcal{G}(\mathfrak{g},\mathfrak{h})\equiv\\{p\in\text{Fun}(\mathfrak{h}^{*},G^{H})\
|\ p\text{ is smooth}\\},$ (6)

the so-called group of dynamical $(\mathfrak{g},\mathfrak{h})$ gauge
transformations, where $H\leq G$ is the Lie subgroup of $G$ such that
$\text{Lie}(H)=\mathfrak{h}$, defines an action given by

$p\rhd
r=\text{Ad}(p)\otimes\text{Ad}(p)\left(r+\overline{\eta}^{p}-\overline{\eta}^{p}_{21}\right)\qquad\text{
for }r\in\text{Dyn}(\mathfrak{g},\mathfrak{h},K),$ (7)

where
$\overline{\eta}^{p}:\mathfrak{h}^{*}\to\mathfrak{h}\otimes\mathfrak{g}^{\mathfrak{h}}$
is the dual of the $\mathfrak{g}^{\mathfrak{h}}$-valued 1-form
$\eta^{p}=p^{-1}dp$, i.e. explicitly

$\overline{\eta}^{p}=\sum_{i=1}^{\dim\mathfrak{h}}h_{i}\otimes
p^{-1}\frac{\partial p}{\partial h^{i}}$

in terms of the dual bases $\\{h_{i}\\}_{i=1,\cdots,\dim\mathfrak{h}}$ and
$\\{h^{i}\\}_{i=1,\cdots,\dim\mathfrak{h}}$ for $\mathfrak{h}$ and
$\mathfrak{h}^{*}$, respectively.

The moduli space of Cartan classical dynamical $r$-matrices associated to
$(\mathfrak{g},K)$ is defined by

$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g},K)=\bigsqcup_{\mathfrak{h}\in[\mathfrak{C}_{\mathfrak{g}}^{\text{Ad}}]}\mathcal{M}(\mathfrak{g},\mathfrak{h},K)$
(8)

where

$\mathcal{M}(\mathfrak{g},\mathfrak{h},K)\equiv\text{Dyn}(\mathfrak{g},\mathfrak{h},K)/\mathcal{G}(\mathfrak{g},\mathfrak{h}),$

such that the full set of Cartan classical dynamical $r$-matrices (5) can be
generated from it via (i) dynamical $(\mathfrak{g},\mathfrak{h})$ gauge
transformations (7) over $\mathcal{M}(\mathfrak{g},\mathfrak{h},K)$ for each
$\mathfrak{h}\in[\mathfrak{C}_{g}^{\text{Ad}}]$ and (ii) the $G$-action (4)
over $\text{Dyn}(\mathfrak{g},\mathfrak{h},K)$ for each $g\in G$ and
$\mathfrak{h}\in[\mathfrak{C}_{g}^{\text{Ad}}]$.

As will be explained below, the Poisson structures of gauge-fixed moduli
spaces of $G$-flat connections (character varieties) over Riemann surfaces are
in bijection with the moduli space of Cartan dynamical $r$-matrices
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g},K)$. The decomposition in (8) implies
that in order to have a full-description of these Poisson structures, it is
sufficient to determine classical dynamical $r$-matrices associated to
$(\mathfrak{g},\mathfrak{h},K)$ for representatives $\mathfrak{h}$ of each
conjugacy class in $\mathfrak{C}^{\text{Ad}}_{\mathfrak{g}}$, up to dynamical
$(\mathfrak{g},\mathfrak{h})$ gauge transformations.

###  1.3 Example

To illustrate the appearance of dynamical $r$-matrices in a simple setting,
consider the character variety

$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}\equiv\\{A\in\text{Hom}(\pi_{1}(\Sigma_{0,4}),\text{SL}(2,\mathbb{R}))|A(\ell_{i})\in\mathcal{C}_{i}\text{
for }i=1,2,3,4\\}/\text{SL}(2,\mathbb{R})$

where $\Sigma_{0,4}\equiv\mathbb{S}^{2}-\\{4\text{pts}\\}$ is the four
punctured 2-sphere, $\\{\ell_{i}\\}_{i=1,\cdots,4}$ is the (homotopy type) set
of generators of its fundamental group consisting of four loops (each going
around one of the punctures once) and the $\mathcal{C}_{i}$’s are fixed
conjugacy classes of $\text{SL}(2,\mathbb{R})$. For simplicity we assume the
conjugacy classes are two dimensional (over $\mathbb{R}$) generated by
elements of the form $e^{s_{i}J_{0}}$ for $i=1,2,3,4$, where
$\\{J_{a}\\}_{a=0,1,2}$ is the standard basis of the real Lie algebra
$\mathfrak{sl}(2,\mathbb{R})$ satisfying

$[J_{a},J_{b}]=\epsilon_{abc}J^{c},$

and the coefficients are raised or lowered using the Minkowskian metric
$\text{diag}(1,-1,-1)$.

The canonical Poisson structure over
$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}$ can be
seen as a reduction of the Poisson structure over the extended space

$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\text{ext}}\equiv\underbrace{\text{SL}(2,\mathbb{R})_{1}\times\cdots\times\text{SL}(2,\mathbb{R})_{4}}_{\text{4
times}},$

with Poisson bivector (see [15]) given by

$\Pi^{0,4}_{\text{ext}}(r)\equiv r^{ab}\sum_{1\leq i\leq j\leq
4}(R_{a}^{i}+L_{a}^{i})\wedge(R_{b}^{j}+L_{b}^{j}),$ (9)

where $r=r^{ab}J_{a}\otimes
J_{b}\in\mathfrak{sl}(2,\mathbb{R})\otimes\mathfrak{sl}(2,\mathbb{R})$ is a
classical $r$-matrix, i.e. a solution of the Classical Yang-Baxter Equation
(CYBE)

$[[r,r]]=0,$ (10)

such that its symmetric part $K\equiv r+r^{21}$ is
$\mathfrak{sl}(2,\mathbb{R})$-invariant.

In the expression above we have adopted the notation of $R_{a}^{i}$ and
$L_{a}^{i}$ to indicate the right and left fundamental vector fields generated
by the Lie algebra element $J_{a}$ and associated to the $i$-th copy of
$\text{SL}(2,\mathbb{R})$, respectively.

The Poisson space
$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}$ is
obtained from
$(\mathcal{P}_{\text{SL}(2,\mathbb{R}),\text{ext}}^{0,4},\Pi_{\text{ext}}^{0,4})$
by ($i$) restricting each copy of $\text{SL}(2,\mathbb{R})$ to the
corresponding conjugacy class, ($ii$) imposing the topological condition that
the product of the elements in each $4$-tuple must be the identity element of
$\text{SL}(2,\mathbb{R})$, and ($iii$) identifying conjugate-equivalent
$4$-tuples. This approach, developed originally by Fock and Rosly (see e.g.
[16], [15], [17]), exhibits explicitly how the character variety
$\mathcal{P}^{0,4}$ can be realized as a constrained system.

The dimension of $\mathcal{P}_{\text{SL}(2,\mathbb{R}),\text{ext}}^{0,4}$ is
$3\times 4=12$, while the dimension of
$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}$ is

$2=12-10=12-\underbrace{4\times(1)}_{(i)}-\underbrace{3}_{(ii)}-\underbrace{3}_{(iii)}$

where the underbrace symbols indicate which of the three types of constraints
indicated above is responsible of the corresponding dimensional reduction.
Since the constraint functions over $\mathcal{P}^{0,4}_{\text{ext}}$
associated to ($i$) are Poisson functions with respect to
$\Pi_{\text{ext}}^{0,4}$ (see [18]), as an intermediate step we have the
partially-constrained space given by

$\mathcal{P}_{\text{SL}(2,\mathbb{R}),\text{ext(cc)}}^{0,4}\equiv\mathcal{C}_{1}\times\mathcal{C}_{2}\times\mathcal{C}_{3}\times\mathcal{C}_{4}$

of dimension $8=4\times 2$, with Poisson structure given still by the bivector
$\Pi_{\text{ext}}^{0,4}$, in such a way that the fully reduced space is
obtained by imposing the remaining $6=3+3$ constraints (topological and
conjugation equivalence constraints). The label $(\text{cc})$ is used to
indicate each copy of $\text{SL}(2,\mathbb{R})$ has been restricted to the
conjugacy class of the corresponding puncture.

The quantization of character varieties (see e.g. [19]) has been an active
topic of research in the recent years, both in mathematics and physics, since
it provides (e.g.) quantum group representations of the mapping class group of
Riemann surfaces and a quantization scheme for Chern-Simons theory. The fact
that character varieties can be realized as constrained Poisson spaces implies
one needs to deal with the constraints at some point along the way to
quantization. Even thought the standard approach is to impose the constraints
at the quantum level (see e.g. [20], [21]), there are technical advantages to
incorporating the constraints at the classical level and then quantizing the
reduced theory111The two approaches are widely assumed to be equivalent, but
there is no proof of this assumption..

At the classical level the (first class) constraints of any constrained
Poisson space can be gauge fixed, which amounts to reducing the space to a
sector of the original one with the help of auxiliary (second class)
constraints and defining over it a new Poisson structure (the so-called Dirac
brackets, see [22] and [23]), in such a way that the constraints become
Poisson functions (for details see [24], [25], [26]). In the particular case
of $\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}$, the
three topological constraints ($ii$) can be gauge fixed á la Dirac via three
auxiliary constraint functions defined just on the part
$\mathcal{C}_{1}\times\mathcal{C}_{2}$ of the space (see [5] for details),
getting a $5=1+2+2$ dimensional intermediate space

$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\},GF(1,2)}=H_{0}\times\mathcal{C}_{3}\times\mathcal{C}_{4}$

where $H_{0}$ is the Cartan group generated by the Lie algebra generator
$J_{0}$, with Poisson structure given explicitly for $F,\tilde{F}\in
C^{\infty}(\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R}))$ by

$\begin{split}\\{F,\varphi\\}&=\sum_{i=3,4}(R_{0}^{i}+L_{0}^{i})F,\\\
\\{F,\tilde{F}\\}(\varphi)&=\Pi_{\text{ext}}^{0,4}(r(\varphi))(dF,d\tilde{F})=r^{ab}(\varphi)\sum_{3\leq
i\leq j\leq
4}(R^{i}_{a}+L_{a}^{i})\wedge(R_{b}^{j}+L_{b}^{j})(dF,d\tilde{F}),\end{split}$
(11)

where $\varphi$ is a variable parametrizing the dual of the Lie subalgebra
$\mathfrak{h}_{0}=\text{Lie}(H_{0})$ and

$r:\mathfrak{h}_{0}^{*}\to\mathfrak{sl}(2,\mathbb{R})\otimes\mathfrak{sl}(2,\mathbb{R})$

is now an $\mathfrak{h}_{0}$-equivariant solution of the CDYBE (1) such that
$\text{Sym}(r(\varphi))=\text{Sym}(r)\equiv K$, i.e. a classical dynamical
$r$-matrix for the triple $(\mathfrak{sl}(2,\mathbb{R}),\mathfrak{h}_{0},K)$.
In fact, the Jacobi identity for the brackets of the form
$\\{F,\\{\tilde{F},\varphi\\}\\}$ and $\\{F,\\{\tilde{F},\overline{F}\\}\\}$
(for $F,\tilde{F},\overline{F}\in
C^{\infty}(\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R}))$) implies the
$\mathfrak{h}_{0}$-equivariance of $r$ and $\text{CDYB}(r)=0$, respectively.

Hence after performing the gauge fixing, an explicit realization of the
Poisson space
$\mathcal{P}^{0,4}_{\text{SL}(2,\mathbb{R}),\\{\mathcal{C}_{i}\\}}$ is
obtained by imposing the now Poisson constraint

$g_{4}^{-1}g_{3}^{-1}=e^{\varphi J_{0}},$

obtaining in this way the two dimensional Atiyah-Bott phase space of
$\text{SL}(2,\mathbb{R})$-flat connections over the four-punctured 2-sphere
(see e.g. [27], [28]). Analogously, if instead conjugacy classes generated by
elements of the form $e^{s_{i}J_{1}}$ are considered, the Poisson structure of
the gauge-fixed phase space will be defined in terms of a classical dynamical
$r$-matrix for the triple $(\mathfrak{sl}(2,\mathbb{R}),\mathfrak{h}_{1},K)$,
where $\mathfrak{h}_{1}$ is the Lie subalgebra generated by $J_{1}$.

This example helps to understand how the Poisson structures of the moduli
space $\mathcal{P}^{0,4}$ are determined by the moduli space of Cartan
classical dynamical $r$-matrices
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{sl}(2,\mathbb{R}),K)$. Indeed, due to
(8) and the fact any Cartan subalgebra of $\mathfrak{sl}(2,\mathbb{R})$ is
either conjugate to $\mathfrak{h}_{0}$ or to $\mathfrak{h}_{1}$, the problem
of describing the Poisson structures of $\mathcal{P}^{0,4}$ reduces to finding
two classical dynamical $(\mathfrak{sl}(2,\mathbb{R}),\mathfrak{h},K)$
$r$-matrices, one for $\mathfrak{h}=\mathfrak{h}_{0}$ and for
$\mathfrak{h}=\mathfrak{h}_{1}$.

###  1.4 Chern-Simons formulation of 3d gravity

Character varieties of the type

$\mathcal{P}^{g,n}_{G,\\{C_{i}\\}}\equiv\\{A\in\text{Hom}(\pi_{1}(\Sigma_{g,n}),G)|A(\ell_{i})\in\mathcal{C}_{i}\text{
for }i=1,\cdots,n\\}/G$

appear in the Chern-Simons formulation of 3d gravity for 3-manifolds of the
form $\mathcal{M}^{3}\cong\mathbb{R}\times\Sigma_{g,n}$ (stationary
spacetimes) and five possible Lie groups $G$. Depending on the signature
(Euclidean or Lorentzian) and the sign of the cosmological constant
$\Lambda_{C}$, the possible five Lie groups $G$ (local isometry groups of the
possible spacetime models of General Relativity in three dimensions) are

$\Lambda_{C}$ | Euclidean | Lorentzian
---|---|---
0 | $\text{SU}(2)\mathrel{\triangleright}\joinrel<\mathbb{R}^{3}$ | $\text{SL}(2,\mathbb{R})\mathrel{\triangleright}\joinrel<\mathbb{R}^{3}$
$>0$ | $\text{SU}(2)\times\text{SU}(2)$ | $\text{SL}(2,\mathbb{C})_{\mathbb{R}}$
$<0$ | $\text{SL}(2,\mathbb{C})_{\mathbb{R}}$ | $\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$
Table 1:  Local isometry groups of pure 3d gravity.

and so the possible associated Lie algebras $\mathfrak{g}$ are

$\Lambda_{C}$ | Euclidean | Lorentzian
---|---|---
0 | $\mathfrak{iso}(3)$ | $\mathfrak{iso}(2,1)$
$>0$ | $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ | $\mathfrak{so}(3,1)$
$<0$ | $\mathfrak{so}(3,1)$ | $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$
Table 2:  Lie algebras of the previous local isometry groups.

These five Lie algebras can be described in a unified way as follows. They are
six dimensional real Lie algebras generated by
$\\{J_{0},J_{1},J_{2},P_{0},P_{1},P_{2}\\}$ with commutation relations given
by

$[J_{a},J_{b}]=\epsilon_{abc}J^{c},\qquad[J_{a},P_{b}]=\epsilon_{abc}P^{c},\qquad[P_{a},P_{c}]=\lambda\epsilon_{abc}J^{c}$
(12)

where $\lambda=-c^{2}\Lambda_{C}$ and the indices are raised or lowered using
the metric

$\eta_{ab}:=\text{diag}\left(1,-\frac{|c|^{2}}{c^{2}},-\frac{|c|^{2}}{c^{2}}\right)$

with $c\in i\mathbb{R}$ and $c\in\mathbb{R}$ for the Euclidean and Lorentzian
cases, respectively, where for the latter it is interpreted as the speed of
light. These Lie algebras are denoted by $\mathfrak{g}_{\lambda}$, since by
picking the right metric, i.e. $\text{diag}(1,1,1)$ for the Euclidean or
$\text{diag}(1,-1,-1)$ for the Lorentzian, the commutation relations depend
only on the parameter $\lambda$. Similarly, we denote the corresponding Lie
groups in Table 1 by $G_{\lambda}$.

For any Lie group $G$, the Poisson structure of the moduli space
$\mathcal{P}_{G,\\{C_{i}\\}}^{g,n}$ depends on the choice of a non-degenerate
symmetric Ad-invariant bilinear form over the Lie algebra
$\mathfrak{g}=\textit{Lie}(G)$. In the case $G={SL}(2,\mathbb{R})$ considered
above, this dependence translates into the (equivalent) choice of the element
$K$ in $(S^{2}\mathfrak{sl}(2,\mathbb{R}))^{\mathfrak{sl}(2,\mathbb{R})}$. For
the Lie groups we are interested in this paper $G_{\lambda}$, the space of
non-degenerate symmetric Ad-invariant bilinear forms over the Lie algebra
$\mathfrak{g}_{\lambda}$ has two generators [29]

$t(J_{a},J_{b})=0,\qquad t(J_{a},P_{a})=c^{2}\eta_{ab},\qquad
t(P_{a},P_{b})=0,$ (13)

and

$s(J_{a},J_{b})=\eta_{ab},\qquad s(J_{a},P_{a})=0,\qquad
s(P_{a},P_{b})=\lambda\eta_{ab}.$ (14)

The bilinear form $t$ is usually called the standard (gravity) pairing, since
when it is used in the Chern-Simons formulation of three dimensional gravity
the action reduces to the Einstein-Hilbert action; meanwhile $s$ is commonly
referred as the exotic pairing. Nevertheless, both actions provide the same
phase space, i.e. the same equations of motions, but equipped with different
symplectic/Poisson structures.

Following the same reductions presented above for $G=\text{SL}(2,\mathbb{R})$
and the four punctured 2-sphere, the canonical Poisson structure over the

$4n+12g-12=n\times(6)+2g\times(6)-\underbrace{n\times(2)}_{(i)}-\underbrace{6}_{(ii)}-\underbrace{6}_{(iii)}$

dimensional space $\mathcal{P}^{g,n}_{G_{\lambda},\\{\mathcal{C}_{i}\\}}$, for
any of the Lie groups $G_{\lambda}$ presented above, can be recovered starting
from the $6n+12g$ dimensional extended space

$\mathcal{P}_{G_{\lambda},\text{ext}}^{g,n}\equiv\underbrace{(G_{\lambda})_{1}\times\cdots\times(G_{\lambda})_{n}}_{\text{n-times}}\times\underbrace{(G_{\lambda})_{A1}\times(G_{\lambda})_{B1}\times\cdots\times(G_{\lambda})_{Ag}\times(G_{\lambda})_{Bg}}_{\text{2g-times}}$

equipped with the Fock-Rosly Poisson structure [15]

$\begin{split}\Pi^{g,n}_{\text{ext}}(r)=&\sum_{i=1}^{n}r^{ab}\left(\frac{1}{2}R_{a}^{i}\wedge
R_{b}^{i}+\frac{1}{2}L_{a}^{i}\wedge L_{b}^{i}+R^{i}_{a}\wedge
L_{b}^{i}\right)+\sum_{1\leq i<j\leq
n}r^{ab}(R_{a}^{i}+L_{a}^{i})\wedge(R_{b}^{j}+L_{b}^{j})\\\ &+\sum_{1\leq
i<j\leq
g}r^{ab}\left(R_{a}^{A_{i}}+L_{a}^{A_{i}}+R_{a}^{B_{i}}+L_{a}^{B_{i}}\right)\wedge\left(R_{b}^{A_{i}}+L_{b}^{A_{i}}+R_{b}^{B_{i}}+L_{b}^{B_{i}}\right)\\\
&+\sum_{i=1}^{n}\sum_{j=1}^{g}r^{ab}\left(R_{a}^{M_{i}}+L_{a}^{M_{i}}\right)\wedge\left(R_{b}^{A_{j}}+L_{b}^{A_{j}}+R_{b}^{B_{j}}+L_{b}^{B_{j}}\right)\\\
&+\sum_{j=1}^{g}r^{ab}\bigg{[}\frac{1}{2}(R_{a}^{A_{j}}\wedge
R_{b}^{A_{i}}+L_{a}^{A_{i}}\wedge L_{b}^{A_{i}}+R_{a}^{B_{i}}\wedge
R_{b}^{B_{i}}+L_{a}^{B_{i}}\wedge L_{b}^{B_{i}})\\\
&+R_{a}^{A_{i}}\wedge(R_{b}^{B_{i}}+L_{b}^{A_{i}}+L_{b}^{B_{i}})+R_{a}^{B_{i}}\wedge(L_{b}^{A_{i}}+L_{b}^{B_{i}})+L_{a}^{A_{i}}\wedge
L_{b}^{B_{i}}\bigg{]},\end{split}$ (15)

where $r\in\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda}$ is a classical
$r$-matrix such that $\text{Sym}(r)=K_{\alpha\beta}$, where
$K_{\alpha\beta}\in(S^{2}\mathfrak{g}_{\lambda})^{\mathfrak{g}_{\lambda}}$ is
the symmetric element

$K_{\alpha\beta}\equiv\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}(J_{a}\otimes
P^{a}+P_{a}\otimes J^{a})-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}(\lambda
J_{a}\otimes J^{a}+P_{a}\otimes P^{a})$ (16)

associated to the $\mathfrak{g}_{\lambda}$ Ad-invariant symmetric bilinear
form

$(\cdot,\cdot)_{\alpha,\beta}=\alpha t(\cdot,\cdot)+\beta s(\cdot,\cdot)$ (17)

with $t$ and $s$ given by (13) and (14), respectively, and
$\alpha,\beta\in\mathbb{R}$. The non-degeneracy of
$(\cdot,\cdot)_{\alpha\beta}$ is equivalent to the condition
$\alpha^{2}-\lambda\beta^{2}\neq 0$ (see e.g [30]).

Exactly as before, given that the constraints reducing the first $n$-copies of
$G_{\lambda}$ to conjugacy classes $\mathcal{C}_{\lambda}$222For simplicity we
consider just conjugacy classes generated by elements of the form $e^{\mu
J_{0}+sP_{0}}$. are Poisson functions with respect to $\Pi_{\text{ext}}^{g,n}$
(see [18] again for details), we have the partially-constrained space given by

$\mathcal{P}_{G_{\lambda},\text{ext(cc)}}^{g,n}\equiv\underbrace{(\mathcal{C}_{\lambda})_{1}\times\cdots\times(\mathcal{C}_{\lambda})_{n}}_{\text{n-times}}\times\underbrace{(G_{\lambda})_{A1}\times(G_{\lambda})_{B1}\times\cdots\times(G_{\lambda})_{Ag}\times(G_{\lambda})_{Bg}}_{\text{2g-times}}.$

In this case the gauge fixing procedure, via four auxiliary constraint
functions defined again over the product of the first two conjugacy classes
$(\mathcal{C}_{\lambda})_{1}\times(\mathcal{C}_{\lambda})_{2}$, will provide
the intermediate space

$\mathcal{P}_{G_{\lambda},\\{\mathcal{C}_{i}\\},GF(1,2)}^{g,n}=H_{\lambda,0}\times\underbrace{(\mathcal{C}_{\lambda})_{3}\times\cdots\times(\mathcal{C}_{\lambda})_{n}}_{(n-2)\text{-times}}\times\underbrace{(G_{\lambda})_{A1}\times(G_{\lambda})_{B1}\times\cdots\times(G_{\lambda})_{Ag}\times(G_{\lambda})_{Bg}}_{2g\text{-times}}$

of dimension $4n+12g-6$, where $H_{\lambda,0}$ is the Cartan subgroup with Lie
subalgebra $\mathfrak{h}_{\lambda,0}$ generated by $J_{0}$ and $P_{0}$ (see
[5]). Similarly to the $\text{SL}(2,\mathbb{R})$ case, the reduced Poisson
structure (i.e the Dirac brackets) will be given for $F,\tilde{F}\in
C^{\infty}(G_{\lambda}^{n-2+2g})$ by

$\begin{split}\\{\gamma,\psi\\}&=0,\\\
\\{F,\gamma\\}&=(R_{J_{0}}+L_{J_{0}})F,\\\
\\{F,\psi\\}&=(R_{P_{0}}+L_{P_{0}})F,\\\
\\{F,\tilde{F}\\}(\gamma,\psi)&=\Pi_{\text{ext}}^{g,n}(r(\gamma,\psi))(dF,d\tilde{F}),\end{split}$
(18)

where $\mathfrak{h}_{\lambda,0}$ is parametrized via $\gamma J_{0}+\psi P_{0}$
and

$r:\mathfrak{h}_{\lambda,0}^{*}\to\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda}$

is a $\mathfrak{h}$-equivariant solution of the CDYBE (1) with
$\text{Sym}(r(\gamma,\psi))=\text{Sym}(r)=K_{\alpha\beta}$.

As shown in Appendix B, the Lie algebras $\mathfrak{g}_{\lambda}$ have at most
four conjugacy classes of Cartan subalgebras: For $\mathfrak{so}(4)$,
$\mathfrak{so}(3,1)$ and $\mathfrak{iso}(3)$ the set
$\mathfrak{C}_{\mathfrak{g}_{\lambda}}^{\text{Ad}}$ is a singleton, being all
the Cartan subalgebras conjugate to the algebra $\mathfrak{h}_{\lambda,0}$.
For $\mathfrak{iso}(2,1)$ the set
$\mathfrak{C}^{\text{Ad}}_{\mathfrak{g}_{\lambda}}$ has cardinality two, since
any Cartan subalgebra is conjugated to one of the non-conjugate Cartan
subalgebras $\mathfrak{h}_{\lambda,0}$ or $\mathfrak{h}_{\lambda,1}$
(generated by $J_{1}$ and $P_{1}$). Finally for $\mathfrak{so}(2,2)$,
$\mathfrak{C}_{\mathfrak{g}_{\lambda}}^{\text{Ad}}$ has cardinality four since
any Cartan subalgebra is conjugated to $\mathfrak{h}_{\lambda,0}$,
$\mathfrak{h}_{\lambda,1}$, $\mathfrak{h}_{01}^{\pm}$ (generated by
$J_{0}+P_{0}$ and $J_{1}-P_{1}$) or $\mathfrak{h}_{01}^{\mp}$ (generated by
$J_{0}-P_{0}$ and $J_{1}+P_{1}$). Hence, depending on the Lie algebra
$\mathfrak{g}_{\lambda}$ and the conjugacy classes considered
$\\{\mathcal{C}_{i}\\}$, the Poisson structures of the gauge-fixed phase space
are in correspondence with classical dynamical $r$-matrix for the triples
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$ with
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$ (and also
$\mathfrak{h}_{\lambda}=\mathfrak{h}^{\pm}_{01}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}^{\mp}_{01}$ for $\mathfrak{so}(2,2)$).

The rest of the paper is organized as follows: In Section 2 we present a
systematical treatment (following the same fashion of [31]) of the CDYBE for
the Lie algebras $\mathfrak{g}_{\lambda}$, using as a main tool the fact these
Lie algebras can be realized as generalized complexifications of
$\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. In Section 3 a full
description of the set
$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ and the
moduli space
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ of Cartan
classical dynamical $r$-matrices is presented, including also some dynamical
generalizations of well-known solutions of the CYBE. Section 4 is devoted to
showing that the classical dynamical $r$-matrices found in the previous
section are gauge equivalent to a family of classical dynamical $r$-matrices
studied by Feher, Gabor, Marshall, Palla and Pusztai in the setting of gauge-
fixed WZNW models. Finally, in the Appendices we include some digressions
regarding the action of $G$ over the sets of classical dynamical $r$-matrices,
the Cartan subalgebras of the Lie algebras $\mathfrak{g}_{\lambda}$ and the
Weierstrass factorization theorem.

##  2 Structure of the CDYBE for $\mathfrak{g}_{\lambda}$

###  2.1 The Lie algebras $\mathfrak{g}_{\lambda}$ as generalised
complexifications

Here we solve the CDYBE for the Lie algebras $\mathfrak{g}_{\lambda}$ (and any
Lie subalgebra $\mathfrak{h}_{\lambda}$) by considering the most general
Ansatz in the CDYBE associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$ and solving
the resulting equations.

The realization of the Lie algebras $\mathfrak{g}_{\lambda}$ as the real form
of a generalized complexification of $\mathfrak{su}(2)$ or
$\mathfrak{sl}(2,\mathbb{R})$, depending on the signature, has proved useful
for describing the classical $r$-matrices and Poisson-Lie structures
associated to the local isometry groups of 3d gravity (see e.g. [30]). The Lie
algebras $\mathfrak{iso}(3)$, $\mathfrak{so}(3,1)$ and $\mathfrak{so}(4)$ can
be constructed via generalized complexification of the real Lie algebra
$\mathfrak{su}(2)$. Analogously, the Lie algebras $\mathfrak{iso}(2,1)$,
$\mathfrak{so}(3,1)$ and $\mathfrak{so}(2,2)$ can be obtained from
$\mathfrak{sl}(2,\mathbb{R})$ using the same construction.

This generalized complexification requires the introduction of a formal
parameter $\theta$ such that $\theta^{2}=\lambda$ and to set

$P_{a}=\theta J_{a}\qquad\text{for}\qquad a=0,1,2,$ (19)

where $\\{J_{a}\\}$ are precisely the generators of $\mathfrak{su}(2)$ or
$\mathfrak{sl}(2,\mathbb{R})$, i.e.

$[J_{a},J_{b}]=\epsilon_{abc}J^{c}.$

using the metrics $\text{diag}(1,1,1)$ or $\text{diag}(1,-1,-1)$ to raised or
lowered indices in the $\mathfrak{su}(2)$ or $\mathfrak{sl}(2,\mathbb{R})$
cases, respectively.

Formally, the main ingredient of the construction is a ring denoted by
$\mathbb{R}_{\lambda}$, which is obtained by adjoining a formal element
$\theta$ such that $\theta^{2}=\lambda$.

######  Definition 1.

[32]. Let $\lambda\in\mathbb{R}$. Then we denote by

$\mathbb{R}_{\lambda}\equiv\\{(x+\theta y)\ |\ x,y\in\mathbb{R},\
\theta^{2}=\lambda\\},$ (20)

the ring homeomorphic to $(\mathbb{R}^{2},+)$ as an abelian group and with
product given by

$(x+\theta y)\cdot(z+\theta w)=(xz+\lambda yw)+\theta(xw+yz).$

for $x,y,z,w\in\mathbb{R}$.

The construction of $\mathbb{R}_{\lambda}$ mimics the one of the complex
numbers $\mathbb{C}$ by adjoining a formal element $i$ to $\mathbb{R}$ such
that $i^{2}=-1$. Indeed it becomes a field in the case $\lambda<0$, isomorphic
to $\mathbb{C}$. For the other cases, the rings posses zero divisors and are
isomorphic to the dual and hyperbolic numbers in the cases $\lambda=0$ and
$\lambda>0$, respectively.

Then for $\mathfrak{g}=\mathfrak{su}(2)$ or
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$, the generalized complexification
$\mathfrak{g}\otimes\mathbb{R}_{\lambda}$ is isomorphic to one of the five Lie
algebras $\mathfrak{g}_{\lambda}$, via the identification established in (19):

######  Lemma 1.

[32]. For $\lambda\in\mathbb{R}$ we have the following isomorphisms

$\mathfrak{su}(2)\otimes\mathbb{R}_{\lambda}\cong\begin{cases}\mathfrak{iso}(3)&\quad\lambda=0\\\
\mathfrak{so}(4)&\quad\lambda>0\\\
\mathfrak{so}(3,1)&\quad\lambda<0\end{cases},\qquad\qquad\mathfrak{sl}(2,\mathbb{R})\otimes\mathbb{R}_{\lambda}\cong\begin{cases}\mathfrak{iso}(2,1)&\quad\lambda=0\\\
\mathfrak{so}(2,2)&\quad\lambda>0\\\
\mathfrak{so}(3,1)&\quad\lambda<0\end{cases}$

In the next subsection the utility of this algebraic observation will become
apparent, and how it allows us to split the CDYBE into simpler pieces. Also,
since the CDYBE involves partial derivatives it will result useful to
introduce the notion of differentiability in the ring $\mathbb{R}_{\lambda}$,
using the fact its construction resembles the one of the complex numbers
$\mathbb{C}$.

Over the ring $\mathbb{R}_{\lambda}$, the generalised complexified variable is
defined as

$z=\psi+\theta\gamma$ (21)

and its conjugate is given by

$\overline{z}=\psi-\theta\gamma.$ (22)

The generalized differentials of these pair of conjugate variable are defined,
analogous to the complex differentials, by

$dz=d\psi+\theta d\gamma\qquad\text{and}\qquad d\overline{z}=d\psi-\theta
d\gamma.$

In the cases when $\lambda\neq 0$, the generalized analogues of the complex
(anti-)holomorphic derivatives are given by

$\partial_{z}=\frac{1}{2}\left(\frac{\partial}{\partial\psi}+\frac{1}{\theta}\frac{\partial}{\partial\gamma}\right)\qquad\text{and}\qquad\partial_{\bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial\psi}-\frac{1}{\theta}\frac{\partial}{\partial\gamma}\right),$

satisfying

$dz(\partial_{z})=1,\quad d\bar{z}(\partial_{z})=0,\quad
d\bar{z}(\partial_{\bar{z}})=1,\quad dz(\partial_{\bar{z}})=0.$

Explicitly, by considering a generalized function $w=b+\theta
c:\mathbb{R}_{\lambda}\to\mathbb{R}_{\lambda}$, its generalized partial
derivatives are given by

$\partial_{z}w=\frac{1}{2}\left(\frac{\partial b}{\partial\psi}+\frac{\partial
c}{\partial\gamma}+\theta\frac{\partial
c}{\partial\psi}+\frac{1}{\theta}\frac{\partial
b}{\partial\gamma}\right)\qquad\text{and}\qquad\partial_{\bar{z}}w=\frac{1}{2}\left(\frac{\partial
b}{\partial\psi}-\frac{\partial c}{\partial\gamma}+\theta\frac{\partial
c}{\partial\psi}-\frac{1}{\theta}\frac{\partial b}{\partial\gamma}\right).$
(23)

Hence, a generalized function $w$ is $\mathbb{R}_{\lambda}$-holomorphic, i.e

$\partial_{\bar{z}}w=0$ (24)

if and only if

$\frac{\partial b}{\partial\psi}=\frac{\partial
c}{\partial\gamma}\quad\text{and}\quad\frac{\partial
b}{\partial\gamma}=\lambda\frac{\partial c}{\partial\psi}$

and consequently the generalized partial derivative of an
$\mathbb{R}_{\lambda}$-holomorphic function becomes a generalized total
derivative, given by

$\frac{dw}{\partial z}\equiv\frac{\partial
b}{\partial\psi}+\theta\frac{\partial c}{\partial\psi}$ (25)

In the next section, when finding explicitly solutions of the CDYBE for
$\mathfrak{g}_{\lambda}$, this generalized notion of holomorphicity for
$\mathbb{R}_{\lambda}$ will play a key role.

######  Remark 1.

We have not formally defined (anti-)holomorphic derivatives for functions of
$\mathbb{R}_{\lambda}$ here, and are aware that the presence of zero-divisors
makes the definition via limits of difference-quotients tricky. However, we
will derive all differential equations in this paper using conventional real
calculus, and use the ring $\mathbb{R}_{\lambda}$ as a convenient tool for
collecting pairs of real equations into one generalised complex one.

###  2.2 Casimirs and the CDYBE for $\mathfrak{g}_{\lambda}$

In terms of the generalized complexification (19), the Casimir (16) can be
written as

$K_{\alpha\beta}=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}(J_{a}\otimes\theta
J^{a}+\theta J_{a}\otimes
J^{a})-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}(\lambda J_{a}\otimes
J^{a}+\theta J_{a}\otimes\theta J^{a})$ (26)

and consequently, by extending linearly the Classical Yang-Baxter map over the
ring $\mathbb{R}_{\lambda}$, we obtain

$\begin{split}\Omega_{\alpha\beta}&=\text{CYB}(K_{\alpha\beta})\\\
&=\mu\epsilon_{abc}(\lambda J^{a}\otimes J^{b}\otimes J^{c}+J^{a}\otimes\theta
J^{b}\otimes\theta J^{c}+\theta J^{a}\otimes J^{b}\otimes\theta J^{c}+\theta
J^{a}\otimes\theta J^{b}\otimes J^{c})\\\ &\quad+\nu\epsilon_{abc}(\theta
J^{a}\otimes\theta J^{b}\otimes\theta J^{c}+\lambda\theta J^{a}\otimes
J^{b}\otimes J^{c}+\lambda J^{a}\otimes\theta J^{b}\otimes J^{c}+\lambda
J^{a}\otimes J^{b}\otimes\theta J^{c})\\\
&=[\mu(\lambda\text{id}\otimes\text{id}\otimes\text{id}+\text{id}\otimes\theta\otimes\theta+\theta\otimes\text{id}\otimes\theta+\theta\otimes\theta\otimes\text{id})]\epsilon_{abc}J^{a}\otimes
J^{b}\otimes J^{c}\\\
&\quad+[\nu(\theta\otimes\theta\otimes\theta)+\nu\lambda(\theta\otimes\text{id}\otimes\text{id}+\text{id}\otimes\theta\otimes\text{id}+\text{id}\otimes\text{id}\otimes\theta)]\epsilon_{abc}J^{a}\otimes
J^{b}\otimes J^{c}\end{split}$

where

$\mu\equiv\frac{\alpha^{2}+\lambda\beta^{2}}{(\alpha^{2}-\lambda\beta^{2})^{2}},\qquad\nu\equiv-\frac{2\alpha\beta}{(\alpha^{2}-\lambda\beta^{2})^{2}}.$
(27)

Here it is important that, even though the Lie algebras
$\mathfrak{g}_{\lambda}$ can be recovered via the generalized
complexifications $\mathfrak{su}(2)\otimes\mathbb{R}_{\lambda}$ and
$\mathfrak{sl}(2,\mathbb{R})\otimes\mathbb{R}_{\lambda}$, any tensor product
of $\mathfrak{g}_{\lambda}$ with itself is taken over $\mathbb{R}$.

Since the goal is to obtain classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$
$r$-matrices, we are looking for $r\in
C^{1}(\mathfrak{h}_{\lambda}^{*})\otimes(\mathfrak{g}_{\lambda}\wedge\mathfrak{g}_{\lambda})$
such that

$\text{CDYB}(r)=-\Omega_{\alpha\beta}.$ (28)

Analogously to [31], the most general antisymmetric classical dynamical
$r$-matrix is of the form

$\begin{split}r(\bm{\gamma},\bm{\psi})&=A_{ab}(\bm{\gamma},\bm{\psi})J^{a}\otimes
J^{b}+B_{ab}(\bm{\gamma},\bm{\psi})P^{a}\otimes
J^{b}-B_{ab}(\bm{\gamma},\bm{\psi})J^{b}\otimes
P^{a}+C_{ab}(\bm{\gamma},\bm{\psi})P^{a}\otimes P^{b}\\\ &=J^{a}\otimes
A(\bm{\gamma},\bm{\psi})J_{a}+P^{a}\otimes
B(\bm{\gamma},\bm{\psi})J_{a}-B(\bm{\gamma},\bm{\psi})J_{a}\otimes
P^{a}+P^{a}\otimes C(\bm{\gamma},\bm{\psi})P_{a}\\\ &=(\text{id}\otimes
A(\bm{\gamma},\bm{\psi})+\theta\otimes
B(\bm{\gamma},\bm{\psi})-B(\bm{\gamma},\bm{\psi})\otimes\theta+\theta\otimes\theta
C(\bm{\gamma},\bm{\psi}))J^{a}\otimes J_{a}\end{split}$ (29)

where $A,B,C\in
C^{1}(\mathfrak{h}^{*}_{\lambda})\otimes\text{L}(\mathfrak{g},\mathfrak{g})$,
with $\mathfrak{g}=\mathfrak{su}(2)$ for Euclidean or
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ for Lorentzian signature, such that
$A$ and $C$ are skew-symmetric, i.e.
$A_{ab}(\bm{\gamma},\bm{\psi})=-A_{ba}(\bm{\gamma},\bm{\psi})$ and
$C_{ab}(\bm{\gamma},\bm{\psi})=-C_{ba}(\bm{\gamma},\bm{\psi})$. The skew-
symmetry of these linear maps of $\mathfrak{g}$, implies there exist vector
functions $v(\bm{\alpha},\bm{\psi})$ and $w(\bm{\alpha},\bm{\psi})$ in
$C^{1}(\mathfrak{h}_{\lambda}^{*},\mathbb{R}^{3})$ (for
$\mathfrak{g}=\mathfrak{su}(2)$) or in
$C^{1}(\mathfrak{h}_{\lambda}^{*},\mathbb{M}^{1,2})$ (for
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$), such that

$A(\bm{\gamma},\bm{\psi})(\cdot)=[v(\bm{\gamma},\bm{\psi}),\ \cdot\
]\qquad\text{and}\qquad
C(\bm{\gamma},\bm{\psi})(\cdot)=[w(\bm{\gamma},\bm{\psi}),\ \cdot\ ],$

where the coordinates $(\bm{\gamma},\bm{\psi})$ parametrizes the dual Lie
subalgebra $\mathfrak{h}_{\lambda}^{*}$ via

$\gamma_{0}J_{0}^{*}+\gamma_{1}J_{1}^{*}+\gamma_{2}J_{2}^{*}+\psi_{0}P_{0}^{*}+\psi_{1}P_{1}^{*}+\psi_{2}P_{2}^{*}.$
(30)

Then, by plugging this general ansatz of $r(\bm{\gamma},\bm{\psi})$ into the
CDYBE (28) and using some algebra, we obtain the first main result of the
present paper, which consists of a set of four equations for the linear maps
$A,B,C$, which must be satisfied in order to conclude (29) is a solution of
the Classical Dynamical Yang-Baxter equation (28).

######  Theorem 1.

Let $\mathfrak{h}_{\lambda}$ be a Lie subalgebra of $\mathfrak{g}_{\lambda}$,
with $\mathfrak{h}^{*}_{\lambda}$ parametrized like in (30). A function $r\in
C^{1}(\mathfrak{h}^{*}_{\lambda})\otimes(\mathfrak{g}_{\lambda}\wedge\mathfrak{g}_{\lambda})$
given by

$\begin{split}r_{\text{d}}(\bm{\psi},\bm{\gamma})&=K_{\alpha\beta}+r(\bm{\psi},\bm{\gamma})\\\
&=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}(J_{a}\otimes P^{a}+P_{a}\otimes
J^{a})-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}(\lambda J_{a}\otimes
J^{a}+P_{a}\otimes P^{a})+J^{a}\otimes A(\bm{\psi},\bm{\gamma})J_{a}\\\
&\qquad+P^{a}\otimes
B(\bm{\psi},\bm{\gamma})J_{a}-B(\bm{\psi},\bm{\gamma})J_{a}\otimes
P^{a}+P^{a}\otimes C(\bm{\psi},\bm{\gamma})P_{a}\end{split}$ (31)

is a solution of the CDYBE (28) if and only if the linear maps $A,B,C\in
C^{1}(\mathfrak{h}_{\lambda}^{*})\otimes\text{L}(\mathfrak{g}_{\lambda},\mathfrak{g}_{\lambda}$),
with $A=[v^{A}(\bm{\psi},\bm{\gamma}),\cdot\ ]$ and
$C=[v^{C}(\bm{\psi},\bm{\gamma}),\cdot\ ]$ skew-symmetric, satisfy the
following set of four equations:

$-\frac{1}{2}\text{tr}(A^{2})+\frac{\lambda}{2}[\text{tr}(B)^{2}-\text{tr}(B^{2})]-\text{div}_{\bm{\gamma}}(v)=-\mu\lambda$
(32)

$-\text{tr}(CB)-\text{div}_{\bm{\psi}}(w)=-\nu$ (33)

$[B-\text{tr}(B)\text{id}](B+B^{t})+\frac{1}{2}[\text{tr}(B)^{2}-\text{tr}(B^{2})]\text{id}-CA+\lambda(C^{2}-\frac{1}{2}\text{tr}(C^{2})\text{id})+\text{curl}_{\bm{\psi}}(B)-\text{grad}_{\bm{\gamma}}(w)=-\mu\text{id}$
(34)

$-A(B+B^{t})+(B^{t}-\text{tr}(B)\text{id})(\lambda
C-A)-\text{tr}(AB)\text{id}+\text{curl}_{\bm{\gamma}}(B^{t})-\text{grad}_{\bm{\psi}}(v)=-\lambda\nu\text{id}$
(35)

where $[\text{grad}(v)]_{ab}=(\text{grad}(v_{b}))_{a}$ and
$[\text{curl}(M)]_{ab}=\text{curl}(\text{Row}_{a}(M))_{b}$, such that the
subindices $\bm{\gamma}$ and $\bm{\psi}$ indicate if the differential
operators are computed with respect to the variables
$(\gamma_{0},\gamma_{1},\gamma_{2})$ or $(\psi_{0},\psi_{1},\psi_{2})$,
respectively.

###### Proof.

In [33], the contraction of the CYBE with a generic element $X\otimes Y\otimes
Z\in\mathfrak{g}_{\lambda}^{\otimes 3}$ was found to be useful to determine
classical $r$-matrices of $\mathfrak{g}_{\lambda}$ with symmetric part
$K_{\alpha\beta}$ (equations (4.11)-(4.18)). Following the same approach, the
contraction

$\langle\text{alt}[d(r(\bm{\psi},\bm{\gamma}))],X\otimes Y\otimes Z\rangle=0$

can be decompose into the following eight independent terms in
$(\mathbb{R}_{\lambda})^{\otimes_{\mathbb{R}}3}$

  * •

$\text{id}\otimes\text{id}\otimes\text{id}$

$\begin{split}0&=\langle J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\gamma_{b}}(A)J_{a},Z\rangle-\langle
J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\gamma_{b}}(A)J_{a},Z\rangle+\langle
J_{a},X\rangle\langle\partial_{\gamma_{b}}(A)J_{a},Y\rangle\langle
J^{b},Z\rangle\\\ &=\langle J^{b},X\rangle\langle
Y,\partial_{\gamma_{b}}(A^{t})(Z)\rangle-\langle J^{b},Y\rangle\langle
X,\partial_{\gamma_{b}}(A^{t})(Z)\rangle+\langle
X,\partial_{\gamma_{b}}(A^{t})(Y)\rangle\langle J^{b},Z\rangle\\\
&=\langle\partial_{\gamma_{a}}(A)([J^{a},[Y,X]]),Z\rangle+\langle\langle
X,\partial_{\gamma_{a}}(A^{t})(Y)\rangle J^{a},Z\rangle\end{split}$ (36)

  * •

$\text{id}\otimes\theta\otimes\theta$

$\begin{split}0&=\langle\partial_{\psi_{b}}(B)J^{a},X\rangle\langle
J^{b},Y\rangle\langle
J_{a},Z\rangle-\langle\partial_{\psi_{b}}(B)J^{a},X\rangle\langle
J_{a},Y\rangle\langle J^{b},Z\rangle+\langle J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\gamma_{b}}(C)J_{a},Z\rangle\\\
&=\langle\partial_{\psi_{b}}(B^{t})X,Z\rangle\langle J^{i},Y\rangle-\langle
Y,\partial_{\psi_{b}}(B^{t})X\rangle\langle J^{b},Z\rangle+\langle
J^{b},X\rangle\langle Y,\partial_{\gamma_{b}}(C^{t})Z\rangle\\\
&=\langle[Y,[\partial_{\psi_{b}}(B^{t})X,J^{b}]],Z\rangle+\langle\langle
J^{b},X\rangle\partial_{\gamma_{b}}(C)Y,Z\rangle\end{split}$ (37)

  * •

$\theta\otimes\text{id}\otimes\theta$

$\begin{split}0&=-\langle
J^{b},X\rangle\langle\partial_{\psi_{b}}(B)J^{a},Y\rangle\langle
J_{a},Z\rangle+\langle
J^{a},X\rangle\langle\partial_{\psi_{b}}(B)J_{a},Y\rangle\langle
J^{b},Z\rangle-\langle J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\gamma_{b}}(C)J_{a},Z\rangle\\\ &=-\langle
J^{b},X\rangle\langle\partial_{\psi_{b}}(B^{t})Y,Z\rangle+\langle
X,\partial_{\psi_{b}}(B^{t})Y\rangle\langle J^{b},Z\rangle-\langle
X,\partial_{\gamma_{b}}(C^{t})Z\rangle\langle J^{b},Y\rangle\\\
&=\langle[X,[J^{a},\partial_{\psi_{a}}(B^{t})Y]],Z\rangle-\langle\langle
J^{b},Y\rangle\partial_{\gamma_{a}}(C)X,Z\rangle\end{split}$ (38)

  * •

$\theta\otimes\theta\otimes\text{id}$

$\begin{split}0&=\langle J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\psi_{b}}(B)J_{a},Z\rangle-\langle
J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\psi_{b}}(B)J_{a},Z\rangle+\langle
J^{a},X\rangle\langle\partial_{\gamma_{b}}(C)J_{a},Y\rangle\langle
J^{b},Z\rangle\\\ &=\langle J^{a},X\rangle\langle
Y,\partial_{\psi_{a}}(B^{t})Z\rangle+\langle J^{a},Y\rangle\langle
X,\partial_{\psi_{a}}(B^{t})Z\rangle-\langle
X,\partial_{\gamma_{a}}(C^{t})Y\rangle\langle J^{a},Z\rangle\\\
&=\langle\partial_{\psi_{a}}(B)([J^{b},[Y,X]]),Z\rangle-\langle\langle
X,\partial_{\gamma_{a}}(C^{t})Y\rangle J^{a},Z\rangle\end{split}$ (39)

  * •

$\theta\otimes\theta\otimes\theta$

$\begin{split}0&=\langle J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\psi_{b}}(C)J_{a},Z\rangle-\langle
J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\psi_{b}}(C)J_{a},Z\rangle+\langle
J^{a},X\rangle\langle\partial_{\psi_{a}}(C)J_{a},Y\rangle\langle
J^{a},Z\rangle\\\ &=\langle J^{b},X\rangle\langle
Y,\partial_{\psi_{b}}(C^{t})Z\rangle-\langle J^{b},Y\rangle\langle
X,\partial_{\psi_{b}}(C^{t})Z\rangle+\langle\langle
X,\partial_{\psi_{b}}(C^{t})Y\rangle J^{b},Z\rangle\\\
&=\langle\partial_{\psi_{a}}(C)([J^{a},[Y,X]]),Z\rangle+\langle\langle
X,\partial_{\psi_{a}}(C^{t})Y\rangle J^{a},Z\rangle\end{split}$ (40)

  * •

$\theta\otimes\text{id}\otimes\text{id}$

$\begin{split}0&=\langle J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\psi_{b}}(A)J_{a},Z\rangle-\langle
J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\gamma_{b}}(B)J_{a},Z\rangle+\langle
J^{a},X\rangle\langle\partial_{\gamma_{b}}(B)J_{a},Y\rangle\langle
J^{b},Z\rangle\\\ &=\langle J^{a},X\rangle\langle
Y,\partial_{\psi_{a}}(A^{t})Z\rangle+\langle J^{a},Y\rangle\langle
X,\partial_{\gamma_{a}}(B^{t})Z\rangle-\langle
X,\partial_{\gamma_{a}}(B^{t})Y\rangle\langle J^{a},Z\rangle\\\
&=\langle\langle
J^{a},X\rangle\partial_{\psi_{a}}(A)Y,Z\rangle-\langle[Y,[J^{a},\partial_{\gamma_{a}}(B)X]],Z\rangle\end{split}$
(41)

  * •

$\text{id}\otimes\theta\otimes\text{id}$

$\begin{split}0&=\langle J^{a},X\rangle\langle
J^{b},Y\rangle\langle\partial_{\psi_{b}}(A)J_{a},Z\rangle+\langle
J^{b},X\rangle\langle
J^{a},Y\rangle\langle\partial_{\gamma_{b}}(B)J_{a},Z\rangle-\langle\partial_{\gamma_{b}}(B)J^{a},X\rangle\langle
J_{a},Y\rangle\langle J^{b},Z\rangle\\\ &=-\langle
X,\partial_{\psi_{a}}(A^{t})Z\rangle\langle J^{a},Y\rangle+\langle\langle
J^{a},X\rangle Y,\partial_{\gamma_{a}}(B^{t})Z\rangle-\langle
Y,\partial_{\gamma_{a}}(B^{t})X\rangle\langle J^{a},Z\rangle\\\
&=\langle[X,[\partial_{\gamma_{a}}(B)Y,J^{a}]],Z\rangle-\langle\langle
J^{a},Y\rangle\partial_{\psi_{a}}(A)X,Z\rangle\end{split}$ (42)

  * •

$\text{id}\otimes\text{id}\otimes\theta$

$\begin{split}0&=\langle
J^{a},X\rangle\langle\partial_{\psi_{b}}(A)J_{a},Y\rangle\langle
J^{b},Z\rangle-\langle
J^{b},X\rangle\langle\partial_{\gamma_{b}}(B)J^{a},Y\rangle\langle
J_{a},Z\rangle+\langle\partial_{\gamma_{b}}(B)J^{a},X\rangle\langle
J^{b},Y\rangle\langle J_{a},Z\rangle\\\ &=\langle
X,\partial_{\psi_{a}}(A^{t})Y\rangle\langle J^{a},Z\rangle-\langle
J^{a},X\rangle\langle\partial_{\gamma_{a}}(B^{t})Y,Z\rangle+\langle\partial_{\gamma_{a}}(B^{t})X,Z\rangle\langle
J^{a},Y\rangle\\\
&=\langle\partial_{\gamma_{a}}(B^{t})([J^{a},[Y,X]]),Z\rangle+\langle\langle
X,\partial_{\psi_{a}}(A^{t})Y\rangle J^{a},Z\rangle\end{split}$ (43)

Hence combining the corresponding terms among (4.11)-(4.18) in [31] and
(36)-(43), we conclude the contraction

$\langle\text{CDYBE}(r(\bm{\psi},\bm{\gamma})),X\otimes Y\otimes
Z\rangle=-\langle\Omega_{\alpha\beta},X\otimes Y\otimes Z\rangle$ (44)

can be decomposed into a set of eight terms, which reduces to the following of
four independent coupled partial differential equations for the linear maps
$A,B,C$

$\begin{split}&[A(X),A(Y)]-A^{2}([X,Y])+\lambda(B([X,B^{t}(Y)]+[B^{t}(X),Y])+[B^{t}(X),B^{t}(Y)])\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\partial_{\gamma_{a}}(A)([J^{a},[X,Y]])+\langle\partial_{\gamma_{a}}(A)(X),Y\rangle
J^{a}=-\mu\lambda[X,Y]\end{split}$

$\begin{split}&B^{t}C([X,Y])+[B(X),C(Y)]+[C(X),B(Y)]-C([B(X),Y]+[X,B(Y)])\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\partial_{\psi_{a}}(C)([J^{a},[X,Y]])+\langle\partial_{\psi_{a}}(C)(X),Y\rangle
J^{a}=-\nu[X,Y]\end{split}$

$\begin{split}&[B(X),B(Y)]-B([X,B(Y)]+[X,B(Y)])-AC([X,Y])+\lambda[C(X),C(Y)]\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\partial_{\psi_{a}}(B)([J^{a},[X,Y]])+\langle\partial_{\gamma_{a}}(C)(X),Y\rangle
J^{a}=-\mu[X,Y]\end{split}$

$\begin{split}&B^{t}A([X,Y])-[B^{t}(X),A(Y)]-[A(X),B^{t}(Y)]+\lambda
C([X,B^{t}(Y)]+[B^{t}(X),Y])\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\partial_{\gamma_{a}}(B^{t})([J^{a},[X,Y]])+\langle\partial_{\psi_{a}}(A)(X),Y\rangle
J^{a}=-\lambda\nu[X,Y]\end{split}$

Finally, using adjugates (exactly like in [33] and [31]) we derive the 4
equations (32)-(35). ∎

The equations (32)-(35) are the dynamical generalization of (4.39) in [31]. It
is immediate to notice that if we assume the linear maps are constant, i.e.
$A,B,C\in\text{L}(\mathfrak{g}_{\lambda},\mathfrak{g}_{\lambda})\subset
C^{1}(\mathfrak{h}_{\lambda}^{*})\otimes\text{L}(\mathfrak{g}_{\lambda},\mathfrak{g}_{\lambda})$,
then the former set of equations reduces to the later, as expected.

##  3 Classical dynamical $r$-matrices for $\mathfrak{g}_{\lambda}$

###  3.1 Classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$ $r$-matrices

In the previous theorem we find a set of equations (32)-(35), equivalent to
the CDYBE (28) for any Lie subalgebra $\mathfrak{h}_{\lambda}$ of
$\mathfrak{g}_{\lambda}$, nevertheless, as mentioned in the introduction we
are interested in the case when it is a Cartan subalgebra. Since any Cartan
Lie subalgebra of $\mathfrak{g}_{\lambda}$ is conjugate-equivalent to
$\mathfrak{h}_{\lambda,0}$ and/or $\mathfrak{h}_{\lambda,1}$ (also to
$\mathfrak{h}^{\pm}_{\lambda,01}$ or $\mathfrak{h}^{\mp}_{\lambda,01}$ only
for $\mathfrak{so}(2,2)$), for our purposes it is enough to determine the set
$\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$
for the cases where $\mathfrak{h}_{\lambda}$ is $\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda,1}$ (and also $\mathfrak{h}^{\pm}_{01}$ and
$\mathfrak{h}^{\pm}_{10}$ for $\mathfrak{so}(2,2)$). This gives us the full
moduli space
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ and then,
via dynamical gauge transformations, the full space
$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ can be
generated.

For all $\lambda\in\mathbb{R}$, the Cartan subalgebras
$\mathfrak{h}_{\lambda}$ are Abelian subalgebras of $\mathfrak{g}_{\lambda}$.
Therefore the $\mathfrak{h}_{\lambda}$-equivariance condition (2) of a
classical dynamical $r$-matrix reduces to

$[r_{\text{d}}(x),h\otimes 1+1\otimes h]=0$ (45)

for all $x\in\mathfrak{h}_{\lambda}^{*}$ and $h\in\mathfrak{h}_{\lambda}$.

Using the notation introduced for the antisymmetric part $r$ of $r_{\text{d}}$
in (31), we now determine the implications of the
$\mathfrak{h}_{\lambda}$-equivariance condition for the linear maps
$[v,\cdot]$, $B$ and $[w,\cdot]$ in (29), which we write more explicitly as

$r(\psi,\gamma)=\epsilon_{abc}v^{c}(\psi,\gamma)J^{a}\otimes
J^{b}+B_{ab}(\psi,\gamma)(P^{a}\otimes J^{b}-J^{b}\otimes
P^{a})+\epsilon_{abc}w^{c}(\psi,\gamma)P^{a}\otimes P^{b}.$ (46)

As we shall see shortly, this condition only has interesting solutions for the
Cartan subalgebras $\mathfrak{h}_{\lambda,0}$ and $\mathfrak{h}_{\lambda,1}$.
In the cases $\mathfrak{h}_{\lambda}=\mathfrak{h}_{01}^{\pm}$ (arising in
$\mathfrak{g}_{\lambda}=\mathfrak{so}(2,2)$), the equivariance condition
implies that all the coefficients in (46) vanish except for $B_{01}$,
$B_{10}$, $v^{2}$ and $w^{2}$, and additionally $B_{01}=B_{10}=v^{2}=-w^{2}$.
This means that the most general $\mathfrak{h}^{\pm}_{01}$-invariant element
in
$\mathcal{F}(\mathfrak{h}^{\pm*}_{01})\otimes\mathfrak{so}(2,2)\otimes\mathfrak{so}(2,2)$
is of the form

$K_{\alpha\beta}+f(\psi,\gamma)(J_{0}+P_{0})\wedge(J_{1}-P_{1})$ (47)

with $f\in\mathcal{F}(\mathfrak{h}^{\pm*}_{01})$ arbitrary. Analogously the
most general $\mathfrak{h}^{\mp}_{01}$-invariant element in
$\mathcal{F}(\mathfrak{h}^{\mp*}_{01})\otimes\mathfrak{so}(2,2)\otimes\mathfrak{so}(2,2)$
is of the form

$K_{\alpha\beta}+g(\psi,\gamma)(J_{0}-P_{0})\wedge(J_{1}+P_{1})$ (48)

with $g\in\mathcal{F}(\mathfrak{h}^{\mp*}_{01})$ arbitrary. However, neither
of these lead to interesting solutions of the CDYBE. By plugging (47) or (48)
into the CDYBE, the equations (32) and (33) reduce in both cases to
$0=\mu\lambda$ and $0=\nu$, respectively, showing that
$\text{Dyn}(\mathfrak{so}(2,2),\mathfrak{h}_{\lambda},K_{\alpha\beta})=\emptyset$
for $\mathfrak{h}_{\lambda}=\mathfrak{h}^{\pm}_{01}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}^{\mp}_{01}$. Therefore, the discussion of
$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ is reduced
to the determination of
$\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})$
and
$\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,1},K_{\alpha\beta})$,
to which we now turn.

######  Lemma 2.

The most general $\mathfrak{h}_{\lambda,0}$-equivariant element
$r\in\mathcal{F}(\mathfrak{h}_{\lambda,0}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
is of the form

$r(\psi_{0},\gamma_{0})=K_{\alpha\beta}+c(\psi_{0},\gamma_{0})(\lambda
J^{1}\wedge J^{2}+P^{1}\wedge P^{2})+f(\psi_{0},\gamma_{0})(P^{0}\wedge
J^{0})+b(\psi_{0},\gamma_{0})(P^{1}\wedge J^{2}-J^{1}\wedge P^{2})$ (49)

with $b,c,f\in\mathcal{F}(\mathfrak{h}_{\lambda,0}^{*})$ arbitrary functions.

Similarly, the most general $\mathfrak{h}_{\lambda,1}$-equivariant element
$r\in\mathcal{F}(\mathfrak{h}_{\lambda,1}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
is of the form

$r(\psi_{1},\gamma_{1})=K_{\alpha\beta}+c(\psi_{1},\gamma_{1})(\lambda
J^{2}\wedge J^{0}+P^{2}\wedge P^{0})+f(\psi_{1},\gamma_{1})(P^{1}\wedge
J^{1})+b(\psi_{1},\gamma_{1})(P^{2}\wedge J^{0}-J^{2}\wedge P^{0})$ (50)

with $b,c,f\in\mathcal{F}(\mathfrak{h}_{\lambda,1}^{*})$ arbitrary.

###### Proof.

For the $\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ case, by direct
computation we have for the first term in the skew-symmetric part of (46)

$[J_{0}\otimes 1+1\otimes J_{0},\epsilon_{abc}v^{c}J^{a}\otimes
J^{b}]=v^{i}(J^{0}\wedge J^{i}),$

while for the second

$\begin{split}[J_{0}\otimes 1+1\otimes J_{0},B_{ab}(P^{a}\otimes
J^{b}-J^{b}\otimes P^{a})]&=\epsilon_{ij}B_{i0}(P^{j}\wedge
J^{0})+\epsilon_{ij}B_{0i}(P^{0}\wedge J^{j})\\\
&\quad+(\epsilon_{ij}B_{ik}+\epsilon_{\ell k}B_{j\ell})(P^{j}\wedge
J^{k}),\end{split}$

and for the third

$\begin{split}[J_{0}\otimes 1+1\otimes J_{0},\epsilon_{abc}w^{c}P^{a}\otimes
P^{b}]&=w^{i}(P^{0}\wedge P^{i}).\end{split}$

Thus, in order to have equivariance with respect to the generator $J_{0}$ of
$\mathfrak{h}_{\lambda,0}$, we need

$0=v^{i}(J^{0}\wedge J^{i})+\epsilon_{ij}B_{i0}(P^{j}\wedge
J^{0})+\epsilon_{ij}B_{0i}(P^{0}\wedge
J^{j})+(\epsilon_{ij}B_{ik}+\epsilon_{\ell k}B_{j\ell})(P^{j}\wedge
J^{k})+w^{i}(P^{0}\wedge P^{i})$ (51)

where the indices $i,j$ just take the values $1$ or $2$. Therefore, the
equivariance with respect to $J_{0}$ requires:

  * •

$v^{i}=w^{i}=0$ for $i=1,2$,

  * •

$B_{i0}=B_{0i}=0$ for $i=1,2$,

  * •

$B_{12}=-B_{21}$,

  * •

$B_{11}=B_{22}$.

Analogously, by straightforward computation we get for the first term in the
skew-symmetric part of (46)

$[P_{0}\otimes 1+1\otimes P_{0},\epsilon_{abc}v^{c}J^{a}\otimes
J^{b}]=v^{0}(P^{1}\wedge J^{1}+P^{2}\wedge J^{2})+v^{1}(J^{0}\wedge
P^{1})+v^{2}(J^{0}\wedge P^{2}),$

for the second

$[P_{0}\otimes 1+1\otimes P_{0},B_{ab}(P^{a}\otimes J^{b}-J^{b}\otimes
P^{a})]=\epsilon_{ji}B_{0i}P^{j}\wedge P^{0}+(B_{11}+B_{22})P^{1}\wedge
P^{2}+\lambda\epsilon_{ij}B_{ib}J^{j}\wedge J^{b},$

and for the last one

$[P_{0}\otimes 1+1\otimes P_{0},\epsilon_{abc}w^{c}P^{a}\otimes P^{b}]=\lambda
w^{0}(J^{1}\wedge P^{1}+J^{2}\wedge P^{2})+\lambda w^{1}(P^{0}\wedge
J^{1})+\lambda w^{2}(P^{0}\wedge J^{2})$

So, the equivariance with respect to the generator $P_{0}$ is equivalent to
the equation

$\begin{split}0&=(\lambda w^{0}-v^{0})(J^{1}\wedge P^{1}+J^{2}\wedge
P^{2})+(\lambda w^{i}-v^{i})(J^{0}\wedge P^{i})+\epsilon_{ji}B_{0i}P^{j}\wedge
P^{0}\\\ &\quad+(B_{11}+B_{22})P^{1}\wedge
P^{2}+\lambda\epsilon_{ij}B_{ib}J^{j}\wedge J^{b}\end{split}$ (52)

where again the indices $i$, $j$ run from $1$ to $2$, while $b$ from $0$ to
$2$. This equation translates into the following set of additional conditions:

  * •

$v^{a}=\lambda w^{a}$ for $a=0,1,2$,

  * •

$B_{11}+B_{22}=0$.

Consequently, by incorporating the set of constraints derived above and
considering the $\mathfrak{g}_{\lambda}$-invariance of $K_{\alpha\beta}$, we
conclude that an element
$r\in\mathcal{F}(\mathfrak{h}_{\lambda,0}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
results to be $\mathfrak{h}_{\lambda,0}$-equivariant if and only if it is of
the form (49), with the identification

$f\equiv B_{00},\qquad b\equiv B_{12},\qquad c\equiv w^{0}.$

The conditions for the equivariance for the case
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$ are derived in an completely
analogous way, concluding an element
$r\in\mathcal{F}(\mathfrak{h}_{\lambda,1}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
results to be $\mathfrak{h}_{\lambda,1}$-equivariant if and only if it is of
the form (50), but now with the identifications $f\equiv B_{11}$, $b\equiv
B_{20}$ and $c\equiv w^{1}$. ∎

Having at hand the form of the most general
$\mathfrak{h}_{\lambda}$-equivariant element of
$\mathcal{C}^{1}(\mathfrak{h}_{\lambda}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$,
for both $\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ (49) and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$ (50), we proceed to
determine all the classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$ $r$-matrices
by solving the CDYBE (1) using it now as Ansatz, deriving in this way the
second main result of this paper.

######  Theorem 2.

Let $(\psi_{C},\gamma_{C})\in\mathbb{R}^{2}$ constants. The classical
dynamical $(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$
$r$-matrices for $\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$ are necessarily elements in
$C^{1}(\mathfrak{h}_{\lambda}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
of the form

$r_{d}(\psi_{0},\gamma_{0})=K_{\alpha\beta}+f(\psi_{0},\gamma_{0})(P^{0}\wedge
J^{0})+b(\psi_{0},\gamma_{0})(P^{1}\wedge J^{2}-P^{2}\wedge
J^{1})+c(\psi_{0},\gamma_{0})(\lambda J^{1}\wedge J^{2}+P^{1}\wedge P^{2})$

and

$r_{d}(\psi_{1},\gamma_{1})=K_{\alpha\beta}+f(\psi_{1},\gamma_{1})(P^{1}\wedge
J^{1})+b(\psi_{1},\gamma_{1})(P^{2}\wedge J^{0}-J^{2}\wedge
P^{0})+c(\psi_{1},\gamma_{1})(\lambda J^{2}\wedge J^{0}+P^{2}\wedge P^{0}),$

respectively, such that $f\in C^{1}(\mathfrak{h}_{\lambda}^{*})$,

$\begin{split}b(\psi,\gamma)&=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}B\left(\frac{\alpha(\psi-\psi_{C})-\beta(\gamma-\gamma_{C})}{\alpha^{2}-\lambda\beta^{2}},\frac{\alpha(\gamma-\gamma_{C})-\lambda\beta(\psi-\psi_{C})}{\alpha^{2}-\lambda\beta^{2}}\right)\\\
&\quad-\lambda\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}C\left(\frac{\alpha(\psi-\psi_{C})-\beta(\gamma-\gamma_{C})}{\alpha^{2}-\lambda\beta^{2}},\frac{\alpha(\gamma-\gamma_{C})-\lambda\beta(\psi-\psi_{C})}{\alpha^{2}-\lambda\beta^{2}}\right)\end{split}$
(53)

and

$\begin{split}c(\psi,\gamma)&=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}C\left(\frac{\alpha(\psi-\psi_{C})-\beta(\gamma-\gamma_{C})}{\alpha^{2}-\lambda\beta^{2}},\frac{\alpha(\gamma-\gamma_{C})-\lambda\beta(\psi-\psi_{C})}{\alpha^{2}-\lambda\beta^{2}}\right)\\\
&\quad-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}B\left(\frac{\alpha(\psi-\psi_{C})-\beta(\gamma-\gamma_{C})}{\alpha^{2}-\lambda\beta^{2}},\frac{\alpha(\gamma-\gamma_{C})-\lambda\beta(\psi-\psi_{C})}{\alpha^{2}-\lambda\beta^{2}}\right)\end{split}$
(54)

where for $\mathfrak{h}_{\lambda,0}$

$B(\Psi,\Gamma)=\begin{cases}\frac{\sin(2\Psi)}{\cos(2\Psi)+\cosh\left(2\sqrt{|\lambda|}\Gamma\right)}\\\
\tan\left(\Psi\right)\\\
\frac{\sin(2\Psi)}{\cos(2\Psi)+\cos\left(2\sqrt{\lambda}\Gamma\right)}\end{cases}\quad
C(\Psi,\Gamma)=\begin{cases}\frac{1}{\sqrt{|\lambda|}}\frac{\sinh\left(2\sqrt{|\lambda|}\Gamma\right)}{\cos(2\Psi)+\cosh\left(2\sqrt{|\Lambda|}\Gamma\right)}\qquad&\lambda<0\\\
\frac{\Gamma}{2\cos^{2}\left(\Psi\right)}\qquad&\lambda=0\\\
\frac{1}{\sqrt{\lambda}}\frac{\sin\left(2\sqrt{\lambda}\Gamma\right)}{\cos(2\Psi)+\cos(2\sqrt{\lambda}\Gamma)}\qquad&\lambda>0\end{cases},$
(55)

while for $\mathfrak{h}_{\lambda,1}$, we have two types of solutions:

  1. (i)

Non-constant coefficients:

$B(\Psi,\Gamma)=\begin{cases}\frac{\sinh(2\Psi)}{\cosh(2\Psi)+\cos\left(2\sqrt{|\lambda|}\Gamma\right)}\\\
\tanh\left(\Psi\right)\\\
\frac{\sinh(2\Psi)}{\cosh(2\Psi)+\cosh\left(2\sqrt{\lambda}\Gamma\right)}\end{cases}\qquad
C(\Psi,\Gamma)=\begin{cases}\frac{1}{\sqrt{|\lambda|}}\frac{\sin\left(2\sqrt{|\lambda|}\Gamma\right)}{\cosh(2\Psi)+\cos\left(2\sqrt{|\Lambda|}\Gamma\right)},\qquad&\lambda<0\\\
\frac{\Gamma}{2\cosh^{2}\left(\Psi\right)},\qquad&\lambda=0\\\
\frac{1}{\sqrt{\lambda}}\frac{\sinh\left(2\sqrt{\lambda}\Gamma\right)}{\cosh(2\Psi)+\cosh(2\sqrt{\lambda}\Gamma)},\qquad&\lambda>0\end{cases}$
(56)

  2. (ii)

or constant coefficients

$B(\Psi,\Gamma)=\pm 1\quad\text{and}\quad C(\Psi,\Gamma)=0$ (57)

###### Proof.

By plugging (49) in (32)-(35) we conclude that classical dynamical
($\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta}$)
$r$-matrices are elements in
$C^{1}(\mathfrak{h}_{\lambda,0}^{*})\otimes(\mathfrak{g}_{\lambda}\otimes\mathfrak{g}_{\lambda})$
of the form

$K_{\alpha\beta}+f(\psi_{0},\gamma_{0})(P^{0}\wedge
J^{0})+b(\psi_{0},\gamma_{0})(P^{1}\wedge J^{2}-P^{2}\wedge
J^{1})+c(\psi_{0},\gamma_{0})(\lambda J^{1}\wedge J^{2}+P^{1}\wedge P^{2})$

where the coefficients satisfy the equations

$\displaystyle\lambda c^{2}+b^{2}-\frac{\partial c}{\partial\gamma_{0}}$
$\displaystyle=-\mu,$ (58a) $\displaystyle 2bc-\frac{\partial
c}{\partial\psi_{0}}$ $\displaystyle=-\nu,$ (58b) $\displaystyle\lambda
c^{2}+b^{2}-\frac{\partial b}{\partial\psi_{0}}$ $\displaystyle=-\mu,$ (58c)
$\displaystyle\lambda\frac{\partial c}{\partial\psi_{0}}$
$\displaystyle=\frac{\partial b}{\partial\gamma_{0}}.$ (58d)

These equations can be understood as the
$\mathfrak{h}_{\lambda,0}$-equivariant reduction of the set (32)-(35) for the
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ case, and can be rewritten
in the more compact way

$\begin{split}\frac{\partial b}{\partial\psi_{0}}&=\lambda
c^{2}+b^{2}+\mu=\frac{\partial c}{\partial\gamma_{0}},\\\ \frac{\partial
b}{\partial\gamma_{0}}&=\lambda(2bc+\nu)=\lambda\frac{\partial
c}{\partial\psi_{0}}.\end{split}$ (59)

Analogously, by inserting (50) in (32)-(35) we get the classical dynamical
$r$-matrices associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,1},K_{\alpha\beta})$ have the
form

$K_{\alpha\beta}+f(\psi_{1},\gamma_{1})(P^{1}\wedge
J^{1})+b(\psi_{1},\gamma_{1})(P^{2}\wedge J^{0}-J^{2}\wedge
P^{0})+c(\psi_{1},\gamma_{1})(\lambda J^{2}\wedge J^{0}+P^{2}\wedge P^{0}),$

such that

$\displaystyle-\lambda c^{2}-b^{2}-\frac{\partial c}{\partial\gamma_{1}}$
$\displaystyle=-\mu,$ (60a) $\displaystyle-2bc-\frac{\partial
c}{\partial\psi_{1}}$ $\displaystyle=-\nu,$ (60b) $\displaystyle-\lambda
c^{2}-b^{2}-\frac{\partial b}{\partial\psi_{1}}$ $\displaystyle=-\mu,$ (60c)
$\displaystyle\lambda\frac{\partial c}{\partial\psi_{1}}$
$\displaystyle=\frac{\partial b}{\partial\gamma_{1}}$ (60d)

or more compactly,

$\begin{split}\frac{\partial b}{\partial\psi_{1}}&=-(\lambda
c^{2}+b^{2})+\mu=\frac{\partial c}{\partial\gamma_{1}},\\\ \frac{\partial
b}{\partial\gamma_{1}}&=\lambda(-2bc+\nu)=\lambda\frac{\partial
c}{\partial\psi_{1}}.\end{split}$ (61)

If we consider the generalized complexified variable
$z_{0}=\psi_{0}+\theta\gamma_{0}$ (over $\mathbb{R}_{\lambda}$) in the sense
of (21) and construct the function over $\mathbb{R}_{\lambda}$ given by

$w(z_{0})=b(z_{0})+\theta c(z_{0}),$

then from (23), we deduce the equations (59) can be re-expressed over the ring
$\mathbb{R}_{\lambda}$ as

$\displaystyle\partial_{\overline{z}_{0}}w$ $\displaystyle=0,$ (62a)
$\displaystyle\partial_{z_{0}}w$ $\displaystyle=w^{2}+(\mu+\theta\nu).$ (62b)

As indicated above in (24), equation (62a) can be understood as a generalized
holomorphy condition of the function $w$, i.e. it just depends on the
generalized variable $z_{0}$ but not on $\overline{z}_{0}$ (precisely like
holomorphic functions in complex analysis); and due to this property, as
explained in (25), the equation (62b) is simply

$\frac{dw}{dz_{0}}=w^{2}+(\mu+\theta\nu)$ (63)

Using the same argument, equations (61) can be re-written over
$\mathbb{R}_{\lambda}$ as

$\displaystyle\partial_{\overline{z}_{1}}w$ $\displaystyle=0,$ (64a)
$\displaystyle\qquad\partial_{z_{1}}w$ $\displaystyle=-w^{2}+(\mu+\theta\nu)$
(64b)

but now $w(z_{1})=b(z_{1})+\theta c(z_{1})$ with
$z_{1}=\psi_{1}+\theta\gamma_{1}$ and, again, the holomorphy condition implies

$\frac{dw}{dz_{1}}=-w^{2}+(\mu+\theta\nu).$ (65)

Taking into account

$\mu+\theta\nu=\frac{1}{(\alpha+\theta\beta)^{2}},$

then solving the CDYBE for the triples
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$ with
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$, amounts to solve the
following two non-linear generalized complex ODEs

$(\alpha+\theta\beta)\frac{d}{dz}((\alpha+\theta\beta)w)=\pm((\alpha+\theta\beta)w)^{2}+1.$
(66)

For $\mathfrak{h}_{\lambda,0}$ the solution is given by

$w(z_{0})=\frac{1}{\alpha+\theta\beta}\tan\left(\frac{z_{0}-C}{\alpha+\theta\beta}\right),$
(67)

where

$C=\psi_{C}+\theta\gamma_{C}$

is a constant, while for $\mathfrak{h}_{\lambda,1}$ the solution could be
dependent of $z_{1}$

$w(z_{1})=\frac{1}{\alpha+\theta\beta}\tanh\left(\frac{z_{1}-C}{\alpha+\theta\beta}\right)$
(68)

with $C\in\mathbb{R}_{\lambda}$ again a constant, or independent of the
complexified variable $z_{1}$

$w(z_{1})=\pm\frac{1}{\alpha+\theta\beta}=\pm\frac{\alpha-\theta\beta}{\alpha^{2}-\lambda\beta^{2}}.$
(69)

Finally, separating (67) into its real-part and $\theta$-part we get $b$ (53)
and $c$ (54), respectively, with $B$ and $C$ as given in (55). Proceeding in
the same way for (68) or (69), we get a similar result but now with B and C as
given by (56) or by (57), respectively. See the appendix D to see in detail
how this splitting into real and $\theta$ parts works for every
$\lambda\in\mathbb{R}$. ∎

In particular, for $\beta=0$, the most general classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha 0})$ $r$-matrix is
given by

$r_{d}^{\beta=0}(\psi_{0},\gamma_{0})=K_{\alpha
0}+f(\psi_{0},\gamma_{0})(P^{0}\wedge
J^{0})+b(\psi_{0},\gamma_{0})(P^{1}\wedge J^{2}-P^{2}\wedge
J^{1})+c(\psi_{0},\gamma_{0})(\lambda J^{1}\wedge J^{2}+P^{1}\wedge P^{2})$
(70)

where $f\in C^{1}(\mathfrak{h}_{\lambda,0}^{*})$ is an arbitrary function,

$b(\psi_{0},\gamma_{0})=\frac{1}{\alpha}\begin{cases}\frac{\sin\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)}{\cos\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)+\cosh\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)},\qquad&\lambda<0\\\
\tan\left(\frac{\psi_{0}-\psi_{C}}{\alpha}\right),\qquad&\lambda=0\\\
\frac{\sin\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)}{\cos\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)+\cos\left(\frac{2\sqrt{\lambda}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)},\qquad&\lambda>0\end{cases}$
(71)

and

$c(\psi_{0},\gamma_{0})=\frac{1}{\alpha}\begin{cases}\frac{1}{\sqrt{|\lambda|}}\frac{\sinh\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)}{\cos\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)+\cosh\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)},\qquad&\lambda<0\\\
\frac{\gamma_{0}-\gamma_{C}}{2\cos^{2}\left(\frac{\psi_{0}-\psi_{C}}{\alpha}\right)},\qquad&\lambda=0\\\
\frac{1}{\sqrt{\lambda}}\frac{\sin\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)}{\cos\left(\frac{2}{\alpha}(\psi_{0}-\psi_{C})\right)+\cos\left(\frac{2\sqrt{\lambda}}{\alpha}\left(\gamma_{0}-\gamma_{C}\right)\right)},\qquad&\lambda>0\end{cases}.$
(72)

Similarly, for $\beta=0$, the most general classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,1},K_{\alpha 0})$ $r$-matrix is
given by

$r_{d}^{\beta=0}(\psi_{1},\gamma_{1})=K_{\alpha\beta}+f(\psi_{1},\gamma_{1})(P^{1}\wedge
J^{1})+b(\psi_{1},\gamma_{1})(P^{2}\wedge J^{0}-J^{2}\wedge
P^{0})+c(\psi_{1},\gamma_{1})(\lambda J^{2}\wedge J^{0}+P^{2}\wedge P^{0}),$
(73)

where $f\in C^{1}(\mathfrak{h}_{\lambda,1}^{*})$ is an arbitrary function,

$b(\psi_{1},\gamma_{1})=\frac{1}{\alpha}\begin{cases}\frac{\sinh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)}{\cosh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)+\cos\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)},\qquad&\lambda<0\\\
\tanh\left(\frac{\psi_{1}-\psi_{C}}{\alpha}\right),\qquad&\lambda=0\\\
\frac{\sinh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)}{\cosh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)+\cosh\left(\frac{2\sqrt{\lambda}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)},\qquad&\lambda>0\end{cases}$
(74)

and

$c(\psi_{0},\alpha_{0})=\frac{1}{\alpha}\begin{cases}\frac{1}{\sqrt{|\lambda|}}\frac{\sin\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)}{\cosh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)+\cos\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)},\qquad&\lambda<0\\\
\frac{\gamma_{1}-\gamma_{C}}{2\cosh^{2}\left(\frac{\psi_{1}-\psi_{C}}{\alpha}\right)},\qquad&\lambda=0\\\
\frac{1}{\sqrt{\lambda}}\frac{\sinh\left(\frac{2\sqrt{|\lambda|}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)}{\cosh\left(\frac{2}{\alpha}(\psi_{1}-\psi_{C})\right)+\cosh\left(\frac{2\sqrt{\lambda}}{\alpha}\left(\gamma_{1}-\gamma_{C}\right)\right)},\qquad&\lambda>0\end{cases}.$
(75)

These correspond to extensions, for both signatures and any value of the
cosmological constant $\Lambda_{C}$, of the classical dynamical $r$-matrices
found in [5] (Lemma 4.11) for $\Lambda_{C}=0$ in the Lorentzian case (with
$\psi_{C}=\gamma_{C}=0$, $\alpha=2$ and $f\equiv 0$).

Theorem 2 provides all the elements in
$\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})$
for $\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$, parametrized in terms of
$(\psi_{C},\gamma_{C})\in\mathbb{R}^{2}$ and $f\in
C^{1}(\mathfrak{h}_{\lambda}^{*})$ in both cases. In order to achieve a full
description of the moduli space of classical dynamical $r$-matrices
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g},K_{\alpha,\beta})$ we first need to
consider the quotients

$\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda},K_{\alpha\beta})/\mathcal{G}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda})$

for both $\mathfrak{h}_{\lambda,0}$ and $\mathfrak{h}_{\lambda,1}$ (if it is
the case).

######  Lemma 3.

The space of orbits of classical dynamical
($\mathfrak{g}_{\lambda}$,$\mathfrak{h}_{\lambda}$,$K_{\alpha\beta}$)
$r$-matrices with respect to the action of dynamical
($\mathfrak{g}_{\lambda}$,$\mathfrak{h}_{\lambda}$) gauge transformations is
parametrized by pairs $(\psi_{C},\gamma_{C})\in\mathbb{R}^{2}$, for both
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ and
$\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,1}$.

###### Proof.

Theorem 2 states that any classical dynamical
($\mathfrak{g}_{\lambda}$,$\mathfrak{h}_{\lambda,i}$,$K_{\alpha\beta}$)
$r$-matrix are of the form

$r_{d}(\psi_{0},\gamma_{0})=K_{\alpha\beta}+f(\psi_{0},\gamma_{0})(P^{0}\wedge
J^{0})+b(\psi_{0},\gamma_{0})(P^{1}\wedge J^{2}-P^{2}\wedge
J^{1})+c(\psi_{0},\gamma_{0})(\lambda J^{1}\wedge J^{2}+P^{1}\wedge P^{2})$

and

$r_{d}(\psi_{1},\gamma_{1})=K_{\alpha\beta}+f(\psi_{1},\gamma_{1})(P^{1}\wedge
J^{1})+b(\psi_{1},\gamma_{1})(P^{2}\wedge J^{0}-J^{2}\wedge
P^{0})+c(\psi_{1},\gamma_{1})(\lambda J^{2}\wedge J^{0}+P^{2}\wedge P^{0}),$

for $i=0$ and $i=1$, respectively, with
$f:\in\mathcal{F}(\mathfrak{h}_{\lambda,i}^{*})$ an arbitrary function and
$b,c:\mathfrak{h}_{\lambda,i}^{*}\to\mathbb{R}$ given by (53) and (54).

All the elements of
$\mathcal{G}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i})$ are of the form

$g(\psi_{i},\gamma_{i})=\exp(m(\psi_{i},\gamma_{i})J_{i}+s(\psi_{i},\gamma_{i})P_{i})$

with $m,s:\mathfrak{h}_{\lambda,i}^{*}\to\mathbb{R}$ smooth maps.

Hence the action of any
$g\in\mathcal{G}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i})$ over any
element
$r_{d}\in\text{Dyn}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i},K_{\alpha\beta})$
is explicitly given by

$\begin{split}g\rhd
r_{d}&=(\text{Ad}(g)\otimes\text{Ad}(g))\left[r_{d}-\left(\frac{\partial
s}{\partial\gamma_{i}}-\frac{\partial m}{\partial\psi_{i}}\right)P^{i}\wedge
J^{i}\right]\\\ &=r_{d}-\left(\frac{\partial
s}{\partial\gamma_{i}}-\frac{\partial m}{\partial\psi_{i}}\right)P^{i}\wedge
J^{i}.\end{split}$

Since, by the Poincaré Lemma, for every function
$f\in\mathcal{F}(\mathfrak{h}_{\lambda,i}^{*})$ there exist smooth functions
$s,m:\mathfrak{h}_{\lambda,i}^{*}\to\mathbb{R}$ such that

$\frac{\partial s}{\partial\gamma_{i}}-\frac{\partial m}{\partial\psi_{i}}=f,$

we conclude that any classical dynamical
($\mathfrak{g}_{\lambda}$,$\mathfrak{h}_{\lambda,i}$,$K_{\alpha\beta}$)
$r$-matrix is gauge equivalent to

$r_{d}(\psi_{0},\gamma_{0})=K_{\alpha\beta}+b(\psi_{0},\gamma_{0})(P^{1}\wedge
J^{2}-P^{2}\wedge J^{1})+c(\psi_{0},\gamma_{0})(\lambda J^{1}\wedge
J^{2}+P^{1}\wedge P^{2})$ (76)

and

$r_{d}(\psi_{1},\gamma_{1})=K_{\alpha\beta}+b(\psi_{1},\gamma_{1})(P^{2}\wedge
J^{0}-J^{2}\wedge P^{0})+c(\psi_{1},\gamma_{1})(\lambda J^{2}\wedge
J^{0}+P^{2}\wedge P^{0}),$ (77)

for $i=0$ and $i=1$, respectively, with $b$ and $c$ as given in Theorem 2.

Hence, since the functions $b$ and $c$ above are uniquely determined by the
constants $\psi_{C}$ and $\gamma_{C}$, it follows each of the quotients
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i},K_{\alpha\beta})$
is in bijection with $\mathbb{R}^{2}$. ∎

Theorem 2 and Lemma 3 provides a parametrization and complete description of
all the elements in the moduli space of Cartan classical dynamical
$r$-matrices associated to $(\mathfrak{g}_{\lambda},K_{\alpha\beta})$

$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})=\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})\sqcup\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,1},K_{\alpha\beta})$

for every $\lambda,\alpha,\beta\in\mathbb{R}$. This amounts, following the
explanation in Subsection 1.4, to a full description of all the Poisson
structures over the gauge-fixed space of $G_{\lambda}$-flat connections over
Riemann surfaces.

Finally, the set of all the Cartan classical dynamical $r$-matrices associated
to the pair $(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ is then given by

$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})=\left[\bigcup_{g\in
G}\text{Dyn}^{g}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})\right]\sqcup\left[\bigcup_{g\in
G}\text{Dyn}^{g}(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,1},K_{\alpha\beta})\right].$

###  3.2 Dynamical generalizations of classical $r$-matrices for
$\mathfrak{g}_{\lambda}$

A systematic algebraic analysis of the CYBE for $\mathfrak{g}_{\lambda}$ and
the derivation of some particular solutions were considered in [31]. In this
spirit now we examine some particular solutions of the CDYBE for
$\mathfrak{g}_{\lambda}$, using for this purpose the equivalent description
derived in Theorem 1. We recall that even though we are mainly interested in
classical dynamical $r$-matrices (where the equivariance condition is
required), since these appear in the Poisson structure of gauge-fixed
character varieties, the space of solutions of the CDYBE has been studied too
(as mentioned in passing in our Introduction) and also, it is interesting to
see what dynamical generalizations of some well-known solutions of the CYBE
for $\mathfrak{g}_{\lambda}$ look like.

Following the presentation in [31], we consider solutions of the CDBYE (1)
associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})$

$r(\psi_{0},\gamma_{0})=K_{\alpha\beta}+A_{ab}(\psi_{0},\gamma_{0})J^{a}\otimes
J^{b}+B_{ab}(\psi_{0},\gamma_{0})(P^{a}\otimes J^{b}-J^{b}\otimes
P^{a})+C_{ab}(\psi_{0},\gamma_{0})P^{a}\otimes P^{b}$

for the cases where (I) $B$ is diagonal and $C\equiv 0$, (II) $A=\lambda C$
and $B$ is skew-symmetric and (III) $A=C\equiv 0$ and B is skew-symmetric,
obtaining in this way dynamical generalizations of the so-called (see Table 1
in [31] for the terminology) classical doubles, generalised complexifications
and kappa-Poincaré $r$-matrices, respectively.

Here we focus on the $\mathfrak{h}_{\lambda}=\mathfrak{h}_{\lambda,0}$ case,
since for $\mathfrak{h}_{\lambda,1}$ analogous solutions of the CDYBE are
obtained (as we observed in the previous subsection) and then via the
$G_{\lambda}$-action (4) all the solutions presented in this section can be
derived for any Cartan subalgebra of $\mathfrak{g}_{\lambda}$.

(I) Solutions with $B$ diagonal and trivial $C$. If we consider solutions with
$C\equiv 0$, the equation (33) forces $\nu=0$, while the other 3 equations
(32), (34) and (35) reduce to

$\begin{split}\frac{1}{2}\text{tr}(A^{2})-\frac{\lambda}{2}\left[\text{tr}(B)^{2}-\text{tr}(B^{2})\right]+\text{div}_{\bm{\alpha}}(v^{A})&=\mu\lambda\\\
[B-\text{tr}(B)\text{id}](B+B^{t})+\frac{1}{2}[\text{tr}(B)^{2}-\text{tr}(B^{2})]\text{id}+\text{curl}_{\bm{\psi}}(B)&=-\mu\text{id}\\\
-A(B+B^{t})-(B^{t}-\text{tr}(B)\text{id})A-\text{tr}(AB)\text{id}+\text{curl}_{\bm{\alpha}}(B^{t})-\text{grad}_{\bm{\psi}}(v^{A})&=0\end{split}$
(78)

Denoting $v^{A}(\psi_{0},\gamma_{0})$ by
$(a_{0}(\psi_{0},\gamma_{0}),a_{1}(\psi_{0},\gamma_{0}),a_{2}(\psi_{0},\gamma_{0}))$
and taking

$B(\alpha_{0},\psi_{0})=\text{diag}(b_{0}(\psi_{0},\gamma_{0}),b_{1}(\psi_{0},\gamma_{0}),b_{2}(\psi_{0},\gamma_{0})),$

(78) leads to the following system of PDEs

$\displaystyle-\sum_{i=0}^{2}a_{i}^{2}-\lambda(b_{0}b_{1}+b_{0}b_{2}+b_{1}b_{2})+\frac{\partial
a_{0}}{\partial\gamma_{0}}$ $\displaystyle=\mu\lambda,$ (79a)
$\displaystyle\begin{pmatrix}b_{1}b_{2}-b_{0}(b_{1}+b_{2})&0&0\\\
0&b_{0}b_{2}-b_{1}(b_{0}+b_{2})&\frac{\partial b_{1}}{\partial\psi_{0}}\\\
0&-\frac{\partial
b_{2}}{\partial\psi_{0}}&b_{0}b_{1}-b_{2}(b_{0}+b_{1})\end{pmatrix}$
$\displaystyle=-\mu\text{id},$ (79b)
$\displaystyle\begin{pmatrix}-\frac{\partial
a_{0}}{\partial\psi_{0}}&a_{2}(b_{1}-b_{2})-\frac{\partial
a_{1}}{\partial\psi_{0}}&-a_{1}(b_{1}-b_{2})-\frac{\partial
a_{2}}{\partial\psi_{0}}\\\
-a_{2}(b_{0}-b_{2})&0&a_{0}(b_{0}-b_{2})+\frac{\partial
b_{1}}{\partial\gamma_{0}}\\\
a_{1}(b_{0}-b_{1})&-a_{0}(b_{0}-b_{1})-\frac{\partial
b_{2}}{\partial\gamma_{0}}&0\end{pmatrix}$ $\displaystyle=0.$ (79c)

From equations (79b) and (79c) we conclude

$B=\pm\sqrt{\mu}\text{id}$ (80)

and

$A(\psi_{0},\gamma_{0})=A(\gamma_{0}),$ (81)

while equation (79a) reduces to a constraint over the vector $v^{A}$
associated to the anti-symmetric matrix $A$,

$\langle
v^{A}(\gamma_{0}),v^{A}(\gamma_{0})\rangle-\frac{da_{0}}{d\gamma_{0}}=-4\mu\lambda.$
(82)

Hence for diagonal $B$ and trivial $C$, we have found there exist solutions to
the CDYBE for
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})$ of the
form

$\begin{split}r^{\beta=0}_{D}(\gamma_{0})&=\frac{1}{\alpha}(J_{a}\otimes
P^{a}+P_{a}\otimes J^{a})+\epsilon^{abc}v_{a}(\gamma_{0})J_{b}\otimes
J_{c}\pm\sqrt{\mu}(P^{a}\otimes J_{a}-J_{a}\otimes P^{a})\\\
&=\left(\frac{1}{\alpha}\mp\sqrt{\mu}\right)J_{a}\otimes
P^{a}+\left(\frac{1}{\alpha}\pm\sqrt{\mu}\right)P_{a}\otimes
J^{a}+\epsilon^{abc}v_{a}(\gamma_{0})J_{b}\otimes J_{c}\end{split}$ (83)

if $\beta=0$, or

$\begin{split}r^{\alpha=0}_{D}(\gamma_{0})&=\frac{1}{\lambda\beta}(\lambda
J_{a}\otimes J^{a}+P_{a}\otimes
P^{a})+\epsilon^{abc}v_{a}(\gamma_{0})J_{b}\otimes
J_{c}\pm\sqrt{\mu}(P^{a}\otimes J_{a}-J_{a}\otimes P^{a})\end{split}$ (84)

if $\alpha=0$, where the choice of sing depends if we are dealing with the
Euclidean or Lorentzian case, respectively.

The family of classical dynamical $r$-matrices (83) is a dynamical
generalization of the well-known classical $r$-matrices associated to the
standard double bialgebra structures over $\mathfrak{g}_{\lambda}$. Similarly,
the family (84) is the dynamical generalization of the classical $r$-matrices
related to exotic double structures over $\mathfrak{g}_{\lambda}$.

(II) Solutions with $A=\lambda C$ and $B$ skew-symmetric. If now we demand for
solutions of the CDYBE such that $A=\lambda C$, the set (32)-(35) reduces to
the following system of coupled PDEs

$\begin{split}\frac{\lambda}{2}\text{tr}(C^{2})-\frac{1}{2}[\text{tr}(B)^{2}-\text{tr}(B^{2})]+\text{div}_{\bm{\gamma}}(v^{C})&=\mu,\\\
\text{tr}(BC)+\text{div}_{\bm{\psi}}(v^{C})&=\nu,\\\
[B-\text{tr}(B)\text{id}](B+B^{t})+\text{div}_{\bm{\gamma}}(v^{C})\text{id}+\text{curl}_{\bm{\psi}}(B)-\text{grad}_{\bm{\gamma}}(v^{C})&=0,\\\
-\lambda
C(B+B^{t})+\lambda[\text{div}_{\bm{\psi}}(v^{C})\text{id}-\text{grad}_{\bm{\psi}}(v^{C})]+\text{curl}_{\bm{\gamma}}(B^{t})&=0.\end{split}$
(85)

This system is still complicated enough to solve in a general way, reason why
we focused on some specific solutions corresponding to a particular choice of
the maps $C$ and $B$: For $B=[v^{B}(\psi_{0},\gamma_{0}),\ \cdot\ ]$ and
$C=[v^{C}(\psi_{0},\gamma_{0}),\ \cdot\ ]$, with $v^{B}(\psi_{0},\gamma_{0})$
and $v^{C}(\psi_{0},\gamma_{0})$ in
$C^{1}(\mathfrak{h}_{\lambda,0}^{*})\otimes\mathbb{R}^{3}$ or in
$C^{1}(\mathfrak{h}_{\lambda,0}^{*})\otimes\mathbb{M}^{1,2}$, given by

$v^{B}(\psi_{0},\gamma_{0})=(b_{0}(\psi_{0},\gamma_{0}),b_{1}(\psi_{0},\gamma_{0}),b_{2}(\psi_{0},\gamma_{0})\text{
and
}v^{C}(\psi_{0},\gamma_{0})=(c_{0}(\psi_{0},\gamma_{0}),c_{1}(\psi_{0},\gamma_{0}),c_{2}(\psi_{0},\gamma_{0})),$

the set of equations (85) reduces to

$\begin{split}-\lambda\langle v^{C},v^{C}\rangle-\langle
v^{B},v^{B}\rangle+\frac{\partial c_{0}}{\partial\gamma_{0}}&=\mu,\\\
-2\langle v^{B},v^{C}\rangle+\frac{\partial c_{0}}{\partial\psi_{0}}&=\nu,\\\
\frac{\partial b_{i}}{\partial\psi_{0}}-\frac{\partial
c_{i}}{\partial\gamma_{0}}&=0\text{ for }i=0,1,2,\\\ \lambda\frac{\partial
c_{i}}{\partial\psi_{0}}-\frac{\partial b_{i}}{\partial\gamma_{0}}&=0\text{
for }i=0,1,2\end{split}$ (86)

which precisely, reduces to the system (58) associated to classical dynamical
$r$-matrices in the case $v^{B}\parallel\bm{e}_{0}$ and
$v^{C}\parallel\bm{e}_{0}$.

(III) Solutions with $A$ and $C$ trivial and $B$ skew-symmetric. If in the
previous case, we additionally consider $C=0$, then (86) reduces to

$\begin{split}\langle v^{B},v^{B}\rangle&=-\mu\\\ \frac{\partial
b_{0}}{\partial\gamma_{0}}=\frac{\partial
b_{1}}{\partial\gamma_{0}}=\frac{\partial b_{2}}{\partial\gamma_{0}}&=0\\\
\frac{\partial b_{0}}{\partial\psi_{0}}=\frac{\partial
b_{1}}{\partial\psi_{0}}=\frac{\partial
b_{2}}{\partial\psi_{0}}&=0\end{split}$ (87)

In other words, $v^{B}$ is a constant vector function (just an element in
$\mathbb{R}^{3}$ or $\mathbb{M}^{1,2}$) whose (pseudo)norm is given by $-\mu$.
Therefore, we find there are not dynamical generalizations of the classical
Kappa-Poincaré $r$-matrices. Explicitly, since $\nu=0$, there exist just two
possible dynamical $r$-matrices of this type given by

$r_{\kappa P}^{\beta=0}=\frac{1}{\alpha}(J_{a}\otimes P^{a}+P_{a}\otimes
J^{a})+\epsilon_{abc}v^{a}(P^{b}\otimes J^{c}-J^{b}\otimes
P^{c})\quad\text{with}\quad\langle v,v\rangle=-\frac{1}{\alpha^{2}}$ (88)

or

$r_{\kappa P}^{\alpha=0}=\frac{1}{\lambda\beta}(\lambda J_{a}\otimes
J^{a}+P_{a}\otimes P^{a})+\epsilon_{abc}v^{a}(P^{b}\otimes J^{c}-J^{b}\otimes
P^{c})\quad\text{with}\quad\langle v,v\rangle=-\frac{1}{\lambda\beta^{2}}$
(89)

which are precisely the 2 possible cases described in [34] (equations (5.5)
and (5.6)).

Therefore $r_{\kappa P}^{\beta=0}$ and $r_{\kappa P}^{\alpha=0}$ correspond to
classical dynamical
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha 0})$ and
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{0\beta})$ r-matrices,
respectively, if and only if $v\in\text{Span}\\{\bm{e}_{0}\\}$. In other
words, the $\mathfrak{h}_{\lambda,0}$ equivariance condition constrains the
vector defining the classical dynamical $r$-matrix to have the same direction
of the generators ($J_{0}$ and $P_{0}$) of $\mathfrak{h}_{\lambda,0}$.

##  4 The FGMPP dynamical $r$-matrices for $\mathfrak{g}_{\lambda}$

Theorem 2 shows that dynamical generalized complexifications, i.e. with
coefficients solving (86), are the only solutions of the CDYBE that actually
are classical dynamical $r$-matrices and therefore have a natural origin in
the setting of gauge-fixed character varieties (see e.g. [4] and [5]). In this
section we show how these solutions are related (indeed coincide up to
dynamical gauge transformations) with a family of classical dynamical
$r$-matrices found by Feher, Gabor, Marshall, Palla and Pusztai when studing
the (quasi-)Poisson structures of the chiral WZNW model (see [35]).

Alekseev and Meinrenken (AM) [36] found that for any self-dual Lie algebra
$\mathfrak{g}$, i.e. equipped with a non-degenerate $Ad$-invariant symmetric
bilinear form $\langle\cdot,\cdot\rangle$, there exists a canonical classical
dynamical $(\mathfrak{g},\mathfrak{g},K)$ $r$-matrix
$r_{\text{AM}}:\mathfrak{g}^{*}\to\mathfrak{g}\otimes\mathfrak{g}$, where $K$
is the element in $S^{2}(\mathfrak{g})^{\mathfrak{g}}$ associated to
$\langle\cdot,\cdot\rangle$ and $\mathfrak{g}^{*}$ is the dual of
$\mathfrak{g}$ with respect to the bilinear form, given by

$r_{\text{AM}}(x)=K+\rho_{\text{AM}}(x)$ (90)

such that

$\begin{split}\rho_{\text{AM}}:\mathfrak{g}^{*}&\to\mathfrak{g}\wedge\mathfrak{g}\\\
x&\mapsto f(1\otimes\text{ad}[K^{\lor}(x)])(K)\end{split}$ (91)

with $K^{\lor}$ the linear map $\mathfrak{g}^{*}\to\mathfrak{g}$ associated to
$K$ and

$f(z)=\coth(z)-\frac{1}{z}.$ (92)

The adjective canonical was introduced by [14] and extensively used since then
(see e.g. [37]), given that any classical dynamical
$(\mathfrak{g},\mathfrak{g},K)$ is dynamical gauge equivalent to
$r_{\text{AM}}(x-x_{0})$ for a shift $x_{0}\in\mathfrak{g}^{*}$.

Analogously, as explained for gauge-fixed character varieties (18), this
classical dynamical ($\mathfrak{g},\mathfrak{g},K$) $r$-matrix defines a
Poisson structure over

$\mathfrak{g}^{*}\times G$

given by

$\displaystyle\\{t^{a},t^{b}\\}$ $\displaystyle=-f^{ab}_{c}t^{c},$ (93a)
$\displaystyle\\{F,t^{a}\\}$ $\displaystyle=(R_{t_{a}}+L_{t_{a}})F,$ (93b)
$\displaystyle\\{F,\tilde{F}\\}(t)$
$\displaystyle=\Pi_{\text{ext}}^{0,1}(r_{\text{AM}}(t))(dF,d\tilde{F}),$ (93c)

where $F,\tilde{F}\in C^{\infty}(G)$,
$\\{t^{a}\\}_{a=1,\cdots,\text{dim}\mathfrak{g}}$ is a basis of $\mathfrak{g}$
and $\\{t_{a}\\}_{a=1,\cdots,\dim\mathfrak{g}}$ its dual in
$\mathfrak{g}^{*}$.

Then we consider a self-dual Lie subalgebra of $\mathfrak{h}$ of
$\mathfrak{g}$ (i.e. such that the restriction
$\langle\cdot,\cdot\rangle\big{|}_{\mathfrak{h}}$ is non-degenerate) and
decompose (even adapting) the basis
$\\{t^{a}\\}_{a=1,\cdots,\dim\mathfrak{g}}$ of $\mathfrak{g}$ into a basis
$\\{h^{a}\\}_{a=1,\cdots,\dim\mathfrak{h}}$ for $\mathfrak{h}$ and
$\\{\tilde{h}^{a}\\}_{a=1,\cdots,\dim\mathfrak{h}^{\perp}}$ for
$\mathfrak{h}^{\perp}$, since
$\mathfrak{g}=\mathfrak{h}+\mathfrak{h}^{\perp}$. Feher, Gabor, Marshall,
Palla and Pusztai in a series of papers (see ([7], [38], [39], [35], [40],
[41]) considered the reduced subspace of $\mathfrak{g}^{*}\times G$ obtained
by imposing the first class constraints $\tilde{h}^{a}\approx 0$ for
$a=1,\cdots,\dim\mathfrak{h}^{\perp}$. In other words, by seeing
$\\{\tilde{h}^{a}\\}_{a=1,\cdots,\dim\mathfrak{h}^{\perp}}$ as functions over
$\mathfrak{g}^{*}\times G$, the Poisson structure (93) is modified in such a
way these functions become Poisson functions with respect to the new (Dirac)
bracket.

Exactly as before, these new Poisson structure is obtained via gauge fixing,
getting so a Poisson structure

$\\{F,\tilde{F}\\}_{D}=\\{F,\tilde{F}\\}-\\{F_{1},e^{a}\\}C_{ab}^{-1}(t_{\mathfrak{h}})\\{e^{b},\tilde{F}\\}$

over

$\mathfrak{h}^{*}\times G$

(after strongly imposing the constraints $\tilde{h}^{a}=0$ for
$a=1,\cdots,\dim\mathfrak{h}^{\perp}$) where

$C^{ab}(t_{\mathfrak{h}})=\\{t^{a},t^{b}\\}|_{t_{\mathfrak{h}^{\perp}}=0}=C(t_{\mathfrak{h}})=\text{ad}(t_{\mathfrak{h}})|_{\mathfrak{h}^{\perp}}.$

Thus, this gauge fixing amounts to modify (93) by changing the skew-symmetric
part of the AM classical dynamical $r$-matrix by

$\rho_{\text{FGMPP}}(x)=\rho_{\text{AM}}(x)+\frac{1}{\text{ad}(x)},$ (94)

getting in this way a Poisson structure over $\mathfrak{h}^{*}\times G$ with
Dirac brackets given by

$\displaystyle\\{h^{a},h^{b}\\}_{D}$ $\displaystyle=0,$ (95a)
$\displaystyle\\{F,h^{a}\\}_{D}$ $\displaystyle=(R_{h^{a}}+L_{h^{a}})F,$ (95b)
$\displaystyle\\{F,\tilde{F}\\}_{D}(h)$
$\displaystyle=\Pi_{ext}^{0,1}(r_{\text{FGMPP}}(h))(dF,d\tilde{F}),$ (95c)

where the canonical classical dynamical ($\mathfrak{g},\mathfrak{h},\Omega$)
$r$-matrix (denoted here by FGMPP) is given by

$r_{\text{FGMPP}}(x)=K+\rho_{\text{FGMPP}}(x)$ (96)

such that

$\begin{split}\rho_{\text{FGMPP}}:\mathfrak{h}^{*}&\to\mathfrak{h}^{\perp}\wedge\mathfrak{h}^{\perp}\\\
x&\mapsto g(1\otimes\text{ad}[K^{\lor}(x)])(K)\end{split}$ (97)

with

$g(z)=\coth(z).$ (98)

In this section we compute the FGMPP classical dynamical $r$-matrices
associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i},K_{\alpha\beta})$ for
$i=0,1$ and any $\alpha,\beta\in\mathbb{R}$, proving by direct computation the
following.

######  Theorem 3.

The generalized complexified classical dynamical
($\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i},K_{\alpha\beta}$)
$r$-matrices (76) and (77) are gauge equivalent to the FGMPP classical
dynamical $r$-matrices (96) associated to
($\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,i},K_{\alpha\beta}$).

###### Proof.

We start by considering the $i=0$ case. Since

$K_{\alpha\beta}\equiv\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}(J_{a}\otimes
P^{a}+P_{a}\otimes J^{a})-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}(\lambda
J_{a}\otimes J^{a}+P_{a}\otimes P^{a}),$

for a general element $x_{0}=\gamma_{0}J_{0}^{*}+\psi_{0}P_{0}^{*}$ in
$\mathfrak{h}^{*}_{\lambda,0}$ we have

$K^{\lor}_{\alpha\beta}(x_{0})=\xi(\gamma_{0},\psi_{0},\alpha,\beta)J_{0}+\zeta(\gamma_{0},\psi_{0},\alpha,\beta)P_{0}$

with

$\xi(\gamma_{0},\psi_{0},\alpha,\beta)=\psi_{0}\left(\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}\right)-\gamma_{0}\left(\frac{\beta\lambda}{\alpha^{2}-\lambda\beta^{2}}\right)$

and

$\zeta(\gamma_{0},\psi_{0},\alpha,\beta)=\gamma_{0}\left(\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}\right)-\psi_{0}\left(\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}\right).$

Note these are precisely the arguments of the functions $B$ and $C$ in (53)
and (54).

Thus, according to (12), we get

$\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}:\begin{pmatrix}J_{1}\\\
J_{2}\\\ P_{1}\\\ P_{2}\end{pmatrix}\mapsto\xi\begin{pmatrix}-J_{2}\\\
J_{1}\\\ -P_{2}\\\ P_{1}\end{pmatrix}+\zeta\begin{pmatrix}-P_{2}\\\ P_{1}\\\
-\lambda J_{2}\\\ \lambda J_{1}\end{pmatrix}=\begin{pmatrix}-(\xi J_{2}+\zeta
P_{2})\\\ \xi J_{1}+\zeta P_{1}\\\ -(\xi P_{2}+\zeta\lambda J_{2})\\\ \xi
P_{1}+\zeta\lambda J_{1}\end{pmatrix},$

where we have omitted the dependence of $\xi$ and $\zeta$ on
$\gamma_{0},\psi_{0},\alpha$ and $\beta$. Hence, the matrix representation of
$\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}$
in the (standard) basis $\\{J_{1},J_{2},P_{1},P_{2}\\}$ of
$\mathfrak{h}_{\lambda,0}^{\perp}$ is given by

$\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}=\begin{pmatrix}0&\xi&0&\zeta\lambda\\\
-\xi&0&-\zeta\lambda&0\\\ 0&\zeta&0&\xi\\\ -\zeta&0&-\xi&0\end{pmatrix}.$ (99)

As indicated in Appendix C, by the Weierstrass factorization theorem, the
function $g(z)$ is equal to

$g(z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{(\pi n)^{2}+z^{2}}.$ (100)

Therefore, given that

$\frac{\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}}{(\pi
n)^{2}+\left(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}\right)^{2}}=\begin{pmatrix}0&F_{n}(\zeta,\xi,\lambda)&0&\lambda
G_{n}(\zeta,\xi,\lambda)\\\ -F_{n}(\zeta,\xi,\lambda)&0&-\lambda
G_{n}(\zeta,\xi,\lambda)&0\\\
0&G_{n}(\zeta,\xi,\lambda)&0&F_{n}(\zeta,\xi,\lambda)\\\
-G_{n}(\zeta,\xi,\lambda)&0&-F_{n}(\zeta,\xi,\lambda)&0\end{pmatrix}$

where

$F_{n}(\zeta,\xi,\lambda)=\frac{\xi[(\pi
n)^{2}-(\xi^{2}-\lambda\zeta^{2})]}{[(\pi
n)^{2}-(\xi^{2}+\lambda\zeta^{2})]^{2}-(2\zeta\xi)^{2}\lambda},\quad
G_{n}(\alpha,\psi,\lambda)=\frac{\zeta[(\pi
n)^{2}+(\xi^{2}-\lambda\zeta^{2})]}{[(\pi
n)^{2}-(\xi^{2}+\lambda\zeta^{2})]^{2}-(2\zeta\xi)^{2}\lambda}$

and the matrix representation of the inverse of
$\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}$
is

$\frac{1}{\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}}=\frac{1}{\xi^{2}-\lambda\zeta^{2}}\begin{pmatrix}0&-\xi&0&\lambda\zeta\\\
\xi&0&-\lambda\zeta&0\\\ 0&\zeta&0&-\xi\\\ -\zeta&0&\xi&0\end{pmatrix},$

then we obtain the matrix representation of
$g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}})$
is

$g\left(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})]|_{\mathfrak{h}^{\perp}_{\lambda,0}}\right)=\begin{pmatrix}0&F(\zeta,\xi,\lambda)&0&\lambda
G(\zeta,\xi,\lambda)\\\ -F(\zeta,\xi,\lambda)&0&-\lambda
G(\zeta,\xi,\lambda)&0\\\ 0&G(\zeta,\xi,\lambda)&0&F(\zeta,\xi,\lambda)\\\
-G(\zeta,\xi,\lambda)&0&-F(\zeta,\xi,\lambda)&0\end{pmatrix}$ (101)

with

$\begin{split}F(\zeta,\xi,\lambda)&=-\frac{\xi}{\xi^{2}-\lambda\zeta^{2}}+2\sum_{n=1}^{\infty}F_{n}(\zeta,\xi,\lambda)\\\
&=-\frac{\xi}{\xi^{2}-\lambda\zeta^{2}}+\sum_{n=1}^{\infty}\frac{2\xi[(\pi
n)^{2}-(\xi^{2}-\lambda\zeta^{2})]}{[(\pi
n)^{2}-(\xi^{2}+\lambda\zeta^{2})]^{2}-(2\zeta\xi)^{2}\lambda}\\\
&=\frac{1}{2}\left[-\frac{1}{\xi+\theta\zeta}+2\sum_{n=1}^{\infty}\frac{\xi+\theta\zeta}{(\pi
n)^{2}-(\xi+\theta\zeta)^{2}}\right]+\frac{1}{2}\left[-\frac{1}{\xi-\theta\zeta}+2\sum_{n=1}^{\infty}\frac{\xi-\theta\zeta}{(\pi
n)^{2}-(\xi-\theta\zeta)^{2}}\right]\end{split}$

and

$\begin{split}G(\zeta,\xi,\lambda)&=\frac{\zeta}{\xi^{2}-\lambda\zeta^{2}}+2\sum_{n=1}^{\infty}G_{n}(\zeta,\xi,\lambda)\\\
&=\frac{\zeta}{\xi^{2}-\lambda\zeta^{2}}+\sum_{n=1}^{\infty}\frac{2\zeta[(\pi
n)^{2}+(\xi^{2}-\lambda\zeta^{2})]}{[(\pi
n)^{2}-(\xi^{2}+\lambda\zeta^{2})]^{2}-(2\zeta\xi)^{2}\lambda}\\\
&=\frac{1}{2\theta}\left[-\frac{1}{\xi+\theta\zeta}+2\sum_{n=1}^{\infty}\frac{\xi+\theta\zeta}{(\pi
n)^{2}-(\xi+\theta\zeta)^{2}}\right]-\frac{1}{2\theta}\left[-\frac{1}{\xi-\theta\zeta}+2\sum_{n=1}^{\infty}\frac{\xi-\theta\zeta}{(\pi
n)^{2}-(\xi-\theta\zeta)^{2}}\right],\end{split}$

where implicitly we have extended $\mathbb{R}$ to the ring
$\mathbb{R}_{\lambda}$ (20).

As shown in Appendix D, using again the Weierstrass factorization theorem and
properties of $\mathbb{R}_{\lambda}$, the functions $F(\zeta,\xi,\lambda)$ and
$G(\zeta,\xi,\lambda)$ over $\mathbb{R}_{\lambda}$ are given by the following
functions for the different signs of $\lambda$:

  * •

For $\lambda=0$, then

$\displaystyle F(\zeta,\xi,0)$
$\displaystyle=\tan\left(\xi-\frac{\pi}{2}\right),$ (102a) $\displaystyle
G(\zeta,\xi,0)$
$\displaystyle=\frac{\zeta}{2\cos^{2}\left(\xi-\frac{\pi}{2}\right)}$ (102b)

  * •

For $\lambda\neq 0$, then

$\displaystyle F(\zeta,\xi,\lambda)$
$\displaystyle=-\frac{1}{2}\left[\cot\left(\xi+\theta\zeta\right)+\cot\left(\xi-\theta\zeta\right)\right],$
(103a) $\displaystyle G(\zeta,\xi,\lambda)$
$\displaystyle=-\frac{1}{2\theta}\left[\cot\left(\xi+\theta\zeta\right)-\cot\left(\xi-\theta\zeta\right)\right]$
(103b)

which for $\lambda<0$ are given by

$\displaystyle F(\zeta,\xi,\lambda)$
$\displaystyle=\frac{-\sin(2\xi)}{-\cos(2\xi)+\cosh(2\sqrt{|\lambda|}\zeta)},$
(104a) $\displaystyle G(\zeta,\xi,\lambda)$
$\displaystyle=\frac{1}{\sqrt{|\lambda|}}\frac{\sinh(2\sqrt{|\lambda|}\zeta)}{-\cos(2\xi)+\cosh(2\sqrt{|\lambda|}\zeta)},$
(104b)

and for $\lambda>0$

$\displaystyle F(\zeta,\xi,\lambda)$
$\displaystyle=\frac{-\sin(2\xi)}{-\cos(2\xi)+\cos(2\sqrt{\lambda}\zeta)},$
(105a) $\displaystyle G(\zeta,\xi,\lambda)$
$\displaystyle=\frac{1}{\sqrt{\lambda}}\frac{\sin(2\sqrt{\lambda}\zeta)}{-\cos(2\xi)+\cos(2\sqrt{\lambda}\zeta)}.$
(105b)

Finally, replacing (101) in (97), we obtain the skew-symmetric part of the
canonical FGMPP dynamical $r$-matrix associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})$ is given
by

$\begin{split}\rho_{\text{FGMPP}}(x_{0})&=\langle
T_{a},g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})])J_{1}\rangle_{s}T^{a}\otimes
P^{1}+\langle
T_{a},g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})])J_{2}\rangle_{s}T^{a}\otimes
P^{2}\\\ &\quad+\langle
T_{a},g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})])P_{1}\rangle_{s}T^{a}\otimes
J^{1}+\langle
T_{a},g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{0})])P_{2}\rangle_{s}T^{a}\otimes
J^{2}\\\ &=\tilde{b}(\zeta,\xi,\lambda)(J^{1}\otimes P^{2}-J^{2}\otimes
P^{1}-P^{2}\otimes J^{1}+P^{1}\otimes J^{2})\\\
&\qquad+\tilde{c}(\zeta,\xi,\lambda)(P^{1}\otimes P^{2}-P^{2}\otimes
P^{1}-\lambda J^{2}\otimes J^{1}+\lambda J^{1}\otimes J^{2})\end{split}$

where

$\tilde{b}(\zeta,\xi,\lambda)=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}F(\zeta,\xi,\lambda)-\lambda\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}G(\zeta,\xi,\lambda)$
(106)

and

$\tilde{c}(\zeta,\xi,\lambda)=\frac{\alpha}{\alpha^{2}-\lambda\beta^{2}}F(\zeta,\xi,\lambda)-\frac{\beta}{\alpha^{2}-\lambda\beta^{2}}G(\zeta,\xi,\lambda)$
(107)

concluding the canonical FGMPP dynamical $r$-matrix associated to
$(\mathfrak{g}_{\lambda},\mathfrak{h}_{\lambda,0},K_{\alpha\beta})$ is given
by

$r_{\text{FGMPP}}(x_{0})=K_{\alpha\beta}+\tilde{b}(\xi,\zeta,\lambda)(P^{1}\wedge
J^{2}-P^{2}\wedge J^{1})+\tilde{c}(\xi,\zeta,\lambda)(\lambda J^{1}\wedge
J^{2}+P^{1}\wedge P^{2}),$ (108)

which coincides precisely with the solution $r_{d}(\psi_{0},\gamma_{0})$ in
Theorem 2 by choosing $\Psi_{0}=\frac{\pi}{2}\alpha$ and
$\Gamma_{0}=\frac{\pi}{2}\beta$.

In a completely analogous way if we consider a generic element
$x_{1}=\gamma_{1}J_{1}^{*}+\psi_{1}P_{1}^{*}$ in
$\mathfrak{h}^{*}_{\lambda,1}$, we obtain for
$g(\text{ad}[K_{\alpha\beta}^{\lor}(x_{1})]|_{\mathfrak{h}^{\perp}_{\lambda,1}})$
a matrix similar to (101) but with $F$ and $G$ now given by

$\displaystyle F(\zeta,\xi,0)$
$\displaystyle=\tanh\left(\xi-\frac{\pi}{2}\right),$ (109a) $\displaystyle
G(\zeta,\xi,0)$
$\displaystyle=\frac{\zeta}{2\cosh^{2}\left(\xi-\frac{\pi}{2}\right)}$ (109b)

for $\lambda=0$,

$\displaystyle F(\zeta,\xi,\lambda)$
$\displaystyle=\frac{\sinh(2\xi-\pi)}{\cosh(2\xi-\pi)+\cosh(2\sqrt{|\lambda|}\zeta)},$
(110a) $\displaystyle G(\zeta,\xi,\lambda)$
$\displaystyle=\frac{1}{\sqrt{|\lambda|}}\frac{\sin(2\sqrt{|\lambda|}\zeta)}{\cosh(2\xi-\pi)+\cos(2\sqrt{|\lambda|}\zeta)},$
(110b)

for $\lambda<0$, and

$\displaystyle F(\zeta,\xi,\lambda)$
$\displaystyle=\frac{\sinh(2\xi-\pi)}{\cosh(2\xi-\pi)+\cosh(2\sqrt{\lambda}\zeta)},$
(111a) $\displaystyle G(\zeta,\xi,\lambda)$
$\displaystyle=\frac{1}{\sqrt{\lambda}}\frac{\sinh(2\sqrt{\lambda}\zeta)}{\cosh(2\xi-\pi)+\cosh(2\sqrt{\lambda}\zeta)}.$
(111b)

for $\lambda>0$.

Thus, evaluating (97) in this case leads to a classical dynamical $r$-matrix
of the form

$r_{\text{FGMPP}}(x_{1})=K_{\alpha\beta}+\tilde{b}(\xi,\zeta,\lambda)(P^{2}\wedge
J^{0}-J^{2}\wedge P^{0})+\tilde{c}(\xi,\zeta,\lambda)(\lambda J^{2}\wedge
J^{0}+P^{2}\wedge P^{0}),$ (112)

with $\tilde{b}$ and $\tilde{c}$ defined exactly as before, which coincides
with the solution $r_{d}(\psi_{1},\gamma_{1})$ in Theorem 2 by choosing again
$\Psi_{0}=\frac{\pi}{2}\alpha$ and $\Gamma_{0}=\frac{\pi}{2}\beta$. ∎

##  5  Conclusion and outlook

In this paper we gave a full classification of the classical dynamical
$r$-matrices up to gauge equivalence for the Lie algebras
$\mathfrak{g}_{\lambda}$ of the local isometry groups of the maximally
symmetric Euclidean and Lorentzian spaces in three dimensions. The classical
dynamical $r$-matrices for vanishing cosmological constant in the Lorentzian
setting were obtained previously by Meusburger and Schönfeld in [5], but our
results for classical dynamical $r$-matrices for the general triple
($\mathfrak{g}_{\lambda}$, $\mathfrak{h}_{\lambda}$, $K_{\alpha\beta}$) are
new.

It is interesting to compare our results with the analogous study of (non-
dynamical) classical $r$-matrices in [31] for the family
$\mathfrak{g}_{\lambda}$. The general ansatz used here is a generalisation the
one used in that paper, where three families of solutions were identified,
associated with the Lie bialgebra structure of a classical double, a
generalised bicrossproducct (or $\kappa$-Poincaré algebra) or a generalised
complexification (in the same sense as used in this paper) of the standard
$\mathfrak{sl}(2,\mathbb{R})$ bialgebra structure. Here we found that, of
these three, only the dynamical version of the generalised complexification
solves both the dynamical classical Yang-Baxter equation and the equivariance
condition required for dynamical $r$-matrices. In particular, there is no
dynamical version of the $r$-matrices associated to the family of classical
double structures identified in [31]. It is interesting that the equivariance
constraint effectively makes the full classification of classical dynamical
$r$-matrices for $\mathfrak{g}_{\lambda}$ easier than the classification of
non-dynamical $r$-matrices for that family of Lie algebras (which indeed has
not yet been achieved).

While our results can be summarised conveniently as generalised
complexifications of known results for $\mathfrak{sl}(2,\mathbb{R})$ or
$\mathfrak{su}(2)$, our derivations and proofs do not make essential use of
this point of view. In future work, it may be interesting to see which parts
of standard holomorphic function theory extend when $\mathbb{C}$ is replaced
by the ring $\mathbb{R}_{\lambda}$, and if the resulting machinery is
sufficient to establish our results directly by ‘generalised holomorphic
methods’. It would then also be interesting if this point of view can be
usefully adopted in the quantization.

As remarked, the quantization of the Hamiltonian Chern-Simons theory (i.e.
over manifolds homeomorphic to $\mathbb{R}\times\Sigma_{g,n}$) reduces to the
quantization of a constrained Poisson space $\mathcal{P}_{\text{ext}}^{g,n}$,
whose Poisson structure is defined in terms of a classical $r$-matrix $r$. In
the so-called combinatorial quantization approach (see [20] and [21]) the
quantization is performed before imposing the constraints. It relies on the
existence of a quantum $R$-matrix which quantizes $r$, i.e. which has an
expansion

$R(\hbar)=1+\hbar r+\mathcal{O}(\hbar^{2}).$

Meanwhile in the approach presented here, the constraints are imposed before
quantization, leading to the appearance of a classical dynamical $r$-matrix
$r^{d}$ describing the Poisson structure of the (constrained) space
$\mathcal{P}^{g,n}$. Hence a dynamical combinatorial quantization (i.e. where
the classical $r$-matrix is replaced by a classical dynamical $r$-matrix
$r^{d}$, see [42]) is expected to lead to a quantization of the Hamiltonian
Chern-Simons theory. In analogy to the non-dynamical case, this quantization
scheme requires a quantum dynamical $R$-matrix that quantizes $r^{d}$. In our
case, where the Poisson structure is defined in terms of classical dynamical
$r$-matrices gauge equivalent to the classical dynamical $r$-matrix, the
quantization requires the existence of a quantum dynamical $R$-matrix, i.e.

$R(x,\hbar)=1+\hbar r(x)+\mathcal{O}(\hbar^{2}).$

The quantization of the AM classical dynamical $r$-matrix discussed at the
start of Section 4, was addressed by Enriquez and Etingof in [43] and the
extension of this approach to the quantization of the FGMPP classical
dynamical $r$-matrix seems possible and interesting. Further developments in
this direction could lead to a complete formulation of quantum character
varieties, with potentially interesting links to the categorical approaches
[19] and [44]), and a formal understanding of quantum of 3d gravity (see [2]).
Also, extensions of the results presented here to the setting of
supersymmetric Chern-Simons theory (see [45]) and Super Character Varieties
(see [46]) seem interesting. For the family of Lie algebras
$\mathfrak{g}_{\lambda}$ relevant for 3d gravity, the point of view of
generalised complexification, which proved so useful here, may provide
additional insights and connections.

Acknowledgments. We are grateful to Catherine Meusburger for helpful
discussions and explanations regarding the physical interpretation of the
classical dynamical $r$-matrices in the setting of 3d Chern-Simons gravity. We
would also like to thank her for sharing with us unpublished results
concerning extensions (for $\Lambda_{C}\neq 0$ and both signatures) of
classical dynamical $r$-matrices derived previously in [5], which coincide
with (70)-(70) for $\alpha=2$ and $\psi_{C}=\gamma_{C}=0$. The work of J.C.M.P
is supported by a James Watt scholarship from Heriot-Watt University.

## Appendix A: Action of $G$ over the space $\text{Dyn}(\mathfrak{g},K)$

Let $\mathfrak{g}$ be a Lie algebra and
$K\in(S^{2}\mathfrak{g})^{\mathfrak{g}}$. In this Appendix we prove that if
$\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are two conjugate Lie subalgebras
of $\mathfrak{g}$, say $\mathfrak{h}^{\prime}=\text{Ad}(g)(\mathfrak{h})$ for
$g\in G$, then the sets of classical dynamical $r$-matrices
$\text{Dyn}(\mathfrak{g},\mathfrak{h},K)$ and
$\text{Dyn}(\mathfrak{g},\mathfrak{h}^{\prime},K)$ are in bijection, via a
natural map constructed using the adjoint action of $G$ over
$\mathfrak{g}\otimes\mathfrak{g}$ and its coadjoint action over
$\mathfrak{g}^{*}$.

Lemma A. Let $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ be conjugate Lie
subalgebras of $\mathfrak{g}$, say by $g\in G$. Then the map

$\begin{split}\text{Dyn}(\mathfrak{g},\mathfrak{h},K)&\to\text{Dyn}(\mathfrak{g},\mathfrak{h}^{\prime},K)\\\
r(x)&\mapsto\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x)\end{split}$
(113)

is well-defined and bijective.

###### Proof.

Take $r\in\text{Dyn}(\mathfrak{g},\mathfrak{h},K)$ and let $g\in G$ such that
$\text{Ad}(g)(\mathfrak{h})=\mathfrak{h}^{\prime}$. Define

$r^{\prime}(x)=\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x)\qquad\text{for
}x\in(\mathfrak{h}^{\prime})^{*}$

where $\text{Ad}^{*}:G\to\text{Aut}(\mathfrak{g}^{*})$ is the coadjoint action
of the Lie group $G$ over the dual of the Lie algebra $\mathfrak{g}^{*}$.

Writing $r$ in terms of a basis $\\{T_{a}\\}_{a=1,\cdots,\dim\mathfrak{g}}$ of
$\mathfrak{g}$, we get $r(x)=r^{ab}(x)T_{a}\otimes T_{b}$ for all
$x\in\mathfrak{h}^{*}$ and by direct computation we obtain

$\begin{split}[r^{\prime}_{12}(x),r^{\prime}_{23}(x)]&=[r^{ab}(\text{Ad}^{*}(g^{-1})x)\text{Ad}(g)T_{a}\otimes\text{Ad}(g)T_{b}\otimes
1,r^{cd}(\text{Ad}^{*}(g^{-1})x)1\otimes\text{Ad}(g)T_{c}\otimes\text{Ad}(g)T_{d}]\\\
&=r^{ab}(\text{Ad}^{*}(g^{-1})x)r^{cd}(\text{Ad}^{*}(g^{-1})x)\text{Ad}(g)T_{a}\otimes[\text{Ad}(g)T_{b},\text{Ad}(g)T_{c}]\otimes\text{Ad}(g)T_{d}\\\
&=\text{Ad}(g)^{\otimes
3}[r_{12}(\text{Ad}^{*}(g^{-1})x),r_{23}(\text{Ad}^{*}(g^{-1})x)].\end{split}$

Similarly

$\begin{split}[r^{\prime}_{13}(x),r^{\prime}_{23}(x)]&=\text{Ad}(g)^{\otimes
3}[r_{13}(\text{Ad}^{*}(g^{-1})x),r_{23}(\text{Ad}^{*}(g^{-1})x)],\\\
[r^{\prime}_{12}(x),r^{\prime}_{13}(x)]&=\text{Ad}(g)^{\otimes
3}[r_{12}(\text{Ad}^{*}(g^{-1})x),r_{13}(\text{Ad}^{*}(g^{-1})x)]\end{split}$

and consequently

$\begin{split}[[r^{\prime}(x),r^{\prime}(x)]]&=\text{}[[\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x),\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x)]]\\\
&=\text{Ad}(g)^{\otimes
3}[[r(\text{Ad}^{*}(g^{-1})x),r(\text{Ad}^{*}(g^{-1})x)]].\end{split}$ (114)

Also, by direct computation

$\begin{split}{h^{\prime}_{i}}^{(1)}\frac{\partial r^{\prime}_{23}}{\partial
h^{\prime
i}}&=(\text{Ad}(g)h_{i})^{(1)}(\text{Ad}(g)\otimes\text{Ad}(g))\frac{\partial
r_{23}}{\partial h^{i}}(\text{Ad}^{*}(g^{-1})x)\\\ &=\text{Ad}(g)^{\otimes
3}h_{i}^{(1)}\frac{\partial r_{23}}{\partial
h^{i}}(\text{Ad}^{*}(g^{-1})x)\end{split}$

and analogously

${h^{\prime}_{i}}^{(2)}\frac{\partial r^{\prime}_{13}}{\partial h^{\prime
i}}=\text{Ad}(g)^{\otimes 3}h_{i}^{(2)}\frac{\partial r_{13}}{\partial
h^{i}}(\text{Ad}^{*}(g^{-1})x),\qquad{h^{\prime}_{i}}^{(3)}\frac{\partial
r^{\prime}_{12}}{\partial h^{\prime i}}=\text{Ad}(g)^{\otimes
3}h_{i}^{(3)}\frac{\partial r_{12}}{\partial h^{i}}(\text{Ad}^{*}(g^{-1})x).$

Hence

$\begin{split}\text{Alt}(dr^{\prime})(x)&=\sum_{i=1}^{\dim\mathfrak{h}^{\prime}}{h^{\prime}_{i}}^{(1)}\frac{\partial
r^{\prime}_{23}}{\partial h^{\prime
i}}(x)-{h^{\prime}_{i}}^{(2)}\frac{\partial r^{\prime}_{13}}{\partial
h^{\prime i}}(x)+{h^{\prime}_{i}}^{(3)}\frac{\partial
r^{\prime}_{12}}{\partial h^{\prime i}}(x)\\\ &=\text{Ad}(g)^{\otimes
3}\left[h_{i}^{(1)}\frac{\partial r_{23}}{\partial
h^{i}}(\text{Ad}^{*}(g^{-1})x)-h_{i}^{(1)}\frac{\partial r_{23}}{\partial
h^{i}}(\text{Ad}^{*}(g^{-1})x)+h_{i}^{(1)}\frac{\partial r_{23}}{\partial
h^{i}}(\text{Ad}^{*}(g^{-1})x)\right]\\\ &=\text{Ad}(g)^{\otimes
3}\text{Alt}(dr)(\text{Ad}^{*}(g^{-1})x)\end{split}$

and so, by combining with (114), we conclude $r^{\prime}$ is indeed a solution
of the CDYBE, i.e.

$\begin{split}\text{CDYB}(r^{\prime})(x)&=[[r^{\prime}(x),r^{\prime}(x)]]+\text{Alt}(dr)(x)\\\
&=\text{Alt}(g)^{\otimes
3}([[r(\text{Ad}^{*}(g^{-1})x),r(\text{Ad}^{*}(g^{-1})x)]]+\text{Alt}(dr)(\text{Ad}^{*}(g^{-1})x))\\\
&=\text{Alt}(g)^{\otimes 3}(\text{CDYBE}(r)(\text{Ad}^{*}(g^{-1})x))\\\
&=-\text{Alt}(g)^{\otimes 3}[[K,K]]\\\ &=-[[K,K]].\end{split}$

Since the subspace $S^{2}(\mathfrak{g})$ of $\mathfrak{g}\otimes\mathfrak{g}$
is invariant under the Adjoint action of the Lie group $G$, the fact
$\text{Sym}(r)=K\in(S^{2}\mathfrak{g})^{\mathfrak{g}}$ implies

$\text{Sym}(r^{\prime})=\text{Sym}(r)=K.$

Finally, take $h^{\prime}\in\mathfrak{h}^{\prime}$ and let $h\in\mathfrak{h}$
be such that $h^{\prime}=\text{Ad}(g)h$. Then we get

$\begin{split}\frac{d}{ds}\bigg{|}_{s=0}r^{\prime}(\text{Ad}^{*}(e^{sh^{\prime}})x)&=\text{Ad}(g)\otimes\text{Ad}(g)\frac{d}{ds}\bigg{|}_{s=0}r(\text{Ad}^{*}(g^{-1})\text{Ad}^{*}(e^{sh^{\prime}})x)\\\
&=\text{Ad}(g)\otimes\text{Ad}(g)\frac{d}{ds}\bigg{|}_{s=0}r(\text{Ad}^{*}(e^{sh})\text{Ad}^{*}(g^{-1})x),\\\
\end{split}$

where we have used
$\text{Ad}^{*}(e^{sh^{\prime}})=\text{Ad}^{*}(g)\text{Ad}^{*}(e^{sh})\text{Ad}^{*}(g^{-1})$,
and

$\begin{split}[r^{\prime}(x),h^{\prime}\otimes 1+1\otimes
h^{\prime}]&=[\text{Ad}(g)\otimes\text{Ad}(g)r(\text{Ad}^{*}(g^{-1})x),\text{Ad}(g)h\otimes
1+1\otimes\text{Ad}(g)h]\\\
&=\text{Ad}(g)\otimes\text{Ad}(g)[r(\text{Ad}^{*}(g^{-1})x),h\otimes
1+1\otimes h].\end{split}$

Therefore

$\frac{d}{ds}\bigg{|}_{s=0}r^{\prime}(\text{Ad}^{*}(e^{sh^{\prime}})x)+[r^{\prime}(x),h^{\prime}\otimes
1+1\otimes h^{\prime}]$

equals

$\text{Ad}(g)\otimes\text{Ad}(g)\left(\frac{d}{ds}\bigg{|}_{s=0}r(\text{Ad}^{*}(e^{sh})\text{Ad}^{*}(g^{-1})x)+[r(\text{Ad}^{*}(g^{-1})x),h\otimes
1+1\otimes h]\right)=0,$

i.e. the $\mathfrak{h}^{\prime}$-equivariance of $r^{\prime}$.

The previous computations show the map (113) is well-defined and is clearly
bijective (the inverse is obtained by replacing $g$ with $g^{-1}$). ∎

## Appendix B: Conjugacy classes of Cartan Subalgebras of
$\mathfrak{g}_{\lambda}$

One of the main results of this paper is the complete description of the set
$\text{Dyn}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ (Theorem
2) and the moduli space
$\mathcal{M}^{\mathfrak{C}}(\mathfrak{g}_{\lambda},K_{\alpha\beta})$ (Lemma 3)
of Cartan classical dynamical $r$-matrices associated to
$(\mathfrak{g}_{\lambda},K_{\alpha\beta})$. Lemma A is essential to achieve
this since reduces the task to finding classical dynamical $r$-matrices for
representatives of $\mathfrak{C}_{\mathfrak{g}_{\lambda}}^{\text{Ad}}$ (the
conjugacy classes of Cartan subalgebras of $\mathfrak{g}_{\lambda}$).

As stated in Subsection 3.1 the cardinality of
$\mathfrak{C}_{\mathfrak{g}_{\lambda}}^{\text{Ad}}$ is at most two for all
$\lambda\in\mathbb{R}$. More precisely, in the case where the cardinal is one
all the Cartan subalgebras are conjugate equivalent to
$\mathfrak{h}_{\lambda,0}$ (the Cartan subalgebra generated by $J_{0}$ and
$P_{0}$), while in the case it is two they could be conjugate equivalent to
$\mathfrak{h}_{\lambda,0}$ or $\mathfrak{h}_{\lambda,1}$ (the Cartan
subalgebra generated by $J_{1}$ and $P_{1}$).

In this Appendix we provide a detailed description of the Cartan subalgebras
of the simple ($\mathfrak{so}(3,1)$), semisimple ($\mathfrak{so}(4)$ and
$\mathfrak{so}(2,2)$) and indecomposable non-solvable ($\mathfrak{iso}(3)$ and
$\mathfrak{iso}(2,1)$) six dimensional real Lie algebras.

### Simple case: $\mathfrak{so}(3,1)$

By $\mathfrak{so}(3,1)_{\lambda}$ ,with $\lambda<0$, we denote the six
dimensional real algebra generated by
$\\{J_{0},J_{1},J_{2},P_{0},P_{1},P_{2}\\}$ such that

$[J_{a},J_{b}]=\epsilon_{abc}J^{c},\qquad[J_{a},P_{b}]=\epsilon_{abc}P^{c},\qquad[P_{a},P_{b}]=\lambda\epsilon_{abc}J^{c}.$

All these Lie algebras are isomorphic to $\mathfrak{so}(3,1)_{-1}$ via the map
$J_{a}\to J_{a}$ and $P_{a}\to\sqrt{|\lambda|}P_{a}$. In standard literature
$\mathfrak{so}_{-1}(3,1)$ is simply denoted by $\mathfrak{so}(3,1)$ and is the
only (up to isomorphism) simple six dimensional real Lie algebra. This follows
from the fact $\mathfrak{so}(3,1)$ is isomorphic to
$\mathfrak{sl}(2,\mathbb{C})$ (viewed as a real Lie algebra of dimension six).

The Lie algebra $\mathfrak{so}(3,1)$ has a representation as linear maps
$\text{End}(\mathbb{R}^{4})$, via the traceless matrices of the form

$m^{i}J_{i}+s^{i}P_{i}=\begin{pmatrix}0&-m^{0}&-m^{1}&s^{2}\\\
m^{0}&0&m^{2}&s^{1}\\\ m^{1}&-m^{2}&0&s^{0}\\\
s^{2}&s^{1}&s^{0}&0\end{pmatrix}.$ (115)

It is straightforward to check that if the eigenvalues of
$m^{i}J_{i}+s^{i}P_{i}$ are all zero then there exists $M\in\text{SO}(3,1)$
such that

$M(m^{i}J_{i}+s^{i}P_{i})M^{-1}\in\mathbb{R}_{\neq 0}(P_{2}-J_{0}),$ (116)

while if at least one is not zero there exists $\tilde{M}\in\text{SO}(3,1)$
such that

$\tilde{M}(m^{i}J_{i}+s^{i}P_{i})\tilde{M}^{-1}\in\\{mJ_{0}+sP_{0}\ |\
(m,s)\in\mathbb{R}^{2},(m,s)\neq(0,0)\\}.$ (117)

Clearly the Lie subalgebra spanned by $J_{2}$ and $P_{2}$ is nilpotent (indeed
it is abelian) and by (117) self-normalizing, then we conclude this Lie
subalgebra is a Cartan subalgebra of $\mathfrak{so}(3,1)$ and so
$\text{rnk}(\mathfrak{so}(3,1))$ is two.

Since the two dimensional Lie subalgebras of $\mathfrak{so}(3,1)$ are Abelian
and the normalizer of $P_{2}-J_{0}$ is spanned by $P_{0}-J_{2}$ and $P_{1}$,
we get that there exist only one conjugacy class of Cartan subalgebras of
$\mathfrak{so}(3,1)$ with representative $\text{Span}\\{J_{0},P_{0}\\}$.

### Semisimple cases: $\mathfrak{so}(4)$ and $\mathfrak{so}(2,2)$

In the case when $\lambda>0$, the Lie algebras $\mathfrak{g}_{\lambda}$

$[J_{a},J_{b}]=\epsilon_{abc}J^{c},\qquad[J_{a},P_{b}]=\epsilon_{abc}P^{c},\qquad[P_{a},P_{b}]=\lambda\epsilon_{abc}J^{c}$

result to be semi-simple for both the Euclidean and Lorentzian cases. To see
this we consider the following six alternative generators

$J_{a}^{\pm}=\frac{1}{2}\left(J_{a}\pm\frac{1}{\sqrt{\lambda}}P_{a}\right)\qquad\text{for
}a=0,1,2.$

Since this new set of generators satisfies the commutation relations

$[J^{\pm}_{a},J^{\pm}_{b}]=\epsilon_{abc}(J^{\pm})^{c},\qquad[J^{+}_{a},J^{-}_{b}]=0,\qquad\text{for
}a,b=0,1,2,$

then we conclude in the Euclidean case

$\mathfrak{so}(4)\cong\mathfrak{su}(2)\oplus\mathfrak{su}(2),$ (118)

while in the Lorentzian case

$\mathfrak{so}(2,2)\cong\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R});$
(119)

showing explicitly that for $\lambda>0$ the Lie algebras
$\mathfrak{g}_{\lambda}$ could be factorised as the direct sum of two simple
three dimensional Lie subalgebras.

Proposition A [47]. Let $\mathfrak{g}$ be a semisimple real Lie algebra, with
simple decomposition given by

$\mathfrak{g}=\oplus_{i=1}^{n}\mathfrak{g}_{n}$

and $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$. Then $\mathfrak{h}$
is decomposable as a direct sum

$\mathfrak{h}=\oplus_{i=1}^{n}\mathfrak{h}_{n}$

such that

  1. 1.

$\mathfrak{h}_{i}=\mathfrak{h}\cap\mathfrak{g}_{i}$ for all $i=1,\cdots,n$.

  2. 2.

$\mathfrak{h}_{i}$ is a Cartan subalgebra of the simple Lie algebra
$\mathfrak{g}_{i}$ for $i=1,\cdots,n$.

This proposition indicates that the set of Cartan subalgebras of a semisimple
algebra $\mathfrak{g}$ is contained in the set of subalgebras generated by
taking the direct sum of Cartan subalgebras of the simple factors of
$\mathfrak{g}$. Therefore, since the adjoint actions of $\text{SO}(4)$ and
$\text{SO}(2,2)$ factor through the actions of each copy of $\text{SU}(2)$ and
$\text{SL}(2,\mathbb{R})$ over each factor of
$\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ and
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$, respectively,
the conjugacy classes of Cartan subalgebras of $\mathfrak{so}(4)$ and
$\mathfrak{so}(2,2)$ are given by the direct sum of conjugacy classes of
Cartan subalgebras of $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$,
respectively. Hence, the problem reduces to determining first the conjugacy
classes of Cartan subalgebras of real forms of $\mathfrak{sl}(2,\mathbb{C})$,
which are known to have rank one: in the case of $\mathfrak{su}(2)$ there
exists just one conjugacy class of Cartan subalgebras with representative
spanned by $J_{0}$, while $\mathfrak{sl}(2,\mathbb{R})$ has two conjugacy
classes of Cartan subalgebras with representatives spanned by $J_{0}$ and
$J_{1}$.

In the Euclidean case there is clearly just one conjugacy class of Cartan Lie
subalgebras with

$\text{Span}\\{J_{0}^{+}\\}\oplus\text{Span}\\{J_{0}^{-}\\}\cong\text{Span}\\{J_{0},P_{0}\\},$

as a representative. Meanwhile, in the Lorentzian one can show that there
exist four conjugacy classes of Cartan subalgebras, with

$\text{Span}\\{J_{0}^{+}\\}\oplus\text{Span}\\{J_{0}^{-}\\}\cong\text{Span}\\{J_{0},P_{0}\\},$
$\text{Span}\\{J_{1}^{+}\\}\oplus\text{Span}\\{J_{1}^{-}\\}\cong\text{Span}\\{J_{1},P_{1}\\},$
$\text{Span}\\{J_{0}^{+}\\}\oplus\text{Span}\\{J_{1}^{-}\\}\cong\text{Span}\\{J_{0}+P_{0},J_{1}-P_{1}\\},$

and

$\text{Span}\\{J_{1}^{+}\\}\oplus\text{Span}\\{J_{0}^{-}\\}\cong\text{Span}\\{J_{1}+P_{1},J_{0}-P_{0}\\}$

as representatives.

### Semidirect sum cases: $\mathfrak{iso}(3)$ and $\mathfrak{iso}(2,1)$

Finally in the $\lambda=0$, the Lie algebras $\mathfrak{g}_{\lambda}$

$[J_{a},J_{b}]=\epsilon_{abc}J^{c},\qquad[J_{a},P_{b}]=\epsilon_{abc}P^{c},\qquad[P_{a},P_{b}]=0$

are isomorphic to the semidirect sums
$\mathfrak{so}(3)\ltimes_{\text{ad}^{*}}\mathfrak{so}(3)^{*}$ and
$\mathfrak{sl}(2,\mathbb{R})\ltimes_{\text{ad}^{*}}\mathfrak{sl}(2,\mathbb{R})^{*}$
for the Euclidean and Lorentzian cases, respectively.

Proposition B [48]. Let $\mathfrak{k}$ be a Lie algebra that acts over a
nilponent Lie algebra $V$ via $\phi:\mathfrak{k}\to\text{Der}(V)$. If
$\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{k}$, then

$\mathfrak{h}\ltimes_{\phi}V^{0}(\mathfrak{h})$

is a Cartan subalgebra of the Lie algebra
$\mathfrak{g}\equiv\mathfrak{k}\ltimes_{\phi}V$, where

$V^{0}(\mathfrak{h})\equiv\\{v\in V\ |\ \forall x\in\mathfrak{h},\ \exists
n\in\mathbb{Z}_{>0}\text{ such that }\phi(x)^{n}v=0\\}.$

For our cases of interest, $\mathfrak{k}=\mathfrak{so}(3)$ or
$\mathfrak{k}=\mathfrak{sl}(2,\mathbb{R})$ with
$V=\mathbb{R}^{3}\cong\mathfrak{so}(3)^{*}$ or
$V=\mathbb{R}^{1,2}\cong\mathfrak{sl}(2,\mathbb{R})^{*}$, respectively, such
that the action $\phi$ is given by the coadjoint action, i.e.

$\begin{split}\text{ad}^{*}:\mathfrak{k}&\to\text{Der}(\mathfrak{k}^{*})\\\
x&\mapsto[x,\cdot]\end{split}$

in both cases.

From this proposition it follows immediately

$\langle J_{i}\rangle\ltimes_{\text{ad}^{*}}\langle P_{i}\rangle\qquad\text{
for }i=0,1,2$

are Cartan subalgebras of $\mathfrak{iso}(3)$ and $\mathfrak{iso}(2,1)$,
showing in this way both Lie algebras have rank two.

Proposition C. The Cartan subalgebras of the Lie algebras $\mathfrak{iso}(3)$
and $\mathfrak{iso}(2,1)$ are generated by sets of the form

$\\{m_{a}P^{a},m_{a}J^{a}+n_{a}P^{a}\ |\
\bm{m},\bm{n}\in\mathbb{R}^{1,2},\bm{m}^{2}\neq 0,\bm{m}\cdot\bm{n}=0\\}$
(120)

###### Proof.

Start considering the most general set of two elements in
$\mathfrak{g}_{\lambda}$ i.e.

$\\{\bm{m}\bm{P}+\bm{n}\bm{J},\bm{k}\bm{P}+\bm{\ell}\bm{J}\\}$ (121)

where we use the notation $\bm{m}\bm{P}$ ($\bm{\ell}\bm{J}$) to denote in a
compact manner the elements $m_{a}P^{a}$ ($\ell_{a}J^{a}$) in the Lie algebra.

Since

$[\bm{m}\bm{P}+\bm{n}\bm{J},\bm{k}\bm{P}+\bm{\ell}\bm{J}]=(\bm{m}\wedge\bm{\ell}+\bm{n}\wedge\bm{k})\bm{P}+(\bm{n}\wedge\bm{\ell})\bm{J}$

in order for (121) to generate a Lie subalgebra, we require:

  1. 1.

$\bm{n}\wedge\bm{\ell}\in\text{Span}\\{\bm{n},\bm{\ell}\\}$: Since
$\langle\bm{n}\wedge\bm{\ell},\bm{n}\rangle=\langle\bm{n}\wedge\bm{\ell},\bm{\ell}\rangle=0$,
this condition holds if and only if $\bm{n}\wedge\bm{\ell}=0$, which implies
(both in the Euclidean and Lorentzian settings) $\bm{\ell}=C\bm{n}$ for some
$C\in\mathbb{R}$.

  2. 2.

$\bm{m}\wedge\bm{\ell}+\bm{n}\wedge\bm{k}\in\text{Span}\\{\bm{m},\bm{k}\\}$:
This condition holds if and only if there exist $A,B\in\mathbb{R}$ such that

$\bm{m}\wedge\bm{\ell}+\bm{n}\wedge\bm{k}=\bm{m}\wedge(C\bm{n})+\bm{n}\wedge\bm{k}=(C\bm{m}-\bm{k})\wedge\bm{n}=A\bm{k}+B\bm{m}$

From above we know $\mathfrak{g}^{(1)}\equiv[\mathfrak{g},\mathfrak{g}]$ is
generated by

$(A\bm{k}+B\bm{m})\bm{P}$

Similarly, by direct computation, we get
$\mathfrak{g}^{(2)}\equiv[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}]$ is
generated by

$((A\bm{k}+B\bm{m})\wedge\bm{n})\bm{P}.$

Indeed, by induction, we have that any term $\mathfrak{g}^{(n)}$ in the lower
central series is one dimensional and generated by

$(\cdots(((A\bm{k}+B\bm{m})\underbrace{\wedge\bm{n})\wedge\bm{n})\wedge\cdots\wedge\bm{n}}_{(n-1)-\text{times}})\bm{P}$

Hence, the condition

$(C\bm{m}-\bm{k})\wedge\bm{n}=0$

is required in order to have a Nilpotent Lie subalgebra. Since the condition
$C\bm{m}-\bm{k}=0$ provides a one dimensional Lie subalgebra, we conclude that
any the two dimensional Nilpotent Lie subalgebras of $\mathfrak{iso}(3)$ or
$\mathfrak{iso}(2,1)$ are generated by sets of the form

$\\{\bm{m}\bm{P}+\bm{n}\bm{J},(C\bm{m}-D\bm{n})\bm{P}+C\bm{n}\bm{J}\ |\
C,D\in\mathbb{R}\\}$ (122)

Finally, assume there exists $\bm{s}P+\bm{t}J$ in the normalizer of the Lie
algebra generated by (122) that does not belong to it. The brackets of this
element with the generators of the Lie algebra are given by

$[\bm{m}\bm{P}+\bm{n}\bm{J},\bm{s}\bm{P}+\bm{t}\bm{J}]=(\bm{m}\wedge\bm{t}+\bm{n}\wedge\bm{s})\bm{P}+(\bm{n}\wedge\bm{t})\bm{J}$

and

$[(C\bm{m}-D\bm{n})\bm{P}+C\bm{n}\bm{J},\bm{s}\bm{P}+\bm{t}\bm{J}]=[(C\bm{m}-D\bm{n})\wedge\bm{t}+(C\bm{n}\wedge\bm{s})]\bm{P}+(C\bm{n}\wedge\bm{t})\bm{J}$

The fact the right hand side belongs to the Lie algebra generated by (122)
implies

$\bm{t}=E\bm{n}\qquad\text{for }E\in\mathbb{R}$

Hence it follows that $\bm{s}\bm{P}+\bm{t}\bm{J}$ must be of the form
$\bm{s}\bm{P}+E\bm{n}\bm{J}$, such that the previous two Lie brackets reduce
then to

$[\bm{m}\bm{P}+\bm{n}\bm{J},\bm{s}\bm{P}+E\bm{n}\bm{J}]=(E\bm{m}\wedge\bm{n}+\bm{n}\wedge\bm{s})\bm{P}$

and

$[(C\bm{m}-D\bm{n})\bm{P}+C\bm{n}\bm{J},\bm{s}\bm{P}+E\bm{n}\bm{J}]=[E(C\bm{m}-D\bm{n})\wedge\bm{n}+(C\bm{n}\wedge\bm{s})]\bm{P}$

Here we split in 2 cases:

  1. 1.

If $\bm{n}^{2}=0$, then we find that if we take

$\bm{s}=E\bm{m}-D\bm{n}+G\bm{m}\wedge\bm{n}$

then

$[\bm{m}\bm{P}+\bm{n}\bm{J},\bm{s}\bm{P}+E\bm{n}\bm{J}]=-G\langle\bm{m},\bm{n}\rangle\bm{n}\bm{P}$

and

$[(C\bm{m}-D\bm{n})\bm{P}+C\bm{n}\bm{J},\bm{s}\bm{P}+E\bm{n}\bm{J}]=-CG\langle\bm{m},\bm{n}\rangle\bm{n}\bm{P}$

concluding that in this case is possible to find an element in the normalizer
that indeed does not belong to the Lie subalgebra generated by (122), say

$(E\bm{m}-F\bm{n}+G\bm{m}\wedge\bm{n})\bm{P}+E\bm{n}\bm{J}$

with $G\neq 0$.

  2. 2.

If $\bm{n}^{2}\neq 0$ and $C\neq 0$, then the only possibility is

$\bm{s}=E\bm{m}-F\bm{n}$

and so an element in the normalizer must be of the form

$(E\bm{m}-F\bm{n})\bm{P}+E\bm{n}\bm{J}$

which clearly belongs to the span of (122).

∎