# Generalized cluster structures on $\operatorname{SL}_{n}^{\dagger}$

Dmitriy Voloshyn and Michael Gekhtman

###### Abstract

We study Poisson varieties
$(\operatorname{SL}_{n}^{\dagger},\pi_{\mathbf{\Gamma}}^{\dagger})$
parameterized by Belavin-Drinfeld triples $\mathbf{\Gamma}$ of type $A_{n-1}$
along with generalized cluster structures
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$ on $\operatorname{SL}_{n}^{\dagger}$
compatible with $\pi_{\mathbf{\Gamma}}^{\dagger}$. As a variety,
$\operatorname{SL}_{n}^{\dagger}$ is a Zariski open subset of
$\operatorname{SL}_{n}$ obtained as the image of the dual Poisson-Lie group
$(\operatorname{SL}_{n}^{*},\pi_{\mathbf{\Gamma}}^{*})$ of
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}})$ under a certain rational map.
We prove that the generalized upper cluster structure of
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$ on $\operatorname{SL}_{n}$ is
naturally isomorphic to $\mathcal{O}(\operatorname{SL}_{n})$. Moreover, for
any connected reductive complex group $G$ and a BD triple $\mathbf{\Gamma}$,
we produce a rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
that is a Poisson isomorphism if the $r_{0}$ parts of
$\pi_{\text{std}}^{\dagger}$ and $\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide;
furthermore, in the case when
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$, we show that
$\mathcal{Q}$ is a birational quasi-isomorphism between
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma}_{\text{std}})$ and
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$. Lastly, for any pair of BD triples
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ of type $A_{n-1}$ comparable in
the natural order, we use the map $\mathcal{Q}$ to construct a birational
quasi-isomorphism between $\mathcal{GC}^{\dagger}(\tilde{\mathbf{\Gamma}})$
and $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$.

00footnotetext: 2010 Mathematics Subject Classification. 53D17,
13F60.00footnotetext: Key words and phrases. Poisson-Lie group, cluster
algebra, Belavin-Drinfeld triple.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Construction of $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$
    3. 1.3 Main results
  2. 2 Background
    1. 2.1 Poisson geometry
    2. 2.2 Generalized cluster algebras
    3. 2.3 Birational quasi-isomorphisms
  3. 3 Poisson properties of the map $\mathcal{Q}$
    1. 3.1 Preliminary identities
    2. 3.2 First computations
    3. 3.3 Pushforward of the diagonal part of $\pi^{\dagger}$
    4. 3.4 Proof of Proposition 1.4
    5. 3.5 An alternative to $\mathcal{Q}$
  4. 4 Birational quasi-isomorphisms
    1. 4.1 Belavin-Drinfeld data for $\operatorname{gl}_{n}(\mathbb{C})$
    2. 4.2 The map $\mathcal{F}$
    3. 4.3 The maps $\mathcal{Q}$ and $\mathcal{F}^{c}$
    4. 4.4 Birational quasi-isomorphisms $\mathcal{G}$ for $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$
    5. 4.5 Birational quasi-isomorphisms $\mathcal{G}$ for $|\Gamma_{1}\setminus\tilde{\Gamma}_{1}|=1$
  5. 5 Proofs of Theorems 1.5 and 1.6 for $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$
    1. 5.1 Completeness
    2. 5.2 Toric action
    3. 5.3 Compatibility
  6. 6 Further properties of $h$-variables
    1. 6.1 Preliminaries
    2. 6.2 Explicit expansion of the $h$-variables
    3. 6.3 Duality of the frozen $h$-variables
    4. 6.4 Semi-invariance of the frozen $h$-variables
  7. 7 The $g$-convention
    1. 7.1 The map $\mathcal{F}^{\mathrm{op}}$
    2. 7.2 Description of $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$
    3. 7.3 Birational quasi-isomorphisms
    4. 7.4 Proofs of Theorems 1.5 and 1.6 for $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$
  8. A Initial quiver for $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$
    1. A.1 The quiver for the trivial BD triple
    2. A.2 The quiver for a nontrivial BD triple (algorithm)
    3. A.3 The quiver for a nontrivial BD triple (explicit)
  9. B Initial quiver for $\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$
    1. B.1 The quiver for the trivial BD triple
    2. B.2 The quiver for a nontrivial BD triple (algorithm)
    3. B.3 The quiver for a nontrivial BD triple (explicit)
  10. C Determinantal identities
  11. D Selected examples
    1. D.1 Cremmer-Gervais in $n=3$, $\gamma:1\mapsto 2$
    2. D.2 Cremmer-Gervais in $n=4$, $\gamma:i\mapsto i-1$
    3. D.3 Negative orientation in $n=5$, $\gamma:1\mapsto 4$, $\gamma:2\mapsto 3$

##  1 Introduction

###  1.1 Overview

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in [13] as an
algebraic framework for studying dual canonical bases and total positivity. S.
Fomin and A. Zelevinsky suggested that many interesting varieties in Lie
theory carry the structure of an (upper) cluster algebra. Some of the known
examples include double Bruhat cells [3], Grassmannians [36], Bott-Samelson
cells [37], connected simply connected simple complex algebraic groups [18],
open Richardson varieties [28].

M. Gekhtman, M. Shapiro and A. Vainshtein noticed in [17] that the problem of
finding cluster structures on varieties can be approached via Poisson
geometry. In the earlier works [17, 18, 24], they showed that the cluster
structures on double Bruhat cells, Grassmannians, connected simply connected
complex simple algebraic groups $G$ are compatible with some Poisson brackets.
They conjectured in [18] that there exist multiple cluster structures on such
groups $G$, each of which is compatible with a Poisson bracket from the
Belavin-Drinfeld class. We refer to this conjecture as the _GSV conjecture_.

Before we state the GSV conjecture in precise terms, let us briefly review the
relevant concepts from the theory of Poisson-Lie groups and the theory of
cluster algebras (a more detailed account can be found in Section 2). Given a
set of simple roots $\Pi$ of $G$, a _Belavin-Drinfeld triple_ (for short, a
_BD triple_) is a triple $\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$
where $\Gamma_{1},\Gamma_{2}\subset\Pi$ and
$\gamma:\Gamma_{1}\rightarrow\Gamma_{2}$ is a nilpotent isometry. The triple
$\mathbf{\Gamma}_{\text{std}}:=\mathbf{\Gamma}$ is called _trivial_ if
$\Gamma_{1}=\Gamma_{2}=\emptyset$. Let $\mathfrak{h}$ be the Cartan subalgebra
of the Lie algebra of $G$ that corresponds to $\Pi$. As was shown by Belavin
and Drinfeld in [1, 2], with some additional continuous parameter
$r_{0}\in\mathfrak{h}\otimes\mathfrak{h}$, the quadruples
$(\Gamma_{1},\Gamma_{2},\gamma,r_{0})$ parameterize the moduli space of the
quasi-triangular Poisson structures on $G$. We will denote a Poisson bivector
associated to $\mathbf{\Gamma}$ (for an arbitrary $r_{0}$) as
$\pi_{\mathbf{\Gamma}}$. The Poisson structure on $G$ induces a Lie algebra
structure on the dual of its Lie algebra; integrating the latter, one obtains
a dual Poisson-Lie group $G^{*}$, which is endowed with its own Poisson
bivector $\pi^{*}_{\mathbf{\Gamma}}$. It is possible to choose a Poisson
bivector $\pi_{\mathbf{\Gamma}}^{D}$ on $D(G):=G\times G$ and the dual $G^{*}$
in such a way that 1) $G$ is the diagonal subgroup of $D(G)$ and $G^{*}$ is an
immersed Lie subgroup of $D(G)$; 2)
$\pi_{\mathbf{\Gamma}}^{D}|_{G}=\pi_{\mathbf{\Gamma}}$ and
$\pi^{D}_{\mathbf{\Gamma}}|_{G^{*}}=\pi^{*}_{\mathbf{\Gamma}}$. Consider the
map

$D(G)\ni(g,h)\mapsto g^{-1}h\in G,$ (1.1)

and define the Poisson bivector $\pi_{\mathbf{\Gamma}}^{\dagger}$ to be the
pushforward of $\pi^{D}_{\mathbf{\Gamma}}$ under (1.1). The restriction of
(1.1) to $G^{*}$ has an open image in $G$, which is denoted as $G^{\dagger}$.
The Poisson structure $\pi_{\mathbf{\Gamma}}^{\dagger}$ serves as a model for
the Poisson structure $\pi_{\mathbf{\Gamma}}^{*}$ inside the group $G$ itself.
When $\mathbf{\Gamma}=\mathbf{\Gamma}_{\text{std}}$, we will denote the
corresponding Poisson bivectors as $\pi_{\text{std}}$, $\pi_{\text{std}}^{D}$
and $\pi_{\text{std}}^{\dagger}$. More generally, one can associate Poisson-
homogeneous bivectors $\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}$ and
$\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}^{D}$ to pairs of Belavin-
Drinfeld triples $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ (see [23, 30,
31]).

Now, let us turn to the cluster algebras theory. In the context of the GSV
conjecture, one starts with an _initial extended cluster_
$(x_{1},x_{2},\ldots,x_{N+M})$, which is a collection of rational functions
such that $\mathbb{C}(G)=\mathbb{C}(x_{1},x_{2},\ldots,x_{N+M})$, $N+M=\dim
G$. The variables $x_{1},\ldots,x_{N}$ are called _cluster variables_ and
$x_{N+1},\ldots,x_{N+M}$ are the _frozen variables_. Given a skew-
symmetrizable $N\times(N+M)$ matrix $B$ with integer entries, for each
$i\in[1,N]$ one can obtain a new extended cluster from the initial one via
substituting the cluster variable $x_{i}$ with a new cluster variable
$x_{i}^{\prime}$ given by

$x_{i}x_{i}^{\prime}=\prod_{b_{ij}>0}x_{j}^{b_{ij}}+\prod_{b_{ij}<0}x_{j}^{-b_{ij}}.$
(1.2)

The above transformation is called a _mutation._ For each new extended
cluster, the matrix $B$ also changes according to a certain pattern. Iterating
the above process, one obtains other extended clusters, and the collection
$\mathcal{C}$ of those is called a _cluster structure_ on $G$. The _cluster
algebra_ $\mathcal{A}_{\mathbb{C}}$ is set to be the polynomial algebra over
$\mathbb{C}$ generated by all cluster and frozen variables. With each extended
cluster $\mathbf{x}$, we associate a ring of Laurent polynomials
$\mathcal{L}_{\mathbb{C}}(\mathbf{x}):=\mathbb{C}[x_{1}^{\pm
1},\ldots,x_{N}^{\pm 1},x_{N+1},\ldots,x_{N+M}]$, and we set the upper cluster
algebra $\bar{\mathcal{A}}_{\mathbb{C}}$ to be the intersection of all
$\mathcal{L}_{\mathbb{C}}(\mathbf{x})$ for $\mathbf{x}\in\mathcal{C}$
(possibly, with an additional localization at some frozen variables). The
famous Laurent phenomenon claims that
$\mathcal{A}_{\mathbb{C}}\subseteq\bar{\mathcal{A}}_{\mathbb{C}}$. We say that
$\mathcal{C}$ is _regular_ if
$\mathcal{A}_{\mathbb{C}}\subseteq\mathcal{O}(G)$ and $\mathcal{C}$ is
_complete_ if $\bar{\mathcal{A}}_{\mathbb{C}}=\mathcal{O}(G)$ (in this case,
we also say that $\bar{\mathcal{A}}_{\mathbb{C}}$ is _naturally isomorphic_ to
$\mathcal{O}(G)$). An extended cluster $(x_{1},\ldots,x_{N+M})$ is called
_log-canonical_ if $\\{x_{i},x_{j}\\}=\omega_{ij}x_{i}x_{j}$, where
$\\{\cdot,\cdot\\}$ is the Poisson bracket that corresponds to the chosen
Poisson bivector and $\omega_{ij}\in\mathbb{C}$. If each extended cluster is
log-canonical, the cluster algebra (or structure) is said to be _compatible_
with the Poisson structure. A generalized cluster structure $\mathcal{GC}$ is
defined in the same way except (1.2) is allowed to have more than two
monomials in the right-hand side along with some coefficients.

##### The GSV conjecture.

In its current form, the conjecture is stated as follows:

> Let $G$ be a connected simple complex Lie group and
> $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ be a pair of Belavin-Drinfeld
> triples defined relative a root system of $G$. Then for each of the Poisson
> varieties $(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$,
> $(D(G),\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}^{D})$ and
> $(G,\pi_{\mathbf{\Gamma}^{c}}^{\dagger})$, there exists a complete and
> compatible generalized cluster structure.

The results of our paper cover the case of
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ for _all_ Belavin-
Drinfeld triples111The reader might notice that the Poisson structure
$\pi_{\mathbf{\Gamma}^{c}}^{\dagger}$ depends only on $\mathbf{\Gamma}^{c}$
and not on the pair $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$. The reason
for that is, the pushforward of
$\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}^{D}$ under the map
$(X,Y)\mapsto X^{-1}Y$ ”forgets” the BD triple $\mathbf{\Gamma}^{r}$.
$\mathbf{\Gamma}:=\mathbf{\Gamma}^{c}$ of type $A_{n-1}$, as well as some
related varieties, such as
$(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ and
$(\operatorname{SL}_{n}^{\dagger},\pi_{\mathbf{\Gamma}}^{\dagger})$. Setting
$\mathbf{\Gamma}^{r}:=(\Gamma_{1}^{r},\Gamma_{2}^{r},\gamma_{r})$ and
$\mathbf{\Gamma}^{c}:=(\Gamma_{1}^{c},\Gamma_{2}^{c},\gamma_{c})$, recall that
the pair $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ is called _aperiodic_ if
the map $\gamma_{r}w_{0}\gamma_{c}^{-1}w_{0}^{-1}$ is nilpotent, where $w_{0}$
is the longest Weyl group element. In type $A_{n-1}$, identifying the set of
simple roots $\Pi$ with the interval of integers $[1,n-1]$, the pair
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ is called _oriented_ if both
$\gamma_{r}$ and $\gamma_{c}$ are increasing maps. The current state of the
GSV conjecture is as follows.

  1. 1)

In the case of
$(\operatorname{SL}_{n},\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$, the
conjecture was verified for aperiodic pairs
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ in [15];

  2. 2)

In the case of
$(D(\operatorname{SL}_{n}),\pi^{D}_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$,
the conjecture was verified for all aperiodic oriented pairs
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ in [31];

  3. 3)

In the case of $(\operatorname{SL}_{n},\pi^{\dagger}_{\mathbf{\Gamma}})$, the
conjecture was verified only for the trivial BD triple in [20]. The results of
the current paper prove the conjecture for all BD triples on
$(\operatorname{SL}_{n},\pi^{\dagger}_{\mathbf{\Gamma}})$;

  4. 4)

As for $(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$ of other Lie
types, the conjecture was verified in [18] only for the trivial pair
$(\mathbf{\Gamma}^{r}_{\text{std}},\mathbf{\Gamma}^{c}_{\text{std}})$.
However, even for the trivial BD triples, constructions in the cases of
$(G,\pi_{\mathbf{\Gamma}}^{\dagger})$ and
$(D(G),\pi^{D}_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$ have not been
discovered yet.

All the results on $\operatorname{SL}_{n}$ can be extended to
$\operatorname{GL}_{n}$. The Poisson-homogeneous structures
$\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}$ and
$\pi^{D}_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}$ were included into the
conjecture in [23, 31]. Generalized cluster mutations appear in the case of
$(\operatorname{SL}_{n},\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$ when
the pair $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ is not aperiodic [22]; in
the case of
$(D(\operatorname{SL}_{n}),\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$
and $(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$, generalized
cluster mutations appear even for the trivial Belavin-Drinfeld triples.

##### Connection to other problems.

We envision a connection between our work and some of the following known
problems in mathematical physics:

  1. 1)

There is a conjecture due to D. Gaiotto on the existence of cluster structures
on Coulomb branches in $\mathcal{N}=4$, $3d$ supersymmetric gauge theories.
The conjecture was verified in the case of quiver gauge theories by A. Shapiro
and G. Schrader in [35]. It would be interesting to see if
$(\operatorname{SL}_{n}^{\dagger},\pi_{\mathbf{\Gamma}}^{\dagger})$ for a
nontrivial $\mathbf{\Gamma}$ can be interpreted as a character variety. For
$\mathbf{\Gamma}=\mathbf{\Gamma}_{\text{std}}$, the question was studied by R.
Brahami in [4] and L. Shen in [38]. There are some thoughts in [21, Section 4]
on a relationship to Fock–Goncharov coordinates on moduli spaces of local
system [10], as well as on a possible interpretation of generalized cluster
mutations. If this is done, one might be able to recognize
$(\operatorname{SL}_{n}^{\dagger},\pi_{\mathbf{\Gamma}}^{\dagger})$ as a
Coulomb branch for some physical theory.

  2. 2)

G. Schrader, A. Shapiro and I. Ip in [26, 34] proposed a quantum cluster
$\mathcal{X}$-realization of the quantum group $U_{g}(\mathfrak{g})$ for
simple complex Lie algebras $\mathfrak{g}$. As a byproduct, they derived a
factorization into quantum dilogarithms of the quantum $R$-matrix associated
with the trivial BD triple. An explicit classification of (quasi-triangular)
quantum $R$-matrices is well-known and given in [9]. We expect the following:
1) there exists a connection between compatible generalized cluster structures
on $(G,\pi_{\mathbf{\Gamma}}^{\dagger})$ and quantum $\mathcal{A}$\- or
$\mathcal{X}$-cluster realizations of $U_{q}(\mathfrak{g})$ (as above, this
requires a realization of $(G,\pi_{\mathbf{\Gamma}}^{\dagger})$ as a character
variety); 2) quantum $R$-matrices for nontrivial BD triples can also be
factorized into quantum dilogarithms (or their twists).

Some of the intrinsic problems of the theory presented in this paper are
listed in our supplementary note [33].

##### Generalized cluster mutations and aperiodicity.

For compatible generalized cluster structures on
$(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$, there is a significant
distinction between the cases of aperiodic and nonaperiodic BD pairs. In the
case of $G=\operatorname{SL}_{n}$ and an aperiodic BD pair, the initial
cluster variables are obtained as trailing minors of the so-called
$\mathcal{L}$-matrices (the definition is given in [15, 23]); however, these
matrices become infinite when the pair is nonaperiodic. Moreover, at least in
the case of
$(\operatorname{SL}_{n},\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$, one
has to consider generalized cluster mutations when
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ is nonaperiodic (for an example,
see [22]). However, we found that in the case of
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$, one does not need
to distinguish between the aperiodic and nonaperiodic cases, and for any BD
triple in our construction, there is always only one generalized cluster
mutation.

##### A new approach.

Compared to the previous papers [19, 23, 31] for nontrivial BD triples, the
proofs in the current paper are significantly simpler. Our new approach, which
was nascent in [23, 31], consists in studying properties of certain Poisson
birational quasi-isomorphisms, which seem to be the right type of equivalences
in the context of the GSV conjecture. These maps (almost) preserve all of the
considered structures on a given variety; hence, once the results are proved
in the case of the trivial Belavin-Drinfeld data, the Poisson birational
quasi-isomorphisms can be used to transfer the results to the case of
nontrivial BD triples.

  1. 1)

Previously, the proofs of compatibility between the chosen Poisson bracket and
the (generalized) cluster structure consisted in direct and involved
computations [19, 23, 31]. In the current paper, given any connected reductive
complex Lie group $G$, any BD triple $\mathbf{\Gamma}$ and a Poisson bivector
$\pi_{\mathbf{\Gamma}}^{\dagger}$ on $G$, we prove that a certain rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism provided the $r_{0}$ parts of the Poisson bivectors
are the same (see Section 3). In the case of
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$, $\mathcal{Q}$ is a
birational quasi-isomorphism. We expect that generalized cluster structures
for nontrivial BD triples in other Lie types can be constructed by means of
the map $\mathcal{Q}$. The compatibility of $\pi_{\mathbf{\Gamma}}^{\dagger}$
with the corresponding generalized cluster structure on $G$ is a consequence
of the Poisson property of $\mathcal{Q}$.

  2. 2)

As for showing that the generalized upper cluster algebra coincides with the
ring of regular functions on $\operatorname{SL}_{n}$, we mainly use the
results of [32] and an inductive argument on the size of Belavin-Drinfeld
triples, which was suggested in [23]. The proof is reduced to verifying
properties of certain marked variables in the case $|\Gamma_{1}|=1$.

  3. 3)

The existence of a global toric action of maximal rank follows from certain
equivariance properties of $\mathcal{Q}$, and the proof is reduced to the
verification of some properties of the marked variables in the case
$|\Gamma_{1}|=1$.

In the case of $(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$, the same
approach is developed in [15]. For every pair of BD triples
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$, the authors constructed a certain
rational map
$\mathbf{h}:(G,\pi_{\text{std}})\dashrightarrow(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$
and showed that $\mathbf{h}$ is Poisson if the $r_{0}$ parts of
$\pi_{\text{std}}$ and $\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}$
coincide. As we mentioned above, there is an obstacle in constructing the
initial extended cluster on
$(G,\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})})$ when
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ is not aperiodic. In terms of
$\mathbf{h}$, the obstacle translates into the fact that $\mathbf{h}^{-1}$ is
not rational for a nonaperiodic pair
$(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$. However, evidence suggests that
by means of $\mathbf{h}^{-1}$, one can transform a cluster structure
compatible with $\pi_{\text{std}}$ into a generalized cluster structure
compatible with $\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}$. It is the
next major question in the research program whether one can combine
$\mathbf{h}$ and $\mathcal{Q}$ into a map that can generate a compatible
generalized cluster structure on the Drinfeld double
$(D(G),\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}^{D})$.

##### Organization of the paper.

In Section 1.2, we provide a description of the initial extended seed in the
_$h$ -convention_ (an alternative construction, which we refer to as the _$g$
-convention_, is given in Section 7). In Section 1.3, we state the main
results of the paper. In Section 2, we provide an expanded background on
Poisson geometry, generalized cluster theory and the theory of birational
quasi-isomorphisms. In Section 3, we construct the rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
for any connected reductive complex Lie group $G$, and then we prove that
$\mathcal{Q}$ is a Poisson isomorphism provided the $r_{0}$ parts of
$\pi_{\mathbf{\Gamma}}^{\dagger}$ and $\pi_{\text{std}}^{\dagger}$ are the
same. In Section 4, we construct various birational quasi-isomorphisms in the
case of $G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$. In Section 5,
we prove our main Theorems 1.5\- 1.6. In Section 6, we derive an explicit
formula for the variables from the initial extended cluster, as well as prove
special Poisson properties of the frozen variables. Some explicit examples of
the generalized cluster structures on $(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
are listed in Appendix D.

##### Acknowledgements.

The first author was supported by the grant IBS-R003-D1. The second author was
supported by the NSF grant 2100785.

###  1.2 Construction of $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$

Let $\Pi$ be a fixed set of simple roots of type $A_{n-1}$ identified with the
interval $[1,n-1]$. In this subsection, given a BD triple
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$, we describe the initial
extended cluster of $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ on
$(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ and
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ in the _$h$
-convention_. The ground field is always assumed to be $\mathbb{C}$. For the
trivial BD triple, the description was provided in [20]. In Section 7 we
provide an alternative construction that we refer to as the _$g$ -convention_.
The initial extended cluster comprises three types of functions:
$c$-functions, $\varphi$-functions and $h$-functions. Only the description of
the $h$-functions depends on the choice of the Belavin-Drinfeld triple. The
description of the initial quiver is provided in Appendix A, and selected
examples are provided in Appendix D.

##### Notation for $\mathcal{GC}$.

The notation $\mathcal{GC}^{\dagger}(\mathbf{\Gamma})$ stands for the
generalized cluster structure on
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ or
$(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ in the $h$\- or
$g$-convention. The lower script $h$ or $g$ will specify the convention. If we
need to specify a variety, we will write
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ or
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$. The same
pattern will be applied for the (upper) generalized cluster algebra, the
ground ring, etc. The notation $(G,\mathcal{GC}^{\dagger}(\mathbf{\Gamma}))$
(with or without the subscript $h$ or $g$) reads as a group
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$ supplied with
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},G)$. The choice of $r_{0}$ is not
relevant for the construction.

##### Description of $\varphi$\- and $c$-functions.

For an element $U\in\operatorname{GL}_{n}$, let us set

$\Phi_{kl}(U):=\begin{bmatrix}(U^{0})^{[n-k+1,n]}&U^{[n-l+1,n]}&(U^{2})^{\\{n\\}}&\cdots&(U^{n-k-l+1})^{\\{n\\}}\end{bmatrix},\
\ k,l\geq 1,\ k+l\leq n;$ (1.3) $s_{kl}:=\begin{cases}(-1)^{k(l+1)}\ &n\
\text{is even},\\\ (-1)^{(n-1)/2+k(k-1)/2+l(l-1)/2}\ &n\ \text{is
odd}.\end{cases}$ (1.4)

Then the $\varphi$-functions are given by

$\varphi_{kl}(U):=s_{kl}\det\Phi_{kl}(U).$ (1.5)

The $\varphi$-functions enjoy the following invariance properties. Given a
unipotent lower triangular matrix $N_{-}$ and a nondegenerate diagonal matrix
$T$,

$\displaystyle\varphi_{kl}(N_{-}U\,\left.N_{-}\right.^{-1})$
$\displaystyle=\varphi_{kl}(U),$ (1.6) $\displaystyle\varphi_{kl}(TUT^{-1})$
$\displaystyle=\chi_{\varphi_{kl}}(T)\varphi_{kl}(U)$

where $\chi_{\varphi_{kl}}$ is a character. The $c$-functions are uniquely
defined via

$\det(I+\lambda U)=\sum_{i=0}^{n}\lambda^{i}s_{i}c_{i}(U)$ (1.7)

where $s_{i}:=(-1)^{i(n-1)}$ and $I$ is the identity matrix. Note that
$c_{0}=I$ and $c_{n}=\det U$.

######  Remark 1.1.

Let
$D(\operatorname{GL}_{n}):=\operatorname{GL}_{n}\times\operatorname{GL}_{n}$
be the Drinfeld double of $\operatorname{GL}_{n}$. For $(X,Y)\in
D(\operatorname{GL}_{n})$, set $U:=X^{-1}Y$. Then the definition of the
$\varphi$\- and $c$-functions is (up to a factor of $\det X$) identical to the
definitions provided in the previous papers [20, 31]. The functions
$\varphi_{kl}$ also coincide with the functions $\psi_{kl}$ described in [20]
for $\operatorname{SL}_{n}^{\dagger}$ in the case of the trivial Belavin-
Drinfeld triple.

##### Description of the $h$-functions.

For any generic $n\times n$ matrix $A$, let us denote by $A_{\oplus}$ and
$A_{-}$ the upper triangular and the unipotent lower triangular matrices,
respectively, such that $A=A_{\oplus}A_{-}$. For a given BD triple
$\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ of type $A_{n-1}$, the map
$\gamma^{*}$ can be extended to a Lie algebra homomorphism
$\gamma^{*}:\mathfrak{n}_{-}(\Gamma_{2})\rightarrow\mathfrak{n}_{-}(\Gamma_{1})$,
where $\mathfrak{n}_{-}(\Gamma_{i})$ are the subalgebras of
$\operatorname{sl}_{n}(\mathbb{C})$ of nilpotent lower triangular matrices
generated by $\Gamma_{i}$, $i\in\\{1,2\\}$. Then $\gamma^{*}$ can be extended
by zero to the map $\gamma^{*}:\mathfrak{n}_{-}\rightarrow\mathfrak{n}_{-}$,
which is still a Lie algebra homomorphism. Lastly, it can be integrated to a
group homomorphism
$\tilde{\gamma}^{*}:\mathcal{N}_{-}\rightarrow\mathcal{N}_{-}$, where
$\mathcal{N}_{-}$ is the subgroup of $\operatorname{SL}_{n}$ of unipotent
lower triangular matrices (see Section 2.1 and Section 4.1 for details).
Define the following sequence of rational maps:

$\mathcal{F}_{0}(U):=U,\ \
\mathcal{F}_{k}(U):=\tilde{\gamma}^{*}([\mathcal{F}_{k-1}(U)]_{-})\cdot U,\ \
U\in\operatorname{GL}_{n},\ k\geq 1.$ (1.8)

We study properties of $\\{\mathcal{F}_{k}\\}_{k\geq 0}$ in Section 4.2. The
sequence stabilizes due to the nilpotency of $\gamma$:

$\mathcal{F}_{\deg\gamma}(U)=\mathcal{F}_{\deg\gamma+1}(U)=\mathcal{F}_{\deg\gamma+2}=\ldots$
(1.9)

(for $\deg\gamma$, see Definition 4.1). Set

$\mathcal{F}(U):=\mathcal{F}_{\deg\gamma}(U),\ \ U\in\operatorname{GL}_{n}.$
(1.10)

######  Definition 1.2.

For $\alpha_{0}\in\Pi\setminus\Gamma_{2}$, set
$\alpha_{t}:=\gamma(\alpha_{t-1})$, $t\geq 1$. The sequence
$S^{\gamma}(\alpha_{0}):=\\{\alpha_{t}\\}_{t\geq 0}$ is called the _$\gamma$
-string_ associated to $\alpha_{0}$.

Note that $\gamma$-strings partition the set of simple roots $\Pi$. For each
$\gamma$-string
$S^{\gamma}(\alpha_{0})=\\{\alpha_{0},\alpha_{1},\ldots,\alpha_{m}\\}$, for
each $i\in[0,m]$ and $j\in[\alpha_{i}+1,n]$, set

$h_{\alpha_{i}+1,j}(U):=(-1)^{\varepsilon_{\alpha_{i}+1,j}}\det[\mathcal{F}(U)]^{[j,n]}_{[\alpha_{i}+1,n-j+\alpha_{i}+1]}\prod_{t\geq
i+1}^{m}\det[\mathcal{F}(U)]^{[\alpha_{t}+1,n]}_{[\alpha_{t}+1,n]}$ (1.11)

where $\varepsilon_{ij}$ is defined as

$\varepsilon_{ij}:=(j-i)(n-i),\ \ 1\leq i\leq j\leq n.$ (1.12)

We refer to the functions $h_{ij}$, $2\leq i\leq j\leq n$, together with
$h_{11}(U):=\det U$ as the _$h$ -functions_. Note that the flag minors of
$\mathcal{F}(U)$ can be expressed as

$(-1)^{\varepsilon_{ij}}\det[\mathcal{F}(U)]_{[i,n-j+i]}^{[j,n]}=\begin{cases}\frac{h_{ij}(U)}{h_{\mu\mu}(U)}\
&\text{if}\ i-1\in\Gamma_{1},\\\ h_{ij}(U)\ &\text{otherwise}\end{cases}$
(1.13)

where $\mu:=\gamma(i-1)+1$ for $i-1\in\Gamma_{1}$. When
$\mathbf{\Gamma}=\mathbf{\Gamma}_{\text{std}}$, we obtain that

$h_{ij}(U)=(-1)^{\varepsilon_{ij}}\det U^{[j,n]}_{[i,n-j+i]},\ \ 2\leq i\leq
j\leq n,$ (1.14)

which coincides with the earlier description given in [20]. We will show in
Section 6.2 that the $h$-functions are regular as elements of
$\mathcal{O}(\operatorname{GL}_{n})$, and we will also derive an explicit
formula in terms of minors of $U\in\operatorname{GL}_{n}$. Moreover, in
Section 5.1.3 we will show that the $h$-functions are irreducible and coprime
with their mutations.

##### Frozen variables.

In the case of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$, the
frozen variables are given by the set

$\\{c_{1},c_{2},\ldots,c_{n-1}\\}\cup\\{h_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{2}\\}\cup\\{h_{11}\\}.$ (1.15)

In the case of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$,
$h_{11}(U)=1$, so this variable is absent. We will show in Section 6.4 that
the zero loci of the frozen variables foliate into unions of symplectic leaves
of the ambient Poisson variety
$(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ or
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$. In the same
section, we will also show that the frozen $h$-variables do not vanish on
$\operatorname{SL}_{n}^{\dagger}$.

##### Initial extended cluster.

The initial extended cluster $\Psi_{0}$ of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ is given
by the set

$\\{h_{ij}\ |\ 2\leq i\leq j\leq n\\}\cup\\{\varphi_{kl}\ |\ k,l\geq 1,\
k+l\leq n\\}\cup\\{c_{1},\ldots,c_{n-1}\\}\cup\\{h_{11}\\}.$ (1.16)

The initial extended cluster of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$ is
obtained from $\Psi_{0}$ via removing $h_{11}$.

##### Ground rings.

The ground rings are defined as follows.

$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{GL}_{n}):=$
$\displaystyle\mathbb{C}[h_{11}^{\pm 1},c_{1},\ldots,c_{n-1},h_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{2}];$ (1.17)
$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{SL}_{n}):=$
$\displaystyle\mathbb{C}[c_{1},\ldots,c_{n-1},h_{i+1,i+1}\ |\
i\in\Pi\setminus\Gamma_{2}];$ (1.18)
$\displaystyle\hat{\mathbb{A}}_{\mathbb{C}}(\mathbf{\Gamma},\operatorname{SL}_{n}^{\dagger}):=$
$\displaystyle\mathbb{C}[c_{1},\ldots,c_{n-1},h_{i+1,i+1}^{\pm 1}\ |\
i\in\Pi\setminus\Gamma_{2}].$ (1.19)

Note that the localizations in the ground rings reflect the fact that the
corresponding frozen variables, as regular functions on the corresponding
varieties, are nonvanishing. In this paper, we prove the results for
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n})$ and
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n})$, and the
results for $\operatorname{SL}_{n}^{\dagger}$ and
$\operatorname{GL}_{n}^{\dagger}$ follow from localizing the ground rings. In
the case of $\operatorname{GL}_{n}^{\dagger}$, the choice of the ground ring
is subtle (see Remark 6.11).

##### A generalized cluster mutation.

In the initial extended cluster, only the variable $\varphi_{11}$ is equipped
with a nontrivial _string_ , which is given by $(1,c_{1},\ldots,c_{n-1},1)$
(see Definition 2.6). The generalized mutation relation for $\varphi_{11}$
reads

$\varphi_{11}\varphi_{11}^{\prime}=\sum_{r=0}^{n}c_{r}\varphi_{21}^{r}\varphi_{12}^{n-r}.$
(1.20)

Other mutations of the initial extended cluster follow the usual pattern from
the theory of cluster algebras of geometric type.

##### Toric action.

Let $\mathfrak{h}$ be the Cartan subalgebra of
$\operatorname{sl}_{n}(\mathbb{C})$. Set

$\mathfrak{h}_{\mathbf{\Gamma}}:=\\{h\in\mathfrak{h}\ |\ \alpha(h)=\beta(h)\
\text{if}\ \gamma^{j}(\alpha)=\beta\ \text{for some}\ j\\}.$ (1.21)

The dimension of $\mathfrak{h}_{\mathbf{\Gamma}}$ is given by

$k_{\mathbf{\Gamma}}:=\dim_{\mathfrak{h}_{\mathbf{\Gamma}}}=|\Pi\setminus\Gamma_{2}|.$
(1.22)

Set $\mathcal{H}_{\mathbf{\Gamma}}$ to be the connected subgroup of
$\operatorname{SL}_{n}$ that corresponds to $\mathfrak{h}_{\mathbf{\Gamma}}$.
We let $\mathcal{H}_{\mathbf{\Gamma}}$ to act upon $\operatorname{GL}_{n}$ and
$\operatorname{SL}_{n}$ via

$(H,U)\mapsto HUH^{-1},\ \ H\in\mathcal{H}_{\mathbf{\Gamma}},\
U\in\operatorname{GL}_{n}.$ (1.23)

In the case of $\operatorname{GL}_{n}$, we will consider the action of
$\mathcal{H}_{\mathbf{\Gamma}}\times\mathbb{C}^{*}$, where $\mathbb{C}^{*}$
acts as a multiplication by scalars:

$(a,U)\mapsto a\cdot U,\ \ a\in\mathbb{C}^{*},\ U\in\operatorname{GL}_{n}.$
(1.24)

In Section 5.2 we will show that the action of $\mathcal{H}_{\mathbf{\Gamma}}$
upon $\operatorname{SL}_{n}$ and the action of
$\mathcal{H}_{\mathbf{\Gamma}}\times\mathbb{C}^{*}$ upon
$\operatorname{GL}_{n}$ induce global toric actions upon the corresponding
generalized cluster structures.

##### The Poisson bracket.

A general formula for the Poisson bracket is given in (2.18). In the cases of
$\operatorname{GL}_{n}$ and $\operatorname{SL}_{n}$, the Poisson bracket that
corresponds to $\pi_{\mathbf{\Gamma}}^{\dagger}$ is given as follows. First,
for a smooth function $f\in C^{\infty}(\operatorname{GL}_{n})$, set

$\nabla_{U}f:=\left(\frac{\partial f}{\partial u_{ij}}\right)_{j,i=1}^{n},\ \
U\in\operatorname{GL}_{n};$ (1.25)

second, equip $\operatorname{gl}_{n}(\mathbb{C})$ and
$\operatorname{sl}_{n}(\mathbb{C})$ with the trace form

$\langle A,B\rangle:=\operatorname{tr}(AB),\ \
A,B\in\operatorname{gl}_{n}(\mathbb{C}).$

Let
$R:=R_{+}:\operatorname{gl}_{n}(\mathbb{C})\rightarrow\operatorname{gl}_{n}(\mathbb{C})$
be the classical $R$-matrix that corresponds to $\mathbf{\Gamma}$ (see formula
(2.7) below). The Poisson bracket on
$(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ is given by

$\\{f,g\\}_{\mathbf{\Gamma}}^{\dagger}=\langle
R([\nabla_{U}f,U]),[\nabla_{U}g,U]\rangle-\langle[\nabla_{U}f,U],\nabla_{U}g\cdot
U\rangle.$ (1.26)

If $\mathbf{\Gamma}$ is trivial, we will denote the corresponding Poisson
bracket as $\\{\cdot,\cdot\\}_{\text{std}}^{\dagger}$. In the case of
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$, the formula is
exactly the same, except one chooses $R$ as a linear endomorphism of
$\operatorname{sl}_{n}(\mathbb{C})$.

##### Casimirs.

For any222In our work on the Drinfeld double [31], the functions
$c_{1},\ldots,c_{n-1}$ defined on
$(D(\operatorname{GL}_{n}),\pi_{\mathbf{\Gamma}}^{D})$ are Casimirs if and
only if $R_{0}(I)=(1/2)I$. However, since $[\nabla_{U}c_{i},U]=0$ for any $i$,
the functions $c_{1},\ldots,c_{n-1}$ are Casimirs for any choice of $R_{0}$
for $(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$. choice of
$r_{0}$ for $(\operatorname{GL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$ and
$(\operatorname{SL}_{n},\pi_{\mathbf{\Gamma}}^{\dagger})$, the functions
$c_{1},c_{2},\ldots,c_{n-1}$, $h_{11}$ are Casimirs of
$\pi_{\mathbf{\Gamma}}^{\dagger}$.

######  Remark 1.3.

Let $(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ be an aperiodic oriented BD
pair and let $\mathcal{GC}^{D}(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ be
the generalized cluster structure on $D(\operatorname{SL}_{n})$ (or on
$D(\operatorname{GL}_{n})$) compatible with
$\pi_{(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})}^{D}$
($\mathcal{GC}^{D}(\mathbf{\Gamma}^{r},\mathbf{\Gamma}^{c})$ was constructed
in our previous work [31]). The initial seed for
$\mathcal{GC}^{\dagger}_{h}(\mathbf{\Gamma}_{\text{std}})$ was recognized as a
subseed of some extended seed in
$\mathcal{GC}^{D}(\mathbf{\Gamma}_{\text{std}},\mathbf{\Gamma}_{\text{std}})$.
It is thus natural to ask whether the initial extended seed of
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ can be obtained as a subseed of
an extended seed from $\mathcal{GC}^{D}(\mathbf{\Gamma},\mathbf{\Gamma})$. We
verified in $n=3$ and $n=4$ for all BD triples $\mathbf{\Gamma}$ that this is
indeed the case. The corresponding mutation sequences are listed in the
supplementary note [33].

###  1.3 Main results

Let $G$ be a connected reductive complex Lie group, $\mathfrak{g}$ be its Lie
algebra with a fixed decomposition
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$ and a
set of simple roots $\Pi$. Let $\mathcal{B}_{+}$ and $\mathcal{N}_{-}$ be the
connected subgroups of $G$ that correspond to
$\mathfrak{n}_{+}\oplus\mathfrak{h}$ and $\mathfrak{n}_{-}$. Decompose a
generic element $U\in G$ as $U=U_{\oplus}U_{-}$,
$U_{\oplus}\in\mathcal{B}_{+}$, $U_{-}\in\mathcal{N}_{-}$. Let
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a Belavin-Drinfeld
triple. Define the rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
as

$\mathcal{Q}(U):=\rho(U)^{-1}U\rho(U),\ \
\rho(U):=\prod_{i=1}^{\rightarrow}(\tilde{\gamma}^{*})^{i}(U_{-}),\ U\in G.$
(1.27)

######  Proposition 1.4.

If the $r_{0}$ components of $\pi_{\text{std}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ are the same, then the rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism.

Section 3 is devoted to the proof of Proposition 1.4. In Section 4.3 we
specialize the group $G$ to
$G\in\\{\operatorname{SL}_{n}(\mathbb{C}),\operatorname{GL}_{n}(\mathbb{C})\\}$
and show that
$\mathcal{Q}:(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}_{\text{std}}))\dashrightarrow(G,\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma}))$
is a birational quasi-isomorphism. In Section 7.3 we construct the map
$\mathcal{Q}^{\mathrm{op}}:(G,\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}_{\text{std}}))\dashrightarrow(G,\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma}))$,
which is a birational quasi-isomorphism in the $g$-convention. In Section 3.5
we prove a version of Proposition 1.4 for $\mathcal{Q}^{\mathrm{op}}$.

######  Theorem 1.5.

Let $\mathbf{\Gamma}$ be any Belavin-Drinfeld triple of type $A_{n-1}$. Then
the generalized cluster structure
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}(\mathbb{C}))$
has the following properties:

  1. 1)

The number of frozen variables is $k_{\mathbf{\Gamma}}+n$, and the exchange
matrix has full rank;

  2. 2)

The generalized cluster structure
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}(\mathbb{C}))$ is
complete; moreover, the initial extended cluster consists of irreducible
elements of $\mathcal{O}(\operatorname{GL}_{n}(\mathbb{C}))$ that are coprime
with their mutations;

  3. 3)

The global toric action of $(\mathbb{C}^{*})^{k_{\mathbf{\Gamma}}+1}$ upon
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}(\mathbb{C}))$ is
induced by the action of $\mathcal{H}_{\mathbf{\Gamma}}\times\mathbb{C}^{*}$
upon $\operatorname{GL}_{n}(\mathbb{C})$;

  4. 4)

For any choice of $r_{0}$,
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{GL}_{n}(\mathbb{C}))$ is
compatible with the Poisson bivector $\pi^{\dagger}_{\mathbf{\Gamma}}$.

A similar theorem is valid in the case of $\operatorname{SL}_{n}$:

######  Theorem 1.6.

Let $\mathbf{\Gamma}$ be any Belavin-Drinfeld triple of type $A_{n-1}$. Then
the generalized cluster structure
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n}(\mathbb{C}))$
has the following properties:

  1. 1)

The number of frozen variables is $k_{\mathbf{\Gamma}}+(n-1)$, and the
exchange matrix has full rank;

  2. 2)

The generalized cluster structure
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n}(\mathbb{C}))$ is
complete; moreover, the initial extended cluster consists of irreducible
elements of $\mathcal{O}(\operatorname{SL}_{n}(\mathbb{C}))$ that are coprime
with their mutations;

  3. 3)

The global toric action of $(\mathbb{C}^{*})^{k_{\mathbf{\Gamma}}}$ upon
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n}(\mathbb{C}))$ is
induced by the action of $\mathcal{H}_{\mathbf{\Gamma}}$ upon
$\operatorname{SL}_{n}(\mathbb{C})$;

  4. 4)

For any choice of $r_{0}$,
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},\operatorname{SL}_{n}(\mathbb{C}))$ is
compatible with the Poisson bivector $\pi^{\dagger}_{\mathbf{\Gamma}}$.

We prove both theorems in Section 5 for
$\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ and in Section 7.4 for
$\mathcal{GC}_{g}^{\dagger}(\mathbf{\Gamma})$.

Given two BD triples $\mathbf{\Gamma}=(\Gamma_{1},\Gamma_{2},\gamma)$ and
$\tilde{\mathbf{\Gamma}}=(\tilde{\Gamma}_{1},\tilde{\Gamma}_{2},\theta)$ of
the same Lie type, let us write $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$
if $\tilde{\Gamma}_{1}\subsetneq\Gamma_{1}$,
$\tilde{\Gamma}_{2}\subsetneq\Gamma_{2}$ and
$\gamma|_{\tilde{\Gamma}_{1}}=\theta$. If
$\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$, then it is possible to choose
the same $r_{0}$ for both $\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$.

######  Proposition 1.7.

Let $\tilde{\mathbf{\Gamma}}\prec\mathbf{\Gamma}$ be two BD triples of type
$A_{n-1}$ and let
$G\in\\{\operatorname{SL}_{n}(\mathbb{C}),\operatorname{GL}_{n}(\mathbb{C})\\}$.
Then there exists a birational quasi-isomorphism
$\mathcal{G}:(G,\mathcal{GC}^{\dagger}(\tilde{\mathbf{\Gamma}}))\dashrightarrow(G,\mathcal{GC}^{\dagger}(\mathbf{\Gamma}))$.
Moreover,
$\mathcal{G}:(G,\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
is a Poisson isomorphism provided the $r_{0}$ parts of
$\pi_{\tilde{\mathbf{\Gamma}}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$ coincide.

If $\tilde{\mathbf{\Gamma}}=\mathbf{\Gamma}_{\text{std}}$, then
$\mathcal{G}=\mathcal{Q}$ in the $h$-convention and
$\mathcal{G}=\mathcal{Q}^{\mathrm{op}}$ in the $g$-convention. In Sections
4.4-4.5 we provide a more explicit formula for $\mathcal{G}$ and prove the
proposition for the $h$-convention. In the $g$-convention, the proof and the
formulas are provided in Section 7.3.

Let $\mathcal{P}_{+}(\Gamma_{2})$ and $\mathcal{P}_{-}(\Gamma_{1})$ be the
parabolic subgroups of $\operatorname{GL}_{n}$ generated by the BD data
$(\Gamma_{1},\Gamma_{2},\gamma)$. Define the subgroup
$\mathcal{D}\subset\mathcal{P}_{+}(\Gamma_{1})\times\mathcal{P}_{-}(\Gamma_{2})$
via

$\mathcal{D}:=\\{(g^{\prime},g)\in\mathcal{P}_{+}(\Gamma_{1})\times\mathcal{P}_{-}(\Gamma_{2})\
|\
{\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}(\mathring{\Pi}_{\Gamma_{1}}(g^{\prime}))=\mathring{\Pi}_{\Gamma_{2}}(g)\\}$
(1.28)

where
$\mathring{\Pi}_{\Gamma_{1}}:\mathcal{P}_{+}(\Gamma_{1})\rightarrow\operatorname{GL}_{n}(\Gamma_{1})$
and
$\mathring{\Pi}_{\Gamma_{2}}:\mathcal{P}_{-}(\Gamma_{2})\rightarrow\operatorname{GL}_{n}(\Gamma_{2})$
are the group projections upon the Levi factors of
$\mathcal{P}_{+}(\Gamma_{1})$ and $\mathcal{P}_{-}(\Gamma_{2})$, and the map
${\mathring{\phantom{\gamma}}\kern-6.3pt\tilde{\gamma}}$ is a certain group
lift of $\gamma$ (see Section 4.1 for details). Define an action of
$\mathcal{D}$ upon $\operatorname{GL}_{n}$ via

$((g^{\prime},g),U)\mapsto g^{\prime}Ug^{-1},\ \
(g^{\prime},g)\in\mathcal{D},\ U\in\operatorname{GL}_{n}.$ (1.29)

######  Proposition 1.8.

Let $\psi$ be a frozen variable in
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},G)$,
$G\in\\{\operatorname{SL}_{n}(\mathbb{C}),\operatorname{GL}_{n}(\mathbb{C})\\}$.
Then the following statements hold:

  1. 1)

The variable $\psi$ is semi-invariant with respect to the action (1.29);

  2. 2)

The variable $\psi$ is log-canonical with any $u_{ij}$;

  3. 3)

The zero locus of $\psi$ foliates into a union of symplectic leaves of
$(G,\pi_{\mathbf{\Gamma}}^{\dagger})$.

In the $h$-convention, we prove Proposition 1.8 in Section 6.4; in the
$g$-convention, we prove it in Section 7.2.

######  Remark 1.9.

By default, we will always assume that the ground field is $\mathbb{C}$.
However, all the above results hold for $\operatorname{SL}_{n}(\mathbb{R})$
and $\operatorname{GL}_{n}(\mathbb{R})$ if one uses the same formula for the
Poisson bracket $\\{\cdot,\cdot\\}_{\mathbf{\Gamma}}^{\dagger}$.

##  2 Background

In this section, we provide a necessary background on Poisson geometry,
generalized cluster structures and birational quasi-isomorphisms. We will also
use some well-known determinantal identities, which are listed in Appendix C.

###  2.1 Poisson geometry

In this section, we briefly recall relevant concepts from Poisson geometry. A
more detailed account can be found in [5], [8] and [30].

##### Poisson-Lie groups.

A _Poisson bracket $\\{\cdot,\cdot\\}$_ on a commutative algebra is a Lie
bracket that satisfies the Leibniz rule. Given a manifold $M$, a _Poisson
bivector field_ on $M$ is a section $\pi\in\Gamma(M,\bigwedge^{2}TM)$ such
that $\\{f,g\\}:=\pi(df\wedge dg)$ is a Poisson bracket on the space of smooth
functions on $M$. The pair $(M,\pi)$ is called a _Poisson manifold_.

######  Definition 2.1.

Given a Lie group $G$ and a Poisson bivector field
$\pi\in\Gamma(G,\Lambda^{2}TG)$, the pair $(G,\pi)$ is called a _Poisson-Lie
group_ if

$\pi_{gh}=[dL_{g}\otimes dL_{g}](\pi_{h})+[dR_{h}\otimes dR_{h}](\pi_{g}),\ \
g,h\in G,$ (2.1)

where $L_{g}$ and $R_{h}$ are the left and right translations by $g$ and $h$,
respectively. Equivalently, if $\\{\cdot,\cdot\\}$ is the Poisson bracket
defined by $\pi$, then the multiplication map from $G\times G$ equipped with
the direct product Poisson structure to $G$ is Poisson.

Let $\mathfrak{g}$ be the Lie algebra of $G$ and
$r\in\mathfrak{g}\otimes\mathfrak{g}$. If $G$ is a connected Lie group, then
the bivector field333If $G$ is simple and complex, then any bivector field
$\pi$ that yields the structure of a Poisson-Lie group on $G$ is of this form
for some $r\in\mathfrak{g}\otimes\mathfrak{g}$.

$\pi_{g}:=[dL_{g}\otimes dL_{g}](r)-[dR_{g}\otimes dR_{g}](r),\ \ g\in G,$
(2.2)

defines the structure of a Poisson-Lie group on $G$ if and only if the
following conditions are satisfied:

  1. 1)

The symmetric part of $r$ is $\operatorname{ad}$-invariant;

  2. 2)

The $3$-tensor $[r,r]:=[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]$ is
$\operatorname{ad}$-invariant, where $(a\otimes b)_{12}=a\otimes b\otimes 1$,
$(a\otimes b)_{13}=a\otimes 1\otimes b$ and $(a\otimes b)_{23}=1\otimes
a\otimes b$, $a,b\in\mathfrak{g}$.

The _Classical Yang-Baxter equation (CYBE)_ is the equation $[r,r]=0$. For
simple complex Lie algebras $\mathfrak{g}$, Belavin and Drinfeld in [1, 2]
classified solutions of the CYBE that have a nondegenerate
$\operatorname{ad}$-invariant symmetric part. The classification was partially
extended by Hodges in [27] to the case of reductive complex Lie algebras
(however, Hodges required the symmetric part of $r$ to be a multiple of the
Casimir element). A full classification of solutions of the CYBE with an
arbitrary nondegenerate $\operatorname{ad}$-invariant symmetric part in the
case of reductive complex Lie algebras was obtained by Delorme in [7].

##### Extensions of the Belavin-Drinfeld maps.

Let $\mathfrak{g}$ be a reductive complex Lie algebra, $\Pi$ be a set of
simple roots and $\langle\,,\,\rangle$ be a nondegenerate invariant symmetric
bilinear form on $\mathfrak{g}$.

######  Definition 2.2.

A _Belavin-Drinfeld triple_ (for conciseness, a _BD triple_) is a triple
$(\Gamma_{1},\Gamma_{2},\gamma)$ with $\Gamma_{1},\Gamma_{2}\subset\Pi$ and
$\gamma:\Gamma_{1}\rightarrow\Gamma_{2}$ a nilpotent444The nilpotency
condition means that for any $\alpha\in\Gamma_{1}$ there exists a number $j$
such that $\gamma^{j}(\alpha)\notin\Gamma_{1}$. isometry.

Decompose $\mathfrak{g}$ as
$\mathfrak{g}=\mathfrak{n}_{+}\oplus\mathfrak{h}\oplus\mathfrak{n}_{-}$, where
$\mathfrak{n}_{+}=\bigoplus_{\alpha>0}\mathfrak{g}_{\alpha}$ and
$\mathfrak{n}_{-}=\bigoplus_{\alpha>0}\mathfrak{g}_{-\alpha}$ are nilpotent
subalgebras, $\mathfrak{g}_{\alpha}$ are root subspaces and $\mathfrak{h}$ is
a Cartan subalgebra. For every positive root $\alpha$, choose
$e_{\alpha}\in\mathfrak{g}_{\alpha}$ and
$e_{-\alpha}\in\mathfrak{g}_{-\alpha}$ such that $\langle
e_{\alpha},e_{-\alpha}\rangle=1$, and set
$h_{\alpha}:=[e_{\alpha},e_{-\alpha}]$. Let $\mathfrak{g}_{\Gamma_{1}}$ and
$\mathfrak{g}_{\Gamma_{2}}$ be the semisimple Lie subalgebras of
$\mathfrak{g}$ generated by $\Gamma_{1}$ and $\Gamma_{2}$. Extend $\gamma$ to
an isomorphism $\mathbb{Z}\Gamma_{1}\xrightarrow{\sim}\mathbb{Z}\Gamma_{2}$,
and then define
$\gamma:\mathfrak{g}_{\Gamma_{1}}\rightarrow\mathfrak{g}_{\Gamma_{2}}$ via
$\gamma(e_{\alpha})=e_{\gamma(\alpha)}$ and
$\gamma(h_{\alpha})=h_{\gamma(\alpha)}$. Let
$\gamma^{*}:\mathfrak{g}_{\Gamma_{2}}\rightarrow\mathfrak{g}_{\Gamma_{1}}$ be
the conjugate of $\gamma$ with respect to the form on $\mathfrak{g}$. Extend
both $\gamma$ and $\gamma^{*}$ by zero to $\mathfrak{g}$ via setting
$\gamma|_{\mathfrak{g}_{\Gamma_{1}}^{\perp}}=0$ and
$\gamma^{*}|_{\mathfrak{g}_{\Gamma_{2}}^{\perp}}=0$. Note that the extensions
$\gamma,\gamma^{*}:\mathfrak{g}\rightarrow\mathfrak{g}$ are still adjoint to
one another.

##### The Belavin-Drinfeld classification.

Continuing in the setup of the previous paragraph, let
$\pi_{>}:\mathfrak{g}\rightarrow\mathfrak{n}_{+}$ and
$\pi_{<}:\mathfrak{g}\rightarrow\mathfrak{n}_{-}$ be the orthogonal
projections, and let $\Omega_{>}$ and $\Omega_{<}$ be the $2$-tensors that
correspond to $\pi_{>}$ and $\pi_{<}$ under the form on $\mathfrak{g}$. In
addition, let $\Omega_{0}$ be the restriction of the Casimir element $\Omega$
to $\mathfrak{h}\otimes\mathfrak{h}$. Consider an element
$r_{0}\in\mathfrak{h}\otimes\mathfrak{h}$ that satisfies the following system
of equations:

$\displaystyle r_{0}+r_{0}^{t}=\Omega_{0};$ (2.3)
$\displaystyle[\gamma(\alpha)\otimes 1](r_{0})+[1\otimes\alpha](r_{0})=0,\ \
\alpha\in\Gamma_{1}$ (2.4)

where $(a\otimes b)^{t}=b\otimes a$, $a,b\in\mathfrak{g}$.

######  Theorem 2.3 (Belavin & Drinfeld).

Under the above setup, if

$r=r_{0}+\left[\frac{1}{1-\gamma^{*}}\otimes
1\right](\Omega_{>})-\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right](\Omega_{<})$
(2.5)

where $r_{0}$ is any solution of (2.3)-(2.4), then $r$ satisfies CYBE.
Moreover,

$r+r^{t}=\Omega.$ (2.6)

Conversely, if $r$ is a solution of the CYBE such that $r$ satisfies (2.6),
then $r$ assumes the form (2.5) for a suitable decomposition of
$\mathfrak{g}$, for some Belavin-Drinfeld triple and some choice of the root
vectors $e_{\alpha}$.

The element $r$ from the theorem is called a _classical $r$-matrix_. If
$\mathfrak{g}$ is simple, then Theorem 2.3 provides a full classification of
solutions $r\in\mathfrak{g}\otimes\mathfrak{g}$ of the CYBE with nondegenerate
$\operatorname{ad}$-invariant symmetric parts. Sometimes we will use the
notation $r_{\mathbf{\Gamma}}$ along with $\pi_{\mathbf{\Gamma}}$ for the
Poisson bivector on $G$ defined by $r_{\mathbf{\Gamma}}$ via the formula
(2.2).

Let us define $R_{+}:\mathfrak{g}\rightarrow\mathfrak{g}$ and
$R_{-}:\mathfrak{g}\rightarrow\mathfrak{g}$ as linear maps via $\langle
R_{+}(x),y\rangle=\langle r,x\otimes y\rangle$ and $\langle
R_{-}(x),y\rangle=-\langle r,y\otimes x\rangle$, $x,y\in\mathfrak{g}$. Then
$R_{+}-R_{-}=\operatorname{id}$ and

$R_{+}=\frac{1}{1-\gamma}\pi_{>}-\frac{\gamma^{*}}{1-\gamma^{*}}\pi_{<}+R_{0}\pi_{0}.$
(2.7)

The system (2.3)-(2.4) can be rewritten in terms of a linear map
$R_{0}:\mathfrak{h}\rightarrow\mathfrak{h}$, $\langle
R_{0}(x),y\rangle=\langle r_{0},x\otimes y\rangle$, $x,y\in\mathfrak{h}$, via

$\displaystyle R_{0}+R_{0}^{*}=\operatorname{id};$ (2.8) $\displaystyle
R_{0}(h_{\alpha}-\gamma(h_{\alpha}))=h_{\alpha},\ \ \alpha\in\Gamma_{1}$ (2.9)

where $R_{0}^{*}$ denotes the adjoint of $R_{0}$ in $\mathfrak{h}$ with
respect to $\langle\cdot,\cdot\rangle$.

##### The Drinfeld Double.

Let $G$ be a reductive connected complex Poisson-Lie group endowed with a
nondegenerate symmetric invariant bilinear form on $\mathfrak{g}$ and with a
Poisson bivector field $\pi_{\mathbf{\Gamma}}$ defined by a classical
$r$-matrix $r_{\mathbf{\Gamma}}$ relative some Belavin-Drinfeld triple
$\mathbf{\Gamma}$. Let $R_{+}$ and $R_{-}$ be defined from
$r_{\mathbf{\Gamma}}$ as in the previous paragraph, and set
$\mathfrak{d}:=\mathfrak{g}\oplus\mathfrak{g}$ to be the direct sum of Lie
algebras. Define a nondegenerate symmetric invariant bilinear form on
$\mathfrak{d}$ as

$\langle(x_{1},y_{1}),(x_{2},y_{2})\rangle=\langle x_{1},x_{2}\rangle-\langle
y_{1},y_{2}\rangle,\ \ x_{1},\,x_{2},\,y_{1},\,y_{2}\in\mathfrak{g}.$ (2.10)

As a vector space, $\mathfrak{d}$ splits into the direct sum of the following
isotropic Lie subalgebras:

$\mathfrak{g}^{\delta}:=\\{(x,x)\ |\ x\in\mathfrak{g}\\},\ \
\mathfrak{g}^{*}:=\\{(R_{+}(x),R_{-}(x))\ |\ x\in\mathfrak{g}\\}$ (2.11)

where $R_{+}$ and $R_{-}$ are defined from $r_{\mathbf{\Gamma}}$. Set
$R^{\mathfrak{d}}_{+}:=P_{\mathfrak{g}^{*}}$ where $P_{\mathfrak{g}^{*}}$ is
the projection of $\mathfrak{d}$ onto $\mathfrak{g}^{*}$, and let
$r^{\mathfrak{d}}\in\mathfrak{d}\otimes\mathfrak{d}$ be the $2$-tensor that
corresponds to $R^{\mathfrak{d}}_{+}$. Then $R_{+}^{\mathfrak{d}}$ yields the
structure of a Poisson-Lie group on the Lie group $D(G):=G\times G$ via the
Poisson bivector field

$(\pi^{D}_{\mathbf{\Gamma}})_{(g,h)}:=[dL_{(g,h)}\otimes
dL_{(g,h)}](r^{\mathfrak{d}})-[dR_{(g,h)}\otimes
dR_{(g,h)}](r^{\mathfrak{d}}),\ \ {(g,h)}\in D(G).$ (2.12)

######  Definition 2.4.

The Poisson-Lie group $(D(G),\pi_{\mathbf{\Gamma}}^{D})$ is called the
_Drinfeld double_ of $(G,\pi_{\mathbf{\Gamma}})$.

The Poisson bracket on $D(G)$ can be written in the form

$\\{f_{1},f_{2}\\}_{\mathbf{\Gamma}}^{D}=\langle
R_{+}(E_{L}f_{1}),E_{L}f_{2}\rangle-\langle
R_{+}(E_{R}f_{1}),E_{R}f_{2}\rangle+\langle\nabla_{X}^{R}f_{1},\nabla^{R}_{Y}f_{2}\rangle-\langle\nabla_{X}^{L}f_{1},\nabla_{Y}^{L}f_{2}\rangle$
(2.13)

where $\nabla^{L}f_{i}=(\nabla_{X}^{L}f_{i},-\nabla_{Y}^{L}f_{i})$ and
$\nabla^{R}f_{i}=(\nabla_{X}^{R}f_{i},-\nabla_{Y}^{R}f_{i})$ are the left and
the right gradients, respectively,
$E_{L}f_{i}=\nabla^{L}_{X}f_{i}+\nabla^{L}_{Y}f_{i}$ and
$E_{R}f_{i}=\nabla^{R}_{X}f_{i}+\nabla^{R}_{Y}f_{i}$. We define the gradients
on $G$ as555This convention is opposite to the one in [23] and [30], but in
this way the left gradient is the gradient in the left trivialization, and the
right gradient is the gradient in the right trivialization of the group.

$\langle\nabla^{L}f|_{g},x\rangle:=\left.\frac{d}{dt}\right|_{t=0}f(g\exp(tx)),\
\ \
\langle\nabla^{R}f|_{g},x\rangle:=\left.\frac{d}{dt}\right|_{t=0}f(\exp(tx)g),\
\ g\in G,\ x\in\mathfrak{g}.$ (2.14)

The group $G$ can be identified with the diagonal subgroup $G^{\delta}$ of
$D(G)$, which corresponds to the Lie subalgebra $\mathfrak{g}^{\delta}$. The
Poisson bracket $\\{\cdot,\cdot\\}_{\mathbf{\Gamma}}$ on $G$ can be expressed
as

$\\{f_{1},f_{2}\\}_{\mathbf{\Gamma}}=\langle
R_{+}(\nabla^{L}f_{1}),\nabla^{L}f_{2}\rangle-\langle
R_{+}(\nabla^{R}f_{1}),\nabla^{R}f_{2}\rangle.$ (2.15)

Additionally, the connected (immersed) Poisson-Lie subgroup
$(G^{*},\pi_{\mathbf{\Gamma}}^{*})$ of $D(G)$ that corresponds to
$\mathfrak{g}^{*}$ is called the _dual Poisson-Lie group of $G$_.

##### The Poisson dual $G^{\dagger}$.

In the setup of the previous paragraph, consider the Drinfeld double
$(D(G),\pi^{D}_{\mathbf{\Gamma}})$ along with
$(G^{*},\pi^{*}_{\mathbf{\Gamma}})$ immersed into $D(G)$. Consider the map

$D(G)\ni(g,h)\mapsto g^{-1}h\in G.$ (2.16)

The pushforward of $\pi^{D}_{\mathbf{\Gamma}}$ under the above map is denoted
as $\pi^{\dagger}_{\mathbf{\Gamma}}$, and the image of $G^{*}$ is an open
subset of $G$ denoted as $G^{\dagger}$. The Poisson bivector
$\pi^{\dagger}_{\mathbf{\Gamma}}$ is given by

$\begin{split}(\pi^{\dagger}_{\mathbf{\Gamma}})_{g}=\left[dL_{g}\otimes
dL_{g}-dR_{g}\right.&\left.\otimes dL_{g}-dL_{g}\otimes dR_{g}+dR_{g}\otimes
dR_{g}\right](r)-\\\ -&\left[dL_{g}\otimes dL_{g}-dR_{g}\otimes
dL_{g}\right](\Omega),\ \ g\in G,\end{split}$ (2.17)

and the corresponding Poisson bracket is

$\\{f_{1},f_{2}\\}^{\dagger}_{\mathbf{\Gamma}}=\langle
R_{+}(\nabla^{L}f_{1}-\nabla^{R}f_{1}),\nabla^{L}f_{2}-\nabla^{R}f_{2}\rangle-\langle\nabla^{L}f_{1}-\nabla^{R}f_{1},\nabla^{L}f_{2}\rangle.$
(2.18)

If $\mathbf{\Gamma}$ is a trivial BD triple (that is,
$\Gamma_{1}=\Gamma_{2}=\emptyset$), we will denote the corresponding Poisson
bivector as $\pi_{\text{std}}^{\dagger}$ and the Poisson bracket as
$\\{\cdot,\cdot\\}_{\text{std}}^{\dagger}$.

##### Symplectic foliation and Poisson submanifolds.

Let $(M,\pi)$ be a Poisson manifold. An immersed submanifold $S\subseteq M$ is
called a _Poisson submanifold_ if $\pi|_{S}\in\Gamma(S,\bigwedge^{2}TS)$.
Define a morphism of vector bundles $\pi^{\\#}:TM^{*}\rightarrow TM$ via
$\langle\pi^{\\#}(\xi),\eta\rangle:=\langle\pi,\xi\wedge\eta\rangle$,
$\xi,\eta\in T_{p}^{*}M$, $p\in M$. The Poisson bivector $\pi$ is called
_nondegenerate_ if $\pi^{\\#}$ is an isomorphism of vector bundles. A
_symplectic leaf_ is a maximal (by inclusion) connected Poisson submanifold
$S$ of $M$ for which $\pi|_{S}$ is nondegenerate. It is a theorem that any
Poisson manifold $M$ is a union of its symplectic leaves.

In this paper, we will be interested in symplectic leaves of the zero loci of
frozen variables (see Proposition 1.8). Assume that $M$ is a smooth real or
complex affine variety. Let us recall that an ideal $J\subseteq\mathcal{O}(M)$
is called a _Poisson ideal_ if $\\{\mathcal{O}(M),J\\}\subseteq J$. The
following proposition is a restatement of [29, Remark 2.4] in the setup of the
current paper:

######  Proposition 2.5.

Let $p\in\mathcal{O}(M)$. Then the following conditions are equivalent:

  1. (i)

$p$ is a prime element of $\mathcal{O}(M)$ and the principal ideal $(p)$ is a
Poisson ideal;

  2. (ii)

The zero locus of $p$ is a union of symplectic leaves of $(M,\pi)$.

Proposition 2.5 can also be generalized to the case of not necessarily smooth
affine varieties; see [25, Proposition 2.3].

###  2.2 Generalized cluster algebras

In this subsection, we briefly review the relevant definitions and
propositions from the theory of generalized cluster structures. An expanded
account with proofs can be found in [20]. The concept of a generalized cluster
algebra is based on an earlier definition given in [6] (see [20, Remark 2.1]).
Throughout the subsection, we let $\mathcal{F}$ to be a field of rational
functions in $N+M$ independent variables with coefficients in $\mathbb{Q}$. We
refer to $\mathcal{F}$ as the _ambient field_.

######  Definition 2.6 (Seeds).

An _extended seed_ is a triple $\Sigma:=(\mathbf{x},B,\mathcal{P})$ that
consists of the following data:

  1. 1)

A collection of elements $\mathbf{x}:=(x_{1},x_{2},\ldots,x_{N+M})$ of
$\mathcal{F}$ such that $\mathcal{F}=\mathbb{Q}(x_{1},x_{2},\ldots,x_{N+M})$
(an _extended cluster_). The elements $x_{1},x_{2},\ldots,x_{N}$ are called
_cluster variables_ and the elements $x_{N+1},\ldots,x_{N+M}$ are called
_frozen variables_ ;

  2. 2)

An integer matrix $B:=(b_{ij}\ |\ i\in[1,N],\ j\in[1,N+M])$ such that the
principal part $B^{[1,N]}$ is skew-symmetrizable666Recall that a square matrix
$A$ of size $N\times N$ is _skew-symmetrizable_ if there exists a diagonal
matrix $D$ with positive entries such that $DA$ is skew-symmetric. (an
_exchange matrix_);

  3. 3)

For each $i\in[1,N]$, set $d_{i}$ to be a factor of $\gcd(b_{ij}\ |\
j\in[1,N])$ (the _multiplicity_ of $x_{i}$). For each $i\in[1,N]$, define the
$i$_th string_ $p_{i}:=(p_{ir})_{r\in[0,d_{i}]}$ to be a collection of
monomials $p_{ir}$ in the frozen variables such that $p_{i0}=p_{id_{i}}=1$.
The collection of _strings_ $\mathcal{P}$ is given by
$\mathcal{P}:=\\{p_{i}\\}_{i\in[1,N]}$.

######  Definition 2.7 ($\tau$-monomials.).

Let $\Sigma:=(\mathbf{x},B,\mathcal{P})$ be an extended seed. For $k\in[1,N]$,
the _cluster $\tau$-monomials_ $u_{k;>}$ and $u_{k;<}$ are monomials in the
cluster variables of $\Sigma$ given by

$u_{k;>}:=\prod_{\begin{subarray}{c}1\leq i\leq N,\\\
b_{ki}>0\end{subarray}}x_{i}^{b_{ki}/d_{k}},\ \
u_{k;<}:=\prod_{\begin{subarray}{c}1\leq i\leq N,\\\
b_{ki}<0\end{subarray}}x_{i}^{-b_{ki}/d_{k}}.$ (2.19)

For $k\in[1,N]$ and $r\in[0,d_{k}]$, the _stable $\tau$-monomials_
$v_{k;>}^{[r]}$ and $v_{k;<}^{[r]}$ are monomials in the frozen variables
given by

$v_{k;>}^{[r]}:=\prod_{\begin{subarray}{c}N+1\leq i\leq N+M,\\\
b_{ki}>0\end{subarray}}x_{i}^{\lfloor rb_{ki}/d_{k}\rfloor},\ \
v_{k;<}^{[r]}:=\prod_{\begin{subarray}{c}N+1\leq i\leq N+M,\\\
b_{ki}<0\end{subarray}}x_{i}^{\lfloor-rb_{ki}/d_{k}\rfloor},$ (2.20)

where the product over an empty set by definition equals $1$ and $\lfloor
m\rfloor$ denotes the floor of $m\in\mathbb{Z}$.

######  Definition 2.8 (Generalized cluster mutations).

Let $\Sigma:=(\mathbf{x},B,\mathcal{P})$ be an extended seed. For $k\in[1,N]$,
a _generalized cluster mutation in direction $k$_ of $\Sigma$ is an extended
seed $\Sigma^{\prime}:=(\mathbf{x}^{\prime},B^{\prime},\mathcal{P}^{\prime})$
defined as follows.

  1. 1)

Let $x_{k}^{\prime}$ be an element given by the _generalized exchange
relation_ :

$x_{k}x_{k}^{\prime}:=\sum_{r=0}^{d_{k}}p_{kr}u_{k;>}^{r}v_{k;>}^{[r]}u_{k;<}^{d_{k}-r}v_{k;<}^{[d_{k}-r]}$
(2.21)

where $u_{k;>}$ and $u_{k;<}$ are the cluster $\tau$-monomials and $v_{k;<}$
and $v_{k;<}$ are the stable $\tau$-monomials constructed relative $\Sigma$.
The extended cluster $\mathbf{x}^{\prime}$ is given by
$\mathbf{x}^{\prime}:=(\mathbf{x}\setminus\\{x_{k}\\})\cup\\{x_{k}^{\prime}\\}$.

  2. 2)

The exchange matrix $B^{\prime}:=(b_{ij}^{\prime}\ |\ i\in[1,N],\
j\in[1,N+M])$ is given by

$b_{ij}^{\prime}:=\begin{cases}-b_{ij}\ \ &\text{if }i=k\text{ or }j=k;\\\
b_{ij}+\dfrac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2}\ \
&\text{otherwise.}\end{cases}$ (2.22)

  3. 3)

Define $p_{k}^{\prime}:=(p_{kr})_{r\in[0,d_{k}]}$ as $p_{kr}:=p_{k,d_{k}-r}$,
$r\in[0,d_{k}]$. The collection of strings $\mathcal{P}^{\prime}$ is given by
$\mathcal{P}^{\prime}:=(\mathcal{P}\setminus\\{p_{k}\\})\cup\\{p_{k}^{\prime}\\}$.

The extended seeds $\Sigma$ and $\Sigma^{\prime}$ are also called _adjacent_.

Let us make a few remarks on the above definition:

  1. 1)

A frozen variable $x_{i}$ is called _isolated_ if $b_{ji}=0$ for all
$j\in[1,N]$. The definition of $b_{ij}^{\prime}$ implies that if $x_{i}$ is
isolated in $\Sigma$, it is isolated in $\Sigma^{\prime}$ as well;

  2. 2)

Since $\gcd\\{b_{ij}\ |\ 1\leq j\leq N\\}=\gcd\\{b_{ij}^{\prime}\ |\ 1\leq
j\leq N\\}$, the numbers $d_{1},\ldots,d_{N}$ are the same in $\Sigma$ and
$\Sigma^{\prime}$;

  3. 3)

If the string $p_{k}$ is trivial, then the generalized exchange relation
(2.21) becomes the exchange relation from the theory of cluster algebras of
geometric type:

$x_{k}x_{k}^{\prime}=\prod_{\begin{subarray}{c}1\leq i\leq N+M\\\
b_{ki>0}\end{subarray}}x_{i}^{b_{ki}}+\prod_{\begin{subarray}{c}1\leq i\leq
N+M\\\ b_{ki<0}\end{subarray}}x_{i}^{-b_{ki}}.$ (2.23)

In fact, the generalized cluster structures studied in this paper have only
one nontrivial string, so all the exchange relations except one are given by
(2.23).

  4. 4)

For each $i\in[1,N]$, set $v_{i;>}:=v_{i;>}^{[d_{i}]}$,
$v_{i;<}:=v_{i;<}^{[d_{i}]}$; define

$q_{ir}:=\frac{v_{i;>}^{r}v_{i;<}^{d_{i}-r}}{(v_{i;>}^{[r]}v_{i;<}^{[d_{i}-r]})^{d_{i}}},\
\ \hat{p}_{ir}:=\frac{p_{ir}^{d_{i}}}{q_{ir}},\ \ i\in[1,N],\ r\in[0,d_{i}].$

Then the relation (2.21) can be rewritten as

$x_{k}x_{k}^{\prime}=\sum_{r=0}^{d_{k}}(\hat{p}_{kr}v_{k;>}^{r}v_{k;<}^{d_{k}-r})^{1/d_{k}}u_{k;>}^{r}u_{k;<}^{d_{k}-r}.$
(2.24)

The expression $(\hat{p}_{kr}v_{k;>}^{r}v_{k;<}^{d_{k}-r})^{1/d_{k}}$ is a
monomial in the frozen variables. Note also that the mutation rule for
$\hat{p}_{ir}$ is the same as for $p_{ir}$ (see Part 3 of Definition 2.8).

######  Definition 2.9 (Generalized cluster structures).

Two extended seeds $\Sigma$ and $\Sigma^{\prime}$ are called _mutation
equivalent_ if there’s a sequence of extended seeds
$\Sigma_{1},\ldots,\Sigma_{m}$ where $\Sigma_{1}:=\Sigma$ and
$\Sigma_{m}:=\Sigma^{\prime}$, such that $\Sigma_{i+1}$ and $\Sigma_{i}$ are
adjacent for each $i$. For a fixed extended seed $\Sigma$, the set of all
extended seeds that are mutation equivalent to $\Sigma$ is called a
_generalized cluster structure_ and is denoted as $\mathcal{GC}(\Sigma)$ or
simply as $\mathcal{GC}$. The _rank_ of $\mathcal{GC}$ is the number $N$.

In practice, we fix one extended seed
$\Sigma_{0}:=(\mathbf{x}_{0},B_{0},\mathcal{P}_{0})$ and provide its full
description. In this case, we call $\Sigma_{0}$ the _initial extended seed_
(correspondingly, $\mathbf{x}_{0}$ is the _initial extended cluster_ and
$B_{0}$ is the _initial exchange matrix_). Also, when several generalized
cluster structures are considered at once, we will denote by
$\mathcal{F}(\mathcal{GC})$ the ambient field of $\mathcal{GC}$.

##### Algebras associated to $\mathcal{GC}$.

Let $\mathcal{GC}$ be a generalized cluster structure and
$x_{N+1},\ldots,x_{N+M}$ be its set of frozen variables. Define the rings
$\mathbb{A}:=\mathbb{Z}[x_{N+1},\ldots,x_{N+M}]$ and
$\bar{\mathbb{A}}:=\mathbb{Z}[x_{N+1}^{\pm 1},\ldots,x_{N+M}^{\pm 1}]$.

######  Definition 2.10.

A _ground ring_ $\hat{\mathbb{A}}$ is a subring of $\bar{\mathbb{A}}$ that
contains $\mathbb{A}$.

######  Definition 2.11.

The $\hat{\mathbb{A}}$-algebra $\mathcal{A}:=\mathcal{A}(\mathcal{GC})$
generated by all the cluster variables in $\mathcal{GC}$ is called the
_generalized cluster algebra_.

For a fixed ground ring $\hat{\mathbb{A}}$ and an extended cluster
$\mathbf{x}:=(x_{1},x_{2},\ldots,x_{N+M})$ in $\mathcal{GC}$, define the
$\hat{\mathbb{A}}$-algebras

$\mathcal{A}(\mathbf{x}):=\hat{\mathbb{A}}[x_{1},\ldots,x_{N}],\ \ \
\mathcal{L}(\mathbf{x}):=\hat{\mathbb{A}}[x_{1}^{\pm 1},\ldots,x_{N}^{\pm
1}].$ (2.25)

For an extended seed $\Sigma$ that contains the extended cluster $\mathbf{x}$,
we will also write $\mathcal{L}(\Sigma):=\mathcal{L}(\mathbf{x})$ and
$\mathcal{A}(\Sigma):=\mathcal{A}(\mathbf{x})$.

######  Definition 2.12.

The $\hat{\mathbb{A}}$-algebra given by

$\bar{\mathcal{A}}:=\bar{\mathcal{A}}(\mathcal{GC}):=\bigcap_{\Sigma\in\mathcal{GC}}\mathcal{L}(\Sigma)$
(2.26)

is called the _generalized upper cluster algebra_.

One of the central results of the theory is the _generalized Laurent
phenomenon_ , which states that there is an inclusion
$\mathcal{A}\subseteq\bar{\mathcal{A}}$. For cluster algebras, the result was
first proved in [13, Theorem 3.1]; later, it was extended to generalized
cluster algebras in [6, Theorem 2.5].

##### Upper bounds.

We continue in the setup of the previous paragraph. Let $\mathbb{T}_{N}$ be a
labeled $N$-regular tree. Associate with each vertex an extended seed so that
adjacent extended seeds are adjacent in the tree, and if an extended seed
$\Sigma^{\prime}$ is adjacent to $\Sigma$ in direction $k$, label the
corresponding edge in the tree with number $k$. A _nerve_ $\mathcal{N}$ in
$\mathbb{T}_{N}$ is a subtree on $N+1$ vertices such that all its edges have
different labels (for instance, a star is a nerve).

######  Definition 2.13.

An _upper bound_ $\bar{\mathcal{A}}(\mathcal{N})$ is the
$\hat{\mathbb{A}}$-algebra defined as

$\bar{\mathcal{A}}(\mathcal{N}):=\bigcap_{\Sigma\in
V(\mathcal{N})}\mathcal{L}(\Sigma)$ (2.27)

where $V(\mathcal{N})$ stands for the vertex set of $\mathcal{N}$.

Upper bounds were first defined and studied in [3]. Let $L$ be the number of
isolated frozen variables in $\mathcal{GC}$. For the $i$th nontrivial string
in $\mathcal{P}$, let ${B}(i)$ be a $(d_{i}-1)\times L$ matrix such that the
$r$th row consists of the exponents of the isolated variables in $p_{ir}$. The
following result was proved in [20, Theorem 3.11]:

######  Proposition 2.14.

Assume that the extended exchange matrix has full rank and let
$\operatorname{rank}{B}(i)=d_{i}-1$ for any nontrivial string in
$\mathcal{P}$. Then for any upper bound $\mathcal{N}$ in $\mathcal{GC}$,
$\bar{\mathcal{A}}(\mathcal{N})=\bar{\mathcal{A}}$.

##### Poisson structures in $\mathcal{GC}$.

Let $\\{\cdot,\cdot\\}$ be a Poisson bracket on $\mathcal{F}$ and let
${\mathbf{x}}$ be any extended cluster in $\mathcal{GC}$. We say that
${\mathbf{x}}$ is _log-canonical_ if for each $i,j\in[1,N+M]$,
$\\{x_{i},x_{j}\\}=\omega_{ij}x_{i}x_{j}$, where $\omega_{ij}\in\mathbb{Q}$.
The matrix $\Omega:=(\omega_{ij})_{i,j=1}^{N+M}$ is called the _coefficient
matrix_ of the Poisson bracket with respect to the extended cluster
$\tilde{\mathbf{x}}$.

######  Definition 2.15.

The generalized cluster structure $\mathcal{GC}$ is called _compatible_ with a
Poisson bracket $\\{\cdot,\cdot\\}$ if each extended cluster in $\mathcal{GC}$
is log-canonical.

The following result is the key connection between cluster algebras and
Poisson geometry. Given a log-canonical collection of elements, it shows how
to reconstruct the exchange matrix $B$ from the coefficient matrix $\Omega$.
Its earliest version was proved in [17, Theorem 1.4].

######  Proposition 2.16.

Let $(\tilde{\mathbf{x}},B,\mathcal{P})\in\mathcal{GC}$ be an extended seed
that satisfies the following properties:

  1. 1)

The extended cluster ${\mathbf{x}}$ is log-canonical with respect to the
Poisson bracket;

  2. 2)

For a diagonal matrix with positive entries $D$ such that $DB^{[1,N]}$ is
skew-symmetric, there exists a diagonal $N\times N$ matrix $\Delta$ such that
$B\Omega=\begin{bmatrix}\Delta&0\end{bmatrix}$ and such that $D\Delta$ is a
multiple of the identity matrix;

  3. 3)

The Laurent polynomials $\hat{p}_{ir}$ are Casimirs of the Poisson bracket.

Then any other extended seed in $\mathcal{GC}$ satisfies properties 1, 2 (with
the same $\Delta$) and 3. In particular, $\mathcal{GC}$ is compatible with
$\\{\cdot,\cdot\\}$.

Condition 2 has the following interpretation, which is used in practice. For
each $i\in[1,N]$, define the _$y$ -variable_ of the cluster variable $x_{i}$
as $y_{i}:=y(x_{i}):=\prod_{j=1}^{N+M}x_{j}^{b_{ij}}$. Then Condition 2 is
equivalent to $\\{\log y_{i},\log x_{j}\\}=\delta_{ij}\Delta_{ii}$, where
$\delta_{ij}$ is the Kronecker symbol. Note also that Condition 2 implies that
$B$ has full rank.

#####  $\mathcal{GC}$ on varieties.

Let $V$ be a complex affine variety, $\mathcal{O}(V)$ be its coordinate ring
and $\mathbb{C}(V)$ be its field of rational functions. For an extended
cluster $\mathbf{x}$ in $\mathcal{GC}$, set

$\displaystyle\mathcal{A}_{\mathbb{C}}(\mathbf{x}):=\mathcal{A}(\mathbf{x})\otimes\mathbb{C},\
$
$\displaystyle\mathcal{L}_{\mathbb{C}}(\mathbf{x}):=\mathcal{L}(\mathbf{x})\otimes\mathbb{C};$
(2.28)
$\displaystyle\mathcal{A}_{\mathbb{C}}(\mathcal{GC}):=\mathcal{A}(\mathcal{GC})\otimes\mathbb{C},\
$
$\displaystyle\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC}):=\bar{\mathcal{A}}(\mathcal{GC})\otimes\mathbb{C};$
(2.29)

For the ambient field $\mathcal{F}$ of $\mathcal{GC}$, we also set
$\mathcal{F}_{\mathbb{C}}:=\mathcal{F}\otimes\mathbb{C}$.

######  Definition 2.17.

The pair $(V,\mathcal{GC})$ is a _generalized cluster structure on the affine
variety $V$_ if there is a fixed embedding
$\mathcal{A}_{\mathbb{C}}(\mathbf{x}_{0})\subseteq\mathcal{O}(V)$. A variable
$x$ in $\mathcal{GC}$ is called _regular_ if $x\in\mathcal{O}(V)$; an extended
cluster $\mathbf{x}$ is called _regular_ if for each $x\in\mathbf{x}$, $x$ is
regular; the generalized cluster structure $(V,\mathcal{GC})$ is called
_regular_ if $\mathcal{A}_{\mathbb{C}}(\mathcal{GC})\subseteq\mathcal{O}(V)$.
Finally, $(V,\mathcal{GC})$ is called _complete_ if
$\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC})=\mathcal{O}(V)$.

##### Toric actions.

Let $\mathbf{x}:=(x_{1},x_{2},\ldots,x_{N+M})$ be an extended cluster in some
generalized cluster structure $\mathcal{GC}$, as above. Given an integer
matrix $W:=(w_{ij}\ |\ i\in[1,N+M],\ j\in[1,s])$, consider an action of
$(\mathbb{C}^{*})^{s}$ upon $\mathcal{F}_{\mathbb{C}}$ by field automorphisms:
for every $\mathbf{t}:=(t_{1},t_{2},\ldots,t_{s})\in(\mathbb{C}^{*})^{s}$, the
corresponding field automorphism $\sigma_{\mathbf{t}}$ is given by

$\sigma_{\mathbf{t}}(x_{i})=\left(\prod_{j=1}^{s}t_{j}^{w_{ij}}\right)\cdot
x_{i},\ \ i\in[1,N+M].$ (2.30)

######  Definition 2.18.

A _local toric action_ of rank $s$ upon an extended cluster $\mathbf{x}$ is
the action by field automorphisms given by (2.30) such that $W$ is of rank
$s$. It is called a _global toric action_ if the action is equivariant777In
other words, given any other extended cluster $\mathbf{x}^{\prime}$ mutation
equivalent to $\mathbf{x}$, there exists a matrix $W^{\prime}$ such that the
action is still of the form (2.30). with respect to mutations.

The first version of following proposition was proved in [17, Lemma 2.3] for
cluster algebras of geometric type, and it was later extended to generalized
cluster algebras in [20, Proposition 2.6].

######  Proposition 2.19.

A local toric action with a weight matrix $W$ is a global toric action if
$\tilde{B}W=0$ and the Laurent polynomials $\hat{p}_{ir}$ are invariant with
respect to the action.

When all the strings are trivial, the statement of Proposition 2.19 is _if and
only if_.

##### Quivers.

A _quiver_ is a directed multigraph with no $1$\- and $2$-cycles. Pick an
extended seed $\Sigma:=({\mathbf{x}},{B},\mathcal{P})$ and let
$D:=\operatorname{diag}(d_{1}^{-1},\ldots,d_{N}^{-1})$ be a diagonal matrix
with $d_{i}$’s defined as above. Assume that $DB^{[1,N]}$ is skew-symmetric.
Then the matrix

$\hat{B}:=\begin{bmatrix}DB^{[1,N]}&{B}^{[N+1,N+M]}\\\
-({B}^{[N+1,N+M]})^{T}&0\end{bmatrix}$ (2.31)

is the adjacency matrix of a quiver $Q$, in which each vertex $i$ corresponds
to a variable $x_{i}\in{\mathbf{x}}$. The vertices that correspond to cluster
variables are called _mutable_ , the vertices that correspond to frozen
variables are called _frozen_ , and the vertices that correspond to isolated
variables are called _isolated_. For each $i$, the number $d_{i}$ is called
the _multiplicity_ of the $i$th vertex. If one mutates the extended seed
$\Sigma$, then the quiver of the resulting extended seed can be obtained from
$Q$ via the following steps:

  1. 1)

For each path $i\rightarrow k\rightarrow j$, add an arrow $i\rightarrow j$;

  2. 2)

If there is a pair of arrows $i\rightarrow j$ and $j\rightarrow i$, remove
both;

  3. 3)

Flip the orientation of all arrows going in and out of the vertex $k$.

The above process is called a _quiver mutation in direction $k$_ (or _at
vertex $k$_). Instead of describing the exchange matrix ${B}$, we describe the
corresponding quiver and multiplicities $d_{i}$’s.

###  2.3 Birational quasi-isomorphisms

In this subsection, we state some of the results on birational quasi-
isomorphisms from [32] that are used in the proofs of Theorems 1.5 and 1.6.
The statements are adjusted for generalized cluster structures in the
coordinate rings of complex affine varieties. We assume that for all
generalized cluster structures, the ground rings are chosen so that
$\hat{\mathbb{A}}=\mathbb{A}$ (in other words, we do not localize at the
frozen variables).

##### Markers and related triples.

Let $\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$ be generalized cluster
structures of ranks $N$ and $\tilde{N}$ with $M$ and $\tilde{M}$ frozen
variables, respectively. Assume that $\tilde{M}-M>0$ and
$M+N=\tilde{M}+\tilde{N}$. Let $\operatorname{\mathbf{fr}}_{\mathcal{GC}}$ and
$\operatorname{\mathbf{fr}}_{\widetilde{\mathcal{GC}}}$ be the indices of the
frozen variables in $\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$. For a
bijection $\kappa:[1,N+M]\rightarrow[1,N+M]$, set

$\displaystyle\tilde{\mathcal{I}}(\kappa):=\operatorname{\mathbf{fr}}_{\widetilde{\mathcal{GC}}}\setminus\kappa(\operatorname{\mathbf{fr}}_{\mathcal{GC}}),$
(2.32)
$\displaystyle\mathcal{I}(\kappa):=\kappa^{-1}(\tilde{\mathcal{I}}(\kappa)).$
(2.33)

Denote by $d_{i}$ and $\tilde{d}_{j}$ the multiplicities of the variables with
indices $i$ and $j$ in $\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$,
respectively.

######  Definition 2.20.

A _marker_ $\kappa$ for the pair $(\mathcal{GC},\widetilde{\mathcal{GC}})$ is
a bijection $k:[1,N+M]\rightarrow[1,N+M]$ such that
$\kappa(\operatorname{\mathbf{fr}}_{\mathcal{GC}})\subset\operatorname{\mathbf{fr}}_{\widetilde{\mathcal{GC}}}$
and for any $i\in[1,N]\setminus\mathcal{I}(\kappa)$,
$d_{i}=\tilde{d}_{\kappa(i)}$. For a given marker $\kappa$, the triple
$(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ is called a _related_ triple.
The sets $\mathcal{I}(\kappa)$ and $\tilde{\mathcal{I}}(\kappa)$ are called
the sets of _marked indices_ in $\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$,
respectively.

Assume that $(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ is a related
triple. Fix initial extended clusters
$\mathbf{x}_{0}:=(x_{1},x_{2},\ldots,x_{N+M})$ and
$\tilde{\mathbf{x}}_{0}:=(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{N+M})$
in $\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$, respectively. Define

$\displaystyle\mathbf{x}_{0}(\kappa):=\\{x_{i}\in\mathbf{x}_{0}\ |\
i\in\mathcal{I}(\kappa)\\},$ (2.34)
$\displaystyle\tilde{\mathbf{x}}_{0}(\kappa):=\\{\tilde{x}_{i}\in\tilde{\mathbf{x}}_{0}\
|\ i\in\tilde{\mathcal{I}}(\kappa)\\}.$ (2.35)

######  Definition 2.21.

The elements of $\mathbf{x}_{0}(\kappa)$ and $\tilde{\mathbf{x}}_{0}(\kappa)$
are called the _marked variables_ in $\mathcal{GC}$ and
$\widetilde{\mathcal{GC}}$.

######  Definition 2.22.

An extended cluster $\mathbf{x}$ in $\mathcal{GC}$ is called _marked_ if
$\mathbf{x}$ is obtained by mutating $\mathbf{x}_{0}$ along the sequence

$x_{i_{1}}\rightarrow x_{i_{2}}\rightarrow\cdots\rightarrow x_{i_{m}}$ (2.36)

such that $i_{j}\notin\mathcal{I}(\kappa)$ for all $j\in[1,m]$.

In particular, $\mathbf{x}_{0}$ itself is an extended marked cluster. A marker
$\kappa$ induces a natural bijection888To be more precise, we label extended
marked clusters in $\mathcal{GC}$ and extended clusters in
$\widetilde{\mathcal{GC}}$ by paths (2.36). between extended marked clusters
in $\mathcal{GC}$ and all extended clusters in $\widetilde{\mathcal{GC}}$, as
follows. If $\mathbf{x}$ is obtained from $\mathbf{x}_{0}$ via the sequence
(2.36), we set $\kappa(\mathbf{x})$ to be an extended cluster in
$\widetilde{\mathcal{GC}}$ obtained from $\tilde{\mathbf{x}}_{0}$ via the
sequence

$\tilde{x}_{\kappa(i_{1})}\rightarrow\tilde{x}_{\kappa(i_{2})}\rightarrow\cdots\rightarrow\tilde{x}_{\kappa(i_{m})}.$
(2.37)

######  Definition 2.23.

The extended clusters $\mathbf{x}$ and
$\tilde{\mathbf{x}}:=\kappa(\mathbf{x})$ are called _related_ via $\kappa$.
For any $i\in[1,N+M]$, the variables $x_{i}\in\mathbf{x}$ and
$\tilde{x}_{\kappa(i)}\in\tilde{\mathbf{x}}$ are called _related_ , and if
$i\in[1,N]\setminus\mathcal{I}(\kappa)$, the corresponding $y$-variables
$y_{i}:=y(x_{i})$ and $\tilde{y}_{\kappa(i)}:=y(\tilde{x}_{\kappa(i)})$ are
called _related_.

By default, if $x$ denotes a variable from an extended cluster in
$\mathcal{GC}$, $\tilde{x}$ denotes the corresponding related variable in
$\widetilde{\mathcal{GC}}$.

##### Quasi-isomorphisms

Here, we introduce the notion of a quasi-isomorphism suitable for our
purposes999As we explained in [32], the closest notion in literature to our
version is the one defined by C. Fraser in [11]. The difference is, C. Fraser
assumes $N=\tilde{N}$ and arbitrary $M$ and $\tilde{M}$, whereas we insist
upon $N+M=\tilde{N}+\tilde{M}$ and $\tilde{M}-M>0$. along with some basic
results. We let $(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ to be a
related triple of generalized cluster structures with fixed initial extended
clusters.

######  Definition 2.24.

A _quasi-isomorphism_
$\mathcal{Q}:\mathcal{GC}\rightarrow\widetilde{\mathcal{GC}}$ is a field
isomorphism
$\mathcal{Q}:\mathcal{F}(\mathcal{GC})\rightarrow\mathcal{F}(\widetilde{\mathcal{GC}})$
such that the following conditions hold:

  1. 1)

For any pair $(\mathbf{x},\tilde{\mathbf{x}})$ of related extended clusters,
there exists an integer matrix

$\Lambda(\mathbf{x}):=\left(\lambda_{ij}\ |\ i\in[1,N+M],\
j\in{\mathcal{I}}(\kappa)\right)$ (2.38)

such that $\lambda_{ii}=1$ for each $i\in\mathcal{I}(\kappa)$ and

$\mathcal{Q}(x_{i})=\tilde{x}_{\kappa(i)}\prod_{j\in{\mathcal{I}}(\kappa)\setminus\\{i\\}}\tilde{x}_{\kappa(j)}^{\lambda_{ij}}$
(2.39)

for each $i\in[1,N+M]$.

  2. 2)

For every $i\in[1,N]\setminus\mathcal{I}(\kappa)$, every pair
$(x_{i},\tilde{x}_{\kappa(i)})$ of related variables and the corresponding
hatted strings $\\{\hat{p}_{ir}\\}_{r=0}^{d_{i}}$ and
$\\{\tilde{\hat{p}}_{ir}\\}_{r=0}^{d_{i}}$,
$\mathcal{Q}(\hat{p}_{ir})=\tilde{\hat{p}}_{ir}$, $r\in[0,d_{i}]$.

Note that the above definition is not symmetric: we do not consider
$\mathcal{Q}^{-1}$ as a quasi-isomorphism.

Let $\tilde{B}$ and $B$ denote the initial exchange matrices in
$\tilde{\mathcal{C}}$ and $\mathcal{C}$, and let $\tilde{Q}$ be the initial
quiver in $\tilde{\mathcal{C}}$. If $i,j\in[1,\tilde{N}]$, set

$i\sim j\ \ \Leftrightarrow\ \ i\text{ and }j\text{ can be connected by a path
consisting of mutable vertices in }\tilde{Q}.$ (2.40)

Denote by $[i]$ the equivalence class of $i\in[1,\tilde{N}]$. For integer
matrices $B_{1}$ and $B_{2}$ of the size
$\tilde{N}\times[1,\tilde{N}+\tilde{M}]$, define an equivalence
relation101010Let $Q_{1}$ and $Q_{2}$ be the quivers associated with $B_{1}$
and $B_{2}$. Remove all the frozen variables from $Q_{1}$ and $Q_{2}$. Then up
to flipping the orientation in each connected component of the resulting
quivers, $Q_{1}$ and $Q_{2}$ coincide.  :

$B_{1}\sim B_{2}\ \Leftrightarrow\ \forall i\in[1,\tilde{N}]:\
[B_{1}]_{[i]}=\pm[B_{2}]_{[i]}.$ (2.41)

######  Proposition 2.25.

Let
$\mathcal{Q}:\mathcal{F}(\mathcal{GC})\rightarrow\mathcal{F}(\widetilde{\mathcal{GC}})$
be a field isomorphism with the following properties:

  1. 1)

There exists an integer matrix $\Lambda:=\left(\lambda_{ij}\ |\
i\in[1,N+M],j\in{\mathcal{I}}(\kappa)\right)$ such that $\lambda_{ii}=1$ for
each $i\in\mathcal{I}(\kappa)$ and such that (2.39) holds for each
$x_{i}\in\mathbf{x}_{0}$;

  2. 2)

For every $i\in[1,N]\setminus\mathcal{I}(\kappa)$, every pair
$(x_{i},\tilde{x}_{\kappa(i)})$ of related variables and the corresponding
hatted strings $\\{\hat{p}_{ir}\\}_{r=0}^{d_{i}}$ and
$\\{\tilde{\hat{p}}_{ir}\\}_{r=0}^{d_{i}}$,
$\mathcal{Q}(\hat{p}_{ir})=\tilde{\hat{p}}_{ir}$, $r\in[0,d_{i}]$.

Then $\mathcal{Q}$ is a quasi-isomorphism if and only if

  1. 3)

For each $y$-variable $y_{i}$ in $\mathbf{x}_{0}$,
$i\notin\mathcal{I}(\kappa)$, $\mathcal{Q}(y_{i})=\tilde{y}_{\kappa(i)}$

where $\tilde{B}$ is considered up to the equivalence (2.41).

Let us introduce the following notation: if $A$ is an $n_{1}\times n_{2}$
matrix, $n_{1},n_{2}\leq N+M$, then $\kappa^{-1}(A)$ is an $n_{1}\times n_{2}$
matrix such that $(\kappa^{-1}(A))_{ij}=A_{\kappa(i),\kappa(j)}$. If $B$ is
another matrix of size $n_{2}\times n_{3}$, $n_{3}\leq N+M$, then
$\kappa^{-1}(AB)=\kappa^{-1}(A)\kappa^{-1}(B)$. In addition, let us extend
$\kappa$ to a field isomorphism
$\kappa:\mathcal{F}(\mathcal{GC})\rightarrow\mathcal{F}(\widetilde{\mathcal{GC}})$
via $\kappa(x_{i})=\tilde{x}_{\kappa(i)}$ in the initial extended clusters.
The next proposition establishes a connection between toric actions and quasi-
isomorphisms.

######  Proposition 2.26.

Suppose that $\mathcal{I}(\kappa)=\\{m\\}$ for some $m\in[1,N]$ and that
$\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$ admit global toric actions of
rank $1$. Let $\omega$ and $\tilde{\omega}$ be the weight vectors of the toric
actions in the initial extended clusters. Assume the following conditions:

  1. 1)

For every $i\in[1,\tilde{N}]$ and $r\in[0,d_{i}]$, the Laurent monomials
$\hat{p}_{ir}$ and $\tilde{\hat{p}}_{ir}$ are invariant with respect to the
toric actions and $\kappa(\hat{p}_{ir})=\tilde{\hat{p}}_{\kappa(i)r}$;

  2. 2)

For each $i\in[1,N+M]$,
$(\omega_{i}-\tilde{\omega}_{\kappa(i)})\tilde{\omega}_{\kappa(m)}^{-1}\in\mathbb{Z}$
and $w_{m}=\tilde{w}_{\kappa(m)}$;

  3. 3)

The following equality holds:

$[\kappa^{-1}(\tilde{B})]_{[1,N]\setminus\\{m\\}}^{[1,N+M]\setminus\\{m\\}}=B^{[1,N+M]\setminus\\{m\\}}_{[1,N]\setminus\\{m\\}}.$
(2.42)

Set

$\Lambda_{i}(\mathbf{x}_{0}):=\Lambda_{i}:=\begin{cases}\dfrac{w_{i}-\tilde{w}_{\kappa(i)}}{\tilde{w}_{\kappa(m)}}\
&\text{if}\ i\neq m\\\\[3.0pt] 1\ &\text{if}\ i=m.\end{cases}$ (2.43)

Then $\Lambda$ defines a quasi-isomorphism
$\mathcal{Q}:\mathcal{GC}\rightarrow\widetilde{\mathcal{GC}}$. Moreover, if
$(\mathbf{x},\tilde{\mathbf{x}})$ is a pair of related extended clusters and
$w^{\prime}$ and $\tilde{w}^{\prime}$ are the corresponding weight vectors of
the toric actions, then the corresponding matrix $\Lambda(\mathbf{x})$ is
given by

$\Lambda_{i}(\mathbf{x}):=\begin{cases}\dfrac{w_{i}^{\prime}-\tilde{w}_{\kappa(i)}^{\prime}}{\tilde{w}_{\kappa(m)}}\
&\text{if}\ i\neq m\\\\[3.0pt] 1\ &\text{if}\ i=m.\end{cases}$ (2.44)

An example of a quasi-isomorphism that arises from toric actions is given in
Section 4.5. For other examples, see [31, Proposition 5.3].

##### Birational quasi-isomorphisms.

In addition to the setup of the previous paragraph, we now assume that both
$\mathcal{GC}$ and $\widetilde{\mathcal{GC}}$ are generalized cluster
structures on some complex affine varieties $V$ and $\widetilde{V}$,
respectively. Here, we introduce the notion of a birational quasi-isomorphism.

######  Definition 2.27.

A birational map $\mathcal{Q}:\widetilde{V}\dashrightarrow V$ is called a
_birational quasi-isomorphism_ if the following conditions hold:

  1. 1)

The map $\mathcal{Q}$ induces an isomorphism of the
$\mathbb{C}$-algebras:111111By $\mathbf{x}_{0}(\kappa)^{\pm 1}$ we mean the
localization at each $x_{i}\in\mathbf{x}_{0}(\kappa)$.

$\mathcal{Q}^{*}:\mathcal{O}(V)[\mathbf{x}_{0}(\kappa)^{\pm
1}]\xrightarrow{\sim}\mathcal{O}(\widetilde{V})[\tilde{\mathbf{x}}_{0}(\kappa)^{\pm
1}];$ (2.45)

  2. 2)

The field isomorphism
$\mathcal{Q}^{*}:\mathbb{C}(V)\xrightarrow{\sim}\mathbb{C}(\widetilde{V})$
restricts to a quasi-isomorphism

$\mathcal{Q}^{*}:\mathcal{F}_{\mathbb{C}}(\mathcal{GC})\rightarrow\mathcal{F}_{\mathbb{C}}(\widetilde{\mathcal{GC}}).$

For a subset $A\subseteq\mathbb{C}(V)$, let us define the following
predicates:

$\displaystyle\operatorname{Irr}(A):=A\subseteq\mathcal{O}(V)\ \text{and}\
\forall a\in A:\,a\text{ is irreducible;}$ (2.46)
$\displaystyle\operatorname{Cop}(A):=A\subseteq\mathcal{O}(V)\ \text{and}\
\text{all elements in }A\text{ are pairwise coprime}.$ (2.47)

Likewise, we define $\operatorname{Irr}(\tilde{A})$ and
$\operatorname{Cop}(\tilde{A})$ for
$\tilde{A}\subseteq\mathbb{C}(\widetilde{V})$ by augmenting all the entries in
(2.46)–(2.47) with a tilde. For $i\in[1,N]$, denote by
$\mathbf{x}_{i}^{\prime}$ the mutation of $\mathbf{x}_{0}$ in direction $i$.

######  Proposition 2.28.

Let $V$ and $\widetilde{V}$ be complex affine varieties such that the
coordinate rings $\mathcal{O}(V)$ and $\mathcal{O}(\widetilde{V})$ are
factorial $\mathbb{C}$-algebras, and such that
$\mathcal{O}(V)^{*}=\mathcal{O}(\widetilde{V})^{*}=\mathbb{C}^{*}$. Let
$(V,\mathcal{GC})$ and $(\widetilde{V},\widetilde{\mathcal{GC}})$ be
generalized cluster structures on the varieties with fixed initial extended
seeds $(\mathbf{x}_{0},B,\mathcal{P})$ and
$(\tilde{\mathbf{x}}_{0},\tilde{B},\tilde{\mathcal{P}})$. Assume that
$(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ is a related triple and that
there exists a birational quasi-isomorphism
$\mathcal{Q}:(\widetilde{V},\widetilde{\mathcal{GC}})\dashrightarrow(V,\mathcal{GC})$.
Then the following statements hold:

  1. 1)

If $\operatorname{Irr}(\tilde{\mathbf{x}}_{0})$ and
$\operatorname{Irr}(\mathbf{x}_{0}(\kappa))$ hold, then
$\operatorname{Irr}(\mathbf{x}_{0})$ holds;

  2. 2)

If in addition to Part 1,
$\operatorname{Cop}(\tilde{x}_{i},\tilde{x}^{\prime}_{i})$ holds for each
cluster variable $\tilde{x}_{i}$ in $\tilde{\mathbf{x}}_{0}$, then
$\operatorname{Cop}(x_{i},x^{\prime}_{i})$ holds for each cluster variable
$x_{i}\in\mathbf{x}_{0}\setminus\mathbf{x}_{0}(\kappa)$;

  3. 3)

If in addition to Parts 1-2, $\operatorname{Cop}(x_{i},x_{i}^{\prime})$ holds
for each $x_{i}\in\mathbf{x}_{0}(\kappa)$, then
$\bar{\mathcal{A}}(\mathcal{GC})\subseteq\mathcal{O}(V)$;

  4. 4)

If in addition to Parts 1-3, the matrices $B$ and $\tilde{B}$ are of full
rank, as well as
$\bar{\mathcal{A}}_{\mathbb{C}}(\widetilde{\mathcal{GC}})\supseteq\mathcal{O}(\widetilde{V})$
and for each $i\in\mathcal{I}(\kappa)$,
$\mathcal{L}_{\mathbb{C}}(\mathbf{x}_{i}^{\prime})\supseteq\mathcal{O}(V)$,
then $\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC})=\mathcal{O}(V)$.

The next result corresponds to the case when a pair of complementary
birational quasi-isomorphisms is available. Consider another generalized
cluster structure $(\widehat{V},\widehat{\mathcal{GC}})$ on an affine complex
variety $\widehat{V}$.

######  Definition 2.29.

Assume that $(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ and
$(\mathcal{GC},\widehat{\mathcal{GC}},\kappa^{\prime})$ are related triples
and that there exist birational quasi-isomorphisms
$\mathcal{Q}:(\widetilde{V},\widetilde{\mathcal{GC}})\dashrightarrow(V,{\mathcal{GC}})$
and
$\mathcal{Q}^{\prime}:(\widehat{V},\widehat{\mathcal{GC}})\dashrightarrow(V,{\mathcal{GC}})$
such that $\mathcal{I}(\kappa)\cap\mathcal{I}(\kappa^{\prime})=\emptyset$.
Then $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ are called _complementary_.

######  Proposition 2.30.

Let $V$, $\widetilde{V}$ and $\widehat{V}$ be complex affine varieties such
that the coordinate rings $\mathcal{O}(V)$, $\mathcal{O}(\widetilde{V})$ and
$\mathcal{O}(\widehat{V})$ are factorial $\mathbb{C}$-algebras, and such that
$\mathcal{O}(V)^{*}=\mathcal{O}(\widetilde{V})^{*}=\mathcal{O}(\widehat{V})^{*}=\mathbb{C}^{*}$.
Let $(V,\mathcal{GC})$, $(\widetilde{V},\widetilde{\mathcal{GC}})$ and
$(\widehat{V},\widehat{\mathcal{GC}})$ be generalized cluster structures on
the varieties with fixed initial extended seeds
$(\mathbf{x}_{0},B,\mathcal{P})$,
$(\tilde{\mathbf{x}}_{0},\tilde{B},\tilde{\mathcal{P}})$ and
$(\hat{\mathbf{x}}_{0},\hat{B},\hat{\mathcal{P}})$. Assume that
$(\mathcal{GC},\widetilde{\mathcal{GC}},\kappa)$ and
$(\mathcal{GC},\widehat{\mathcal{GC}},\kappa^{\prime})$ are related triples
and that there exists a pair of complementary birational quasi-isomorphisms
$\mathcal{Q}:(\widetilde{V},\widetilde{\mathcal{GC}})\dashrightarrow(V,\mathcal{GC})$
and
$\mathcal{Q}^{\prime}:(\widehat{V},\widehat{\mathcal{GC}})\dashrightarrow(V,\mathcal{GC})$.
Then the following statements hold:

  1. 1)

If $\operatorname{Irr}(\tilde{\mathbf{x}}_{0})$ and
$\operatorname{Irr}(\hat{\mathbf{x}}_{0})$ hold, then
$\operatorname{Irr}(\mathbf{x}_{0})$ holds as well;

  2. 2)

If in addition to Part 1, for each cluster variable $\tilde{x}_{i}$ in
$\tilde{\mathbf{x}}_{0}(\kappa)$ and $\hat{x}_{j}$ in
$\hat{\mathbf{x}}_{0}(\kappa^{\prime})$,
$\operatorname{Cop}(\tilde{x}_{i},\tilde{x}_{i}^{\prime})$ and
$\operatorname{Cop}(\hat{x}_{j},\hat{x}_{j}^{\prime})$ hold, then for each
cluster variable $x_{k}\in\mathbf{x}_{0}$,
$\operatorname{Cop}(x_{k},x_{k}^{\prime})$; moreover,
$\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC})\subseteq\mathcal{O}(V)$;

  3. 3)

In addition to Parts 1-2, assume that the matrices $B$, $\tilde{B}$ and
$\hat{B}$ are of full rank, as well as
$\bar{\mathcal{A}}_{\mathbb{C}}(\widetilde{\mathcal{GC}})\supseteq\mathcal{O}(\widetilde{V})$
and
$\bar{\mathcal{A}}_{\mathbb{C}}(\widehat{\mathcal{GC}})\supseteq\mathcal{O}(\widehat{V})$.
Then $\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{GC})=\mathcal{O}(V)$.

##  3 Poisson properties of the map $\mathcal{Q}$

Let $G$ be a connected complex reductive Lie group and $\mathfrak{g}$ be its
Lie algebra with a fixed set of simple roots $\Pi$ and a nondegenerate
invariant symmetric bilinear form $\langle\cdot,\cdot\rangle$. We decompose
$\mathfrak{g}$ as
$\mathfrak{g}:=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$,
where $\mathfrak{n}_{\pm}$ are nilpotent subalgebras and $\mathfrak{h}$ is a
Cartan subalgebra. We set
$\mathfrak{b}_{\pm}:=\mathfrak{n}_{\pm}\oplus\mathfrak{h}$ and
$\mathcal{B}_{\pm}$, $\mathcal{H}$ and $\mathcal{N}_{\pm}$ to be the connected
subgroups of $G$ that correspond to $\mathfrak{b}_{\pm}$, $\mathfrak{h}$ and
$\mathfrak{n}_{\pm}$. For a generic element $U\in G$, we set
$U_{\oplus}\in\mathcal{B}_{+}$ and $U_{-}\in\mathcal{N}_{-}$ to be the
elements such that $U=U_{\oplus}U_{-}$. Let
$\mathbf{\Gamma}:=(\Gamma_{1},\Gamma_{2},\gamma)$ be a Belavin-Drinfeld triple
associated with $\Pi$. Since
$\gamma^{*}:\mathfrak{n}_{-}\rightarrow\mathfrak{n}_{-}$ is a Lie algebra
homomorphism, it integrates to a group homomorphism
$\tilde{\gamma}^{*}:\mathcal{N}_{-}\rightarrow\mathcal{N}_{-}$. We set
$\pi_{\mathbf{\Gamma}}^{\dagger}$ to be the Poisson bivector (2.17) associated
with $\mathbf{\Gamma}$ and $\pi_{\text{std}}^{\dagger}$ to be the Poisson
bivector associated with the trivial BD triple. We define the rational map
$\mathcal{Q}:(G,\pi_{\text{std}}^{\dagger})\dashrightarrow(G,\pi_{\mathbf{\Gamma}}^{\dagger})$
via

$\mathcal{Q}(U):=\rho(U)^{-1}U\rho(U),\ \
\rho(U):=\prod_{i=1}^{\rightarrow}\tilde{\gamma}^{*}(U_{-}),\ U\in G.$ (3.1)

Note that since $\gamma^{*}$ is nilpotent, the above product is finite. The
objective of this section is to prove that the map $\mathcal{Q}$ is a Poisson
isomorphism provided the choices of $r_{0}$ in $\pi^{\dagger}_{\text{std}}$
and $\pi^{\dagger}_{\mathbf{\Gamma}}$ coincide. In Section 4, we will show
that $\mathcal{Q}$ is a birational quasi-isomorphism when
$G\in\\{\operatorname{SL}_{n},\operatorname{GL}_{n}\\}$.121212We expect that
for a connected reductive complex group $G$, there exists
$\mathcal{GC}^{\dagger}(\mathbf{\Gamma},G)$ compatible with
$\pi_{\mathbf{\Gamma}}^{\dagger}$ and such that $\mathcal{Q}$ is a birational
quasi-isomorphism as well. In Section 5, we will apply the obtained result in
order to conclude that $\mathcal{GC}_{h}^{\dagger}(\mathbf{\Gamma})$ is
compatible with $\pi_{\mathbf{\Gamma}}^{\dagger}$.

##### The choice of trivialization.

In this section, all computations are performed in the right trivialization of
$G$. In particular, given a smooth map $f:G\rightarrow H$ between Lie groups
$G$ and $H$, its differential $d^{R}f$ in the right trivialization is given by

$d_{U}^{R}f(x):=\left.\frac{d}{dt}\right|_{t=0}f(\exp(tx)U)f(U)^{-1},\ \ U\in
G,\ x\in\mathfrak{g},$ (3.2)

where $I$ is the identity element. The following properties of $d^{R}$ will be
used below: if $G,H,K$ are Lie groups and $g:G\rightarrow H$ and
$h:H\rightarrow K$ are smooth maps, then the product rule for $d_{U}^{R}$
reads $d_{U}^{R}[f\cdot
g](x)=d_{U}^{R}f(x)+\operatorname{Ad}_{f(U)}d_{U}^{R}g(x)$, and the chain rule
$d_{U}^{R}[h\circ f](x)=d_{f(U)}^{R}h\circ d_{U}^{R}f(x)$, $x\in\mathfrak{g}$.
We also set

$\operatorname{Ad}_{U}^{\circ}:=\operatorname{Ad}_{U}-\operatorname{id},\ \
U\in G.$ (3.3)

For a given BD triple $\mathbf{\Gamma}$, the Poisson bivector
$\pi_{\mathbf{\Gamma}}^{\dagger}$ in the right trivialization is given by

$(\pi_{\mathbf{\Gamma}}^{\dagger,R})_{U}:=[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{\mathbf{\Gamma}})-[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega).$
(3.4)

###  3.1 Preliminary identities

In this subsection, we list various auxiliary identities that are referenced
in the proofs below. We let $\pi_{>}$, $\pi_{0}$, $\pi_{<}$, $\pi_{\geq}$,
$\pi_{\leq}$ to be the orthogonal projections upon $\mathfrak{n}_{+}$,
$\mathfrak{h}$, $\mathfrak{n}_{-}$, $\mathfrak{b}_{+}$ and $\mathfrak{b}_{-}$,
respectively, and we let $\Omega_{>}$, $\Omega_{0}$, $\Omega_{<}$,
$\Omega_{\geq}$ and $\Omega_{\leq}$ to be the $2$-tensors that correspond to
the projections with respect to the chosen form on $\mathfrak{g}$.

######  Lemma 3.1.

For any $N\in\mathcal{N}_{-}$, the following identities hold:

$\displaystyle[(\pi_{0}\circ\operatorname{Ad}_{N})\otimes
1](\Omega_{<})=[1\otimes(\pi_{<}\circ\operatorname{Ad}_{N^{-1}})](\Omega_{0});$
(3.5)
$\displaystyle[(\pi_{\geq}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega_{<})+[1\otimes(\pi_{<}\circ\operatorname{Ad}_{N})](\Omega_{0})=\Omega_{<};$
(3.6)
$\displaystyle[\operatorname{Ad}_{N}\otimes\operatorname{Ad}_{N}](\Omega_{>})+[(\pi_{<}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega_{\leq})=\Omega_{>}.$
(3.7)

###### Proof.

Formula (3.5) is straightforward. To verify (3.6), we first use (3.5) to
derive

$\begin{split}[(\pi_{0}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega_{<})=[1\otimes\operatorname{Ad}_{N}]\circ[1\otimes(\pi_{<}\circ\operatorname{Ad}_{N^{-1}})](\Omega_{0})=\Omega_{0}-[1\otimes\operatorname{Ad}_{N}](\Omega_{0});\end{split}$

now, using the latter relation together with
$[(\pi_{>}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega_{<})=\Omega_{<}$,
we see that

$\begin{split}[(\pi_{\geq}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega_{<})+[1\otimes(\pi_{<}&\circ\operatorname{Ad}_{N})](\Omega_{0})=\\\
=&\Omega_{<}+\Omega_{0}-[1\otimes\operatorname{Ad}_{N}](\Omega_{0})+[1\otimes(\pi_{<}\circ\operatorname{Ad}_{N})](\Omega_{0}).\end{split}$

The last three terms cancel each other out, and hence (3.6) holds. Formula
(3.7) follows after rewriting $\Omega_{\leq}$ as
$\Omega_{\leq}=\Omega-\Omega_{>}$ and noticing that
$\Omega_{>}\in\mathfrak{n}_{-}\otimes\mathfrak{n}_{+}$ and
$[(\pi_{<}\circ\operatorname{Ad}_{N})\otimes\operatorname{Ad}_{N}](\Omega)=\Omega_{>}$.
∎

Let us define rational maps

$\displaystyle\mathcal{B}:G\dashrightarrow G,$ $\displaystyle\ \ \ \
\mathcal{B}(U)=U_{\oplus};$ $\displaystyle\mathcal{N}:G\dashrightarrow G,$
$\displaystyle\ \ \ \ \mathcal{N}(U)=U_{-}.$

######  Lemma 3.2.

The following identities hold:

$\displaystyle
d_{U}^{R}\mathcal{N}\operatorname{Ad}_{U}=\operatorname{Ad}_{U_{-}}\pi_{<}+\pi_{<}\operatorname{Ad}_{U_{-}}\pi_{>}+\operatorname{Ad}^{\circ}_{U_{-}}\pi_{0};$
(3.8) $\displaystyle d_{U}^{R}\mathcal{N}\pi_{>}=0.$ (3.9)

###### Proof.

Differentiating $\mathcal{N}(U)=[\mathcal{B}(U)]^{-1}\cdot U$, we see that
$d_{U}^{R}\mathcal{N}(\cdot)=\pi_{<}[\operatorname{Ad}_{\,\left.U_{\oplus}\right.^{-1}}(\cdot)]$.
The identities follow from the latter formula. ∎

###  3.2 First computations

Let $\mathbf{\Gamma}$ be any Belavin-Drinfeld triple of type $A_{n-1}$. In
this subsection, we compute some of the first formulas for the proof that
$\mathcal{Q}$ is a Poisson isomorphism. We will assume that $r_{0}$ is the
same for both $\pi_{\text{std}}^{\dagger}$ and
$\pi_{\mathbf{\Gamma}}^{\dagger}$.

##### The Poisson bivectors.

We will work with the Poisson bivectors $\pi^{\dagger}_{\mathbf{\Gamma}}$ and
$\pi^{\dagger}_{\text{std}}$ in the right trivialization of $G$ in the
following form:

$\begin{split}(\pi^{\dagger,R}_{\mathbf{\Gamma}})_{U}=&[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](r_{0})-[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{0})+\\\
+&[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}]\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes
1\right](\Omega_{>})-[\operatorname{Ad}_{U}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\otimes 1\right](\Omega_{>})-\\\
-&[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right](\Omega_{<})+[\operatorname{Ad}_{U}^{\circ}\otimes
1]\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right](\Omega_{<});\end{split}$
(3.10)
$\begin{split}(\pi^{\dagger,R}_{\text{std}})_{U}=[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{0})-[\operatorname{Ad}_{U}^{\circ}&\otimes\operatorname{Ad}_{U}](\Omega_{0})-\\\
&-[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{<})-[\operatorname{Ad}_{U}^{\circ}\otimes
1](\Omega_{>}).\end{split}$ (3.11)

The objective is to prove that
$\mathcal{Q}_{*}\left((\pi^{\dagger,R}_{\text{std}})_{U}\right)=(\pi^{\dagger,R}_{\mathbf{\Gamma}})_{\mathcal{Q}(U)}$.

######  Lemma 3.3.

The following identities hold:

$\displaystyle\operatorname{Ad}_{\rho(U)^{-1}}\operatorname{Ad}_{U}$
$\displaystyle=\operatorname{Ad}_{\mathcal{Q}(U)}\operatorname{Ad}_{\rho(U)^{-1}};$
(3.12)
$\displaystyle\operatorname{Ad}_{\rho(U)^{-1}}\operatorname{Ad}_{U}^{\circ}$
$\displaystyle=\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\operatorname{Ad}_{\rho(U)^{-1}}.$

###### Proof.

The identities follow from the definition of $\mathcal{Q}$ (see formula
(3.1)). ∎

######  Lemma 3.4.

The differential of the map
$\rho(U)=\prod_{i=1}^{\rightarrow}(\tilde{\gamma}^{*})^{i}(U_{-})$ is given by

$\operatorname{Ad}_{\rho(U)^{-1}}d_{U}^{R}\rho(\cdot)=\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}(d_{U}^{R}\mathcal{N}(\cdot)).$
(3.13)

###### Proof.

Recall that $\rho(U)$ satisfies the relation

$\rho(U)=\tilde{\gamma}^{*}(U_{-})\tilde{\gamma}^{*}(\rho(U)).$ (3.14)

Differentiating (3.14) and applying $\operatorname{Ad}_{\rho(U)^{-1}}$ to both
sides, we see that

$\operatorname{Ad}_{\rho(U)^{-1}}d_{U}^{R}\rho=\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}+\operatorname{Ad}_{\rho(U)^{-1}\tilde{\gamma}^{*}(U_{-})}\gamma^{*}d_{U}^{R}\rho.$
(3.15)

The last term can be rewritten as

$\operatorname{Ad}_{\rho(U)^{-1}\tilde{\gamma}^{*}(U_{-})}\gamma^{*}d_{U}^{R}\rho=\operatorname{Ad}_{\tilde{\gamma}^{*}(\rho(U))}\gamma^{*}d_{U}^{R}\rho=\gamma^{*}\operatorname{Ad}_{\rho(U)^{-1}}d_{U}^{R}\rho.$
(3.16)

Plugging (3.16) back into (3.15) and solving for $d_{U}^{R}\rho$, the result
follows. ∎

######  Lemma 3.5.

The differential of $\mathcal{Q}$ is given by

$d_{U}^{R}\mathcal{Q}=\operatorname{Ad}_{\rho(U)^{-1}}+\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\frac{1}{1-\gamma^{*}}\left[\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}(\cdot)\right].$
(3.17)

###### Proof.

Differentiating the product $\mathcal{Q}(U)=\rho(U)^{-1}U\rho(U)$, we see that

$d_{U}^{R}\mathcal{Q}=-\operatorname{Ad}_{\rho(U)^{-1}}d_{U}^{R}\rho+\operatorname{Ad}_{\rho(U)^{-1}}+\operatorname{Ad}_{\rho(U)^{-1}}\operatorname{Ad}_{U}d_{U}^{R}\rho.$

Rewriting the third term as
$\operatorname{Ad}_{\rho(U)^{-1}}\operatorname{Ad}_{U}d_{U}^{R}\rho=\operatorname{Ad}_{\mathcal{Q}(U)}\operatorname{Ad}_{\rho(U)^{-1}}d_{U}^{R}\rho$
and applying Lemma 3.4 to the first and the third terms, we obtain (3.17). ∎

######  Lemma 3.6.

The following identities hold:

$\displaystyle\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}(\operatorname{Ad}_{U}\pi_{<}(\cdot))$
$\displaystyle=\frac{\gamma^{*}}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{<}(\cdot);$
(3.18)
$\displaystyle\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}(\operatorname{Ad}_{U}\pi_{0}(\cdot))$
$\displaystyle=\frac{1}{1-\gamma^{*}}\pi_{<}[\gamma^{*},\operatorname{Ad}_{\rho(U)^{-1}}]\pi_{0}(\cdot)$
(3.19)

where
$[\gamma^{*},\operatorname{Ad}_{\rho(U)^{-1}}]=\gamma^{*}\operatorname{Ad}_{\rho(U)^{-1}}-\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}$.

###### Proof.

The first identity follows from formula (3.8) and formula (3.14). As for the
second formula, an application of (3.8) yields

$\begin{split}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}\operatorname{Ad}_{U}\pi_{0}=&\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\operatorname{Ad}_{U_{-}}^{\circ}\pi_{0}=\operatorname{Ad}_{\rho(U)^{-1}}\left(\operatorname{Ad}_{\tilde{\gamma}^{*}(U_{-})}\gamma^{*}-\gamma^{*}\right)\pi_{0}=\\\
=&\left(\operatorname{Ad}_{\tilde{\gamma}^{*}(\rho(U)^{-1})}\gamma^{*}-\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\right)\pi_{0}=[\gamma^{*},\operatorname{Ad}_{\rho(U)^{-1}}]\pi_{0}.\qed\end{split}$

######  Lemma 3.7.

The following formula holds:

$\begin{split}-\left[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}\right]&[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{<})=-D_{0}-D_{1}-\\\
=-&\left[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}\right]\left[\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{<})-\\\
-&\left[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\right]\left[\operatorname{Ad}_{\rho(U)^{-1}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{<})\end{split}$
(3.20)

where

$\displaystyle
D_{0}:=\left[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}\right]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{<});$
(3.21) $\displaystyle
D_{1}:=\left[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\right]\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<}).$
(3.22)

###### Proof.

Recall that $\Omega_{<}\in\mathfrak{n}_{+}\otimes\mathfrak{n}_{-}$. One
applies formula (3.17) (the differential of $\mathcal{Q}$) to
$\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}$, and then one
invokes formulas (3.8) and (3.9) (the differential of $\mathcal{N}$) with
(3.18) and (3.12). ∎

######  Lemma 3.8.

The following formula holds:

$\begin{split}-[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}]&[\operatorname{Ad}_{U}^{\circ}\otimes
1](\Omega_{>})=D_{0}-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{>})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[\frac{\gamma^{*}}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{>})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[\frac{1}{1-\gamma^{*}}\pi_{<}[\gamma^{*},\operatorname{Ad}_{\rho(U)^{-1}}]\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{0}),\end{split}$
(3.23)

where $D_{0}$ is defined in (3.21).

###### Proof.

Applying formula (3.17) to $-[\operatorname{Ad}_{U}^{\circ}\otimes 1]$ and
using formula (3.9), we see that

$\begin{split}-[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}][\operatorname{Ad}_{U}^{\circ}\otimes
1](\Omega_{>})=&-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{>})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega).\end{split}$
(3.24)

Since $\Omega=[\operatorname{Ad}_{U}\otimes\operatorname{Ad}_{U}](\Omega)$,
the last term can be expanded as

$\begin{split}[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]&\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}d_{U}^{R}\mathcal{N}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega)=D_{0}+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{>})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\operatorname{Ad}^{\circ}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\right](\Omega_{0}).\end{split}$
(3.25)

Now one can apply (3.18) and (3.19) to (3.25) and combine the result with
formula (3.24). ∎

###  3.3 Pushforward of the diagonal part of $\pi^{\dagger}$

The objective of this subsection is to compute the term

$[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}]\left([\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{0})-[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{0})\right).$
(3.26)

Let us define the orthogonal projections
$\pi_{\Gamma_{1}}:\mathfrak{g}\rightarrow\mathfrak{g}_{\Gamma_{1}}$ and
$\pi_{\hat{\Gamma}_{1}}:\mathfrak{g}\rightarrow\mathfrak{g}_{\Gamma_{1}}^{\perp}$.

######  Lemma 3.9.

The following identities hold:

$\displaystyle[1\otimes(1-\gamma^{*})](r_{0})=[1\otimes\pi_{\hat{\Gamma}_{1}}](r_{0})-[1\otimes\gamma^{*}](\Omega_{0});$
(3.27) $\displaystyle[(1-\gamma^{*})\otimes
1](r_{0})=[\pi_{\Gamma_{1}}\otimes\pi_{\Gamma_{1}}](\Omega_{0})+[\pi_{\hat{\Gamma}_{1}}\otimes
1](r_{0});$ (3.28)
$\displaystyle[(1-\gamma^{*})\otimes(1-\gamma^{*})](r_{0})=[\pi_{\hat{\Gamma}_{1}}\otimes\pi_{\hat{\Gamma}_{1}}](r_{0})+[\pi_{\Gamma_{1}}\otimes\pi_{\Gamma_{1}}](\Omega_{0})-[1\otimes\gamma^{*}](\Omega_{0}).$
(3.29)

###### Proof.

The formulas follow from (2.4). ∎

######  Lemma 3.10.

The following identities hold:

$\displaystyle
d_{U}^{R}\mathcal{Q}\operatorname{Ad}_{U}^{\circ}\pi_{0}=\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\pi_{0}+\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\frac{1}{1-\gamma^{*}}\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}(1-\gamma^{*})\pi_{0};$
(3.30) $\displaystyle
d_{U}^{R}\mathcal{Q}\operatorname{Ad}_{U}\pi_{0}=\operatorname{Ad}_{\rho(U)^{-1}}\pi_{0}+d_{U}\mathcal{Q}\operatorname{Ad}_{U}^{\circ}\pi_{0}.$
(3.31)

###### Proof.

It follows from Lemma 3.2 that
$d^{R}_{U}\mathcal{N}\operatorname{Ad}_{U}^{\circ}\pi_{0}=d^{R}_{U}\mathcal{N}\operatorname{Ad}_{U}\pi_{0}$;
combining this with formulas (3.19), (3.17) and (3.12), we see that

$\begin{split}d_{U}^{R}\mathcal{Q}\operatorname{Ad}_{U}^{\circ}\pi_{0}=\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\operatorname{Ad}_{\rho(U)^{-1}}&\pi_{0}+\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\frac{\gamma^{*}}{1-\gamma^{*}}\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{0}-\\\
-&\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\frac{1}{1-\gamma^{*}}\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{0}.\end{split}$
(3.32)

Now, formula (3.30) follows after noticing that

$\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{0}=\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{0}+\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\pi_{0}$
(3.33)

and combining the terms. As for formula (3.31), it follows from the fact that
$d_{U}^{R}\mathcal{N}\pi_{0}=0$ and formula (3.17). ∎

######  Lemma 3.11.

The following identity holds:

$\begin{split}[d_{U}^{R}&\mathcal{Q}\otimes
d^{R}_{U}\mathcal{Q}][\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{0})=[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}](r_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][1\otimes\gamma^{*}](\Omega_{0})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes
1][\pi_{\Gamma_{1}}\otimes\pi_{\Gamma_{1}}](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes\gamma^{*}](\Omega_{0}).\end{split}$
(3.34)

###### Proof.

It follows from formula (3.30) that

$\begin{split}[d_{U}^{R}\mathcal{Q}&\otimes
d_{U}^{R}\mathcal{Q}][\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{0})=[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}](r_{0})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][1\otimes(1-\gamma^{*})](r_{0})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes
1][(1-\gamma^{*})\otimes 1](r_{0})+\\\
&+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}]\circ\\\
&\hskip 216.81pt\circ[(1-\gamma^{*})\otimes(1-\gamma^{*})](r_{0}).\end{split}$
(3.35)

Now formula (3.34) follows from applying the identities (3.27), (3.28) and
(3.29) to formula (3.35) and combining the terms (in the process, note also
that
$\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{\hat{\Gamma}_{1}}\pi_{0}=0$). ∎

######  Lemma 3.12.

The following identity holds:

$\begin{split}[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}]&\left([\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}^{\circ}](r_{0})-[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{0})\right)=\\\
=&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}](r_{0})-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}](\Omega_{0})-D_{2}-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes 1][\gamma^{*}\otimes
1](\Omega_{0})-\\\ -&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0})\end{split}$ (3.36)

where

$D_{2}:=[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0}).$ (3.37)

###### Proof.

Notice that

$\begin{split}[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}](\Omega_{0}&)+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1][1\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0})=\\\
=&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}](\Omega_{0})+[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0}).\end{split}$
(3.38)

Let us expand $[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}][\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{0})$
via (3.30) and (3.31), and apply (3.38) to the resulting expression. We obtain

$\begin{split}-[d_{U}^{R}\mathcal{Q}\otimes
d_{U}^{R}\mathcal{Q}]&[\operatorname{Ad}_{U}^{\circ}\otimes\operatorname{Ad}_{U}](\Omega_{0})=-[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[1\otimes\frac{1}{1-\gamma^{*}}\right][1\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][1\otimes(1-\gamma^{*})](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes
1][(1-\gamma^{*})\otimes 1](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes
1]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0})-\\\
-&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}]\circ\\\
&\hskip
173.44756pt\circ[(1-\gamma^{*})\otimes(1-\gamma^{*})](\Omega_{0}).\end{split}$
(3.39)

Now, let us indicate how the terms in (3.34) and in (3.39) combine into the
terms of (3.36). The term $-D_{2}$ in (3.36) is obtained via combining the
last term in (3.34) with the last term in (3.39). The fourth and the fifth
terms in (3.36) are obtained via combining the second term in (3.34) with the
second and the third terms in (3.39). To obtain the sixth term in (3.36), one
combines the third term in (3.34) with the fourth term in (3.39), together
with the fact that
$\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{\hat{\Gamma}_{1}}\pi_{0}=0$. The
last term in (3.36) is the penultimate term in (3.39). ∎

######  Lemma 3.13.

The following identity holds:

$\begin{split}-D_{1}-D_{2}=&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
+&[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0}).\end{split}$ (3.40)

where $D_{1}$ is given by (3.22) and $D_{2}$ is given by (3.37).

###### Proof.

For convenience, let us drop the operator
$[\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}\otimes\operatorname{Ad}_{\mathcal{Q}(U)}^{\circ}]$
from the computations. A part of $D_{1}$ can be rewritten in the following
way:

$\begin{split}\left[\frac{1}{1-\gamma^{*}}\right.\left.\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})-\\\
-&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{0}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})-\\\
-&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{>}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<}).\end{split}$
(3.41)

Now let us work with the three terms in the right-hand side of (3.41). To
rewrite the first term, we apply formula (3.8) and (3.18):

$\begin{split}\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<}).\end{split}$
(3.42)

Next, let us notice the following formula that follows from (3.14):

$\frac{\gamma^{*}}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}\,\left.U_{-}\right.^{-1}}\pi_{<}=\frac{1}{1-\gamma^{*}}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}.$
(3.43)

For the second term in the right-hand side of (3.41), we apply formula (3.5):

$\begin{split}-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{0}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\otimes\operatorname{Ad}_{\rho(U)^{-1}}\pi_{<}\operatorname{Ad}_{\,\left.U_{-}\right.^{-1}}](\Omega_{0});\end{split}$
(3.44)

then, we write

$-\operatorname{Ad}_{\rho(U)^{-1}}\pi_{<}\operatorname{Ad}_{\,\left.U_{-}\right.^{-1}}\pi_{0}=\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{0}-\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}\,\left.U_{-}\right.^{-1}}\pi_{0}$

and apply formula (3.43):

$\begin{split}-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{0}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0})-\\\
&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes\gamma^{*}](\Omega_{0}).\end{split}$
(3.45)

To rewrite the third term in the right-hand side of (3.41), we apply the
identity

$[\pi_{>}\operatorname{Ad}_{U_{-}}\otimes
1](\Omega_{<})=[1\otimes\operatorname{Ad}_{\,\left.U_{-}\right.^{-1}}](\Omega_{<})$

together with formula (3.43):

$\begin{split}-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{>}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\otimes\operatorname{Ad}_{\rho(U)^{-1}\,\left.U_{-}\right.^{-1}}](\Omega_{<})=\\\
=&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes\gamma^{*}](\Omega_{<}).\end{split}$
(3.46)

Combining formulas (3.42), (3.44) and (3.46), we obtain

$\begin{split}\left[\frac{1}{1-\gamma^{*}}\right.\left.\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]&[\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})=\\\
=&\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
+&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0})-\\\
-&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes\gamma^{*}](\Omega_{\leq}).\end{split}$
(3.47)

Now, let us turn to $D_{2}$:

$\begin{split}-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right]&[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0})=\\\
=&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{0})+\\\
&+\left[\frac{1}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0})+\\\ &+\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0}).\end{split}$ (3.48)

The negative of (3.47) together with (3.48) combine into

$\displaystyle-\left[\frac{1}{1-\gamma^{*}}\right.\left.\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})-$
(3.49)
$\displaystyle-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0})=$ $\displaystyle\hskip
79.49744pt=-\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+$
$\displaystyle\hskip
82.38774pt\hphantom{=}+\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes\gamma^{*}](\Omega_{<})+$
$\displaystyle\hskip
82.38774pt\hphantom{=}+\left[\frac{1}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][\gamma^{*}\otimes
1](\Omega_{0}).$

Let us write

$\begin{split}\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right]&[\pi_{\Gamma_{1}}\otimes\pi_{\Gamma_{1}}]-\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right]=\\\
=&\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{\hat{\Gamma}_{1}}\otimes\pi_{\hat{\Gamma}_{1}}]+1\otimes
1+\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes
1\right]+\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right],\end{split}$
(3.50)

and let us observe that
$\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\pi_{\hat{\Gamma}_{1}}\pi_{>}=0$, so
(3.49) can be updated as

$\begin{split}&-\left[\frac{1}{1-\gamma^{*}}\right.\left.\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\operatorname{Ad}_{\rho(U)^{-1}}\gamma^{*}\pi_{<}\operatorname{Ad}_{U_{-}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})-\\\
&-\left[\frac{1}{1-\gamma^{*}}\otimes\frac{1}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}][(1-\gamma^{*})\otimes
1](\Omega_{0})=\\\ &\hskip
79.49744pt=[\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
&\hskip
82.38774pt\hphantom{=}+\left[1\otimes\frac{\gamma^{*}}{1-\gamma^{*}}\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
&\hskip 82.38774pt\hphantom{=}+\left[\frac{\gamma^{*}}{1-\gamma^{*}}\otimes
1\right][\pi_{<}\operatorname{Ad}_{\rho(U)^{-1}}\otimes\operatorname{Ad}_{\rho(U)^{-1}}](\Omega_{<})+\\\
&\hskip 82.38774pt\hphantom{=}+\left[\frac{1}{1-\gamma^{*}}\otimes