# Non-Commutative Integrability
of the
Grassmann Pentagram Map

Nicholas Ovenhouse Department of Mathematics, Michigan State University

###### Abstract

The pentagram map is a discrete integrable system first introduced by Schwartz
in 1992. It was proved to be intregable by Schwartz, Ovsienko, and Tabachnikov
in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated
to certain networks embedded in a disc or annulus, and its relation to cluster
algebras. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram
map in terms of these networks, and used the associated Poisson structures to
give a new proof of integrability. In 2011, Mari Beffa and Felipe introduced a
generalization of the pentagram map to certain Grassmannians, and proved it
had a Lax representation. We reinterpret this Grassmann pentagram map in terms
of noncommutative algebra, in particular the double brackets of Van den bergh,
and generalize the approach of Gekhtman et al. to establish a noncommutative
version of integrability.

###### Contents

  1. 0 Introduction
    1. 0.1 Background
    2. 0.2 Structure of the Paper
  2. 1 The “Classical” Pentagram Map
    1. 1.1 Twisted Polygons and the Pentagram Map
    2. 1.2 Corrugated Polygons and Higher Pentagram Maps
    3. 1.3 Coordinates on the Moduli Space
  3. 2 Networks and Poisson Structures
    1. 2.1 Weighted Directed Fat Graphs
    2. 2.2 Poisson Structures on the Space of Edge Weights
  4. 3 Interpreting the Pentagram Map in Terms of Networks
    1. 3.1 The Quiver and Poisson Bracket
    2. 3.2 The $x,y$ Coordinates
    3. 3.3 The $p,q$ Coordinates
    4. 3.4 The Postnikov Moves and the Invariants
  5. 4 The Grassmann Pentagram Map
    1. 4.1 Twisted Grassmann Polygons and the Pentagram Map
    2. 4.2 Corrugated Grassmann Polygons and Higher Pentagram Maps
    3. 4.3 Description of the Moduli Space by Networks
    4. 4.4 Expression for the Pentagram Map
  6. 5 Non-Commutative Poisson Structures
    1. 5.1 Double Brackets
  7. 6 Non-Commutative Networks and Double Brackets
    1. 6.1 Weighted Directed Fat Graphs (Revisited)
    2. 6.2 The Free Skew Field and Mal’cev Neumann Series
    3. 6.3 An Example of a Series Expansion
    4. 6.4 Double Brackets Associated to a Quiver
    5. 6.5 A Formula for the Bracket
    6. 6.6 Goldman’s Bracket and the Twisted Ribbon Surface
    7. 6.7 The $X,Y$ Variables
    8. 6.8 The $P,Q$ Variables
  8. 7 Invariance, Invariants, and Integrability
    1. 7.1 Invariance of the Induced Bracket
    2. 7.2 The Invariants
    3. 7.3 Involutivity of the Invariants
  9. 8 Recovering the Lax Representation
  10. 9 Conclusions and Suggestions for Future Work
  11. 10 Acknowledgments

##  0 Introduction

###  0.1 Background

The pentagram map was first introduced by Schwartz [Sch92] as a transformation
of the moduli space of labeled polygons in $\mathbb{P}^{2}$. After labeling
the vertices of an $n$-gon with the numbers $1,\dots,n$, the diagonals are
drawn which connects each $i$ to $i+2$ (see Figure 1). The intersection points
of these diagonals are taken to be the vertices of a new $n$-gon. Later,
Schwartz, along with Ovsienko and Tabachnikov, generalized the map to the
space of “twisted” $n$-gons in $\mathbb{P}^{2}$, and proved that this
generalized pentagram map is completely integrable in the Liouville sense
[OST10]. In 2011, Max Glick interpreted the pentagram map in terms of
“$Y$-mutations” of a certain cluster algebra [Gli11]. In the literature, there
are other names for $Y$-mutations or $Y$-dynamics. Fomin and Zelevinsky have
called them “coefficient dynamics” [FZ02] [FZ07]. Gekhtman, Shapiro, and
Vainshtein refer to them as “$\tau$-coordinate mutations” [GSV10a]. Fock and
Goncharov refer to them as “$\mathcal{X}$-variable mutations” [FG06].

In 2016, Gekhtman, Shapiro, Vainshtein, and Tabachnikov, building on the work
of both Glick and Postnikov [Pos06], interpreted a certain set of coordinates
on the space of twisted polygons as weights on some directed graph, and the
pentagram map in these coordinates as a certain sequence of “Postnikov moves”
applied to this graph [GSTV16]. They used their previous work on the Poisson
geometry of the space of edge weights of such graphs [GSV09] [GSV10b] to give
a new proof of the integrability of the pentagram map.

Recently, Mari Beffa and Felipe considered a generalization of the pentagram
map, where the twisted polygons were taken to be in the Grassmannian instead
of projective space [FMB15], and they demonstrated a Lax representation for
this version of the pentagram map, establishing integrability for this
generalized version. The purpose of the present paper is to interpret this
Grassmannian pentagram map as a transformation of a set of matrix-valued
variables (and more generally as a formal noncommutative rational
transformation), and use ideas from non-commutative Poisson geometry – namely
the “double brackets” of Van den Bergh [Ber08] and “$H_{0}$-Poisson
structures” of Crawley-Boevey[CB11] – to formulate a non-commutative version
of integrability, generalizing the approach of Gekhtman, Shapiro, Vainshtein,
and Tabachnikov, using weighted directed graphs. This non-commutative
algebraic structure is a generalization of both the original pentagram map and
this more recent Grassmann version. Furthermore, the integrability of both
examples follows from this non-commutative structure, by projecting to
representation spaces of a certain non-commutative algebra.

###  0.2 Structure of the Paper

The structure of the paper is as follows. The first three sections are a
condensed review of the basics of the pentagram map. We mainly follow the
notations and conventions of Gekhtman, Shapiro, Vainshtein, and Tabachnikov
[GSV09] [GSV10b] [GSTV16]. The later sections are an introduction to the
Grassmannian pentagram map, and the development of the new theory. We now give
a more detailed description of the sections.

In section 1, we review the definition of “twisted” polygons and the pentagram
map. We also construct a system of coordinates on the moduli space of twisted
polygons (following [GSTV16]), and express the pentagram map in these
coordinates, where it is seen to be a relatively simple rational expression.

In section 2, we define the weighted directed graphs that we will be using,
and review the associated Poisson structure on the space of edge weights
introduced in [GSV10b]. We also review the notion of “boundary measurements”
for such graphs, and the “Postnikov moves” and “gauge transformations” which
leave the boundary measurements invariant [Pos06].

In section 3, we review the observation from [GSTV16] that after identifying
the coordinates from section 1 with certain weights on some graph, the
pentagram map can be interpreted as a sequence of Postnikov moves and gauge
transformations. This observation allows us to find invariants of the
pentagram map, defined in terms of the boundary measurements. These invariants
also turn out to be in involution with respect to the Poisson structure,
giving the integrability of the pentagram map.

In section 4, we introduce the notion of twisted polygons in the Grassmannian,
and define the pentagram map for these twisted Grassman polygons, following
the presentation of Mari Beffa and Felipe [FMB15]. We then identify the moduli
space with a quotient of the space of matrix-valued weights on certain graphs,
and use this identification to coordinatize the moduli space. Next, we write
the pentagram map in terms of the matrix weights, in an attempt to mimic,
formally, the approach of [GSTV16]. We show that in these (non-commutative)
variables, the pentagram map is a non-commutative version of the formula from
section 1.

In section 5, we review the definitions and properties of “double brackets” on
non-commutative algebras (due to Van den Bergh [Ber08]), and “$H_{0}$-Poisson
structures” (due to Crawley-Boevey [CB11]). This is the formalism we will use
in place of a usual Poisson structure in this non-commutative setting.

In section 6, we consider the same directed graph from section 2, but we
consider non-commutative weights. We also consider the analogues of the
boundary measurements, Postnikov moves, and gauge transformations in this non-
commutative setting. To make rigorous sense of the Postnikov moves, we
introduce the free skew field in a set of formal non-commuting indeterminates.
We then define a double bracket on the free skew field generated by the edges.
This algebra acts as a sort of “space of non-commutative edge weights”. The
relationship with Goldman’s bracket on character varieties [Gol86] is
discussed, giving the double bracket (and its induced brackets) a geometric
interpretation in terms of intersection numbers on a certain surface. Finally,
we find a set of elements/variables in the free skew field which are non-
commutative analogues to the coordinates from section 1. The same sequence of
Postnikov moves as in the classical case transforms these variables precisely
by the formula from section 4 for the Grassman pentagram map.

In section 7, analogous to the classical case, we find noncommutative
“invariants” of the map which are defined in terms of the boundary
measurements. They are in fact invariant only modulo commutators. We also show
that these invariants are, in some sense, involutive with respect to the non-
commutative Poisson structure, giving a non-commutative version of
integrability.

In section 8, we discuss how our noncommutative Poisson structures induce
usual “commutative” Poisson structures on the moduli space of representations
of a certain free algebra. We review a discussion from section 4 which
identifies the moduli space of polygons with a subset of this representation
space. Thus, the moduli space of polygons inherits a Poisson structure. The
invariants obtained by Mari Beffa and Felipe are the traces of our non-
commutative invariants, and they form a Liouville integrable system under the
induced Poisson bracket.

Finally, in section 9, we pose some lingering questions and directions for
further research.

##  1 The “Classical” Pentagram Map

###  1.1 Twisted Polygons and the Pentagram Map

We refer to the pentagram map as the “classical” pentagram map to distinguish
it from the Grassmann pentagram map (the subject of the later sections of the
paper). First we give the basic idea, and then extend to so-called “twisted”
polygons. The map was originally introduced by Schwartz [Sch92], but here we
mainly follow the notations and conventions from [GSTV16]. Consider an $n$-gon
in $\mathbb{P}^{2}=\mathbb{RP}^{2}$, with vertices labeled $p_{1}$ through
$p_{n}$. Draw the diagonals of the $n$-gon which connect $p_{i}$ to $p_{i+2}$
(with indices read cyclically). Label the intersection of the lines
$\overline{p_{i}p_{i+2}}$ and $\overline{p_{i+1}p_{i+3}}$ as $q_{i}$. Then the
pentagram map, which we denote by $T$, sends the first labeled polygon $P$ to
the labeled polygon $Q=T(P)$ whose vertices are $q_{i}$. An example for $n=6$
is shown below in Figure 1.

Figure 1: The pentagram map for a
hexagon$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$$p_{6}$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$q_{5}$$q_{6}$

More generally, we define “twisted polygons”.

######  Definition 1.

A _twisted $n$-gon_ in $\mathbb{P}^{2}$ is a bi-infinte sequence
$(p_{i})_{i\in\mathbb{Z}}$ of points in $\mathbb{P}^{2}$ such that
$p_{i+n}=Mp_{i}$ for all $i$, where $M$ is some projective transformation in
$\mathrm{PGL}_{3}(\mathbb{R})$, referred to as the _monodromy_ of the twisted
$n$-gon.

######  Remark.

The usual notion of an $n$-gon can be thought of as the case where
$M=\mathrm{Id}$, and the sequence is periodic. In this case we call the
polygon a _closed $n$-gon_.

The pentagram map, $T$, can be defined analogously for twisted $n$-gons. We
always assume twisted $n$-gons are generic in the sense that no three
consecutive points $p_{i},p_{i+1},p_{i+2}$ are collinear.

######  Definition 2.

Two twisted $n$-gons $(p_{i})$ and $(q_{i})$ said to be _projectively
equivalent_ if there is some $G\in\mathrm{PGL}_{3}(\mathbb{R})$ so that
$q_{i}=Gp_{i}$ for every $i$. We will denote by $\mathcal{P}_{n}$ the moduli
space of twisted $n$-gons under projective equivalence.

######  Remark.

The pentagram map commutes with projective transformations in the sense that
for $G\in\mathrm{PGL}_{3}(\mathbb{R})$, the polygons $T(P)$ and $T(G\cdot P)$
are projectively equivalent. Thus it induces a well-defined map
$T\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$.

There is the following important result about this map:

######  Theorem 1.

[OST10] The pentagram map $T\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ is
completely integrable, in the Liouville sense. That is, there exists a
$T$-invariant Poisson structure on $\mathcal{P}_{n}$ and sufficiently many
independent functions $f_{i}\in C^{\infty}(\mathcal{P}_{n})$ for which
$\\{f_{i},f_{j}\\}=0$.

###  1.2 Corrugated Polygons and Higher Pentagram Maps

In [GSTV16], the authors define _generalized higher pentagram maps_ , which we
define and discuss now. Instead of working in $\mathbb{P}^{2}$, we generalize
to $\mathbb{P}^{k-1}$, with the usual pentagram map being the specialization
to $k=3$. In the same way as before, we define twisted polygons in
$\mathbb{P}^{k-1}$.

######  Definition 3.

A _twisten $n$-gon_ in $\mathbb{P}^{k}$ is a bi-infinite sequences of points
$(p_{i})_{i\in\mathbb{Z}}$ in $\mathbb{P}^{k-1}$ with monodromy
$M\in\mathrm{PGL}_{k}$ so that $p_{i+n}=Mp_{i}$ for all $i$. Define
$\mathcal{P}_{k,n}$ to be the set of all projective equivalence classes of
twisted $n$-gons in $\mathbb{P}^{k-1}$ with the genericity condition that any
consecutive $k$ points do not lie in a proper projective subspace. Keeping
with the previous notation, we define $\mathcal{P}_{n}:=\mathcal{P}_{3,n}$.

######  Definition 4.

A _corrugated polygon_ is a twisted $n$-gon in $\mathbb{P}^{k-1}$ with the
additional property that $p_{i}$, $p_{i+1}$, $p_{i+k-1}$, $p_{i+k}$ span a
projective plane for all $i$. In other words, for any lift $v_{i}$ of $p_{i}$
to $\mathbb{R}^{k}$, the four vectors $v_{i}$, $v_{i+1}$, $v_{i+k-1}$,
$v_{i+k}$ span a $3$-dimensional subspace. In particular, when $k=3$, any
twisted polygon is automatically corrugated.

######  Definition 5.

Define $\mathcal{P}^{0}_{k,n}$ to be the subset of corrugated polygons with
the property that for each $i$, any $3$ of the $4$ points $p_{i}$, $p_{i+1}$,
$p_{i+k-1}$, $p_{i+k}$ are not collinear. In other words, for any lift
$v_{i}$, and any $i$, any $3$ of the $4$ vectors $v_{i}$, $v_{i+1}$,
$v_{i+k-1}$, $v_{i+k}$ are linearly independent.

Define $\mathscr{L}_{i}$ to be the line containing $p_{i}$ and $p_{i+k-1}$.
Then since $p_{i}$, $p_{i+1}$, $p_{i+k-1}$, $p_{k}$ span a projective plane in
$\mathbb{P}^{k-1}$, the lines $\mathscr{L}_{i}$ and $\mathscr{L}_{i+1}$ must
intersect. We call the intersection point $q_{i}$. We can then define a
pentagram map $T\colon\mathcal{P}^{0}_{k,n}\to\mathcal{P}_{k,n}$ which sends
$(p_{i})$ to $(q_{i})$. Note the codomain is not $\mathcal{P}^{0}_{k,n}$,
since the image my be degenerate.

###  1.3 Coordinates on the Moduli Space

Next we will construct a system of coordinates on $\mathcal{P}_{n}$, following
the presentation in [GSTV16]. The following construction will also give a
system of coordinates on $\mathcal{P}^{0}_{k,n}$ for the generalized higher
pentagram maps, but we present only the case $k=3$, for the sake of
simplicity. Given a twisted $n$-gon $(p_{i})$, we lift it to a bi-infinite
sequence of vectors $(v_{i})$ in $\mathbb{R}^{3}$. That is, $v_{i}$ projects
to $p_{i}$ under the canonical map
$\mathbb{R}^{3}\setminus\\{0\\}\to\mathbb{P}^{2}$. The genericity assumption
guarantees that for each $i$, $\\{v_{i},v_{i+1},v_{i+2}\\}$ is a basis for
$\mathbb{R}^{3}$. Thus we have for each $i$ a linear dependence relation:

$v_{i+3}=a_{i}v_{i}+b_{i}v_{i+1}+c_{i}v_{i+2}$ (1)

We call the lift “twisted” if $v_{i+n}=Av_{i}$, where $A\in\mathrm{GL}_{3}$ is
any matrix representing the monodromy. If the lift is twisted, then the
sequences $a_{i}$, $b_{i}$, and $c_{i}$ are $n$-periodic, and none of the
$a_{i}$, $b_{i}$, or $c_{i}$ are zero. The following result is proved in more
generality in [GSTV16], but we prove it here for the sake of presentation.

######  Proposition 1.

[GSTV16] The lift $(v_{i})$ can be chosen so that $c_{i}=1$ for all $i$, and
the remaining coefficients are $n$-periodic.

###### Proof.

We first start by choosing a twisted lift $(v_{i})$, which is always possible.
Any other lift $(\hat{v}_{i})$ differs by rescaling. That is, there are non-
zero constants $\lambda_{i}$ so that $\hat{v}_{i}=\lambda_{i}v_{i}$. Then
Equation 1 becomes:

$\hat{v}_{i+3}=\left(a_{i}\frac{\lambda_{i+3}}{\lambda_{i}}\right)\hat{v}_{i}+\left(b_{i}\frac{\lambda_{i+3}}{\lambda_{i+1}}\right)\hat{v}_{i+1}+\left(c_{i}\frac{\lambda_{i+3}}{\lambda_{i+2}}\right)\hat{v}_{i+2}$

We want to choose $\lambda_{i}$ so that
$c_{i}\frac{\lambda_{i+3}}{\lambda_{i+2}}=1$ for all $i$. Re-arranging this
equation, and re-indexing gives the recurrence
$\lambda_{i+1}=\frac{\lambda_{i}}{c_{i-2}}$. We may therefore set
$\lambda_{0}=1$, and then define the rest by this recurrence.

Solving this recurrence gives $\lambda_{i+n}=\frac{\lambda_{i}}{c_{1}\cdots
c_{n}}$. The factors of $c_{1}\cdots c_{n}$ therefore cancel in the
expressions of the new coefficients, and since $a_{i}$, $b_{i}$, $c_{i}$ were
periodic, so are the new coefficients. ∎

With the previous result in mind, we change notation and let $x_{i}:=b_{i}$
and $y_{i}:=a_{i}$, so that Equation 1 becomes

$v_{i+3}=y_{i}v_{i}+x_{i}v_{i+1}+v_{i+2}$ (2)

######  Lemma 1.

[GSTV16] $\dim\mathcal{P}^{0}_{k,n}=2n$, with the $x_{i}$, $y_{i}$ being a
system of coordinates.

Next we will write the pentagram map in these coordinates, and see that it is
a rational map. Since we consider the pentagram map acting on labeled
polygons, we will abuse notation slightly and write $T(p_{i})=q_{i}$ for the
individual vertices of the polygon. Taking the abuse a step further, if we
have lifts $(v_{i})$ of $(p_{i})$ and $(w_{i})$ of $(q_{i})=(T(p_{i}))$, we
will also write $T(v_{i})=w_{i}$. To see how the pentagram map transforms the
$x_{i}$ and $y_{i}$ coordinates, we will look at how the pentagram map acts on
a lift. First let’s introduce some notation. As before, let $\mathscr{L}_{i}$
be the line between $p_{i}$ and $p_{i+2}$. Recall that the pentagram map is
given by $q_{i}=T(p_{i})=\mathscr{L}_{i}\cap\mathscr{L}_{i+1}$. Let
$\mathscr{P}_{i}$ be the plane in $\mathbb{R}^{3}$ which projects onto
$\mathscr{L}_{i}$. Then any vector in the intersection
$\mathscr{P}_{i}\cap\mathscr{P}_{i+1}$ can be taken to be the lift of the
image $w_{i}=T(v_{i})$. In particular, re-arrangeing Equation 2 gives a
candidate which can be written in two ways:

$v_{i+3}-x_{i}v_{i+1}=y_{i}v_{i}+v_{i+2}\in\mathscr{P}_{i}\cap\mathscr{P}_{i+1}$
(3)

######  Proposition 2.

[GSTV16] The pentagram map $T\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ is given
in the $x_{i},y_{i}$ coordinates by

$\displaystyle x_{i}$ $\displaystyle\mapsto
x_{i}\,\frac{x_{i+2}+y_{i+3}}{x_{i}+y_{i+1}}$ $\displaystyle y_{i}$
$\displaystyle\mapsto y_{i+1}\,\frac{x_{i+2}+y_{i+3}}{x_{i}+y_{i+1}}$

###### Proof.

The lifts of the image polygon’s vertices will also satisfy the linear
dependence relation as in Equation 1:

$T(v_{i+3})=Y_{i}T(v_{i})+X_{i}T(v_{i+1})+Z_{i}T(v_{i+2})$

In the above equation, substitute for $T(v_{i})$ and $T(v_{i+2})$ the
expressions from Equation 3 belonging to $\mathscr{P}_{i+1}$, and substitute
for $T(v_{i+1})$ and $T(v_{i+3})$ the corresponding expressions belonging to
$\mathscr{P}_{i}$. Doing so, one obtains the equation

$v_{i+5}+y_{i+3}v_{i+3}=\left(X_{i}y_{i+1}-Y_{i}x_{i}\right)v_{i+1}+\left(X_{i}+Y_{i}-Z_{i}x_{i+2}\right)v_{i+3}+Z_{i}v_{i+5}$

Comparing coefficients on both sides, we can solve to get that

$\displaystyle Z_{i}$ $\displaystyle=1$ $\displaystyle X_{i}$
$\displaystyle=x_{i}\frac{x_{i+2}+y_{i+3}}{x_{i}+y_{i+1}}$ $\displaystyle
Y_{i}$ $\displaystyle=y_{i+1}\frac{x_{i+2}+y_{i+3}}{x_{i}+y_{i+1}}$

∎

##  2 Networks and Poisson Structures

In this section, we review the relevant definitions and constructions needed
to formulate the pentagram map in terms of edge-weighted directed graphs.
First, we review the types of weighted directed graphs we will be working
with, and an important class of transformations of such graphs. Then we review
the Poisson structures introduced in [GSV09][GSV10b] on the space of weights
of such graphs.

###  2.1 Weighted Directed Fat Graphs

######  Definition 6.

A _fat graph_ is a graph, together with a prescribed cyclic ordering of the
half-edges incident to each vertex.

In particular, any graph embedded onto a two-dimensional oriented surface is
naturally a fat graph, with the orientation induced from that of the surface.
Later on, we will only be considering such graphs which are embedded on a
surface, so all graphs will assumed to be fat graphs, even if not explicitly
stated.

######  Definition 7.

A _quiver_ (or _directed graph_) is a tuple $Q=(Q_{0},Q_{1},s,t)$ where
$Q_{0}$ is the set of vertices of the underlying graph, $Q_{1}$ the set of
edges, and $s,t\colon Q_{1}\to Q_{0}$ are the “source” and “target” maps,
indicating the direction of the arrows.

######  Definition 8.

We will use the term _network_ to mean a weighted directed fat graph. A
weighting means an assignment $Q_{1}\to\mathbb{R}$ of a real number to each
edge.

Postnikov considered what he called _perfect planar networks_ [Pos06], which
are networks embedded in a disk such that

  1. 1.

All vertices on the boundary are univalent

  2. 2.

All internal vertices are trivalent and are neither sources nor sinks

  3. 3.

All edge weights are positive real numbers

Note that the second condition, that internal vertices are trivalent, and are
not sources or sinks, implies that they are one of two types: either a vertex
has one incoming and two outgoing edges, or it has one outgoing and two
incoming edges. When we draw networks, we will picture the former as white
vertices, and the latter as black. This is pictured in Figure 2.

Figure 2: Types of vertices in a perfect network

######  Definition 9.

[Pos06] For an acyclic perfect network $Q$, let $a_{1},\dots,a_{k}$ denote the
sources on the boundary, and let $b_{1},\dots,b_{\ell}$ be the sinks on the
boundary. The _boundary measurement_ $b_{ij}$ is defined to be the sum of the
weights of all paths from $a_{i}$ to $b_{j}$, where the weight of a path is
the product of the weights of its edges. We arrange the boundary measurements
into a $k\times\ell$ matrix $\mathscr{B}(Q)=(b_{ij})$, called the _boundary
measurement matrix_ of $Q$.

Later on, we will be interested only in a particular acyclic network, so we
will not discuss the more general definition for when $Q$ has directed cycles.

Postnikov described several local transformations of networks which leave the
boundary measurements invariant. The moves are pictured in Figure 3, and are
labeled with the corresponding edge weights.

Figure 3: Local Postnikov moves( I
)$a$$b$$c$$d$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$d\,\frac{c}{\Delta}$$\Delta$$d\,\frac{a}{\Delta}$$d\,\frac{b}{\Delta}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$(\Delta:=b+adc)$(
II )$a$$b$$c$$x$$y$$a$$b$$c$$\frac{x}{b}$$yb$( III
)$a$$b$$c$$x$$y$$a$$b$$c$$xb$$\frac{y}{b}$

We refer to the type I move as the “square move”, the type II move as “white-
swapping”, and the type III move as “black-swapping”. It is an easy exercise
to check that these three moves do not change the boundary measurements from a
source to a sink in these pictures.

There is another type of local move called a _gauge transformation_ , which
changes the edge weights but not the underlying graph. Let $\lambda$ be any
non-zero number. Then at any vertex we may multiply all incoming weights by
$\lambda$ and all outgoing weights by $\lambda^{-1}$. This obviously does not
change the boundary measurements. The group generated by all gauge
transformations is called the _gauge group_.

From now on, we will only consider networks where the sources and sinks are
not interlaced, so that all sources can be pictured on the left side, and all
sinks on the right. The reason for restricting to these networks is that they
have the following nice property. One may consider “concatenating” two such
networks in a disk, by glueing segments of their boundaries in a way that each
sink on one is identified with a source of the other. This is pictured in
Figure 4. The resulting edge after identification is weighted by the product
of the two edges. If the sources and sinks are not interlaced, as mentioned
above, then the boundary measurement matrix of the resulting network after
concatenation is the product of the two boundary measurement matrices.

We may also consider weighted directed networks embedded on an annulus
[GSV10b]. Later, when we talk about the pentagram map, it will be these
annular networks that we will consider. For simplicity, we will again only
consider acyclic networks. Similar to the disk case, consider _perfect_
networks, meaning any internal vertices are trivalent (and not sources or
sinks), and that all boundary vertices are univalent. We also assume all
vertices on the inner boundary component are sources, and all vertices on the
outer boundary component are sinks. Then in a similar way as before, we define
a boundary measurement matrix, whose entries are the sums of path weights from
a given source to a given sink. Since the Postnikov moves are local, and do
not take into account the global topology of the surface on which the network
is embedded, they may also be applied in this case, and of course they still
do not change the boundary measurements.

As in the disk case, we may consider concatenation of two annular networks, by
identifying the outer boundary circle of one with the inner boundary circle of
the other, in a way that each sink is identified with a source. If we assume,
as mentioned above, that all sources are on the inner boundary circle and all
sinks on the outer, then the resulting boundary measurement matrix is the
product of the two boundary measurement matrices.

If there are the same number of sources and sinks, then it will sometimes be
convenient to glue the inner and outer boundary circles together so that pairs
of sources and sinks are identified, to obtain a network on the torus.

Lastly, we define the _modified edge weights_ , which are elements of the
Laurent polynomial ring $\mathbb{R}[\lambda^{\pm}]$ in the indeterminate
$\lambda$, defined as follows. Choose an oriented curve $\rho$, called the
_cut_ , which connects the inner and outer boundary components of the annulus.
Suppose an oriented edge $\alpha\in Q_{1}$ has weight
$x_{\alpha}\in\mathbb{R}$. It may happen that $\alpha$ intersects the cut. If
$i$ is an intersection point, define $\varepsilon_{i}=1$ if the tangent
vectors to $\alpha$ and $\rho$ at $i$ form an oriented basis of the tangent
space, and define $\varepsilon_{i}=-1$ if they have the opposite orientation.
Then the modified edge weight of $\alpha$ is defined as

$x_{\alpha}\lambda^{\varepsilon_{i}}$

If we use the modified edge weights, then the boundary measurements become
Laurent polynomials in $\lambda$. From now on, we always assume the boundary
measurements are in terms of modified edge weights. We will use the notation
$\mathscr{B}_{Q}(\lambda)$ to emphasize that the entries are functions of
$\lambda$.

######  Remark.

As mentioned above, if there are the same number of sources and sinks, we may
glue the two boundary circles together to obtain a network on the torus. The
two boundary circles become a single loop on the torus, which we call the
_rim_. If we choose a new rim, and cut the torus along this new rim, we get a
new (possibly different) network on an annulus. The boundary measurement
matrices differ by a re-factorization, which we will now explain. If we draw
the new rim on the old annulus, we can realize the annulus as the
concatenation of two annular networks $Q^{\prime}$ and $Q^{\prime\prime}$, one
on each side of the new rim. Let $A$ and $B$ be the boundary measurement
matrices of $Q^{\prime}$ and $Q^{\prime\prime}$ respectively. Then the
boundary measurement matrix is $AB$. Glueing into a torus and then cutting
along the new rim amounts to concatenating the pictures of $Q^{\prime}$ and
$Q^{\prime\prime}$ in the opposite order. The new boundary measurement matrix
is thus $BA$. It was observed by Izosimov [Izo18] that many different
generalizations of the pentagram map may be described in terms of these types
of matrix re-factorizations in Poisson Lie groups.

###  2.2 Poisson Structures on the Space of Edge Weights

Given a perfect network in an annulus, as in the last section, we may forget
the particular choice of edge weights and consider the underlying quiver
$Q=(Q_{0},Q_{1})$. The _space of edge weights_ of the quiver $Q$, denoted
$\mathscr{E}_{Q}:=(\mathbb{R}^{*})^{Q_{1}}$, is the space of all possible
choices of non-zero weights. Gekhtman, Shapiro, and Vainshtein defined a
family of Poisson structures on $\mathscr{E}_{Q}$ [GSV09], which we will
review now.

The Poisson structures are defined locally at each vertex, and then the local
Poisson brackets are shown to be “compatible” with concatenation (this will be
made more precise later), and so they can be combined to give a global Poisson
bracket. This is outlined in [GSV09] and [GSV10b], but we present the
construction here in detail, since we will mimic it very closely later on when
we define a non-commutative analogue. Recall that for networks on an annulus,
we require internal vertices are trivalent, and that they are neither sources
nor sinks, and so they are either white or black, as mentioned before:

$x$$z$$y$$a$$b$$c$

Let $\mathscr{E}_{\circ}$ be the space $(\mathbb{R}^{*})^{3}$ with coordinates
$x,y,z$. We will think of the variables $x,y,z$ as representing the edge
weights at a white vertex picture above. Similarly, let
$\mathscr{E}_{\bullet}=(\mathbb{R}^{*})^{3}$ with coordinates $a,b,c$
corresponding to a black vertex. Of course they are diffeomorphic, but we will
define different Poisson brackets on them. Choose any
$w_{1},w_{2},w_{3}\in\mathbb{R}$, and define a _log-canonical_ Poisson bracket
$\\{-,-\\}_{\circ}$ on $\mathscr{E}_{\circ}$ by

$\\{x,y\\}_{\circ}=w_{1}xy,\quad\\{x,z\\}_{\circ}=w_{2}xz,\quad\\{y,z\\}_{\circ}=w_{3}yz$

Similarly, for scalars $k_{1},k_{2},k_{3}\in\mathbb{R}$, define
$\\{-,-\\}_{\bullet}$ on $\mathscr{E}_{\bullet}$ by

$\\{a,b\\}_{\bullet}=k_{1}ab,\quad\\{a,c\\}_{\bullet}=k_{2}ac,\quad\\{b,c\\}_{\bullet}=k_{3}bc$

Next we will consider the operation of glueing/concatenating these local
pictures together, as described before. This means identifying part of the
boundary of one picture with part of another, in such a way that a sink is
identified with a source. We then erase the common glued boundary, and remove
the common identified vertex, identifying the two incident edges (which have a
consistent orientation by construction). The weight of the new edge is
associated with the product of the weights of the edges being glued. Figure 4
illustrates glueing a black and white local picture:

Figure 4: Concatenation of Local
Pictures$a$$b$$c$$x$$z$$y$$\rightsquigarrow$$w$$b$$c$$z$$y$

The definitions of $\\{-,-\\}_{\circ}$ and $\\{-,-\\}_{\bullet}$ are
“compatible” with this glueing procedure in a way that we now make precise.
Recall that $\mathscr{E}_{Q}=(\mathbb{R}^{*})^{Q_{1}}$ is the space of edge
weights. We can associate to it the algebra $E_{Q}:=\mathbb{R}(Q_{1})$, which
is the field of rational functions with indeterminates corresponding to the
arrows of the quiver. We also define
$\mathscr{H}_{Q}:=(\mathbb{R}^{*})^{Q_{1}\coprod Q_{1}}$ to be the _space of
half-edge weights_. This is an assignment of a non-zero number to each half-
edge. Each half edge is associated to the vertex which it is incident to. For
an arrow $\alpha\in Q_{1}$, we call the corresponding half edges $\alpha_{s}$
and $\alpha_{t}$, for the source and target ends of the arrow. The algebra of
functions $H_{Q}$ corresponding to $\mathscr{H}_{Q}$ is then the field of
rational functions in twice as many variables, corresponding to $\alpha_{s}$
and $\alpha_{t}$ for $\alpha\in Q_{1}$.

Recall that $Q_{0}$ denotes the set of vertices of the quiver $Q$. We can
partition this into $Q_{0}=V_{\circ}\cup V_{\bullet}\cup
V_{\partial}=V_{i}\cup V_{\partial}$, where $V_{\circ}$ is the set of white
vertices, $V_{\bullet}$ the black vertices, and $V_{\partial}$ the boundary
vertices, and $V_{i}=V_{\circ}\cup V_{\bullet}$ is the set of internal
vertices. Note that since all interal vertices are trivalent, and all boundary
vertices are univalent, the set of half-edges is in bijection with
$V_{\partial}\coprod V_{i}\coprod V_{i}\coprod V_{i}$. Recall we defined the
spaces
$\mathscr{E}_{\circ}\cong\mathscr{E}_{\bullet}\cong(\mathbb{R}^{*})^{3}$. We
now additionally define $\mathscr{E}_{\partial}\cong\mathbb{R}^{*}$ with
trivial Poisson bracket, corresponding to boundary vertices. Then the space of
half-edges can be thought of as

$\mathscr{H}_{Q}\cong\prod_{\ast\in
V_{\partial}}\mathscr{E}_{\partial}\,\times\,\prod_{\circ\in
V_{\circ}}\mathscr{E}_{\circ}\,\times\,\prod_{\bullet\in
V_{\bullet}}\mathscr{E}_{\bullet}$

Under this identification, we realize the algebra $H_{Q}$ as the tensor
product

$H_{Q}\cong\bigotimes_{\ast\in
V_{\partial}}E_{\partial}\,\otimes\,\bigotimes_{\circ\in
V_{\circ}}E_{\circ}\,\otimes\,\bigotimes_{\bullet\in V_{\bullet}}E_{\bullet}$

From the Poisson brackets defined above on the algebras $E_{\circ}$,
$E_{\bullet}$, and $E_{\partial}$, we get a natural induced Poisson bracket on
$H_{Q}$, where $\\{\alpha,\beta\\}=0$ if $\alpha$ and $\beta$ are half-edges
which are not incident to a common vertex, and the boundary half-edges are
casimirs.

We define a “glueing” map $g\colon\mathscr{H}_{Q}\to\mathscr{E}_{Q}$,
represented by Figure 4, given by $(\alpha_{s},\alpha_{t})_{\alpha\in
Q_{1}}\mapsto(\alpha_{s}\alpha_{t})_{\alpha\in Q_{1}}$. The earlier claim that
the local brackets on $\mathscr{E}_{\circ}$ and $\mathscr{E}_{\bullet}$ are
compatible with glueing is made precise by the following.

######  Proposition 3.

There is a unique Poisson bracket on $\mathscr{E}_{Q}$ such that the glueing
map $g\colon\mathscr{H}_{Q}\to\mathscr{E}_{Q}$ is a Poisson morphism.

###### Proof.

The statement that $g$ is a Poisson map is equivalent to the pull-back map
$g^{*}\colon E_{Q}\to H_{Q}$ being a homomorphism of Poisson algebras. Note
that for $\alpha\in Q_{1}$, the map $g^{*}$ is given by
$g^{*}(\alpha)=\alpha_{s}\alpha_{t}$. To be a Poisson algebra homomorphism
would mean that for any $\alpha,\beta\in Q_{1}$,

$\\{g^{*}(\alpha),g^{*}(\beta)\\}_{H_{Q}}=g^{*}\left(\\{\alpha,\beta\\}_{E_{Q}}\right)$

We claim that $\\{\alpha,\beta\\}_{E_{Q}}$ is determined by this property. If
$\alpha$ and $\beta$ do not share a common vertex, then obviously
$\\{\alpha,\beta\\}=0$. There are still many cases to consider, depending on
which end (source or target) of each $\alpha$ and $\beta$ meet at the common
vertex, and where in the cyclic ordering those half-edges are at that vertex.
For example, let’s consider the picture in Figure 4, and try to define
$\\{w,y\\}_{E_{Q}}$. We only see one end of $y$ in the figure, so let’s say
$g^{*}(y)=y_{s}y_{t}$, where $y_{s}$ is the end we see at the white vertex.
Then the condition that $g^{*}$ be Poisson means we must have

$\displaystyle g^{*}\left(\\{w,y\\}_{E_{Q}}\right)$
$\displaystyle=\\{g^{*}(w),g^{*}(y)\\}_{H_{Q}}$
$\displaystyle=\\{ax,y_{s}y_{t}\\}_{H_{Q}}$
$\displaystyle=ay_{t}\\{x,y_{s}\\}_{H_{Q}}$
$\displaystyle=w_{1}\,axy_{s}y_{t}$ $\displaystyle=w_{1}g^{*}(w)g^{*}(y)$
$\displaystyle=g^{*}\left(w_{1}\,wy\right)$

Since $g^{*}$ is injective, this uniquely defines $\\{w,y\\}_{E_{Q}}$. The
calculations for all other cases are similarly simple. We see that for
$\alpha,\beta\in Q_{1}$ with a common vertex, the bracket
$\\{\alpha,\beta\\}_{E_{Q}}$ is given by the same expression as the bracket of
the corresponding half-edges in $H_{Q}$. ∎

We will also want to consider the _doubled quiver_.

######  Definition 10.

The _doubled quiver_ , $\overline{Q}$, is defined to have the same vertex set
as $Q$, whereas the edge set of $\overline{Q}$ is the disjoint union of two
copies of $Q_{1}$. For each arrow $\alpha\in Q_{1}$, there are two arrows
$\alpha$ and $\alpha^{*}$ in $\overline{Q}_{1}$. The arrow $\overline{\alpha}$
is called the “opposite” arrow of $\alpha$, with $s(\alpha^{*})=t(\alpha)$ and
$t(\alpha^{*})=s(\alpha)$.

We associate $\alpha^{*}$ with the function $\alpha^{-1}$ on
$\mathscr{E}_{Q}$. Then a path in $\overline{Q}$ may be represented as a
Laurent monomial, which is the product of the edge weights along the path
(allowing inverses). Since Poisson brackets extend uniquely to localizations,
we may extend the Poisson bracket defined above to all Laurent polynomials
(and indeed any rational functions) on $\mathscr{E}_{Q}$.

We will now be interested in describing the Poisson bracket of two paths,
thought of as rational functions on $\mathscr{E}_{Q}$. To do so, it will be
convenient to introduce the constants

$\displaystyle A_{\bullet}$ $\displaystyle=k_{1}-k_{2}-k_{3}$ $\displaystyle
A_{\circ}$ $\displaystyle=w_{1}-w_{2}-w_{3}$

Consider two paths $f$ and $g$. Whenever the paths meet (share at least one
edge in common), then there is a corresponding maximal subpath which $f$ and
$g$ share. If the paths go in the same direction on the common subpath, we
will say they are _parallel_ on that common subpath. If $f$ and $g$ are
parallel on a subpath, which is a proper subpath of both $f$ and $g$, then
there are two possibilities, which are depicted in Figure 5. In the figure,
$f$ is the blue path, $g$ is the red path. In the first situation (the left
image in the figure), the paths are said to “touch”, and in the second
situation, they are said to “cross”. The other possibility is that one of the
paths (either $f$ or $g$) is a subpath of the other. In this case, if $f$ is a
subpath of $g$, then we may write $g=fg^{\prime}$, where $g^{\prime}$ is the
rest of $g$. We extend the notions of “touching” and “crossing” to the
situation where $g=fg^{\prime}$ by saying $f$ and $g$ “touch” on the subpath
$f$ if $g^{\prime}$ and $f$ touch, and similarly for “crossing”.

Figure 5: Two parallel paths sharing a common
subpath$\cdots$“touching”$\cdots$“crossing”

We will denote by $f\cap g$ the set of all maximal common subpaths of $f$ and
$g$. We will define a function $\varepsilon(f,g)$ on the set $f\cap g$, and
for $x\in f\cap g$, we denote the value by $\varepsilon_{x}(f,g)$. It depends
on whether $f$ and $g$ touch or cross, and on the colors of the vertices at
the endpoints of the common subpath. The values are given in Table 1.

Table 1: Definition of $\varepsilon(f,g)$ for parallel paths
$\begin{array}[]{|c|c|c|c|}\hline\cr\mathrm{type}&\mathrm{left~{}endpt}&\mathrm{right~{}endpt}&\varepsilon_{x}(f,g)\\\
\hline\cr\mathrm{touch}&\bullet&\bullet&0\\\ \mathrm{touch}&\circ&\circ&0\\\
\mathrm{touch}&\circ&\bullet&A_{\circ}+A_{\bullet}\\\
\mathrm{touch}&\bullet&\circ&-(A_{\circ}+A_{\bullet})\\\
\mathrm{cross}&\bullet&\bullet&-2A_{\bullet}\\\
\mathrm{cross}&\circ&\circ&2A_{\circ}\\\
\mathrm{cross}&\circ&\bullet&A_{\circ}-A_{\bullet}\\\
\mathrm{cross}&\bullet&\circ&A_{\circ}-A_{\bullet}\\\ \hline\cr\end{array}$

We define $\varepsilon(f,g)$ to be skew-symmetric, so that
$\varepsilon(g,f)=-\varepsilon(f,g)$ if the roles are switched. Also, there
are other cases of intersecting paths which we have not considered. If we
allow paths in $\overline{Q}$, then two paths can meet with opposing
orientations on a common subpath. This can be obtained from the local pictures
above by changing one of the paths to its inverse. Since Poisson brackets
extend uniquely to localizations, this will not be an issue.

We then have the following formula:

######  Proposition 4.

Suppose $f$ and $g$ are two paths. Then

$\\{f,g\\}=\sum_{*\in f\cap g}\varepsilon_{*}(f,g)\,fg$

###### Proof.

Since $f$ and $g$ are monomials in the coordinate variables, it is obvious
that $\\{f,g\\}=c\,fg$ for some constant $c$. We must prove that
$c=\sum\varepsilon_{*}(f,g)$. Each time $f$ and $g$ share a proper subpath on
which they are parallel, it will look like one of the two pictures in Figure
5. We can then write the paths as $f=f_{0}axwyb$ and $g=g_{0}cxwyd$, where
$xwy$ is the common subpath, $a$,$x$, and $c$ are the edges incident to the
“left” vertex (where the paths come together), and $y$,$b$, and $d$ are the
edges incident to the “right” vertex (where the paths diverge at the end of
the common subpath). Then after expanding $\\{f,g\\}$ by the Leibniz rule, we
get contributions from $\\{a,c\\}$, $\\{a,x\\}$, and $\\{x,c\\}$ at the left
vertex, and $\\{y,d\\}$, $\\{b,y\\}$, and $\\{b,d\\}$ from the right vertex.
Obviously the contributions from the common subpath cancel, since if $p$ and
$q$ are two consecutive edges in the common subpath, we will get a
contribution from both $\\{p,q\\}$ and $\\{q,p\\}$. There are of course other
terms in the expansion coming from $f_{0}$ and $g_{0}$, but they will
correspond to other common subpaths in the same way. It is therefore enough to
consider the six terms mentioned above (3 each for the beginning and end
vertices) for each common subpath.

In general, we must consider 36 different cases, since each endpoint could be
either white or black, and there are three orientations for the arrows
incident to each of the endpoint vertices. A simple calculation shows that the
orientations at the vertices are in fact irrelevant. Therefore we only have 4
cases, depending on the colors of the vertices. As mentioned above, by the
Leibniz rule, we can add the contributions from the endpoint vertices
separately.

First consider the left endpoint. For all three choices of orientations, a
black left endpoint will give a contribution of $-A_{\bullet}$ to the
coefficient, and a white left endpoint will give a contibution of $A_{\circ}$.
For the right endpoint, a black vertex gives $A_{\bullet}$, and a white gives
$-A_{\circ}$. Combining all the possibilities together gives the list of
values in the table for $\varepsilon_{x}(f,g)$.

We assumed so far that the common subpaths were parallel. If the paths are
oriented in opposite directions on a common subpath, then it corresponds to
reversing one of the paths in Figure 5. Then we can extend the table for
$\varepsilon_{x}(f,g)$ by the rule $\\{f,g^{-1}\\}=-g^{-2}\\{f,g\\}$.

We also assumed so far that the subpaths were proper. It remains to consider
when one of the paths is a subpath of the other. For instance, let’s say $f$
is contained in $g$. Then we can write $g$ as $g=fg^{\prime}$. Then by the
Leibniz rule, $\\{f,g\\}=\\{f,fg^{\prime}\\}=f\\{f,g^{\prime}\\}$. It must be
(since all vertices are trivalent) that the beginning of $g^{\prime}$ is a
common subpath with $f$, and so this is the term corresponding to the common
subpath of $f$ and $g$ which is all of $f$. The rest of the common subpaths of
$f$ and $g^{\prime}$ are just usual proper common subpaths of $f$ and $g$, as
discussed above. ∎

######  Remark.

Note that this result depends only on the parameters $A_{\circ}$ and
$A_{\bullet}$, and not on the actual values of $w_{1}$, $w_{2}$, $w_{3}$,
$k_{1}$, $k_{2}$, $k_{3}$. In particular, we could choose
$w_{1}=w_{2}=k_{1}=k_{2}=0$, and $w_{3}=-A_{\circ}$, $k_{3}=-A_{\bullet}$, and
the formula from the proposition would be the same. From now on, we will
assume this is the case (that all but $w_{3}$ and $k_{3}$ are zero).

##  3 Interpreting the Pentagram Map in Terms of Networks

We now use the constructions from the previous section to realize the
pentagram map as a sequence of Postnikov moves and gauge transformations of a
particular network in an annulus. The properties of the boundary measurements
will give us invariants of the pentagram map, which also turn out to be
involutive with respect to the Poisson structures described in the previous
section. This section summarizes the main points of [GSTV16].

###  3.1 The Quiver and Poisson Bracket

We now look at a very specific example of the quivers and Poisson structures
discussed previously. The quiver $Q_{k,n}$ (or just $Q_{n}$ if $k=3$) is
embedded in an annulus (or cylinder), and the case $k=3$, $n=5$ is shown below
in Figure 6. In general, there are $k$ sources and $k$ sinks, the number of
square faces is $n$, and they are connected as in the figure. More
specifically, the bottom right of the $i^{\mathrm{th}}$ square face connects
to the top left of the $(i+1)^{\mathrm{st}}$, and the top right of the
$i^{\mathrm{th}}$ square face is connected to the bottom left of the
$(i+2)^{\mathrm{nd}}$, with the indices understood cyclically. This is how the
labels of the sinks on the right side are chosen. The top and bottom edges of
the boundary rectangle are identified, giving a cylinder. We take the cut to
be the top and bottom edges which we identify. The sources and sinks are
labeled on the left and right edges, which become the inner and outer boundary
circles, respectively. As was mentioned in the previous sections, we will
sometimes consider the network on a torus by also identifying the left and
right edges (up to a twist, indicated by the labels).

Figure 6: The network $Q_{3,5}$$3$$2$$1$$1$$3$$2$

As was mentioned in the previous section, the Poisson bracket depends only on
the parameters $A_{\circ}$ and $A_{\bullet}$. So we may assume that all
$w_{i}$ and $k_{i}$ are zero except $w_{3}=-A_{\circ}$ and
$k_{3}=-A_{\bullet}$. From now on we will consider the specific choice
$w_{3}=k_{3}=\frac{1}{2}$. Note that when two paths meet in this network, they
must come together at a black vertex, and separate at a white vertex. This
means for two paths $f$ and $g$, that $\varepsilon_{x}(f,g)=\pm 1$ when $f$
and $g$ touch, and $\varepsilon_{x}(f,g)=0$ when $f$ and $g$ cross.

###  3.2 The $x,y$ Coordinates

Next, we will apply gauge transformations to this network and consider some
other functions on $\mathscr{E}_{Q}$. We start by defining our notation for
the edge weights. We will consider the toric network obtained by glueing the
boundary components together. So the $n^{\mathrm{th}}$ square connects to the
$1^{\mathrm{st}}$ square. We label the edge weights around each square face as
follows:

$\alpha_{i}$$\beta_{i}$$\gamma_{i}$$\delta_{i}$$\varepsilon_{i}$$\overline{\varepsilon}_{i}$$\overline{\varepsilon}_{i-1}$$\varepsilon_{i-2}$

Because of our choice of coefficients $A_{\circ}=A_{\bullet}=-\frac{1}{2}$,
the brackets between these coordinates are given by

$\displaystyle\\{\beta_{i},\alpha_{i}\\}$
$\displaystyle=\frac{1}{2}\beta_{i}\alpha_{i}$ $\displaystyle\hskip
28.45274pt\\{\beta_{i},\gamma_{i}\\}$
$\displaystyle=\frac{1}{2}\gamma_{i}\beta_{i}$
$\displaystyle\\{\alpha_{i},\overline{\varepsilon}_{i-1}\\}$
$\displaystyle=\frac{1}{2}\overline{\varepsilon}_{i-1}\alpha_{i}$
$\displaystyle\hskip 28.45274pt\\{\gamma_{i},\varepsilon_{i}\\}$
$\displaystyle=\frac{1}{2}\gamma_{i}\varepsilon_{i}$

After applying guage transformations at the corners of each square face, we
can obtain the following edge weights:

$\alpha_{i}\overline{\varepsilon}_{i-1}^{-1}$$\beta_{i}$$\varepsilon_{i}^{-1}\gamma_{i}$$\overline{\varepsilon}_{i-1}\delta_{i}\varepsilon_{i}$

The effect of these gauge transformations is that all the edges other than
those bounding the square faces have been set to $1$. These new weights, which
are monomials in the original weights, we will call by
$a_{i},b_{i},c_{i},d_{i}$:

$\displaystyle a_{i}$
$\displaystyle=\alpha_{i}\overline{\varepsilon}_{i-1}^{-1}$
$\displaystyle\hskip 28.45274ptb_{i}$ $\displaystyle=\beta_{i}$ $\displaystyle
c_{i}$ $\displaystyle=\varepsilon_{i}^{-1}\gamma_{i}$ $\displaystyle\hskip
28.45274ptd_{i}$
$\displaystyle=\overline{\varepsilon}_{i-1}\delta_{i}\varepsilon_{i}$

It is easily checked that the Poisson brackets of these monomials are given by

$\displaystyle\\{b_{i},a_{i}\\}$ $\displaystyle=\frac{1}{2}b_{i}a_{i}$
$\displaystyle\hskip 28.45274pt\\{b_{i},c_{i}\\}$
$\displaystyle=\frac{1}{2}b_{i}c_{i}$ $\displaystyle\\{a_{i},d_{i}\\}$
$\displaystyle=\frac{1}{2}a_{i}d_{i}$ $\displaystyle\hskip
28.45274pt\\{c_{i},d_{i}\\}$ $\displaystyle=\frac{1}{2}c_{i}d_{i}$

We will now apply further gauge transformations, in order to set as many edge
weights as possible equal to $1$. It is possible, after further gauge
transformations, to set most of the weights equal to 1, except the bottom and
left edges of each square face, and a few other edges. The result (again for
$n=5$) is pictured in Figure 7.

Figure 7: The $n=5$ quiver after gauge
transformations$3$$2$$1$$1$$3$$2$$x_{1}$$y_{1}$$x_{2}$$y_{2}$$x_{3}$$y_{3}$$x_{4}$$y_{4}$$x_{5}$$y_{5}$$z$$z$$z$

The weights in the new quiver after gauge transformations are given as follows

$\displaystyle x_{i}$ $\displaystyle=\frac{a_{i}}{c_{i-1}d_{i-1}c_{i-2}}$
$\displaystyle y_{i}$
$\displaystyle=\frac{b_{i}}{c_{i}d_{i}c_{i-1}d_{i-1}c_{i-2}}$ $\displaystyle
z$ $\displaystyle=\prod_{k=1}^{n}d_{k}c_{k}$

Thinking of the quiver as being on a torus, all of these monomials represent
loops. They are depicted in Figure 8. The blue loop is $x_{4}$, the red loop
is $y_{4}$, and the green loop is $z$.

Figure 8: The weights $x_{i},y_{i},z$ represented as cycles$3$$2$$1$$1$$3$$2$

It is an easy calculation to see that $z$ is a Casimir of the Poisson bracket,
and so the bracket descends to the level surface in $\mathscr{E}_{Q}$ defined
by $z=1$. We will now consider just edge weights lying on this hypersurface.
In this case, we have that all edge weights except those labeled by $x_{i}$
and $y_{i}$ are equal to $1$. Therefore, this level surface has coordinates
given by $x_{i}$ and $y_{i}$, and so it is $2n$-dimensional. We will associate
these coordinates with the $x_{i},y_{i}$ coordinates on $\mathcal{P}_{n}$
introduced in section 1. Their brackets are given by

$\displaystyle\\{x_{i+1},x_{i}\\}$ $\displaystyle=x_{i+1}x_{i}$
$\displaystyle\hskip 28.45274pt\\{y_{i},x_{i}\\}$ $\displaystyle=y_{i}x_{i}$
$\displaystyle\\{y_{i+1},y_{i}\\}$ $\displaystyle=y_{i+1}y_{i}$
$\displaystyle\hskip 28.45274pt\\{y_{i},x_{i-1}\\}$
$\displaystyle=y_{i}x_{i-1}$ $\displaystyle\\{y_{i+2},y_{i}\\}$
$\displaystyle=y_{i+2}y_{i}$ $\displaystyle\hskip
28.45274pt\\{x_{i},y_{i-1}\\}$ $\displaystyle=x_{i}y_{i-1}$
$\displaystyle\hskip 28.45274pt\\{x_{i},y_{i-2}\\}$
$\displaystyle=x_{i}y_{i-2}$

###  3.3 The $p,q$ Coordinates

It will be convenient to also consider a different set of functions on
$\mathscr{E}_{Q}$. We return to the $a,b,c,d$ coordinates before the guage
transformations which gave the $x,y$ coordinates. We define the _face weights_
$p_{i}$ and $q_{i}$ as the counter-clockwise paths around the faces of the
quiver, taking inverses when the orientation disagrees:

$\displaystyle p_{i}$ $\displaystyle=\frac{b_{i}}{a_{i}c_{i}d_{i}}$
$\displaystyle\hskip 28.45274ptq_{i}$
$\displaystyle=\frac{c_{i-2}d_{i-1}a_{i+1}}{b_{i}}$

These are related to the $x,y$ coordinates by

$\displaystyle p_{i}$ $\displaystyle=\frac{y_{i}}{x_{i}}$ $\displaystyle\hskip
28.45274ptq_{i}$ $\displaystyle=\frac{x_{i+1}}{y_{i}}$

Define the quiver $Q^{\prime}$, dual to $Q$, to have vertices corresponding to
the faces of $Q$, and arrows corresponding to arrows of $Q$ connecting
vertices of different colors, directed so that white vertices are on the left
and black vertices on the right. The vertices of $Q^{\prime}$ are labeled by
$p_{i}$ and $q_{i}$, with arrows $p_{i}\to q_{i-1}$, $p_{i}\to q_{i+2}$,
$q_{i}\to p_{i}$, and $q_{i+1}\to p_{i}$. This is pictured in Figure 9, with
the dual quiver drawn in blue.

Figure 9: The dual quiver
$Q^{\prime}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}q_{i-1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}q_{i}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}q_{i+2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}q_{i+1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{i}$

The brackets of the $p$’s and $q$’s is “log-canonical”, with the skew-
symmetric coefficient matrix given by the adjacency matrix of the dual quiver
$Q^{\prime}$. That is,

$\displaystyle\\{q_{i},p_{i}\\}$ $\displaystyle=q_{i}p_{i}$
$\displaystyle\hskip 28.45274pt\\{q_{i+1},p_{i}\\}$
$\displaystyle=q_{i+1}p_{i}$ $\displaystyle\\{p_{i},q_{i-1}\\}$
$\displaystyle=p_{i}q_{i-1}$ $\displaystyle\hskip
28.45274pt\\{p_{i},q_{i+2}\\}$ $\displaystyle=p_{i}q_{i+2}$

###  3.4 The Postnikov Moves and the Invariants

Now we will describe a sequence of Postnikov moves which will transform this
quiver (considered as being on the torus) into an isomorphic quiver. The new
edge weights obtained after this sequence will be the expressions for the
pentagram map on $\mathcal{P}_{n}$ in the $x,y$ coordinates given in section
1. We start with the quiver as in Figure 7, where all weights are $1$ except
the $x_{i}$ and $y_{i}$ weights on the bottom and sides of the square faces.
We then apply the following moves, in order:

  1. 1.

Perform the “square move” at each of the $n$ square faces

  2. 2.

Perform the “white-swap” at each white-white edge

  3. 3.

Perform the “black-swap” at each black-black edge

After this sequence of moves, the underlying directed graph is isomorphic to
the one we started with. However, the edge weights will not be of the same
form. It remains to perform gauge transformations (as we did in the previous
section) so that again all weights are $1$ except the bottom and left of each
square face. After these gauge transformations, and after choosing a
particular choice of graph isomorphism, the $x,y$ weights transform as

$\displaystyle x_{i}$ $\displaystyle\mapsto
x_{i}\,\frac{x_{i+2}+y_{i+2}}{x_{i}+y_{i}}$ $\displaystyle y_{i}$
$\displaystyle\mapsto y_{i+1}\,\frac{x_{i+3}+y_{i+3}}{x_{i+1}+y_{i+1}}$

This is almost (but not quite) the same as the expression for the pentagram
map on $\mathcal{P}_{n}$ given in section 1. But after making the change of
variables $y_{i}\mapsto y_{i-1}$, the formulas agree. So it only differs by a
shift of indices in the $y$-variables.

As discussed before, the Postnikov moves and gauge transformations performed
above do not change the boundary measurements. However, since the quiver was
considered on a torus, we may have had some of the vertices or edges “wrap
around” from the right side to the left side when doing the “white-swap” and
“black-swap” moves. This corresponds to cutting a piece off the right side of
the picture, and glueing it back onto the left side. As discussed earlier,
this changes the boundary measurement matrix by a re-factorization
$B_{1}B_{2}\mapsto B_{2}B_{1}$. The boundary measurement matrix
$\mathscr{B}_{Q}(\lambda)$ is therefore changed only up to conjugation, since
$B_{2}B_{1}=B_{2}(B_{1}B_{2})B_{2}^{-1}$.

Thus the components of the characteristic polynomial
$\chi(t)=\mathrm{det}(\mathrm{Id}+tB)$ are unchanged by the sequence of moves
described above. Recall that we consider the elements of the boundary
measurement matrix to be the modified edge weights, which are Laurent
polynomials in the variable $\lambda$. So $\chi$ is a function of both
$\lambda$ and $t$, which is polynomial in $t$, and Laurent in $\lambda$. We
denote the coefficients (which are functions of the edge weights of the
quiver) by $I_{ij}$:

$\chi(\lambda,t)=\sum_{i,j}I_{ij}\lambda^{i}t^{j}$

The discussion above implies that $I_{ij}$ are invariants of the pentagram
map. Furthermore, we have:

######  Theorem 2.

[GSTV16] Let $I_{ij}$ be as defined above. Then
$(a)$ $I_{ij}$ are invariant under the pentagram map $(b)$
$\\{I_{ij},I_{k\ell}\\}=0$ for all $i,j,k,\ell$ $(c)$ The pentagram map is
completely integrable

We will also consider the dynamics in the $p,q$ coordinates described before.
A simple calculation using the relations between $p,q$ and $x,y$ shows that
the pentagram map transforms the $p,q$ coordinates by

$\displaystyle q_{i}$ $\displaystyle\mapsto\frac{1}{p_{i+1}}$ $\displaystyle
p_{i}$ $\displaystyle\mapsto
q_{i+2}\frac{(1+p_{i})(1+p_{i+3})}{(1+p_{i+1}^{-1})(1+p_{i+2}^{-1})}$

In these coordinates we can interpret the Postnikov moves as cluster
transformations. To see this, consider the dual quiver $Q^{\prime}$ described
before. Interpret the $p,q$ variables as the initial cluster of a cluster
algebra whose exchange matrix $B=(b_{ij})$ is defined by the dual quiver
$Q^{\prime}$. We use the mutation formula

$\mu_{k}(\tau_{i})=\begin{cases}\frac{1}{\tau_{i}}&\mathrm{if}~{}i=k\\\\[5.16663pt]
\tau_{i}(1+\tau_{k})^{b_{ik}}&\mathrm{if}~{}b_{ik}>0\\\\[5.16663pt]
\tau_{i}(1+\tau_{k}^{-1})^{b_{ik}}&\mathrm{if}~{}b_{ik}<0\end{cases}$

These are referred to as cluster $\mathcal{X}$-variables (as opposed to
cluster $\mathcal{A}$-variables) by Fock and Goncharov [FG06]. They are also
called “$\tau$-coordinates” by Gekhtman, Shapiro, and Vainshtein [GSV10a].
This type of cluster algebra is called a _cluster Poisson algebra_ , since it
inherits a natural Poisson bracket which is “compatible” with mutation (see
[GSV10a]), given by

$\\{\tau_{i},\tau_{j}\\}=b_{ij}\tau_{i}\tau_{j}$

Using the dual quiver $Q^{\prime}$ for the exchange matrix $B$, this cluster
Poisson bracket in the $p,q$ variables coincides with the bracket on
$\mathscr{E}_{Q}$. Additionally, the square moves in the sequence giving the
pentagram map coincide with mutations at the $p$-vertices. The sequence of
mutations, $\mu$, that mutates at each $p_{i}$ once gives

$\displaystyle\widetilde{p}_{i}$ $\displaystyle=\mu(p_{i})=\frac{1}{p_{i}}$
$\displaystyle\hskip 28.45274pt\widetilde{q}_{i}$
$\displaystyle=\mu(q_{i})=q_{i}\,\frac{(1+p_{i+1})(1+p_{i-2})}{(1+p_{i}^{-1})(1+p_{i-1}^{-1})}$

The white-swap and black-swap Postnikov moves give a graph isomorphic to the
original, which exchanges the $\widetilde{p}$ and $\widetilde{q}$ face
weights. That is, the new square face weights $p_{i}^{*}$ and the new
octagonal face weights $q_{i}^{*}$ are

$\displaystyle p_{i}^{*}$
$\displaystyle=\widetilde{q}_{i}=q_{i}\,\frac{(1+p_{i+1})(1+p_{i-2})}{(1+p_{i}^{-1})(1+p_{i-1}^{-1})}$
$\displaystyle q_{i}^{*}$ $\displaystyle=\widetilde{p}_{i}=\frac{1}{p_{i}}$

The pentagram map coincides with this formula up to the permutation of the
variables which shifts the indices by $p^{*}_{i}\mapsto p^{*}_{i+2}$ and
$q^{*}_{i}\mapsto q^{*}_{i+1}$, More technically, we have

$\displaystyle T(p_{i})$ $\displaystyle=\widetilde{q}_{i+1}$
$\displaystyle\hskip 28.45274ptT(q_{i})$ $\displaystyle=\widetilde{p}_{i+2}$

It is a simple calculation to verify that the brackets
$\\{T(p_{i}),T(q_{j})\\}$ have exactly the same form as $\\{p_{i},q_{j}\\}$.

##  4 The Grassmann Pentagram Map

###  4.1 Twisted Grassmann Polygons and the Pentagram Map

Mari Beffa and Felipe studied a generalization of the pentagram map [FMB15] to
the Grassmann manifold. The exposition and notation in this section is largely
borrowed from that paper, with minor variations. The projective plane
$\mathbb{P}^{2}$ coincides with the Grassmannian $\mathrm{Gr}(1,3)$ of
$1$-dimensional subspaces of $\mathbb{R}^{3}$. A natural generalization would
be to consider $\mathrm{Gr}(N,3N)$, in which the previous case is just when
$N=1$. In actuality, Mari Beffa and Felipe considered more generally
$\mathrm{Gr}(N,kN)$, which generalizes $\mathbb{P}^{k-1}$, but we will focus
here on the case $k=3$.

Consider the set $\mathrm{Mat}_{3N\times N}$ of $3N$-by-$N$ real matrices.
There are two natural multiplication actions: on the left by
$\mathrm{GL}_{3N}$, and on the right by $\mathrm{GL}_{N}$. If we call
$\mathcal{M}_{N}\subset\mathrm{Mat}_{3N\times N}$ the subset of rank-$N$
matrices, then we will identify $\mathrm{Gr}(N,3N)$ with the orbit space
$\mathcal{M}_{N}/\mathrm{GL}_{N}$ by this right action, since two matrices
will be equivalent if they have the same column-span. Then $\mathrm{Gr}(N,3N)$
carries a natural left action by $\mathrm{GL}_{3N}$, induced by the action on
$\mathrm{Mat}_{3N\times N}$.

######  Definition 11.

A _twisted Grassmann $n$-gon_ (or just “polygon” if $n$ is understood) is a
bi-infinite sequence $(p_{i})_{i\in\mathbb{Z}}$ of points in
$\mathrm{Gr}(N,3N)$, with the property that $p_{i+n}=Mp_{i}$ for some
projective transformation $M\in\mathrm{PGL}_{3N}$ called the _monodromy_.

As in the classical case, we will only consider twisted polygons satsifying
some nondegeneracy condition. More specifically, choose a lift $(V_{i})$ of
$(p_{i})$ to $\mathcal{M}_{N}$. Define the block column matrices
$\mathbf{V}_{i}=(V_{i}V_{i+1}V_{i+2})$. We require for any $i$ that
$\mathbf{V}_{i}$ is nonsingular, or equivalently, that the combined columns of
$V_{i},V_{i+1},V_{i+2}$ form a basis of $\mathbb{R}^{3N}$. If this
nondegeneracy condition is satisfied, then we obtain something analogous to
the linear dependence relations given in Equation 1. For each $i$, there are
matrices $A_{i},B_{i},C_{i}\in\mathrm{GL}_{N}$ so that

$V_{i+3}=V_{i}A_{i}+V_{i+1}B_{i}+V_{i+2}C_{i}$ (4)

For the remainder of the paper, we will use the abbreviated phrase _twisted
polygon_ to mean a twisted Grassmann polygon. We will use the notation
$\mathcal{GP}_{n,N}$ to denote the moduli space of twisted Grassmann $n$-gons
in $\mathrm{Gr}(N,3N)$, up to the action of $\mathrm{PGL}_{3N}$. We will use
the adjective “classical” when we wish to distinguish the $\mathbb{P}^{2}$
case ($N=1$). Next we will describe the pentagram map on Grassmann polygons.

Let $(p_{i})$ be a twisted polygon in $\mathrm{Gr}(N,3N)$. Since $p_{i}$ and
$p_{i+2}$ are $N$-dimensional subspaces with trivial intersection, they span a
$2N$-dimenional (codimension $N$) subspace, which we call $\mathscr{L}_{i}$.
Then the intersection $\mathscr{L}_{i}\cap\mathscr{L}_{i+1}$ is a codimension
$2N$ (dimension $N$) subspace, which is again an element of
$\mathrm{Gr}(N,3N)$. We define the pentagram map to be the map that sends
$(p_{i})$ to $(q_{i})$, where $q_{i}=\mathscr{L}_{i}\cap\mathscr{L}_{i+1}$. We
will also refer to the pentagram map as
$T\colon\mathcal{GP}_{n,N}\to\mathcal{GP}_{n,N}$. Abusing notation as in the
classical case, we also write the map as if it were defined on individual
vertices $T(p_{i})=q_{i}$, and also write the map as if it is defined on lifts
$T(V_{i})=W_{i}$ if $(V_{i})$ and $(W_{i})$ are lifts of $(p_{i})$ and
$(q_{i})$.

###  4.2 Corrugated Grassmann Polygons and Higher Pentagram Maps

Analogous to the classical case, we can also define _generalized higher
pentagram maps_ in the Grassmann case. This is a generalization from
$\mathrm{Gr}(N,3N)$ to $\mathrm{Gr}(N,kN)$. A twisted Grassmann polygon in
$\mathrm{Gr}(N,kN)$ is a sequence $(p_{i})$ with monodromy
$M\in\mathrm{PGL}_{kN}$ so that $p_{i+n}=Mp_{i}$ for all $i$. We denote the
set of all equivalence classes of twisted polygons (up to the action of
$\mathrm{PGL}_{kN}$) by $\mathcal{GP}_{k,n,N}$. In keeping with the notation
of the previous section, we write just $\mathcal{GP}_{n,N}$ for $k=3$.

Given a polygon $P=(p_{i})$, we define the subspaces
$\mathscr{L}_{i}:=p_{i}+p_{i+k-1}$. We say that a twisted Grassmann polygon is
_corrugated_ if $\mathscr{L}_{i}+\mathscr{L}_{i+1}$ is a $3N$-dimensional
subspace of $\mathbb{R}^{kN}$ for all $i$. Let $\mathcal{GP}^{0}_{k,n,N}$
denote the set of classes of corrugated polygons with the following additional
properties, all of which hold generically:

  * •

Any $3$ of the $4$ subspaces $p_{i}$, $p_{i+1}$, $p_{i+k-1}$, $p_{i+k}$ span a
$3N$-dimensional subspace.

  * •

$\mathscr{L}_{i}\cap\mathscr{L}_{i+1}\cap\mathscr{L}_{i+2}=0$

  * •

$\dim(\mathscr{L}_{i}\cap\mathscr{L}_{i+k-1})=N$

We define the generalized higher pentagram map
$T\colon\mathcal{GP}^{0}_{k,n,N}\to\mathcal{GP}_{k,n,N}$ as follows. The
“corrugated” property guarantees that

$\dim(\mathscr{L}_{i}\cap\mathscr{L}_{i+1})=\dim\mathscr{L}_{i}+\dim\mathscr{L}_{i+1}-\dim(\mathscr{L}_{i}+\mathscr{L}_{i+1})=2N+2N-3N=N$

So $q_{i}:=\mathscr{L}_{i}\cap\mathscr{L}_{i+1}$ is again an element of
$\mathrm{Gr}(N,kN)$, and we define the pentagram map to be
$(p_{i})\mapsto(q_{i})$.

######  Proposition 5.

If $P=(p_{i})\in\mathcal{GP}^{0}_{k,n,N}$, then its image under $T$ is
corrugated.

###### Proof.

We need to show that for the image $T(P)=Q=(q_{i})$, the span
$q_{i}+q_{i+1}+q_{i+k-1}+q_{i+k}$ is $3N$-dimensional. Define subspaces
$\mathscr{K}_{i}:=q_{i}+q_{i+1}$. Then we are trying to show that
$\dim(\mathscr{K}_{i}+\mathscr{K}_{i+k-1})=3N$. Note that by definition,
$q_{i}$ and $q_{i+1}$ are both subspaces of $\mathscr{L}_{i+1}$, and so
$\mathscr{K}_{i}\subseteq\mathscr{L}_{i+1}$. The assumption that
$\mathscr{L}_{i}\cap\mathscr{L}_{i+1}\cap\mathscr{L}_{i+2}=0$ ensures that
$\dim\mathscr{K}_{i}=2N$, and so in fact $\mathscr{K}_{i}=\mathscr{L}_{i+1}$.
Then by shifting indices we also get that
$\mathscr{K}_{i+k-1}=\mathscr{L}_{i+k}$. It is always true that
$\mathscr{L}_{i+1}\cap\mathscr{L}_{i+k}$ contains $p_{i+k}$, and so its
dimension is at least $N$. The third assumption, however, gaurantees that the
intersection is exactly $p_{i+k}$. We then have that

$\dim(\mathscr{K}_{i}+\mathscr{K}_{i+k-1})=\dim\mathscr{K}_{i}+\dim\mathscr{K}_{i+k-1}-\dim(\mathscr{K}_{i}\cap\mathscr{K}_{i+k-1})=3N$

∎

######  Remark.

It is worth pointing out that the image of the map is not necessarily in the
set $\mathcal{GP}^{0}_{k,n,N}$. The above proof shows that the image is
corrugated, but it does not necessarily satisfy the genericity assumptions.
The set on which the map may be iterated is thus the complement of countably
many subsets of codimension 1 (and hence non-empty).

###  4.3 Description of the Moduli Space by Networks

For simplicity of presentation, we will for the remainder of the paper
restrict to considering the case $k=3$, where all polygons are automatically
corrugated. In this section, we model the moduli space $\mathcal{GP}_{n,N}$
using the networks $Q_{n}$ from Section 3. We will use variations of this
network on three surfaces: the cylinder $S^{1}\times[0,1]$ (with two boundary
components), the infinite cylinder $S^{1}\times(0,1)$, and the torus
$S^{1}\times S^{1}$.

First, we establish some notation. As before, we let $\mathcal{M}_{N}$ be the
space of all $3N$-by-$N$ matrices of full rank. It is an open subset of
$\mathrm{Mat}_{3N\times N}$. In particular, for a twisted polygon
$P\in\mathcal{GP}_{n,N}$, a lift of $P$ will be a point in
$\mathcal{M}_{N}^{\mathbb{Z}}$. We denote by
$\mathcal{L}_{n,N}\subset\mathcal{M}_{N}^{\mathbb{Z}}$ the subset of lifts of
twisted $n$-gons, and we let $\mathcal{TL}_{n,N}$ denote the set of lifts
which are “twisted” (the lifts satisfy $V_{i+n}=MV_{i}$). Since a twisted lift
is determined by the first $n$ of the $V_{i}$ and the monodromy, we may view
$\mathcal{TL}_{n,N}$ as an open subset of
$\mathcal{M}_{N}^{n}\times\mathrm{GL}_{3N}$. The dimension is thus
$3N^{2}(n+3)$.

Recall the networks $Q_{n}$ defined in Section 3, which are embedded on the
cylinder. We will also consider the infinite networks $\widetilde{Q}_{n}$,
embedded on the infinite cylinder, which are just infinitely many copies of
$Q_{n}$ concatenated together, according to the labels of the sources and
sinks. Finally, we also consider the networks $\widehat{Q}_{n}$, on the torus,
which are obtained by glueing the boundary components of the clyinder to each
other according to the labels of the sources and sinks in $Q_{n}$.

Let $\mathcal{E}_{n,N}$ be the space of all possible choices of
$\mathrm{GL}_{N}$ weights on the edges which are the left, bottom, and right
of each square face of the network $Q_{n}$. Clearly there is a bijection
$\mathcal{E}_{n,N}\cong\mathrm{GL}_{N}^{3n}$. Similarly, let
$\widetilde{\mathcal{E}}_{n,N}$ be space of $\mathrm{GL}_{N}$-weights on the
same edges, but on the infinite network $\widetilde{Q}_{n}$. Then of course
$\widetilde{\mathcal{E}}_{n,N}\cong\mathrm{GL}_{N}^{\mathbb{Z}}$.

We now define _gauge transformations_ on the networks $\widetilde{Q}_{n}$. At
any vertex, we may choose some $A\in\mathrm{GL}_{N}$, and multiply all
incoming edge weights at that vertex by $A$ on the right, and all outgoing
edge weights by $A^{-1}$ on the left. Clearly, the weight of any path passing
through this vertex is unchanged by this action. We let $\mathcal{G}$ be the
group generated by these transformations. If $S$ is the set of vertices, then
clearly $\mathcal{G}\cong\mathrm{GL}_{N}^{S}$.

We would like to think of the gauge group acting on the space
$\widetilde{\mathcal{E}}_{n,N}$. However, this space consists only of those
assignments of weights in which all edges other than the left, right, and
bottom of each square face have weight equal to $\mathrm{Id}_{N}$. The gauge
transformations described above do not, in general, preserve this property. So
we will instead restrict to the subgroup
$\mathcal{G}^{\circ}\subset\mathcal{G}$ which preserves
$\widetilde{\mathcal{E}}_{n,N}$. We will now describe a system of generators
for $\mathcal{G}^{\circ}$. The top edge of each square face has six “nearby”
nontrivial weights – three to the left, and three to the right. For a matrix
$X\in\mathrm{GL}_{N}$, the corresponding generator of $\mathcal{G}^{\circ}$
will multiply the three weights to the left by $X$ on the right, and multiply
the three weights to the right by $X^{-1}$ on the left. This is pictured in
Figure 10.

Figure 10: A generator of the gauge subgroup
$\mathcal{G}^{\circ}$$A_{i-2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}A_{i-1}$$A_{i}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}A_{i+1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{i-1}$$B_{i}$$B_{i+1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{i+2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{i-1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{i}$$C_{i+1}$$C_{i+2}$$\rightsquigarrow$$A_{i-2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}A_{i-1}X$$A_{i}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}X^{-1}A_{i+1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{i-1}X$$B_{i}$$B_{i+1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}X^{-1}B_{i+2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{i-1}X$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}X^{-1}C_{i}$$C_{i+1}$$C_{i+2}$

Also, the subgroup $\mathcal{G}^{\bullet}\subset\mathcal{G}^{\circ}$
consisting of gauge transformations which are $n$-periodic can be considered
to act on $\mathcal{E}_{n,N}$, thought of as the space of weights for
$\widehat{Q}_{n}$ on the torus, since $\mathcal{E}_{n,N}$ is naturally
identified with the subset of $\widetilde{\mathcal{E}}_{n,N}$ with
$n$-periodic weights.

We will now consider several quotients of $\mathcal{TL}_{n,N}$ by different
group actions. First, we consider the action of $\mathbb{R}^{*}$ on
$\mathcal{TL}_{n,N}$ by dilations. For $\lambda\in\mathbb{R}^{*}$, the action
is given by $V_{i}\mapsto\lambda V_{i}$, and the monodromy matrix is
unchanged. This obviously does not change the polygon which the lift
represents. We let $\widetilde{\mathcal{TL}}_{n,N}$ denote the quotient by
this action. Since the action is free and proper, we have

$\dim\widetilde{\mathcal{TL}}_{n,N}=\dim\mathcal{TL}_{n,N}-1=3N^{2}(n+3)-1$

Now we may define an action of $\mathrm{PGL}_{3N}$ on
$\widetilde{\mathcal{TL}}_{n,N}$. Let $A\in\mathrm{GL}_{3N}$ be a
representative of $\overline{A}\in\mathrm{PGL}_{3N}$, and let
$(V_{1},\dots,V_{n},M)\in\mathcal{TL}_{n,N}$. Then $A$ acts in the usual way
by $V_{i}\mapsto AV_{i}$ and $M\mapsto AMA^{-1}$. The action by conjugation is
well-defined on $\mathrm{PGL}_{3N}$, since scalar matrices act trivially. The
left action $V_{i}\mapsto AV_{i}$ is also well-defined, since
$\overline{A}=\overline{\lambda A}$ and
$\overline{V_{i}}=\overline{\lambda^{-1}V_{i}}$. This action is also free and
proper, so

$\dim\widetilde{\mathcal{TL}}_{n,N}/\mathrm{PGL}_{3N}=\dim\widetilde{\mathcal{TL}}_{n,N}-\dim\mathrm{PGL}_{3N}=3nN^{2}$

######  Theorem 3.

The spaces $\mathcal{TL}_{n,N}$ and $\mathcal{E}_{n,N}$ are related in the
following ways:

$(a)$ There is a bijective map
$\widetilde{\mathcal{TL}}_{n,N}/\mathrm{PGL}_{3N}\to\mathcal{E}_{n,N}$ $(b)$
There is a bijective map
$\widetilde{\mathcal{L}}_{n,N}/\mathrm{PGL}_{3N}\to\widetilde{\mathcal{E}}_{n,N}$.
$(c)$ Under these identifications, the actions of $\mathcal{G}^{\circ}$ on
$\widetilde{\mathcal{E}}_{n,N}$ and $\mathcal{G}^{\bullet}$ on
$\mathcal{E}_{n,N}$ correspond to changing the lift of a fixed polygon. $(d)$
$\mathcal{GP}_{n,N}\cong\mathcal{E}_{n,N}/\mathcal{G}^{\bullet}\cong\widetilde{\mathcal{E}}_{n,N}/\mathcal{G}^{\circ}$.

###### Proof.

$(a)$ Given a twisted lift $(V_{i})$ with monodromy $M$, we have for each $i$
matrices $A_{i}$, $B_{i}$, and $C_{i}$ in $\mathrm{GL}_{N}$ so that

$V_{i+3}=V_{i}A_{i}+V_{i+1}B_{i}+V_{i+2}C_{i}$ (5)

The fact that the lift is twisted guarantees that the $A_{i}$, $B_{i}$, and
$C_{i}$ are periodic. Placing $A_{i-1}$ on the left edge of each square face,
$B_{i}$ on the bottom, and $C_{i}$ on the right defines a point in
$\mathcal{E}_{n,N}$. This gives a map
$\mathcal{TL}_{n,N}\to\mathcal{E}_{n,N}$.

It is clear that two equivalent polygons have the same $A_{i}$, $B_{i}$,
$C_{i}$ coefficients for both the $\mathbb{R}^{*}$ and the $\mathrm{PGL}_{3N}$
actions. Therefore the proposed map
$\widetilde{\mathcal{TL}}_{n,N}/\mathrm{PGL}_{3N}\to\mathcal{E}_{n,N}$ is
well-defined. Now, supposing that $\overline{V}$ and $\overline{W}$ in
$\widetilde{\mathcal{TL}}_{n,N}$ have the same $A_{i}$, $B_{i}$, $C_{i}$
coefficients, we wish to show that there is some
$\overline{g}\in\mathrm{PGL}_{3N}$ so that
$\overline{W}=\overline{g}\overline{V}$. This will show that the map is
injective. We first define the “transfer matrices” $L_{i}$:

$L_{i}:=\left(\begin{array}[]{ccc}0&0&A_{i}\\\ \mathrm{Id}_{N}&0&B_{i}\\\
0&\mathrm{Id}_{N}&C_{i}\end{array}\right)$

Now, if we form the block-column matrices $\mathbf{V}_{i}$, whose columns are
the combined columns of $V_{i}$, $V_{i+1}$, and $V_{i+2}$, then Equation 5 can
be written succinctly as $\mathbf{V}_{i+1}=\mathbf{V}_{i}L_{i}$. By the
nondegeneracy assumption on lifts, $\mathrm{det}(\mathbf{V}_{i})\neq 0$ for
all $i$. For any two lifts $V$ and $W$ (even of two different polygons), there
are matrices $g_{i}\in\mathrm{GL}_{3N}$ so that $W_{i}=g_{i}V_{i}$ for all
$i$. We claim that if $V$ and $W$ have the same $A_{i}$,$B_{i}$,$C_{i}$
coefficients, then all $g_{i}$ are the same. To see this, note that $V$ and
$W$ will have the same transfer matrices $L_{i}$. That is,
$\mathbf{V}_{i+1}=\mathbf{V}_{i}L_{i}$ and
$\mathbf{W}_{i+1}=\mathbf{W}_{i}L_{i}$. Let $G_{i}$ be the block diagonal
matrices with $g_{i}$, $g_{i+1}$, $g_{i+2}$ as the diagonal blocks. Then the
equations $W_{i}=g_{i}V_{i}$ can be written as
$\mathbf{W}_{i}=G_{i}\mathbf{V}_{i}$. Then we have

$\mathbf{W}_{i+1}=\mathbf{W}_{i}L_{i}=G_{i}\mathbf{V}_{i}L_{i}=G_{i}\mathbf{V}_{i+1}$

But also, by definition of the $G_{i}$, we have
$\mathbf{W}_{i+1}=G_{i+1}\mathbf{V}_{i+1}$. So it must be that $g_{i+1}=g_{i}$
for all $i$. We have shown that for two lifts $V$ and $W$ in the same fiber of
the map $\mathcal{TL}_{n,N}\to\mathcal{E}_{n,N}$, there is some
$g\in\mathrm{GL}_{3N}$ so that $W=gV$. In terms of the elements of
$\widetilde{\mathcal{TL}}_{n,N}$, we can say that
$\overline{W}=\overline{g}\overline{V}$ for the corresponding element
$\overline{g}\in\mathrm{PGL}_{3N}$. This shows the map is injective.

To see that the map is surjective, let $A_{i}$, $B_{i}$, $C_{i}$ be an
arbitrary choice of weights in $\mathcal{E}_{n,N}$. We will construct a lift
which maps to this choice of weights. We will define $V_{1}$, $V_{2}$, and
$V_{3}$ so that the matrix $\mathbf{V}_{1}=(V_{1}V_{2}V_{3})$ is the identity
matrix $\mathrm{Id}_{3N}$. We then define the rest of the $V_{i}$ by Equation
5. Any other equivalent polygon is given by a different choice of
$\mathbf{V}_{1}$, which differs only by the $\mathrm{PGL}_{3N}$ action, giving
the same point in $\widetilde{\mathcal{TL}}_{n,N}/\mathrm{PGL}_{3N}$. This
gives an inverse to the map described above.

$(b)$ There is the obvious map
$\mathcal{L}_{n,N}\to\widetilde{\mathcal{E}}_{n,N}$ which sends a lifted
polygon $V$ to the $A_{i},B_{i},C_{i}$ defined above. If the lift is not
twisted, these are not necessarily periodic. Two lifted polygons $V$ and $W$
have the same $A_{i}$, $B_{i}$, $C_{i}$ coefficients if and only if
$W_{i}=gV_{i}$ for all $i$ and some fixed $g\in\mathrm{GL}_{3N}$, as described
above. The proof is the same as in part $(a)$. This shows that the induced map
$\widetilde{\mathcal{L}}_{n,N}/\mathrm{PGL}_{3N}\to\widetilde{\mathcal{E}}_{n,N}$
is bijective.

$(c)$ To see the equivalence of the gauge action of $\mathcal{G}^{\circ}$ and
changing lifts of a polygon, we will use the following visualization
technique. We will associate to the top edge of each square face the lifted
vertex $V_{i}$. The left edge will be labelled by $A_{i-3}$, the bottom with
$B_{i-2}$, and the right by $C_{i-2}$. Then the relation
$V_{i+3}=V_{i}B_{i}+V_{i+1}A_{i}+V_{i+2}C_{i}$ can be visualized in the
following way. We sum over all paths from $V_{j}$ to $V_{i}$ which do not pass
over another $V_{k}$ in between. We see that there are only 3 such paths,
which start at $V_{i}$, $V_{i+1}$, and $V_{i+2}$. The weights of the paths are
the matrix coefficients which we multiply by on the right.

The claim is that the action of the generators of $\mathcal{G}^{\circ}$ and
$\mathcal{G}^{\bullet}$ described above correspond precisely to changing to a
different lift of the same twisted polygon. So consider changing one lifted
vertex $V_{i}$ to $\widehat{V}_{i}=V_{i}G$ for some $G\in\mathrm{GL}_{N}$.
Each $V_{i}$ is involved in exactly four versions of Equation 5. Since
$V_{i}=\widehat{V}_{i}G^{-1}$, these four relations for the new lift become:

$\displaystyle V_{i+3}$
$\displaystyle=\widehat{V}_{i}(G^{-1}B_{i})+V_{i+1}A_{i}+V_{i+2}C_{i}$
$\displaystyle V_{i+2}$
$\displaystyle=V_{i-1}B_{i-1}+\widehat{V}_{i}(G^{-1}A_{i-1})+V_{i+1}C_{i-1}$
$\displaystyle V_{i+1}$
$\displaystyle=V_{i-2}B_{i-2}+V_{i-1}A_{i-2}+\widehat{V}_{i}(G^{-1}C_{i-2})$
$\displaystyle\widehat{V}_{i}$
$\displaystyle=V_{i-3}(B_{i-3}G)+V_{i-2}(A_{i-3}G)+V_{i-1}(C_{i-3}G)$

These are precisely how the weights change in Figure 10 by taking $X=G$.

$(d)$ Recall that $\mathcal{GP}_{n,N}$ is the quotient of $\mathcal{TL}_{n,N}$
by the actions of $\mathrm{GL}_{3N}$ and of changing the lift. By part $(a)$,
we identify $\mathcal{E}_{n,N}$ with the space of twisted lifts, up to the
actions of $\mathbb{R}^{*}$ and $\mathrm{PGL}_{3N}$. The $\mathbb{R}^{*}$
action is one way of changing the lift, and corresponds to a subgroup of
$\mathcal{G}^{\bullet}$. This subgroup is the diagonal embedding of the scalar
matrices $\\{\lambda\mathrm{Id}_{N}\\}$ into $\mathrm{GL}_{N}^{n}$. By part
$(c)$, the action of $\mathcal{G}^{\bullet}$ on $\mathcal{E}_{n,N}$ is
equivalent to chaging the lift. We may thus identify $\mathcal{GP}_{n,N}$ with
$\mathcal{E}_{n,N}/\mathcal{G}^{\bullet}$. Similarly, we may describe it also
as $\mathcal{GP}_{n,N}\cong\widetilde{\mathcal{E}}_{n,N}/\mathcal{G}^{\circ}$.
∎

We now use the gauge actoin to choose a convenient normalization, which we
will use to coordinatize the moduli space. The following is an analogue of
Proposition 1.

######  Proposition 6.

The lift $(V_{i})$ of $(p_{i})$ can be chosen so that $C_{i}=\mathrm{Id}_{N}$
for all $i$.

###### Proof.

We start by choosing a twisted lift $V_{i}$. This guarantees that the $A_{i}$,
$B_{i}$, and $C_{i}$ are periodic. Any other lift $\widehat{V}_{i}$ differs
from $V_{i}$ by a change of basis. That is, there are
$G_{i}\in\mathrm{GL}_{N}$ so that $\widehat{V}_{i}=V_{i}G_{i}$. Equivalently,
$V_{i}=\widehat{V}_{i}G_{i}^{-1}$. Substituting this into Equation 4, we
obtain

$\widehat{V}_{i+3}=\widehat{V}_{i}\left(G_{i}^{-1}A_{i}G_{i+3}\right)+\widehat{V}_{i+1}\left(G_{i+1}^{-1}B_{i}G_{i+3}\right)+\widehat{V}_{i+2}\left(G_{i+2}^{-1}C_{i}G_{i+3}\right)$

We want to prove that we can always ensure that
$G_{i+2}^{-1}C_{i}G_{i+3}=\mathrm{Id}$. In other words, we desire
$G_{i+3}=C_{i}^{-1}G_{i+2}$. Re-indexing gives $G_{i+1}=C_{i-2}^{-1}G_{i}$.
Now we may choose for instance $G_{0}=\mathrm{Id}$, and determine the rest by
this recurrence.

Equivalently, we may think of this in terms of the weights in
$\widetilde{\mathcal{E}}_{n,N}$ and the gauge action of $\mathcal{G}^{\circ}$.
Solving the recurrence above means first choosing a position (a square face)
in the network, and then applying elementary gauge transformations to the left
and right to cancel the weights on the right edges of each square face. ∎

The lift obtained in the above proposition is identified with a choice of
weights in $\widetilde{\mathcal{E}}_{n,N}$ for which the only non-trivial
weights are on the left and bottom edges of each square face. We call the
weights on the left edges $X_{i}$, and the ones on the bottom $Y_{i}$, so that
Equation 4 becomes

$V_{i+3}=V_{i}Y_{i}+V_{i+1}X_{i}+V_{i+2}$ (6)

######  Remark.

It is worth pointing out that although in the proof of the above proposition,
we started with a twisted lift, the resulting weights in
$\widetilde{\mathcal{E}}_{n,N}$ are _not_ periodic, meaning the lift
guaranteed by the proposition is not twisted. However, the $X_{i}$, $Y_{i}$
weights are _almost_ periodic in the following sense. Define the matrix which
is the cyclic product of the $C_{i}$’s:

$Z:=C_{n-1}C_{n}C_{1}\cdots C_{n-2}$

Then shifting the indices by $n$ corresponds to conjugation by $Z$. That is,

$\displaystyle X_{i+n}$ $\displaystyle=Z^{-1}\,X_{i}\,Z$ $\displaystyle
Y_{i+n}$ $\displaystyle=Z^{-1}\,Y_{i}\,Z$

So although the weights themselves are not periodic, their conjugacy classes
_are_ periodic.

As a consequence of this normalization, we have a new model for the moduli
space.

######  Theorem 4.

There is a bijection between the moduli space $\mathcal{GP}_{n,N}$ and
$\mathrm{GL}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$, the space of
$(2n+1)$-tuples of matrices up to simultaneous conjugation. Therefore we have
that $\dim\mathcal{GP}_{n,N}=2nN^{2}+1$.

###### Proof.

We would like to say that the collection of matrices $X_{i},Y_{i},Z$ for
$1\leq i\leq n$ defines a map $\mathcal{GP}_{n,N}\to\mathrm{GL}_{N}^{2n+1}$.
However, this is not quite well-defined. This depended on our choice of lift.
The gauge action does not always preserve this collection of $X_{i},Y_{i},Z$
matrices, but in general changes them all by simultaneous conjugation.
Therefore the induced map
$\mathcal{GP}_{n,N}\to\mathrm{GL}_{N}^{2n+1}/\mathrm{GL}_{N}$ is well-defined.

The map is injective, since the $X_{i}$’s, $Y_{i}$’s and $Z$ determine the
class of a polygon. To see this, choose any elements for $V_{1}$, $V_{2}$, and
$V_{3}$. We may for instance choose them so that
$\mathbf{V}_{1}=\mathrm{Id}_{3N}$. Then the rest of the $V_{i}$ are determined
by the relations $V_{i+3}=V_{i}Y_{i}+V_{i+1}X_{i}+V_{i+2}$. However, the
$X_{i}$’s, $Y_{i}$’s and $Z$ are only defined up to simultaneous conjugation.
Choosing some matrix $A$ and conjugating all the $X_{i}$, $Y_{i}$ and $Z$ by
$A$, however, changes the lift by $V_{i}\mapsto V_{i}A$ for all $i$. This just
changes to a different lift of the same polygon, and so the choice of
representatives of the $X_{i}$, $Y_{i}$, and $Z$ do not matter. Also, any
other choice of the initial $V_{1}$, $V_{2}$, $V_{3}$ differs by the left
action of $\mathrm{PGL}_{3N}$, and so gives an equivalent polygon. This gives
an inverse to the map
$\mathcal{GP}_{n,N}\to\mathrm{GL}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$. ∎

######  Remark.

The space $\mathrm{GL}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$ parameterizes
isomorphism classes of $N$-dimensional representations of the free associative
algebra on $2n+1$ generators. Later, we will think of the matrices $X_{i}$,
$Y_{i}$, and $Z$ as formal non-commutative variables (generators of a free
algebra), and we will identify our moduli space $\mathcal{GP}_{n,N}$ with the
moduli space of $N$-dimensional representations of this algebra.

With this description of the moduli space, we can now give some convenient
coordinates. There is the following classical theorem of Procesi:

######  Theorem 5.

[Pro76] Any polynomial invariant of an $n$-tuple of matrices
$(A_{1},\dots,A_{n})$, under the action of simultaneous conjugation, is a
polynomial in the functions $\mathrm{tr}(A_{i_{1}}\cdots A_{i_{k}})$, where
$A_{i_{1}}\cdots A_{i_{k}}$ ranges over all non-commutative monomials in the
matrices $A_{1},\dots,A_{n}$.

As mentioned above, we identify $\mathcal{GP}_{n,N}$ with the quotient
$\mathrm{GL}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$. We consider the
coordinate ring to be the subring of invariants of polynomial functions on
$\mathrm{GL}_{N}^{2n+1}$. Procesi’s theorem says that this coordinate ring is
generated by the traces of monomials in $2n+1$ matrices. As coordinates for
the moduli space $\mathcal{GP}_{n,N}$, we may therefore take any algebraically
independent family of traces of size $2nN^{2}+1$.

######  Example.

Consider the case of $\mathrm{GL}_{2}^{3}/\mathrm{Ad}\,\mathrm{GL}_{2}$, the
space of triples $(X,Y,Z)$ of $2\times 2$ matrices up to simultaneous
conjugation. A theorem of Sibiirski [Sib68] says that the ring of invariants
is minimally generated by the ten functions

$\begin{array}[]{ccc}\mathrm{tr}(X)&\mathrm{tr}(Y)&\mathrm{tr}(Z)\\\
\mathrm{tr}(X^{2})&\mathrm{tr}(Y^{2})&\mathrm{tr}(Z^{2})\\\
\mathrm{tr}(XY)&\mathrm{tr}(XZ)&\mathrm{tr}(YZ)\\\
&\mathrm{tr}(XYZ)&\end{array}$

However, the dimension of the space is $9$. We claim that the first $9$ (all
but $\mathrm{tr}(XYZ)$) can be taken as a system of local coordinates. The
tangent space to $\mathrm{GL}_{2}^{3}$ may be identified with
$\mathrm{Mat}_{2}^{3}$. The gradients of the ten functions above are given by

$\begin{array}[]{ccc}(\mathrm{Id}_{2},0,0)&(0,\mathrm{Id}_{2},0)&(0,0,\mathrm{Id}_{2})\\\
(X^{T},0,0)&(0,Y^{T},0)&(0,0,Z^{T})\\\
(Y^{T},X^{T},0)&(Z^{T},0,X^{T})&(0,Z^{T},Y^{T})\\\
&((YZ)^{T},(ZX)^{T},(XY)^{T})&\end{array}$

The first $9$ are clearly linearly independent for generic choices of $X$,
$Y$, and $Z$. The fact that the last is a linear combination of the first $9$
is equivalent to the statement that there are coefficients $c_{1},\dots,c_{9}$
for which

$\displaystyle ZY$ $\displaystyle=c_{1}\mathrm{Id}_{2}+c_{4}X+c_{7}Y+c_{8}Z$
$\displaystyle XZ$ $\displaystyle=c_{2}\mathrm{Id}_{2}+c_{7}X+c_{5}Y+c_{9}Z$
$\displaystyle YX$ $\displaystyle=c_{3}\mathrm{Id}_{2}+c_{8}X+c_{9}Y+c_{6}Z$

Of course, if $X$, $Y$, and $Z$ are linearly independent, then the set
$\\{\mathrm{Id}_{2},X,Y,Z\\}$ is a linear basis of the set of $2\times 2$
matrices, and we can expand the products $YX$, $ZY$, and $XZ$ in terms of this
basis. The claim above is that when we do this, three pairs of coefficients
will be equal — the $X$-coefficient of $YX$ is equal to the $Z$-coefficient of
$ZY$, etc. The fact that these pairs of coefficients will be equal follows
from the Jacobi identity:

$[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0$

###  4.4 Expression for the Pentagram Map

In this section, we see how the pentagram map transforms the $X_{i}$, $Y_{i}$,
and $Z$ matrices. The map is only defined generically, since we need to assume
certain matrices are invertible.

######  Proposition 7.

The pentagram map transforms the $X_{i}$, $Y_{i}$, and $Z$ by

$\displaystyle X_{i}$
$\displaystyle\mapsto(X_{i}+Y_{i+1})^{-1}X_{i}(X_{i+2}+Y_{i+3})$
$\displaystyle Y_{i}$
$\displaystyle\mapsto(X_{i}+Y_{i+1})^{-1}Y_{i+1}(X_{i+2}+Y_{i+3})$
$\displaystyle Z$ $\displaystyle\mapsto Z$

###### Proof.

The proof is essentially the same as that of Proposition 2.

By re-arranging Equation 6, we see that for a lift of the image, we may take
for $T(V_{i})$ the subspace spanned by the columns of

$V_{i+3}-V_{i+1}X_{i}=V_{i}Y_{i}+V_{i+2}$ (7)

Since the lift of the image under the pentagram map is again a twisted
polygon, it satisfies Equation 4:

$T(V_{i+3})=T(V_{i})A_{i}+T(V_{i+1})B_{i}+T(V_{i+2})C_{i}$

We substitute the expressions from Equation 7 into the equation above, using
either the left-hand or right-hand side, according to the same convention used
in the proof of Proposition 2. Doing so, we obtain

$V_{i+5}+V_{i+3}Y_{i+3}=V_{i+1}\left(Y_{i+1}B_{i}-X_{i}A_{i}\right)+V_{i+3}\left(A_{i}+B_{i}-X_{i+2}C_{i}\right)+V_{i+5}C_{i}$

Comparing coefficients of each $V_{i}$, we conclude that $C_{i}=\mathrm{Id}$,
and that

$\displaystyle X_{i}A_{i}$ $\displaystyle=Y_{i+1}B_{i}$ $\displaystyle
Y_{i+3}+X_{i+2}$ $\displaystyle=A_{i}+B_{i}$

Assuming that each $X_{i}+Y_{i+1}$ is invertible, these equations can be
solved for $A_{i}$ and $B_{i}$ to give the desired result. To see that $Z$ is
unchanged, it is a simple check that a shift of indices by $n$ still
corresponds to conjugation by the same $Z$. ∎

Notice that the expressions in the proposition are a non-commutative version
of the expressions obtained in the classical case, in Proposition 2. For the
remainder of the paper, we attempt to generalize much of what was done in the
previous expository sections to noncommutative variables, using elements of a
noncommutative ring in place of the commutative coordinates on
$\mathscr{E}_{Q}$, and we introduce a noncommutative Poisson structure which
mimics the classical counterpart in [GSTV16].

##  5 Non-Commutative Poisson Structures

###  5.1 Double Brackets

In this section, we discuss the basic constructions and definitions introduced
by Van den Bergh [Ber08]. Throughout, $\mathbb{K}$ is a field, and $A$ is an
associative $\mathbb{K}$-algebra. Unadorned tensor products are assumed to be
over $\mathbb{K}$. We will assume the characteristic of $\mathbb{K}$ is zero,
but we do not necessarily assume it is algebraically closed.

If $A$ is an associative algebra, then $A^{\otimes n}$ is an $A$-bimodule in
the obvious way:

$x\cdot(a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n-1}\otimes a_{n})\cdot
y=xa_{1}\otimes a_{2}\otimes\cdots\otimes a_{n-1}\otimes a_{n}y$

We refer to this as the _outer bimodule structure_ on $A^{\otimes n}$.

######  Definition 12.

A _double bracket_ on $A$ is a $\mathbb{K}$-bilinear map

$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}\colon A\times A\to A\otimes
A$

which satisfies:

  1. $(1)$

$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is a derivation in the
second argument with respect to the outer bimodule structure:

$\left\\{\\!\\!\left\\{a,bc\right\\}\\!\\!\right\\}=\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\}(1\otimes
c)+(b\otimes 1)\left\\{\\!\\!\left\\{a,c\right\\}\\!\\!\right\\}$

  2. $(2)$

$\left\\{\\!\\!\left\\{b,a\right\\}\\!\\!\right\\}=-\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\}^{\tau}$,
where $(x\otimes y)^{\tau}:=y\otimes x$

Note that these properties imply that
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ also satisfies

$\left\\{\\!\\!\left\\{ab,c\right\\}\\!\\!\right\\}=(1\otimes
a)\left\\{\\!\\!\left\\{b,c\right\\}\\!\\!\right\\}+\left\\{\\!\\!\left\\{a,c\right\\}\\!\\!\right\\}(b\otimes
1)$

######  Definition 13.

A double bracket $\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is called
a _double Poisson bracket_ if it additionally satisfies a version of the
Jacobi identity:

$0=\sum_{k=0}^{2}\sigma^{k}\circ(\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}\otimes\mathrm{Id})\circ(\mathrm{Id}\otimes\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\})\circ\sigma^{-k}$

The right-hand side is an operator on $A\otimes A\otimes A$, and $\sigma$ is
the permutation operator which sends $x\otimes y\otimes z$ to $z\otimes
x\otimes y$.

For an algebra $A$, let $\mu\colon A\otimes A\to A$ be the multilpication map.
If $A$ has a double bracket
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$, then we define another
operation $\\{-,-\\}\colon A\times A\to A$ by composing with $\mu$:

$\\{a,b\\}:=\mu(\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\})$

######  Proposition 8.

[Ber08] Suppose $\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is a
double bracket on $A$ (not necessarily a double Poisson bracket). The induced
bracket $\\{-,-\\}$ has the following properties

  1. 1.

$\\{a,bc\\}=\\{a,b\\}c+b\\{a,c\\}$ (Leibniz in the $2^{\mathrm{nd}}$ argument)

  2. 2.

$\\{ab,c\\}=\\{ba,c\\}$ (cyclic in the $1^{\mathrm{st}}$ argument)

  3. 3.

$\\{a,b\\}\equiv-\\{b,a\\}\;\mathrm{mod}\;[A,A]$ (skew mod commutators)

###### Proof.

Suppose that
$\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\}=\sum_{i}\omega_{i}\otimes\bar{\omega}_{i}$
and
$\left\\{\\!\\!\left\\{a,c\right\\}\\!\\!\right\\}=\sum_{i}\eta_{i}\otimes\bar{\eta}_{i}$.
Then using the Leibniz rule for
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ in the second argument,
and composing with multiplication, we get

$\\{a,bc\\}=\sum_{i}\omega\bar{\omega}_{i}c+\sum_{i}b\eta_{i}\bar{\eta}_{i}=\\{a,b\\}c+b\\{a,c\\}$

This proves the first identity. The third identity follows from the fact that

$\\{a,b\\}+\\{b,a\\}=\sum_{i}[\omega_{i},\bar{\omega_{i}}]$

For the second identity, suppose that
$\left\\{\\!\\!\left\\{b,c\right\\}\\!\\!\right\\}=\sum_{i}\zeta_{i}\otimes\bar{\zeta}_{i}$.
Then on the one hand, we have

$\left\\{\\!\\!\left\\{ab,c\right\\}\\!\\!\right\\}=\sum_{i}\eta_{i}b\otimes\bar{\eta}_{i}+\sum_{i}\zeta_{i}\otimes
a\bar{\zeta}_{i}$

On the other hand, we have

$\left\\{\\!\\!\left\\{ba,c\right\\}\\!\\!\right\\}=\sum_{i}\eta_{i}\otimes
b\bar{\eta}_{i}+\sum_{i}\zeta_{i}a\otimes\bar{\zeta}_{i}$

Obviously both will be the same after composing with the multiplication map. ∎

######  Definition 14.

For an associative algebra $A$, define the _cyclic space_ , denoted
$A^{\natural}$, to be the vector space quotient $A/[A,A]$, by the linear span
of all commutators.

Note that since $[A,A]$ is not in general an ideal, this is not necessarily an
algebra. For a vector subspace $V\leq A$, we let $V^{\natural}$ denote its
image under the projection. Similarly, for an element $a\in A$, we use the
notation $a^{\natural}$ to denote its image under the projection. Note that
conjugate elements are equivalent, since $xyx^{-1}-y=[xy,x^{-1}]$. From this
it follows that in $A^{\natural}$, monomials are equivalent up to cyclic
permutation, since for any $x_{1},\dots,x_{n}\in A$, we have $x_{n}x_{1}\cdots
x_{n-1}=x_{n}(x_{1}\cdots x_{n})x_{n}^{-1}$.

The following follows easily from the properties above.

######  Proposition 9.

[Ber08] Suppose $\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is a
double bracket on $A$. Then there is a well-defined bilinear skew-symmetric
map $\left<-,-\right>\colon A^{\natural}\times A^{\natural}\to A^{\natural}$,
given by

$\left<a^{\natural},b^{\natural}\right>:=\\{a,b\\}^{\natural}$

The next few results indicate the extra structure which $\\{-,-\\}$ and
$\left<-,-\right>$ inherit if
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ additionally satisfies the
double Jacobi identity.

######  Proposition 10.

[Ber08] Suppose that $\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is a
double Poisson bracket. Then

  1. 1.

$\\{-,-\\}$ is a “Loday bracket” on $A$. That is, it satisfies a version of
the Jacobi identity:

$\\{a,\\{b,c\\}\\}=\\{\\{a,b\\},c\\}+\\{b,\\{a,c\\}\\}$

  2. 2.

$\left<-,-\right>$ is a Lie bracket on $A^{\natural}$.

A more general notion is given by what William Crawley-Boevey calls an
$H_{0}$-Poisson structure:

######  Definition 15.

[CB11] An _$H_{0}$ -Poisson strcuture_ on an associative algebra $A$ is a Lie
bracket $[-,-]$ on $A^{\natural}$ such that each $[a,-]$ is induced by a
derivation of $A$.

In particular, the induced bracket $\left<-,-\right>$ of a double Poisson
bracket $\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}$ is an
$H_{0}$-Poisson structure by Proposition 8 and Proposition 10. However, there
are $H_{0}$-Poisson structures which do not arise from double Poisson
brackets. Later, we will define a double bracket which is _not_ double
Poisson, but still has the property that the induced bracket
$\left<-,-\right>$ on $A^{\natural}$ is a Lie bracket. This will be an example
of an $H_{0}$-Poisson structure which comes from a double bracket, but not a
double Poisson bracket.

There is a theorem of Crawley-Boevey which says that $H_{0}$-Poisson
structures on $A$ induce usual “commutative” Poisson structures on the
representation space
$\mathrm{Rep}_{N}(A):=\mathrm{Hom}(A,\mathrm{Mat}_{N})/\mathrm{Ad}\,\mathrm{GL}_{N}$.

######  Theorem 6.

[CB11] If $A$ is an associative algebra, and the bracket $\left<-,-\right>$ on
$A^{\natural}$ defines an $H_{0}$-Poisson structure, then there is a unique
Poisson structure on $\mathrm{Rep}_{N}(A)$ such that the trace map is a Lie
algebra homomorphism. That is,

$\\{\mathrm{tr}(a),\mathrm{tr}(b)\\}=\mathrm{tr}\left<a^{\natural},b^{\natural}\right>$

With this theorem in mind, we now describe the plan of the remainder of the
paper. Recall that by Theorem 4, our moduli space $\mathcal{GP}_{n,N}$ is
identified with $\mathrm{GL}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$. We will
consider our algebra $A$ to be a free algebra with $2n+1$ generators,
corresponding to the $X_{i}$, $Y_{i}$, and $Z$. Since any homomorphism with
domain $A$ is determined by the images of the generators, we see that
$\mathrm{Rep}_{N}(A)\cong\mathrm{Mat}_{N}^{2n+1}/\mathrm{Ad}\,\mathrm{GL}_{N}$.
We may therefore interpret $\mathrm{tr}(X_{i})$, $\mathrm{tr}(Y_{i})$, and
$\mathrm{tr}(Z)$ as functions on $\mathcal{GP}_{n,N}$. We will spend the next
couple sections defining non-commutative Poisson structures on this free
algebra. The theorem above will then induce a Poisson structure on our moduli
space.

##  6 Non-Commutative Networks and Double Brackets

###  6.1 Weighted Directed Fat Graphs (Revisited)

Let $Q=(Q_{0},Q_{1})$ be a quiver as in section 2, which is embedded in an
annulus, with all sources on the inner boundary circle, all sinks on the outer
boundary circle, and all internal vertices trivalent. Define the algebra
$A=A_{Q}=\mathbb{Q}\left<\alpha~{}|~{}\alpha\in Q_{1}\right>$ to be the free
associative algebra generated by the arrows of the quiver. Assuming, as
before, that $Q$ is acyclic, we can define the boundary measurement matrix
$\mathscr{B}_{Q}(\lambda)=(b_{ij}(\lambda))$, where $b_{ij}(\lambda)$ is the
sum of the weights of all paths from source $i$ to sink $j$. Here, the weight
of a path is the product of the weights, in order from left to right. We again
assign an indeterminate $\lambda$ to the cut, and consider the boundary
measurements to be Laurent polynomials in $A[\lambda^{\pm}]$, where $\lambda$
commutes with elements in $A$.

We can define _non-commutative gauge transformations_. For any Laurent
monomial $t$ in the generators of $A$, we can multiply all incoming arrows at
one vertex by $t$ on the right, and multiply all outgoing arrows at that
vertex by $t^{-1}$ on the left. This will obviously not change the boundary
measurements.

We can also define non-commutative versions of the the Postnikov moves, which
preserve the boundary measurements. They are pictured in Figure 11. As in the
commutative case, $\Delta:=b+adc$. We must take special care at this point to
say what we mean by expressions such as $(b+acd)^{-1}$. We do so now.

Figure 11: non-commutative Postnikov moves( I
)$a$$b$$c$$d$$e_{1}$$e_{2}$$e_{3}$$e_{4}$$dc\Delta^{-1}$$\Delta$$\Delta^{-1}ad$$dc\Delta^{-1}bc^{-1}$$e_{1}$$e_{2}$$e_{3}$$e_{4}$(
II )$a$$b$$c$$x$$y$$a$$b$$c$$b^{-1}x$$by$( III
)$a$$b$$c$$x$$y$$a$$b$$c$$xb$$yb^{-1}$

###  6.2 The Free Skew Field and Mal’cev Neumann Series

In order to make sense of expressions of the form $(x+y)^{-1}$, we need an
appropriate notion of “noncommutative rational functions”. This will be the
_free skew field_ , which we will define below. Then, we will discuss how the
free skew field can be identified with a certain subset of noncommutative
formal power series.

For a set $X=(x_{1},\dots,x_{n})$ of formal noncommuting variables, the _free
skew field_ or _universal field of fractions_ , which we will denote by
$\mathcal{F}_{\mathbb{Q}}(X)$, or just $\mathcal{F}(X)$, is a division algebra
over $\mathbb{Q}$ characterized by the following universal property:

There is an injective homomorphism
$i\colon\mathbb{Q}\left<X\right>\hookrightarrow\mathcal{F}_{\mathbb{Q}}(X)$
from the free associative algebra on $X$ into $\mathcal{F}_{\mathbb{Q}}(X)$
such that for any homomorphism $\varphi\colon\mathbb{Q}\left<X\right>\to D$
into a division ring $D$, there is a unique subring
$\mathbb{Q}\left<X\right>\subset\mathcal{R}_{\varphi}\subset\mathcal{F}_{\mathbb{Q}}(X)$
and a homomorphism $\psi\colon\mathcal{R}\to D$ with $\varphi=\psi\circ i$,
and such that if $0\neq a\in\mathcal{R}$ and $\psi(a)\neq 0$, then
$a^{-1}\in\mathcal{R}$.

The explicit construction of $\mathcal{F}(X)$ is a bit complicated (see
[GGRW02] or [Coh77] for details), but informally it consists of noncommutative
rational expressions in the variables $x_{1},\dots,x_{n}$, under some suitable
notion of equivalence. For example, $\mathcal{F}(X)$ contains expressions such
as $(1+x)^{-1}$ and $w(x+y)^{-1}z$.

Another way to embed the free algebra into a division ring is by so-called
_Mal’cev Neumann series_ (also called Hahn-Mal’cev-Neumann series). Suppose
that the free group generated by $X$ is given some order relation compatible
with multiplication. This means that if $f\leq g$, then $hf\leq hg$ for any
$h$. Given an ordering of the free group, the ring of Mal’cev Nemuann series
is the subset of all formal series over the free group which have well-ordered
support. This is a division ring which contains $\mathbb{Q}\left<X\right>$.
Jaques Lewin proved [Lew74] that the free skew field $\mathcal{F}(X)$ is
isomorphic to the subfield of the ring of Mal’cev Neumann series generated by
the variables $x_{1},\dots,x_{n}$. In the next section, we show an example of
expanding an element of the free skew field as a series.

Since $A=A_{Q}=\mathbb{Q}\left<Q_{1}\right>$ is a free algebra, it is a
subalgebra of $\mathcal{F}(Q_{1})$. So, when we consider the noncommutative
Postnikov “square move”, we interpret the expressions $(b+adc)^{-1}$ as
elements of the free skew field $\mathcal{F}(Q_{1})$. If we choose an order
relation compatible with multiplication, then we may also view these
expressions as noncommutative Mal’cev Neumann series. Thus, from now on, given
a quiver, we will work more generally with the skew field $\mathcal{F}(Q_{1})$
rather than the free algebra $\mathbb{Q}\left<Q_{1}\right>$.

###  6.3 An Example of a Series Expansion

Consider the free skew field on two variables $x$ and $y$. We will write the
element $(x+y)^{-1}$ as a noncommutative power series. As mentioned above, we
need to put an order on the free group generated by $x$ and $y$ to determine
the ring of Mal’cev Neumann series. We will use the order induced by the
_Magnus embedding_ of the free group into the ring $\mathbb{Z}[[x,y]]$ of
formal power series in non-commuting variables $x$ and $y$ [MKS76].

More specifically, we embed the free group into $\mathbb{Z}[[x,y]]$ by the map
$\alpha\mapsto 1+\alpha$ and $\alpha^{-1}\mapsto\sum_{i\geq
0}(-1)^{i}\alpha^{i}$ where $\alpha$ is either $x$ or $y$. This embeds the
free group as a subgroup of the multiplicative group of all power series with
constant term $1$. Choosing an order of the variables, say $x<y$, this
determines a graded lexicographic order on monomials/words in
$\mathbb{Z}[[x,y]]$. Then we define an order on $\mathbb{Z}[[x,y]]$ where
$f<g$ if the coefficient of $f$ at the first place they disagree is smaller.
This induces an order on the free group. We then define the ring of Mal’cev
Neumann series to be those series which have well-ordered support with respect
to this ordering of the free group.

There are a couple different ways we could try to expand $(x+y)^{-1}$ as a
series. We could factor out either of the variables, and then expand as a
geometric series. For example, we could first factor out $x$ to get

$(x+y)^{-1}=(x(1+x^{-1}y))^{-1}=(1+x^{-1}y)^{-1}x^{-1}$

Then we expand $(1+x^{-1}y)^{-1}$ as a geometric series to get

$(x+y)^{-1}=(1-x^{-1}y+x^{-1}yx^{-1}y+\cdots)x^{-1}=x^{-1}-x^{-1}yx^{-1}+x^{-1}yx^{-1}yx^{-1}+\cdots$

To see if the support is well-ordered, we need to go through the above
construction. Under the Magnus embedding, we have

$\displaystyle x^{-1}$ $\displaystyle\mapsto 1-x+x^{2}-x^{3}+\cdots$
$\displaystyle x^{-1}yx^{-1}$ $\displaystyle\mapsto 1-2x+y+3x^{2}-xy-
yx+\cdots$ $\displaystyle x^{-1}yx^{-1}yx^{-1}$ $\displaystyle\mapsto
1-3x+2y+6x^{2}+y^{2}-3xy-3yx+\cdots$ $\displaystyle\vdots$
$\displaystyle(x^{-1}y)^{n}x^{-1}$ $\displaystyle\mapsto 1-(n+1)x+ny+\cdots$

The sequence of terms in our series expansion for $(x+y)^{-1}$ all differ at
the coefficient for $x$. We see that the coefficients are decreasing in the
sequence $-1,-2,-3,\cdots,-(n+1),\cdots$. There is thus no lowest term in this
sequence, and so it is _not_ well-ordered. This expansion is then _not_ a
Mal’cev Neumann series for our chosen ordering.

If, however, we factor out $y$ instead of $x$, things will work out. Then we
get

$(x+y)^{-1}=y^{-1}(1+xy^{-1})^{-1}=y^{-1}-y^{-1}xy^{-1}+y^{-1}xy^{-1}xy^{-1}+\cdots$

Then the general term, under the Magnus embedding, will be

$y^{-1}(xy^{-1})^{n}\mapsto 1+nx-(n+1)y+\cdots$

Now the sequence of $x$ coefficients is the increasing sequence
$0,1,2,3,\cdots,n,\cdots$. Therefore the first term $y^{-1}$ is the lowest in
the sequence, and so the support is well-ordered. This expansion is thus a
Mal’cev Neumann series.

###  6.4 Double Brackets Associated to a Quiver

We now define a family of double brackets on the skew field
$\mathcal{F}_{Q}:=\mathcal{F}(Q_{1})$ which generalize the Poisson structures
on the space of edge weights $\mathcal{E}_{Q}$ described in section 2. We
again define it locally, and then describe the concatenation procedure by
which they can be glued together. We start with the two local pictures of
white and black vertices:

$x$$z$$y$$a$$b$$c$

Let $\mathcal{F}_{\circ}=\mathcal{F}(x,y,z)$ be the free skew field (over
$\mathbb{Q}$) on 3 generators. We will think of the variables $x,y,z$ as
representing the edge weights on the white vertex picture above. Similarly,
let $\mathcal{F}_{\bullet}=\mathcal{F}(a,b,c)$ correspond to the black vertex.
Of course they are isomorphic as associative $\mathbb{Q}$-algebras, but we
will define different brackets on them. Choose any
$w_{1},w_{2},w_{3}\in\mathbb{Q}$, and define a double bracket
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}_{\circ}$ on
$\mathcal{F}_{\circ}$ by

$\displaystyle\left\\{\\!\\!\left\\{x,y\right\\}\\!\\!\right\\}_{\circ}$
$\displaystyle=w_{1}(1\otimes xy)$
$\displaystyle\left\\{\\!\\!\left\\{x,z\right\\}\\!\\!\right\\}_{\circ}$
$\displaystyle=w_{2}(1\otimes xz)$
$\displaystyle\left\\{\\!\\!\left\\{y,z\right\\}\\!\\!\right\\}_{\circ}$
$\displaystyle=w_{3}(y\otimes z)$

Similarly, for scalars $k_{1},k_{2},k_{3}\in\mathbb{Q}$, define
$\left\\{\\!\\!\left\\{-,-\right\\}\\!\\!\right\\}_{\bullet}$ on
$\mathcal{F}_{\bullet}$ by

$\displaystyle\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\}_{\bullet}$
$\displaystyle=k_{1}(ba\otimes 1)$
$\displaystyle\left\\{\\!\\!\left\\{a,c\right\\}\\!\\!\right\\}_{\bullet}$
$\displaystyle=k_{2}(ca\otimes 1)$
$\displaystyle\left\\{\\!\\!\left\\{b,c\right\\}\\!\\!\right\\}_{\bullet}$
$\displaystyle=k_{3}(c\otimes b)$

Also define $\mathcal{F}_{\partial}=\mathcal{F}(x)=\mathbb{Q}(x)$ to be the
field of rational functions in one variable, corresponding to a univalent
boundary vertex, with trivial double bracket.

These double brackets are again “compatible” with concatenating/glueing in a
similar sense as in the commutative case. Let $H_{Q}$ be the free skew field
generated by the half-edges of $Q$, denoted again by $\alpha_{s}$ and
$\alpha_{t}$ for the source and target ends of $\alpha\in Q_{1}$. Then $H_{Q}$
is the free product of the algebras defined above:

$H_{Q}\cong\left(\mathop{\hbox{\scalebox{1.7}{$\ast$}}}_{\circ\in
V_{\circ}}\mathcal{F}_{\circ}\right)\,\ast\,\left(\mathop{\hbox{\scalebox{1.7}{$\ast$}}}_{\bullet\in
V_{\bullet}}\mathcal{F}_{\bullet}\right)\,\ast\,\left(\mathop{\hbox{\scalebox{1.7}{$\ast$}}}_{\star\in
V_{\partial}}\mathcal{F}_{\partial}\right)$

As is mentioned in [Ber08], for algebras $A$ and $B$ with double brackets,
there is a uniquely defined double bracket on the free product $A\ast B$ such
that $\left\\{\\!\\!\left\\{a,b\right\\}\\!\\!\right\\}=0$ for $a\in A$ and
$b\in B$. Since the free product is the coproduct in the category of
associative algebras (analogously the tensor product is the coproduct for
commutative algebras), this is analogous to the product bracket on
$\mathscr{H}_{Q}$ in the commutative case.

Again we define a glueing map $g^{*}\colon\mathcal{F}_{Q}\to H_{Q}$ by
$g^{*}(\alpha)=\alpha_{s}\alpha_{t}$ for $\alpha\in Q_{1}$. Since the algebras
are non-commutative, this is not actually the pull-back of a
geometric/topological map, but we keep the notation for the sake of analogy.
We have the following result, which mimics the commutative case:

######  Proposition 11.

There is a unique double bracket on $\mathcal{F}_{Q}$ so that the glueing
homomorphism $g^{*}\colon\mathcal{F}_{Q}\to H_{Q}$ satisfies

$\left\\{\\!\\!\left\\{g^{*}(\alpha),g^{*}(\beta)\right\\}\\!\\!\right\\}_{H_{Q}}=(g^{*}\otimes
g^{*})\left(\left\\{\\!\\!\left\\{\alpha,\beta\right\\}\\!\\!\right\\}_{\mathcal{F}_{Q}}\right)$

###### Proof.

As in the commutative case, there are many cases to consider, but all of them
are similar and are simple calculations. We show one as an example to
illustrate the idea. We again look at Figure 4, and try to define
$\left\\{\\!\\!\left\\{w,y\right\\}\\!\\!\right\\}_{\mathcal{F}_{Q}}$. If it
is to satisfy the desired property, we must have

$\displaystyle(g^{*}\otimes
g^{*})(\left\\{\\!\\!\left\\{w,y\right\\}\\!\\!\right\\}_{\mathcal{F}_{Q}})$
$\displaystyle=\left\\{\\!\\!\left\\{g^{*}(w),g^{*}(y)\right\\}\\!\\!\right\\}_{H_{Q}}$
$\displaystyle=\left\\{\\!\\!\left\\{ax,y_{s}y_{t}\right\\}\\!\\!\right\\}_{H_{Q}}$
$\displaystyle=(1\otimes
a)\left\\{\\!\\!\left\\{x,y_{s}\right\\}\\!\\!\right\\}_{H_{Q}}(1\otimes
y_{t})$ $\displaystyle=w_{1}(1\otimes a)(1\otimes xy_{s})(1\otimes y_{t})$
$\displaystyle=w_{1}(1\otimes axy_{s}y_{t})$ $\displaystyle=w_{1}(1\otimes
g^{*}(w)g^{*}(y))$ $\displaystyle=(g^{*}\otimes g^{*})(w_{1}(1\otimes wy))$

This suggests that
$\left\\{\\!\\!\left\\{w,y\right\\}\\!\\!\right\\}_{\mathcal{F}_{Q}}$ must be
defined to be $w_{1}(1\otimes wy)$. As mentioned above, all other cases are
similar. Just as in the commutative case, the double bracket on
$\mathcal{F}_{Q}$ is given by the same expressions as in $H_{Q}$, treating
edges as the corresponding half-edges which meet at a common vertex. ∎

We may also consider the doubled quiver $\overline{Q}$, as before. We then
associate to each opposite arrow $\alpha^{*}$ the element $\alpha^{-1}$ in the
free skew field. The double bracket defined above on generators extends in a
unique way to the free skew field by the formulas

$\displaystyle\left\\{\\!\\!\left\\{\beta,\alpha^{-1}\right\\}\\!\\!\right\\}$
$\displaystyle=-(\alpha^{-1}\otimes
1)\left\\{\\!\\!\left\\{\beta,\alpha\right\\}\\!\\!\right\\}(1\otimes\alpha^{-1})$
$\displaystyle\left\\{\\!\\!\left\\{\alpha^{-1},\beta\right\\}\\!\\!\right\\}$
$\displaystyle=-(1\otimes\alpha^{-1})\left\\{\\!\\!\left\\{\alpha,\beta\right\\}\\!\\!\right\\}(\alpha^{-1}\otimes
1)$

We may then interpret any path in $\overline{Q}$ as a non-commutative Laurent
monomial in $\mathcal{F}_{Q}$.

###  6.5 A Formula for the Bracket

We will primarily be concerned with paths that are closed loops in
$\overline{Q}$. Let $\mathscr{L}\subset\mathcal{F}_{Q}$ be the vector subspace
spanned by all monomials which represent closed loops, and let
$f,g\in\mathscr{L}$ be two elements.

As before, we let $f\cap g$ denote the set of all maximal common subpaths. We
will prove that, as in the commutative case, we only need to consider the ends
of common subpaths in order to compute $\left<f,g\right>$. First we give a
result about the local structure of the double bracket.

######  Lemma 2.

Let $x$ and $y$ be edges in $\overline{Q}$. That is, $x$ and $y$ can either be
arrows in the quiver, or “reverse” arrows. Then

($a$) If $y$ follows $x$ (i.e. $s(y)=t(x)$), then
$\left\\{\\!\\!\left\\{x,y\right\\}\\!\\!\right\\}=\lambda(1\otimes xy)$ for
some $\lambda\in\mathbb{Q}$. ($b$) If $x$ and $y$ have the same source, then
$\left\\{\\!\\!\left\\{x,y\right\\}\\!\\!\right\\}=\lambda(x\otimes y)$ for
some $\lambda\in\mathbb{Q}$. ($c$) If $x$ and $y$ have the same target, then
$\left\\{\\!\\!\left\\{x,y\right\\}\\!\\!\right\\}=\lambda(y\otimes x)$ for
some $\lambda\in\mathbb{Q}$.

###### Proof.

The pictures below show the possibilities at a white vertex. The red are for
part $(a)$, the blue for part $(b)$, and green for part $(c)$. Note that there
are 3 additional possibilities for part $(a)$, given by the reversal/inverse
of the pairs shown in red.

As in the picture from the previous section, call the unique incoming arrow
$x$, and the outgoing arrows $y$ and $z$, in counter-clockwise order from $x$.
Then the three pairs of edges in the picture above for $(a)$ give

$\displaystyle\left\\{\\!\\!\left\\{x,z\right\\}\\!\\!\right\\}$
$\displaystyle=w_{2}(1\otimes xz)$
$\displaystyle\left\\{\\!\\!\left\\{z^{-1},y\right\\}\\!\\!\right\\}$
$\displaystyle=w_{3}(1\otimes z^{-1})(z\otimes y)(z^{-1}\otimes
1)=w_{3}(1\otimes z^{-1}y)$
$\displaystyle\left\\{\\!\\!\left\\{y^{-1},x^{-1}\right\\}\\!\\!\right\\}$
$\displaystyle=-w_{1}(x^{-1}\otimes y^{-1})(xy\otimes 1)(y^{-1}\otimes
x^{-1})=-w_{1}(1\otimes y^{-1}x^{-1})$

For part $(b)$, the pictured pairs give

$\displaystyle\left\\{\\!\\!\left\\{y,z\right\\}\\!\\!\right\\}$
$\displaystyle=w_{3}(y\otimes z)$
$\displaystyle\left\\{\\!\\!\left\\{x^{-1},y\right\\}\\!\\!\right\\}$
$\displaystyle=-w_{1}(1\otimes x^{-1})(1\otimes xy)(x^{-1}\otimes
1)=-w_{1}(x^{-1}\otimes y)$
$\displaystyle\left\\{\\!\\!\left\\{x^{-1},z\right\\}\\!\\!\right\\}$
$\displaystyle=-w_{2}(1\otimes x^{-1})(1\otimes xz)(x^{-1}\otimes
1)=-w_{2}(x^{-1}\otimes z)$

For part $(c)$, the pairs pictured above give

$\displaystyle\left\\{\\!\\!\left\\{x,y^{-1}\right\\}\\!\\!\right\\}$
$\displaystyle=-w_{1}(y^{-1}\otimes 1)(1\otimes xy)(1\otimes
y^{-1})=-w_{1}(y^{-1}\otimes x)$
$\displaystyle\left\\{\\!\\!\left\\{x,z^{-1}\right\\}\\!\\!\right\\}$
$\displaystyle=-w_{2}(z^{-1}\otimes 1)(1\otimes xz)(1\otimes
z^{-1})=-w_{2}(z^{-1}\otimes x)$
$\displaystyle\left\\{\\!\\!\left\\{y^{-1},z^{-1}\right\\}\\!\\!\right\\}$
$\displaystyle=w_{3}(z^{-1}\otimes y^{-1})(y\otimes z)(y^{-1}\otimes
z^{-1})=w_{3}(z^{-1}\otimes y^{-1})$

All other possibilities at a black vertex are verified by similar
calculations. ∎

######  Lemma 3.

Let $f,g\in\mathscr{L}$. The induced bracket $\left<f,g\right>$ depends only
on the endpoints of maximal common subpaths of $f$ and $g$.

###### Proof.

The result can be stated more technically as follows. Suppose that $p$ is an
edge in $f$ and $q$ is an edge in $g$. Expanding
$\left\\{\\!\\!\left\\{f,g\right\\}\\!\\!\right\\}$ with the Leibniz rule,
there will be a term involving
$\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}$. The result says that
unless $p$ and $q$ are incident to a common vertex which is an endpoint of a
maximal common subpath, then this term is either zero, or it cancels with
another term in the Leibniz expansion.

First of all, if $p$ and $q$ are not incident to the same vertex, then
$\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}=0$. Since each vertex is
trivalent, if two paths go through a common vertex, then they must have at
least one edge in common at that vertex. So every non-zero contribution
$\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}$ comes from when $p$ and
$q$ belong to a common subpath of $f$ and $g$. We now argue that even in this
case, $\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}$ is either zero or
cancels unless $p$ and $q$ occur at the ends of a common subpath. For the
remainder of the proof, let $w$ be a maximal common subpath of $f$ and $g$.

We first consider the simplest case, which is when $p$ and $q$ are consecutive
edges in $w$. So suppose that $q$ immediately follows $p$ in $w$. Then we can
write $w=w_{1}pqw_{2}$, and $f$ and $g$ can be written as
$f=f_{0}w_{1}pqw_{2}$ and $g=g_{0}w_{1}pqw_{2}$.

Then after expanding using the Leibniz rules, we get two terms in
$\left\\{\\!\\!\left\\{f,g\right\\}\\!\\!\right\\}$ involving
$\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}$ and
$\left\\{\\!\\!\left\\{q,p\right\\}\\!\\!\right\\}$. By the previous lemma,
$\left\\{\\!\\!\left\\{p,q\right\\}\\!\\!\right\\}=\lambda(1\otimes pq)$ for
some $\lambda$, and
$\left\\{\\!\\!\left\\{q,p\right\\}\\!\\!\right\\}=-\lambda(pq\otimes 1)$.
Then these two terms are given by

$\lambda(g_{0}w_{1}p\otimes f_{0}w_{1})(1\otimes pq)(qw_{2}\otimes
w_{2})=\lambda g_{0}w_{1}pqw_{2}\otimes f_{0}w_{1}pqw_{2}$ $\mathrm{and}$
$-\lambda(g_{0}w_{1}\otimes f_{0}w_{1}p)(pq\otimes 1)(w_{2}\otimes
qw_{2})=-\lambda g_{0}w_{1}pqw_{2}\otimes f_{0}w_{1}pqw_{2}$

Obviously, these terms cancel.

There is also the case that the path $w$ intersects itself, so it may be
possible that $p$ and $q$ share a common vertex, but do not occur
consecutively in the path. There are three cases, corresponding to parts
$(a)$, $(b)$, and $(c)$ in the previous lemma.

First we consider part $(a)$ of the previous lemma. So suppose $p$ and $q$ are
consecutive arrows in the quiver, but they occur non-consecutively in the
path. Let $r$ be the third edge incident to the common vertex of $p$ and $q$,
oriented outward. There are three cases:

  1. 1.

$p$ and $q$ occur consecutively twice, so $w=w_{1}pqw_{2}pqw_{3}$.

  2. 2.

The path follows $r$ after $p$, and $p$ before $q$, so
$w=w_{1}prw_{2}pqw_{3}$.

  3. 3.

The path follows $r$ after $p$, and $r^{-1}$ before $q$, so
$w=w_{1}prw_{2}r^{-1}qw_{3}$.

In cases $(2)$ and $(3)$, the vertex in question is the end of another common
subpath, and so we don’t consider these cases. In case $(1)$, the brackets of
the first $p$ and first $q$ cancel by the previous discussion, since they are
consecutive edges in the path. The terms coming from the bracket of the first
$p$ with the second $q$ and vice versa are given by

$\lambda(g_{0}w_{1}pqw_{2}pqw_{2}pqw_{3}\otimes
f_{0}w_{1}pqw_{3})~{}\mathrm{and}~{}-\lambda(g_{0}w_{1}pqw_{2}pqw_{2}pqw_{3}\otimes
f_{0}w_{1}pqw_{3})$

All other cases correspond to cases $(b)$ and $(c)$ from the previous lemma.
All these cases imply that the vertex which $p$ and $q$ share is the end of
some other common subpath, and so we don’t need to consider these. We have
thus covered all cases, and the lemma is proved. ∎

Write $f$ and $g$ as products of edges: $f=f_{1}\cdots f_{n}$ and
$g=g_{1}\cdots g_{m}$. Then by the Leibniz rule:

$\left\\{\\!\\!\left\\{f,g\right\\}\\!\\!\right\\}=\sum_{i,j}(g_{1}\cdots
g_{j-1}\otimes f_{1}\cdots
f_{i-1})\left\\{\\!\\!\left\\{f_{i},g_{j}\right\\}\\!\\!\right\\}(f_{i+1}\cdots
f_{n}\otimes g_{j+1}\cdots g_{m})$

The result of the preceding lemma says that this sum is only over the pairs
$f_{i},g_{j}$ which are incident to a common vertex which is the end of a
maximal common subpath of $f$ and $g$. So we will now compute what happens at
these vertices.

As in the commutative case, there appear to be 36 cases to consider, since the
two endpoints of a common subpath could each have one of two colors and one of
three orientations. It happens again, however, that the orientations do not
affect the outcome, which we formulate now as a lemma.

######  Lemma 4.

The contribution to $\left<f,g\right>$ coming from a common subpath $w\in
f\cap g$ depends only on the colors of the endpoint vertices, and not on the
orientations.

###### Proof.

Let us first consider the beginning of a common subpath $w$, where the paths
$f$ and $g$ first come together. Then we may write $f=f_{1}\cdots
f_{i}w_{1}\cdots f_{n}$ and $g=g_{1}\cdots g_{j}w_{1}\cdots g_{m}$, where
$w_{1}=f_{i+1}=g_{j+1}$ is the first edge in the common subpath $w$, and
$f_{i}$ and $g_{j}$ are incident to the same vertex as $w_{1}$. There are
three terms in the expansion of
$\left\\{\\!\\!\left\\{f,g\right\\}\\!\\!\right\\}$ coming from this vertex,
corresponding to $w$, given by

$(g_{1}\cdots g_{j-1}\otimes f_{1}\cdots
f_{i-1})\left\\{\\!\\!\left\\{f_{i},g_{j}\right\\}\\!\\!\right\\}(w_{1}f_{i+2}\cdots
f_{n}\otimes w_{1}g_{j+2}\cdots g_{m})$ $(g_{1}\cdots g_{j}\otimes f_{1}\cdots
f_{i-1})\left\\{\\!\\!\left\\{f_{i},w_{1}\right\\}\\!\\!\right\\}(w_{1}f_{i+2}\cdots
f_{n}\otimes g_{j+2}\cdots g_{m})$ $(g_{1}\cdots g_{j-1}\otimes f_{1}\cdots
f_{i})\left\\{\\!\\!\left\\{w_{1},g_{j}\right\\}\\!\\!\right\\}(f_{i+2}\cdots
f_{n}\otimes w_{1}g_{j+2}\cdots g_{m})$

By Lemma 2, all three of these terms give the simple tensor

$g_{1}\cdots g_{j}w_{1}f_{i+2}\cdots f_{n}\otimes f_{1}\cdots
f_{i}w_{1}g_{j+2}\cdots g_{m},$

but with different coefficients. Which coefficient goes with which term
depends on the orientation of the vertex. The following picture illustrates
the three possibilities at a black vertex, with $f_{i}w_{1}$ in blue and
$g_{j}w_{1}$ in red.

$(i)$$(ii)$$(iii)$

The coefficients for the three pictures above are given in the following
table:

$\begin{array}[]{|c|c|c|c|}\hline\cr\mathrm{Case}&\left\\{\\!\\!\left\\{f_{i},g_{j}\right\\}\\!\\!\right\\}&\left\\{\\!\\!\left\\{f_{i},w_{1}\right\\}\\!\\!\right\\}&\left\\{\\!\\!\left\\{w_{1},g_{j}\right\\}\\!\\!\right\\}\\\
\hline\cr(i)&k_{3}(g_{j}\otimes f_{i})&-k_{1}(1\otimes
f_{i}w_{1})&k_{2}(g_{j}w_{1}\otimes 1)\\\ \hline\cr(ii)&k_{2}(g_{j}\otimes
f_{i})&k_{3}(1\otimes f_{i}w_{1})&-k_{1}(g_{j}w_{1}\otimes 1)\\\
\hline\cr(iii)&-k_{1}(g_{j}\otimes f_{i})&k_{2}(1\otimes
f_{i}w_{1})&k_{3}(g_{j}w_{1}\otimes 1)\\\ \hline\cr\end{array}$

In all three cases, the three terms are the same simple tensor, so the
coefficients add up. As we can see from the table, the combined coefficient is
always $-A_{\bullet}=k_{2}+k_{3}-k_{1}$. Similarly, for all three orientations
at a white vertex, we will get the same simple tensor with a coefficient of
$A_{\circ}=w_{1}-w_{2}-w_{3}$. If we consider the endpoint vertex of the
common subpath, we will again get the same simple tensor for all three terms.
If the vertex is black, we will get a coefficient of $A_{\bullet}$, and if it
is white, we will get $-A_{\circ}$. These are exactly the same as the values
for $\varepsilon_{w}(f,g)$ in the commutative case. ∎

Let $f_{w}g_{w}$ denote the loop which follows $f$ starting with the path $w$,
followed by $g$ starting at $w$. Then putting the previous lemmas together
gives

######  Theorem 7.

Let $f,g\in\mathscr{L}$. Then the induced bracket in
$\mathcal{F}_{Q}^{\natural}$ is given by

$\left<f,g\right>=\sum_{x\in f\cap g}\varepsilon_{x}(f,g)f_{x}g_{x}$

In particular, $\mathscr{L}^{\natural}$ is closed under $\left<-,-\right>$.

Moreover, we have the following important observation.

######  Theorem 8.

The induced bracket $\left<-,-\right>$ makes $\mathscr{L}^{\natural}$ a Lie
algebra.

###### Proof.

We only need to verify the Jacobi identity. So let $f,g,h\in\mathscr{L}$. Then

$\left<f,\left<g,h\right>\right>=\sum_{i\in g\cap
h}\varepsilon_{i}(g,h)\left<f,g_{i}h_{i}\right>=\sum_{i\in g\cap
h}\varepsilon_{i}(g,h)\left(\sum_{j\in f\cap
g}\varepsilon_{j}(f,g)f_{j}(g_{i}h_{i})_{j}+\sum_{k\in f\cap
h}\varepsilon_{k}(f,h)f_{k}(g_{i}h_{i})_{k}\right)$

$\left<\left<f,g\right>,h\right>=\sum_{j\in f\cap
g}\varepsilon_{j}(f,g)\left<f_{j}g_{j},h\right>=\sum_{j\in f\cap
g}\varepsilon_{j}(f,g)\left(\sum_{k\in f\cap
h}\varepsilon_{k}(f,h)(f_{j}g_{j})_{k}h_{k}+\sum_{i\in g\cap
h}\varepsilon_{i}(g,h)(f_{j}g_{j})_{i}h_{i}\right)$

$\left<g,\left<f,h\right>\right>=\sum_{k\in f\cap
h}\varepsilon_{k}(f,h)\left<g,f_{k}h_{k}\right>=\sum_{k\in f\cap
h}\varepsilon_{k}(f,h)\left(\sum_{i\in g\cap
h}\varepsilon_{i}(g,h)g_{i}(f_{k}h_{k})_{i}-\sum_{j\in f\cap
g}\varepsilon_{j}(f,g)g_{j}(f_{k}h_{k})_{j}\right)$

The Jacobi identity will hold if the first equation is equal to the sum of the
second and third. Comparing the right-hand side of the first with the sum of
the right-hand sides of the second and third, we see that the Jacobi identity
will hold if the following identities are true:

$f_{j}(g_{i}h_{i})_{j}=(f_{j}g_{j})_{i}h_{i}=g_{i}(f_{j}h_{j})_{i}$

It is an easy check that these identities are indeed true for all cycles. ∎

######  Remark.

This double bracket is _not_ a double Poisson bracket (it does not satisfy the
double Jacobi identity), so it does not immediately guarantee the Jacobi
identity for $\left<-,-\right>$. Since we do have the Jacobi identity, though,
for $\left<-,-\right>$, this makes it into an $H_{0}$-Poisson structure as
defined by Crawley-Boevey [CB11].

###  6.6 Goldman’s Bracket and the Twisted Ribbon Surface

In this section, we give a geometric interpretation of the bracket just
described. We first recall some preliminaries.

Let $G$ be a connected Lie group and $S$ a smooth oriented surface with
fundamental group $\pi:=\pi_{1}(S)$. We consider the space of representations
of $\pi$ in $G$, modulo conjugations, which we call $\mathrm{Rep}_{G}(\pi)$:

$\mathrm{Rep}_{G}(\pi):=\mathrm{Hom}(\pi,G)/G$

Let $f\colon G\to\mathbb{R}$ be any invariant function on $G$, with respect to
conjugation. Then for $\alpha\in\pi$, we can define the function
$f_{\alpha}\colon\mathrm{Rep}_{G}(\pi)\to\mathbb{R}$ by the formula
$f_{\alpha}([\varphi]):=f(\varphi(\alpha))$. In particular, if $G$ is a group
of matrices, we may take $f=\mathrm{tr}$. In this case we will write
$\mathrm{tr}(\alpha)$ for $f_{\alpha}$. Also, if
$G=\mathrm{GL}_{n}(\mathbb{R})$, we will write $\mathrm{Rep}_{n}(\pi)$ instead
of $\mathrm{Rep}_{\mathrm{GL}_{n}(\mathbb{R})}(\pi)$.

In 1984, William Goldman described a symplectic structure on
$\mathrm{Rep}_{G}(\pi)$ which generalizes the Weil-Petersson symplectic
structure on Teichmüller space in the case that
$G=\mathrm{PSL}_{2}(\mathbb{R})$ [Gol84]. Then, in 1986, he studied the
Poisson bracket induced by this symplectic structure, in terms of the
functions $f_{\alpha}$ [Gol86]. Goldman gives explicit formulas for
$\\{f_{\alpha},f_{\beta}\\}$ for various choices of the group $G$, in terms of
the topology of the surface and the intersection of curves representing
$\alpha$ and $\beta$. In particular, when $G=\mathrm{GL}_{n}(\mathbb{R})$ and
$f=\mathrm{tr}$, we have the following.

######  Theorem 9.

[Gol86] The Poisson bracket of the functions $\mathrm{tr}(\alpha)$ on
$\mathrm{Rep}_{n}(\pi)$ is given by

$\\{\mathrm{tr}(\alpha),\mathrm{tr}(\beta)\\}=\sum_{p\in\alpha\cap\beta}\varepsilon_{p}(\alpha,\beta)\,\mathrm{tr}(\alpha_{p}\beta_{p})$

Here, $\alpha_{p}\beta_{p}$ denotes the loop which traverses first $\alpha$,
and then $\beta$, both based at the point $p$, and
$\varepsilon_{p}(\alpha,\beta)$ is the oriented intersection number of the
curves at $p$.

Note the obvious similarity with our formula from Theorem 7. We will now