Skew-Zigzag Algebras

Skew-Zigzag Algebras

Chad COUTURE C. Couture Department of Mathematics and Statistics, University
of Ottawa,
585 King Edward Ave, Ottawa, ON K1N 6N5, Canada<EMAIL_ADDRESS>
Received October 02, 2015, in final form June 17, 2016; Published online June
26, 2016

We investigate the skew-zigzag algebras introduced by Huerfano and Khovanov.
In particular, we relate moduli spaces of such algebras with the cohomology of
the corresponding graph.

zigzag algebra; path algebra; Dynkin diagram; moduli space; graph cohomology

16G20

##  1 Introduction

Zigzag algebras were introduced by Huerfano and Khovanov in [5] in their
categorification of the adjoint representation of simply-laced quantum groups.
The zigzag algebra $A(\Gamma)$ is a quotient of the path algebra of the double
quiver associated to the Dynkin diagram $\Gamma$ of the quantum group in
question. The Grothendieck group of the category of $A(\Gamma)$-modules is
then naturally identified with the weight lattice of $\mathfrak{g}$. Zigzag
algebras have also recently appeared in categorifications of the Heisenberg
algebra. See [2] and [7, Remark 6.2(c)].

In addition to their importance in categorification, zigzag algebras have a
variety of nice features, as pointed out in [5]. Examples of such features
include the following: they have nondegenerate symmetric trace forms and are
quadratic algebras (provided that $\Gamma$ has at least 3 vertices); if
$\Gamma$ is a finite Dynkin diagram, then $A(\Gamma)$ is of finite type and
its indecomposable representations are enumerated by roots of $\mathfrak{g}$;
and if $\Gamma$ is bipartite, then the quadratic dual of the $A(\Gamma)$ is
the preprojective algebra of $\Gamma$ for a sink-source orientation.

_Skew_ -zigzag algebras, also introduced in [5], are a generalisation of
zigzag algebras. They involve coefficients $v_{b,c}^{a}$ for vertices $a$,
$b$, $c$ such that $a$ is connected to both $b$ and $c$. When all coefficients
are equal to one, they recover the zigzag algebras. The goal of the current
paper is to investigate some important properties of skew-zigzag algebras. We
provide proofs of some results mentioned in [5] without proof, in addition to
proving some results that appear to be new.

We begin with a review of the concepts in graph theory necessary for the
current paper in Section 2. In Section 3, we recall the definition of the
skew-zigzag algebras, find an explicit basis for them (see Proposition 3.7),
and prove that they are graded symmetric algebras (see Proposition 3.9). In
Section 4, we describe two moduli spaces of skew-zigzag algebras. The first is
the moduli space of skew-zigzag algebras up to isomorphism preserving
vertices, which we show to be isomorphic to the graph cohomology of $\Gamma$
(see Theorem 4.8), as stated in [5] without proof (see Remark 4.9). The second
is the moduli space of skew-zigzag algebras up to _arbitrary_ isomorphism,
which we show to be isomorphic to the quotient of the graph cohomology of
$\Gamma$ by a natural action of the automorphism group of $\Gamma$ (see
Theorem 4.12). We should note here that we consider the moduli space only as a
group, and do not consider any geometric structure. Finally, in Section 5, we
discuss an alternate definition of skew-zigzag algebras that has appeared in
the literature. We show that this alternate definition is more restrictive
(see Proposition 5.3).

##  2 Graph theory background

Recall that a _graph_ $\Gamma$ is a pair $(V,E)$ where $V$ is a finite set and
$E$ is a set consisting of two element subsets of $V$. The elements of $V$ are
called _vertices_ and the elements of $E$ are called _edges_. Note that this
implies that we consider graphs with no loops or multiple edges. We will often
depict graphs as diagrams, with a node for each vertex and curves between
nodes $a$ and $b$ if $\\{a,b\\}\in E$. If there exist subsets $A,B\subseteq V$
such that $A\sqcup B=V$ and each element of $E$ contains one element of $A$
and one element $B$, then $\Gamma$ is said to be _bipartite_.

A _quiver_ , $\mathcal{Q}$, is a four-tuple
$(\mathcal{Q}_{0},\mathcal{Q}_{1},s,t)$ where $\mathcal{Q}_{0}$ and
$\mathcal{Q}_{1}$ are both finite sets and $s$, $t$ are set maps from
$\mathcal{Q}_{1}$ to $\mathcal{Q}_{0}$. The elements of $\mathcal{Q}_{0}$ are
again called vertices and the elements of $\mathcal{Q}_{1}$ are called
_directed edges_. For each directed edge $f$, we call $s(f)$ and $t(f)$ the
_source_ and _target_ of $f$ (respectively). We will often denote a directed
edge with source $a$ and target $b$ by $(a\,|\,b)$. Throughout this paper we
will consider quivers with no parallel edges, i.e., no directed edges $f_{1}$,
$f_{2}$ such that $s(f_{1})=s(f_{2})$ and $t(f_{1})=t(f_{2})$.

######  Example 2.1 (a graph and a quiver).

The leftmost diagram below represents the graph $(V,E)$ with $V=\\{a,b,c,d\\}$
and $E=\\{\\{a,b\\},\\{b,c\\},\\{a,c\\},\\{a,d\\}\\}$. The rightmost diagram
depicts the quiver $\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1},s,t)$ with
$\mathcal{Q}_{0}=\\{a,b,c,d\\}$,
$\mathcal{Q}_{1}=\\{(a\,|\,b),(a\,|\,d),(b\,|\,c),(c\,|\,d),(d\,|\,b)\\}$ and
$s,t\colon\mathcal{Q}_{1}\to\mathcal{Q}_{0}$ given by

$\displaystyle s((a\,|\,b))=s((a\,|\,d))=a,\qquad s((b\,|\,c))=b,\qquad
s((c\,|\,d))=c,\qquad s((d\,|\,b))=d,$ $\displaystyle
t((a\,|\,b))=t((d\,|\,b))=b,\qquad t((a\,|\,d))=t((c\,|\,d))=d,\qquad
t((b\,|\,c))=c.$ $d$$a$$b$$c$$d$$a$$b$$c$

We define a _path_ in a graph $\Gamma$ to be a sequence of vertices
$(a_{1},a_{2},\dots,a_{n})$ such that $\\{a_{i},a_{i+1}\\}\in E$ for
$i=1,\dots,n-1$. Analogously, we define a path, $P$, in $\mathcal{Q}$ to be a
sequence of directed edges $(f_{1},\dots,f_{n})$ such that the source of
$f_{i+1}$ is equal to the target of $f_{i}$ for $1\leq i\leq n-1$. The source
and the target of $P$ are the source of $f_{1}$ and the target of $f_{n}$
(respectively). Furthermore, for any path $P=(a_{1},a_{2},\dots,a_{n})$
(respectively $P=(f_{1},\dots,f_{m})$), the _length_ , $\ell(P)$, of $P$ is
equal to $n-1$ (respectively $m$). We also consider paths of length 0 which
start and end at the same vertex $a$, called _empty paths_ or _trivial paths_
and denoted $(a)$. We shall use $\mathcal{Q}_{i}$ to denote the paths of
length $i$ in $\mathcal{Q}$. In addition, we shall use
$(a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n})$ $(n\geq 1)$ to denote a path
$(f_{1},\dots,f_{n-1})$ such that $s(f_{i})=a_{i}$ and $t(f_{i})=a_{i+1}$ for
$1\leq i\leq n-1$ in $\mathcal{Q}$. We say that $\Gamma$ is _connected_ if for
any vertices $a$ and $b$, there exits a path $P$ between $a$ and $b$.

######  Example 2.2 (paths).

In the graph of Example 2.1, $(a,b,c,a,d)$ is a path from $a$ to $d$ of length
4. In the quiver of Example 2.1, $(a\,|\,b\,|\,c\,|\,d)$ is a path of length
3. However, $(a\,|\,b\,|\,d)$ is not a path in the quiver of Example 2.1.

We define the _double graph_ of $\Gamma$, denoted $D\Gamma$, to be the quiver
consisting of the vertices of $\Gamma$ and for each edge $\\{a,b\\}$ of
$\Gamma$, $D\Gamma$ has two edges $f_{1}$ and $f_{2}$ with
$s(f_{1})=t(f_{2})=a$ and $s(f_{2})=t(f_{1})=b$.

######  Example 2.3 (double graph).

Let $\Gamma$ be the graph in Example 2.1. Its double graph, $D\Gamma$, is the
following quiver.

$d$$a$$b$$c$

A path $C=(a_{1},a_{2},\dots,a_{n})$ in $\Gamma$ is said to be a _cycle_ if
$a_{1}=a_{n}$. Then we define $V_{C}$ and $E_{C}$ to be the sets
$\\{a_{i}\,|\,1\leq i\leq n-1\\}$ and $\\{\\{a_{i},a_{i+1}\\}\,|\,1\leq i\leq
n-1\\}$ respectively. Similarly, a path
$C=(a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n})$ in $\mathcal{Q}$ is said to be a
cycle if $a_{1}=a_{n}$. Then we define $V_{C}$ and $E_{C}$ to be the sets
$\\{a_{i}\,|\,1\leq i\leq n-1\\}$ and $\\{\\{a_{i},a_{i+1}\\}\,|\,1\leq i\leq
n-1\\}$ respectively. For each vertex $a\in V$, we define the degree of $a$,
denoted $\deg(a)$, to be the cardinality of the set $\\{e\in E\,|\,a\in e\\}$.
Furthermore, for any cycle $C$ (in a quiver or a graph), we define the degree
in $C$ of a vertex $a\in V_{C}$, denoted $\deg_{C}(a)$, to be the cardinality
of the set $\\{e\in E_{C}\,|\,a\in e\\}$. Finally, we say that $C$ is a
_simple cycle_ if for all $a\in V_{C}$ we have $\deg_{C}(a)=2$.

We say that a graph $\Gamma^{\prime}=(V^{\prime},E^{\prime})$ is a _subgraph_
of $\Gamma=(V,E)$ if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$. If
$T$ is a connected graph that does not contain any simple cycles, then $T$ is
said to be a _tree_. Let $T=(V_{T},E_{T})$ be a subgraph of $\Gamma$. If
$V_{T}=V$ and if $T$ is a tree, then $T$ is called a _spanning tree_ of
$\Gamma$. Every connected graph has at least one spanning tree (see, for
instance, [4, Section 1.5]). If $\Gamma=(V,E)$ is a connected graph and
$T=(V_{T},E_{T})$ is a spanning tree of $\Gamma$, then
$\left|{E_{T}}\right|=|V|-1$.

##  3 Skew-zigzag algebras

In this section we introduce our main objects of study, the skew-zigzag
algebras, and prove some basic facts about them that are stated without proof
in the literature. Throughout this section, we fix a field $\Bbbk$ and a
connected graph $\Gamma=(V,E)$.

###  3.1 Definitions

We first recall the definition of the path algebra of a quiver. We refer the
reader to Chapter 2 of [1] for further details. The _path algebra_ of a quiver
$\mathcal{Q}$, denoted $\Bbbk\mathcal{Q}$, is the vector space with basis
consisting of all paths. We define the _concatenation_ of two paths
$(a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n})$ and
$(a_{1}^{\prime}\,|\,a_{2}^{\prime}\,|\,\dots\,|\,a_{m}^{\prime})$ to be the
path
$(a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n}=a_{1}^{\prime}\,|\,a_{2}^{\prime}\,|\,\dots\,|\,a_{m}^{\prime})$
when $a_{n}=a_{1}^{\prime}$ and 0 otherwise. Then, we define the
multiplication of two paths $P_{1}*P_{2}$ to be the concatenation of paths.
The path algebra is an $\mathbb{N}$-graded algebra, i.e.,

$\displaystyle\Bbbk\mathcal{Q}=\bigoplus_{i=0}^{\infty}\Bbbk\mathcal{Q}_{i},$

where each $\Bbbk\mathcal{Q}_{i}$ is the $\Bbbk$-vector space spanned by all
paths of length $i$, and
$\Bbbk\mathcal{Q}_{i}\Bbbk\mathcal{Q}_{j}\subseteq\Bbbk\mathcal{Q}_{i+j}$ for
all $i$, $j$. We call an element of $\Bbbk\mathcal{Q}$ _homogeneous of degree
$i$_ if it lies in $\Bbbk\mathcal{Q}_{i}$. We will be mostly interested in the
case where $\mathcal{Q}=D\Gamma$.

Recall that an ideal $I$ of a graded algebra $A=\bigoplus_{i=0}^{\infty}A_{i}$
is called _graded_ or _homogeneous_ if it is generated by homogeneous elements
of $A$. Equivalently, $I$ is homogeneous if $I=\bigoplus_{i=0}^{\infty}(I\cap
A_{i})$.

######  Definition 3.1 (skew-zigzag coefficients).

A set $v=\big{(}v^{a}_{b,c}\in\Bbbk\,|\,\\{a,b\\},\\{a,c\\}\in E\big{)}$ is a
collection of _skew-zigzag coefficients_ for the graph $\Gamma=(V,E)$ if it
satisfies the following three conditions:

  * •

$v^{a}_{b,b}$=1 for all $\\{a,b\\}\in E$,

  * •

$v^{a}_{b,c}v^{a}_{c,b}=1$ for all $\\{a,b\\},\\{a,c\\}\in E$, and

  * •

$v^{a}_{b,c}v^{a}_{c,d}v^{a}_{d,b}=1$ for all
$\\{a,b\\},\\{a,c\\},\\{a,d\\}\in E$.

From now on, whenever we say that two vertices are _connected_ , we shall mean
that there is an edge between these two vertices. We are now ready to recall
the definition of the algebras defined originally in [5, p. 527].

######  Definition 3.2 (skew-zigzag algebra).

Let $v$ be a collection of skew-zigzag coefficients.

  * •

If $\Gamma$ only contains one vertex, then we define $A_{v}(\Gamma$) to be the
algebra generated by 1 and $X$ with $X^{2}$=0.

  * •

If $\Gamma$ contains two vertices, then we define $A_{v}(\Gamma)$ to be the
quotient algebra of the path algebra of $D\Gamma$ by the two-sided ideal
generated by all paths of length greater than two.

  * •

If $\Gamma$ has at least three vertices, we define $I_{v}$ to be the ideal
generated by

    1. a)

paths of the form $(a_{1}\,|\,a_{2}\,|\,a_{3})$ for all $a_{1}$, $a_{2}$,
$a_{3}$ in $\Gamma$ such that $a_{1}$ is connected to $a_{2}$, $a_{2}$ is
connected to $a_{3}$ and $a_{1}\neq a_{3}$,

    2. b)

elements of the form
$(a_{1}\,|\,a_{2}\,|\,a_{1})-v_{a_{2},a_{3}}^{a_{1}}(a_{1}\,|\,a_{3}\,|\,a_{1})$
for all $a_{1},a_{2},a_{3}\in V$ such that $a_{1}$ is connected to $a_{2}$,
$a_{3}$.

We then define $A_{v}(\Gamma)$ to be the quotient algebra of the path algebra
of $D\Gamma$ by the ideal $I_{v}$.

We call $A_{v}(\Gamma)$ the _skew-zigzag algebra_ of $\Gamma$. When
$v_{a_{2},a_{3}}^{a_{1}}=1$ for all vertices $a_{1}$, $a_{2}$, $a_{3}$ such
that $a_{1}$ is connected to both $a_{2}$ and $a_{3}$, then we call
$A_{v}(\Gamma)$ the _zigzag_ algebra of $\Gamma$ and denote it $A(\Gamma)$.

######  Remark 3.3.

Notice that $I_{v}$ is generated by homogeneous elements of $\Bbbk\Gamma$ and
hence is a graded ideal. Therefore, the skew-zigzag algebra inherits a grading
of its own. More precisely, we have

$\displaystyle
A_{v}(\Gamma)=\bigoplus_{i=0}^{\infty}\Bbbk\mathcal{Q}_{i}/(I_{v}\cap\Bbbk\mathcal{Q}_{i})=\bigoplus_{i=0}^{\infty}(\Bbbk\mathcal{Q}_{i}+I_{v})/I_{v}.$

Moreover, if $\Gamma$ contains at least two vertices, $I_{v}$ is generated by
elements of $\Bbbk\mathcal{Q}_{2}$. Thus $I_{v}$ is contained in
$\bigoplus_{i\geq 2}\Bbbk\mathcal{Q}_{i}$. Consequently, any path of length
less than two cannot sit inside $I_{v}$.

For a path $P$ in $D\Gamma$, we let $[P]$ denote the equivalence class of $P$
in $A_{v}(\Gamma)$. Similarly, we shall use the notation
$[a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n}]$ to denote the equivalence class of
$(a_{1}\,|\,a_{2}\,|\,\dots\,|\,a_{n})$, $n\geq 1$, in $A_{v}(\Gamma)$.

###  3.2 Bases

Our next goal is to describe an explicit basis of the skew-zigzag algebra. We
begin with a few technical results. Throughout this subsection, we fix a
collection $v$ of skew-zigzag coefficients for the connected graph
$\Gamma=(V,E)$.

######  Lemma 3.4.

Let $a,a_{1},\ldots,a_{n}\in V$ with $n\geq 2$, and suppose that $a$ is
connected to $a_{1},\ldots,a_{n}$. Then

$\displaystyle v^{a}_{a_{1},a_{2}}v^{a}_{a_{2},a_{3}}\cdots
v^{a}_{a_{n-1},a_{n}}=v^{a}_{a_{1},a_{n}}.$ (3.1)

###### Proof.

We shall proceed by induction on $n$. For $n=2$, this trivially holds. Now
suppose that (3.1) holds for some integer $n\geq 2$. Let us prove that it
holds for $n+1$. We have

$\displaystyle\prod_{i=1}^{n}v^{a}_{a_{i},a_{i+1}}=\left(\prod_{i=1}^{n}v^{a}_{a_{i},a_{i+1}}\right)v^{a}_{a_{n+1},a_{n-1}}v^{a}_{a_{n-1},a_{n+1}}=\left(\prod_{i=1}^{n-2}v^{a}_{a_{i},a_{i+1}}\right)v^{a}_{a_{n-1},a_{n+1}}=v^{a}_{a_{1},a_{n+1}},$

where the last equality follows from the induction hypothesis. ∎

######  Corollary 3.5.

For all $n\geq 2$, we have

$\displaystyle v^{a}_{a_{1},a_{2}}v^{a}_{a_{2},a_{3}}\cdots
v^{a}_{a_{n-1},a_{n}}v^{a}_{a_{n},a_{1}}=1.$

###### Proof.

Suppose $n\geq 2$. By Lemma 3.4, we have

$\displaystyle v^{a}_{a_{1},a_{2}}v^{a}_{a_{2},a_{3}}\cdots
v^{a}_{a_{n-1},a_{n}}v^{a}_{a_{n},a_{1}}=v^{a}_{a_{1},a_{n}}v^{a}_{a_{n},a_{1}}=1.$
∎

######  Lemma 3.6.

Let $P_{1},\dots,P_{n}$ be paths in a quiver, no two of which have the same
source and target. If $[P_{i}]\neq 0$ for all $i=0,\dots,n$ then
$[P_{1}],\dots,[P_{n}]$ are linearly independent.

###### Proof.

Suppose there exist $\alpha_{1},\dots,\alpha_{n}\in\Bbbk$ such that

$\displaystyle\alpha_{1}[P_{1}]+\cdots+\alpha_{n}[P_{n}]=0.$

Then, for all $i=1,\dots,n$, we obtain

$\displaystyle
0=\alpha_{1}[s(P_{i})][P_{1}][t(P_{i})]+\cdots+\alpha_{i}[s(P_{i})][P_{i}][t(P_{i})]+\cdots+\alpha_{n}[s(P_{i})][P_{n}][t(Pi)]=\alpha_{i}[P_{i}].$

Hence, we must have $\alpha_{i}=0$ for all $i=1,\dots,n$. Thus,
$[P_{1}],\dots,[P_{n}]$ are linearly independent. ∎

We are now in a position to determine a basis of the skew-zigzag algebra. In
particular, this gives us the dimension of the zigzag algebra, which was
stated in [5, Section 3] without proof.

######  Proposition 3.7 (basis of the skew-zigzag algebra).

Recall that $\Gamma=(V,E)$ is a connected graph and $v$ is a collection of
skew-zigzag coefficients.

  * •

If $\Gamma$ only has one vertex, then $\\{1,X\\}$ is a basis for
$A_{v}(\Gamma)$.

  * •

If $\Gamma$ has two vertices, $a$ and $b$, then
$\\{[a],[b],[a\,|\,b],[b\,|\,a],[a\,|\,b\,|\,a],[b\,|\,a\,|\,b]\\}$ is a basis
for $A_{v}(\Gamma)$.

  * •

If $\Gamma$ has three or more vertices, for all $x\in V$ we define $V_{x}$ to
be set of all vertices that are connected to $x$ and we fix a vertex $y_{x}\in
V_{x}$. Then

$\displaystyle J\coloneqq\\{[a],[b\,|\,c],[x\,|\,y_{x}\,|\,x]\,|\,a,x\in
V,\,\\{b,c\\}\in E\\}$ (3.2)

is a basis for $A_{v}(\Gamma)$. In particular, we have $\dim
A_{v}(\Gamma)=2|V|+2|E|$.

###### Proof.

The author would like to thank a referee for bringing into light a much
simpler proof of this proposition.

The first two claims are obvious. Therefore, we assume that $\Gamma$ has at
least three vertices. Recall that the ideal $I_{v}$ is generated by the set

$\displaystyle
X_{v}\coloneqq\big{\\{}(a\,|\,b\,|\,c),(x\,|\,y\,|\,x)-v_{y,z}^{x}(x\,|\,z\,|\,x)\,|\,\\{a,b\\},\\{b,c\\},\\{x,y\\},\\{x,z\\}\in
E,\,a\neq c\big{\\}},$

and hence

$\displaystyle[a\,|\,b\,|\,c]=0,\qquad[x\,|\,y\,|\,x]=v_{y,z}^{x}[x\,|\,z\,|\,x],$
(3.3)

for all $\\{a,b\\},\\{b,c\\},\\{x,y\\},\\{x,z\\}\in E$ with $a\neq c$. Note
that the first equality implies that any path with three consecutive pairwise
distinct vertices is equivalent to zero. Now, consider a path of the form
$(a\,|\,b\,|\,a\,|\,b)$ where $\\{a,b\\}\in E$. Since $\Gamma$ is connected
and has at least three vertices, either $a$ or $b$ is connected to a third
vertex $d\neq a,b$. Suppose $b$ is connected to $d$. (The case that $a$ is
connected to $d$ is analogous.) Then, (3.3) yields

$\displaystyle[a\,|\,b\,|\,a\,|\,b]=[a\,|\,b][b\,|\,a\,|\,b]=v^{b}_{a,d}[a\,|\,b][b\,|\,d\,|\,b]=[a\,|\,b\,|\,d][d\,|\,b]=0.$

Thus, any path of length 3 or greater has an equivalence class equal to 0. So
$A_{v}(\Gamma)$ only contains elements of degree 0, 1 or 2. It is clear that
$\\{[a]\,|\,a\in V\\}$ and $\\{[a\,|\,b]\,|\,\\{a,b\\}\in E\\}$ are bases for
$(A_{v}(\Gamma))_{0}$ and $(A_{v}(\Gamma))_{1}$ respectively since $I_{v}$ is
concentrated in degrees 2 and higher. Now, let $x\in V$. Notice that for any
$a\in V_{x}$, we have $[x\,|\,a\,|\,x]=v^{x}_{a,y_{x}}[x\,|\,y_{x}\,|\,x]$. So
any element in $([x]A_{v}(\Gamma)[x])_{2}$ can be written as some nonzero
scalar times $[x\,|\,y_{x}\,|\,x]$. Thus, $\\{[x\,|\,y_{x}\,|\,x]\,|\,x\in
V\\}$ is a spanning set for $(A_{v}(\Gamma))_{2}$ and so, by Lemma 3.6, it is
a basis of $(A_{v}(\Gamma))_{2}$. Consequently, $J$ is a basis for
$A_{v}(\Gamma)$.

Finally, notice that we have

$\displaystyle J=\\{[a]\,|\,a\in V\\}\sqcup\\{[a\,|\,b]\,|\,\\{a,b\\}\in
E\\}\sqcup\\{[x\,|\,y_{x}\,|\,x]\,|\,x\in V\\}.$

Thus, $\left|{J}\right|=\left|{\\{[a]\,|\,a\in
V\\}}\right|+\left|{\\{[a\,|\,b]\,|\,\\{a,b\\}\in
E\\}}\right|+\left|{\\{[x\,|\,y_{x}\,|\,x]\,|\,x\in V\\}}\right|=2|V|+2|E|$. ∎

###  3.3 Skew-zigzag algebras as Frobenius algebras

We begin by recalling the concept of a Frobenius algebra, referring the reader
to [6] for further details. Let $f$ be a bilinear form

$\displaystyle f\colon\ V\times V\to\Bbbk,$

where $V$ is a vector space of finite dimension over the field $\Bbbk$. A
trace map gives rise to a bilinear form $(x,y)\mapsto\mathrm{tr}(xy)$.

Let $A$ be a $\Bbbk$-algebra. Let $\operatorname{tr}$ be a $\Bbbk$-linear map

$\displaystyle\operatorname{tr}\colon\ A\to\Bbbk.$

We call $\operatorname{tr}$ a _trace_ map. A trace map gives rise to a
bilinear form $(x,y)\mapsto\mathrm{tr}(xy)$.

######  Definition 3.8 (Frobenius and symmetric algebra).

Let $A$ be a finite-dimensional unital associative $\Bbbk$-algebra. If there
exists a nondegenerate trace map $\operatorname{tr}\colon A\to\Bbbk$, then $A$
is said to be a  _Frobenius_ algebra. Moreover, if there exists a
nondegenerate symmetric trace map, then $A$ is said to be a  _symmetric
Frobenius_ algebra or simply a _symmetric algebra_.

Recall that $\Gamma=(V,E)$ is a connected graph. Let $v$ be a collection of
skew-zigzag coefficients, and let $P$ be a path in $D\Gamma$. Throughout this
article, we shall define the source and the target of the equivalence class
$[P]$, $s([P])$ and $t([P])$, to be $[s(P)]$ and $[t(P)]$ respectively. If
$P_{1}$ and $P_{2}$ are both trivial paths or paths of length 1 then we have
$[P_{1}]=[P_{2}]$ if and only if $P_{1}=P_{2}$. If $P_{1}$ and $P_{2}$ are
paths of length 2 then $[P_{1}]$ is a scalar multiple of $[P_{2}]$ if and only
if $s(P_{1})=s(P_{2})$ and $t(P_{1})=t(P_{2})$. Thus $[s(P)]$ and $[t(P)]$ are
well-defined.

If $[P]\neq 0$, we define the length of $[P]$, denoted $\ell([P])$, to be
$\ell(P)$. If $[P]=0$, then we simply define $\ell([P])$ to be 0. Since
$I_{v}$ is generated by homogeneous elements of the same degree, $\ell([P])$
is well-defined.

Let $J$ be as in (3.2) and define the $\Bbbk$-linear map
$\operatorname{tr}\colon A_{v}(\Gamma)\to\Bbbk$ on the elements of $J$ as
follows:

$\displaystyle\operatorname{tr}([P])=\begin{cases}1&\text{if }\ell([P])=2,\\\
0&\text{otherwise.}\end{cases}$

For any path $P=(a\,|\,b\,|\,a)$, we let $v_{P}=v_{b,y_{a}}^{a}$ where $y_{a}$
is defined as in Proposition 3.7.

######  Proposition 3.9.

Recall that $\Gamma=(V,E)$ is a connected graph, and let $v$ be a collection
of skew-zigzag coefficients. Then $A_{v}(\Gamma)$ is a graded Frobenius
algebra. In addition, $A(\Gamma)$ is a graded symmetric algebra.

###### Proof.

As noted in [5, Proposition 1], zigzag algebras are symmetric algebras.
Furthermore, as stated in [5, Section 4.5], skew-zigzag algebras are Frobenius
algebras. This follows from the fact that a basic finite-dimensional algebra
over an algebraically closed field is Frobenius if the socle of any projective
indecomposable module is simple, and the map
$P/\operatorname{rad}(P)\mapsto\operatorname{soc}(P)$ is a bijection onto the
set of isomorphism classes of simple modules. The fact that the Frobenius form
is as defined above then follows from Propositions 1.10.18 and 3.6.14 of [9].
It is also possible to prove directly that the trace map defined above has the
desired properties. ∎

##  4 Moduli spaces of skew-zigzag algebras

In this section we describe the moduli spaces of skew-zigzag algebras up to
various types of isomorphism. We will see that such moduli spaces are related
to the cohomology of the corresponding graph. As noted in the introduction, we
will only consider the group structure, and not any geometric structure, on
the moduli spaces to be introduced below. Throughout this section we fix a
field $\Bbbk$ that contains square roots.

Let $\Gamma=(V,E)$ be a connected graph and $v$ a collection of skew-zigzag
coefficients. Let $P=(a_{1}\,|\,\ldots\,|\,a_{n})$ be a path in $\Gamma$.
Define

$\displaystyle v_{P}^{\rm path}=\prod_{i=2}^{n-1}v_{a_{i-1},a_{i+1}}^{a_{i}}.$

We call $v_{P}^{\rm path}$ the _product of $v$ along $P$_. If $P$ is a cycle,
then we also define

$\displaystyle v_{P}^{\rm
cycle}=v_{a_{n-1},a_{2}}^{a_{1}}\prod_{i=2}^{n-1}v_{a_{i-1},a_{i+1}}^{a_{i}}=v_{a_{n-1},a_{2}}^{a_{1}}v_{P}^{\rm
path},$

and call $v_{P}^{\rm cycle}$ the _cycle product of $v$ along $P$_.
Furthermore, we define $P^{*}=(a_{n}\,|\,\ldots\,|\,a_{1})$. It is easily seen
that $\big{(}v_{P}^{\rm path}\big{)}^{-1}=v_{P^{*}}^{\rm path}$.

Let $P_{1}=(a_{1}\,|\,\ldots\,|\,a_{n})$ and
$P_{2}=(a_{n}\,|\,\ldots\,|\,a_{m})$ be two paths. Notice that we have

$\displaystyle v_{P_{1}P_{2}}^{\rm
path}=\left(\prod_{i=2}^{m-1}v_{a_{i-1},a_{i+1}}^{a_{i}}\right)=\left(\prod_{i=2}^{n-1}v_{a_{i-1},a_{i+1}}^{a_{i}}\right)v_{a_{n-1},a_{n+1}}^{a_{n}}\left(\prod_{i=n+1}^{m-1}v_{a_{i-1},a_{i+1}}^{a_{i}}\right)$
$\displaystyle\hphantom{v_{P_{1}P_{2}}^{\rm path}}{}=v_{P_{1}}^{\rm
path}v_{a_{n-1},a_{n+1}}^{a_{n}}v_{P_{2}}^{\rm path}.$ (4.1)

If, in addition, $P_{1}$ and $P_{2}$ are cycles, then $P_{1}P_{2}$ is also a
cycle and thus,

$\displaystyle v_{P_{1}P_{2}}^{\rm
cycle}=v_{a_{m-1},a_{2}}^{a_{1}}v_{P_{1}P_{2}}^{\rm
path}=v_{a_{m-1},a_{2}}^{a_{1}}v_{a_{n-1},a_{n+1}}^{a_{n}}v_{P_{1}}^{\rm
path}v_{P_{2}}^{\rm path}$ $\displaystyle\hphantom{v_{P_{1}P_{2}}^{\rm
cycle}}{}=v_{a_{m-1},a_{n+1}}^{a_{1}}v_{a_{n-1},a_{2}}^{a_{1}}v_{a_{n-1},a_{n+1}}^{a_{n}}v_{P_{1}}^{\rm
path}v_{P_{2}}^{\rm path}$ $\displaystyle\hphantom{v_{P_{1}P_{2}}^{\rm
cycle}}{}=v_{a_{n-1},a_{2}}^{a_{1}}v_{a_{m-1},a_{n+1}}^{a_{n}}v_{P_{1}}^{\rm
path}v_{P_{2}}^{\rm path}=v_{P_{1}}^{\rm cycle}v_{P_{2}}^{\rm cycle}.$ (4.2)

In particular, the second equality of (4.2) implies that

$\displaystyle v_{P_{1}P_{2}^{*}}^{\rm
cycle}=v_{a_{n+1},a_{2}}^{a_{1}}v_{a_{n-1},a_{m-1}}^{a_{n}}v_{P_{1}}^{\rm
path}v_{P_{2}^{*}}^{\rm path}.$ (4.3)

######  Example 4.1 (product of $v$ along a path and a cycle product).

Consider the graph $\Gamma=(V,E)$ where $V=\\{a,b,c,d\\}$ and
$E=\\{\\{a,b\\},\\{a,d\\},\\{d,c\\},\\{b,c\\},\\{b,d\\}\\}$ and let
$\Bbbk=\mathbb{C}$. Its associated double graph is the following quiver.

$d$$a$$b$$c$

It is easy to show that the following are skew-zigzag coefficients for
$\Gamma$ using straightforward calculations:

$\displaystyle v^{a}_{b,d}=v^{c}_{b,d}=2,\quad
v^{a}_{d,b}=v^{c}_{d,b}=1/2,\quad v^{d}_{a,c}=v^{b}_{a,c}=5,\quad
v^{d}_{c,a}=v^{b}_{c,a}=1/5,\quad v^{d}_{b,c}=v^{b}_{d,c}=7,$ $\displaystyle
v^{d}_{c,b}=v^{b}_{c,d}=1/7,\quad v^{d}_{a,b}=v^{b}_{a,d}=5/7,\quad
v^{d}_{b,a}=v^{b}_{d,a}=7/5,\quad v^{x}_{y,y}=1\quad\text{for all}\ \
\\{x,y\\}\in E.$

Now, consider the cycles $P_{1}=(d\,|\,b\,|\,c\,|\,d)$ and
$P_{2}=(d\,|\,a\,|\,b\,|\,d)$. Then, we have

$\displaystyle v_{P_{1}}^{\rm path}=14,\quad v_{P_{2}}^{\rm path}=5/14,\quad
v_{P_{1}P_{2}}^{\rm path}=1,\quad v_{P_{1}}^{\rm cycle}=2,\quad v_{P_{2}}^{\rm
cycle}=1/2,\quad\text{and}\quad v_{P_{1}P_{2}}^{\rm cycle}=1.$

######  Remark 4.2.

We note that $A_{v}(\Gamma)$ is both a $\Bbbk$-algebra and a
$\Bbbk\mathcal{Q}_{0}$-algebra. Moreover, note that a homomorphism of
$\Bbbk\mathcal{Q}_{0}$-modules is precisely a homomorphism of $\Bbbk$-modules
that fixes the vertices.

######  Lemma 4.3.

Let $\Gamma$ be a connected graph with at least $3$ vertices and $v$ and $u$
be two collections of skew-zigzag coefficients. Suppose that

$\displaystyle\phi\colon\ A_{v}(\Gamma)\to A_{u}(\Gamma),$

is an isomorphism of graded algebras such that $\phi([a])=[a]$ for all $a\in
V$. Then

$\displaystyle u_{P}^{\rm cycle}=v_{P}^{\rm cycle},$

for any cycle $P$.

###### Proof.

For all $\\{a,b\\}\in E$, we have $\phi([a\,|\,b])=\alpha_{a,b}[a\,|\,b]$ for
some $\alpha_{a,b}\in\Bbbk^{*}$. In addition, for any $\\{a,b\\}\in E$ we have

$\displaystyle\phi([a\,|\,b\,|\,a])=\phi([a\,|\,b][b\,|\,a])=\phi([a\,|\,b])\phi([b\,|\,a])=\alpha_{a,b}[a\,|\,b]\alpha_{b,a}[b\,|\,a]=\alpha_{a,b}\alpha_{b,a}[a\,|\,b\,|\,a].$

Let $a,b,c\in V$ be such that $a$ is connected to both $b$ and $c$. Then

$\displaystyle\alpha_{a,b}\alpha_{b,a}u_{b,c}^{a}[a\,|\,c\,|\,a]$
$\displaystyle=\alpha_{a,b}\alpha_{b,a}[a\,|\,b\,|\,a]=\alpha_{a,b}\alpha_{b,a}[a\,|\,b][b\,|\,a]$
$\displaystyle=\phi([a\,|\,b][b\,|\,a])=\phi([a\,|\,b\,|\,a])=\phi(v_{b,c}^{a}[a\,|\,c\,|\,a])=v_{b,c}^{a}\alpha_{a,c}\alpha_{c,a}[a\,|\,c\,|\,a].$

Thus, we have

$\displaystyle
u_{b,c}^{a}=\frac{\alpha_{a,c}\alpha_{c,a}}{\alpha_{a,b}\alpha_{b,a}}v_{b,c}^{a}.$

Therefore, if $P=(a_{1}\,|\,\ldots\,|\,a_{n})$ is a cycle, we have

$\displaystyle u_{P}^{\rm
cycle}=\frac{\alpha_{a_{1},a_{n-1}}\alpha_{a_{n-1},a_{1}}}{\alpha_{a_{1},a_{2}}\alpha_{a_{2},a_{1}}}\left(\prod_{i=2}^{n-2}\frac{\alpha_{a_{i},a_{i-1}}\alpha_{a_{i-1},a_{i}}}{\alpha_{a_{i},a_{i+1}}\alpha_{a_{i+1},a_{i}}}\right)\frac{\alpha_{a_{n-1},a_{n-2}}\alpha_{a_{n-2},a_{n-1}}}{\alpha_{a_{n-1},a_{1}}\alpha_{a_{1},a_{n-1}}}v_{P}^{\rm
cycle}=v_{P}^{\rm cycle},$

as required. ∎

The following proposition shows that the converse to Lemma 4.3 holds.

######  Proposition 4.4.

Let $\Gamma=(V,E)$ be a connected graph with at least $3$ vertices and let $v$
and $u$ be two collections of skew-zigzag coefficients. Then $v_{P}^{\rm
cycle}=u_{P}^{\rm cycle}$ for every cycle $P$ if and only if there exists an
isomorphism of graded algebras $\phi\colon
A_{v}(\Gamma)\xrightarrow{\cong}A_{u}(\Gamma)$ such that $\phi([a])=[a]$ for
all $a\in V$.

###### Proof.

$\Rightarrow\colon$ Fix an edge $\\{a,b\\}\in E$. For any $\\{d,e\\}\in E$,
consider a path $P=(a_{1}\,|\,\ldots\,|\,a_{n})$ with $a_{1}=a$, $a_{n-1}=d$
and $a_{n}=e$. Note that since $\Gamma$ is connected, there is at least one
such path. Now, choose $\alpha_{d,e}=\alpha_{e,d}\in\Bbbk^{*}$ such that

$\displaystyle\alpha_{d,e}^{2}=u_{b,a_{2}}^{a}v_{a_{2},b}^{a}u_{P}^{\rm
path}v_{P^{*}}^{\rm path}.$ (4.4)

We will show that since we have $v_{C}^{\rm cycle}=u_{C}^{\rm cycle}$ for any
cycle $C$, (4.4) is independent of the choice of the path. Let
$P_{1}=(a_{1}\,|\,\ldots\,|\,a_{n})$ and $P_{2}=(b_{1}\,|\,\ldots\,|\,b_{m})$
be two paths such that $a=a_{1}=b_{1}$, $d=a_{n-1}=b_{m-1}$ and
$e=a_{n}=b_{m}$. Since $P_{1}P_{2}^{*}$ is a cycle, we have

$\displaystyle v_{P_{1}P_{2}^{*}}^{\rm
cycle}=v_{b_{2},a_{2}}^{a_{1}}v_{a_{n-1},b_{m-1}}^{a_{n}}v_{P_{1}}^{\rm
path}v_{P_{2}^{*}}^{\rm path}=v_{b_{2},b}^{a}v_{b,a_{2}}^{a}v_{P_{1}}^{\rm
path}v_{P_{2}^{*}}^{\rm path},$

where the first equality follows from (4.3) and the second equality follows
from Lemma 3.4 and the fact that $a_{n-1}=b_{m-1}$. Similarly, we obtain

$\displaystyle u_{P_{1}P_{2}^{*}}^{\rm
cycle}=u_{b_{2},b}^{a}u_{b,a_{2}}^{a}u_{P_{1}}^{\rm path}u_{P_{2}^{*}}^{\rm
path}.$

Since $v_{P_{1}P_{2}^{*}}^{\rm cycle}=u_{P_{1}P_{2}^{*}}^{\rm cycle}$, we have

$\displaystyle v_{b_{2},b}^{a}v_{b,a_{2}}^{a}v_{P_{1}}^{\rm
path}v_{P_{2}^{*}}^{\rm path}=u_{b_{2},b}^{a}u_{b,a_{2}}^{a}u_{P_{1}}^{\rm
path}u_{P_{2}^{*}}^{\rm path}.$

Consequently,

$\displaystyle u_{b,b_{2}}^{a}v_{b_{2},b}^{a}u_{P_{2}}^{\rm
path}v_{P_{2}^{*}}^{\rm path}=u_{b,a_{2}}^{a}v_{a_{2},b}^{a}u_{P_{1}}^{\rm
path}v_{P_{1}^{*}}^{\rm path},$

as required. Thus the coefficients $\alpha_{d,e}$ are path independent. Now,
define a map $\phi$ as follows

$\displaystyle\phi\colon\ \Bbbk D\Gamma\to\Bbbk
D\Gamma,\quad(d)\mapsto(d)\quad\text{for all}\ \ d\in
V,\quad(d\,|\,e)\mapsto\alpha_{d,e}(d\,|\,e)\quad\text{for all}\ \
\\{d,e\\}\in E.$

By definition this map is $\Bbbk$-linear. We then extend the map to longer
paths by requiring it to be an algebra homomorphism. We will now show that
$\phi(I_{v})\subseteq I_{u}$. Now suppose $\\{x,y\\},\\{x,z\\}\in E$. Let
$P_{1}=(a_{1}\,|\,\ldots\,|\,a_{n})$ and $P_{2}=(b_{1}\,|\,\ldots\,|\,b_{m})$
be two paths such that $a=a_{1}=b_{1}$, $x=a_{n-1}=b_{m-1}$, $y=a_{n}$ and
$z=b_{m}$. We have

$\displaystyle\frac{\alpha_{x,y}^{2}}{\alpha_{x,z}^{2}}v_{z,y}^{x}=\frac{u_{b,a_{2}}^{a}v_{a_{2},b}^{a}u_{P_{1}}^{\rm
path}v_{P_{1}^{*}}^{\rm path}}{u_{b,b_{2}}^{a}v_{b_{2},b}^{a}u_{P_{2}}^{\rm
path}v_{P_{2}^{*}}^{\rm
path}}v_{z,y}^{x}=\frac{v_{a_{2},b_{2}}^{a}v_{P_{1}^{*}}^{\rm
path}v_{P_{2}}^{\rm path}}{u_{a_{2},b_{2}}^{a}u_{P_{1}^{*}}^{\rm
path}u_{P_{2}}^{\rm path}}v_{z,y}^{x}.$ (4.5)

The path $P_{2}(z\,|\,x\,|\,y)P_{1}^{*}$ is a cycle. Thus,

$\displaystyle v_{P_{2}(z\,|\,x\,|\,y)P_{1}^{*}}^{\rm
cycle}=v_{a_{2},b_{2}}^{a}v_{P_{2}(z\,|\,x\,|\,y)P_{1}^{*}}^{\rm
path}=v_{a_{2},b_{2}}^{a}v_{x,x}^{z}v_{x,x}^{y}v_{P_{2}}^{\rm
path}v_{(z\,|\,x\,|\,y)}^{\rm path}v_{P_{1}^{*}}^{\rm
path}=v_{a_{2},b_{2}}^{a}v_{P_{2}}^{\rm path}v_{P_{1}^{*}}^{\rm
path}v^{x}_{z,y},$

where the second equality uses (4.1). Similarly, we obtain

$\displaystyle u_{P_{2}(z\,|\,x\,|\,y)P_{1}^{*}}^{\rm
cycle}=u_{a_{2},b_{2}}^{a}u_{P_{2}}^{\rm path}u_{P_{1}^{*}}^{\rm
path}u_{z,y}^{x}.$

Since we have $v_{P_{2}(z\,|\,x\,|\,y)P_{1}^{*}}^{\rm
cycle}=u_{P_{2}(z\,|\,x\,|\,y)P_{1}^{*}}^{\rm cycle}$, we must also have

$\displaystyle v_{a_{2},b_{2}}^{a}v_{P_{2}}^{\rm path}v_{P_{1}^{*}}^{\rm
path}v_{z,y}^{x}=u_{a_{2},b_{2}}^{a}u_{P_{2}}^{\rm path}u_{P_{1}^{*}}^{\rm
path}u_{z,y}^{x}.$ (4.6)

Combining (4.5) and (4.6) gives us

$\displaystyle\frac{\alpha_{x,y}^{2}}{\alpha_{x,z}^{2}}v_{z,y}^{x}=\frac{v_{a_{2},b_{2}}^{a}v_{P_{1}^{*}}^{\rm
path}v_{P_{2}}^{\rm path}}{u_{a_{2},b_{2}}^{a}u_{P_{1}^{*}}^{\rm
path}u_{P_{2}}^{\rm
path}}v_{z,y}^{x}=\frac{u_{a_{2},b_{2}}^{a}u_{z,y}^{x}u_{P_{1}^{*}}^{\rm
path}u_{P_{2}}^{\rm path}}{u_{a_{2},b_{2}}^{a}u_{P_{1}^{*}}^{\rm
path}u_{P_{2}}^{\rm path}}=u_{z,y}^{x}.$

Hence, we can now deduce that $\phi(I_{v})\subseteq I_{u}$, and thus $\phi$
induces an algebra homomorphism

$\displaystyle\bar{\phi}\colon\ A_{v}(\Gamma)\to
A_{u}(\Gamma),\quad\begin{cases}[d]\mapsto[d],&d\in V,\\\
[d\,|\,e]\mapsto\alpha_{d,e}[d\,|\,e],&\\{d,e\\}\in E,\\\
[d\,|\,e\,|\,d]\mapsto\alpha_{d,e}^{2}[d\,|\,e\,|\,d],&\\{d,e\\}\in
E.\end{cases}$

Since $\phi$ is surjective, $\bar{\phi}$ is also surjective. Since
$A_{v}(\Gamma)$ and $A_{u}(\Gamma)$ have the same dimension, $\bar{\phi}$ is
also injective. Consequently, $\bar{\phi}$ is an isomorphism and hence
$A_{v}(\Gamma)\cong A_{u}(\Gamma)$.

The converse of this proposition is Lemma 4.3. ∎

We will now introduce the concept of _graph cohomology_. Let $\Gamma=(V,E)$ be
a connected graph and $D\Gamma=(V,E^{\prime})$ its double graph. Let

$\displaystyle\mathbb{Z}V=\left\\{\sum_{a\in
V}\alpha_{a}a\,|\,\alpha_{a}\in\mathbb{Z}\text{ for all }a\in
V\right\\},\qquad\mathbb{Z}E^{\prime}=\left\\{\sum_{e\in
E^{\prime}}\alpha_{e}e\,|\,\alpha_{e}\in\mathbb{Z}\text{ for all }e\in
E^{\prime}\right\\}.$

Define the map $\delta$ by

$\displaystyle\delta\colon\
\mathbb{Z}E^{\prime}/\\{(a\,|\,b)+(b\,|\,a)\,|\,\\{a,b\\}\in
E\\}\to\mathbb{Z}V,\qquad e\mapsto s(e)-t(e),$

where we extend the map by linearity. For any path
$(a_{1}\,|\,\ldots\,|\,a_{n})$, we associate the element
$\sum\limits_{i=1}^{n-1}(a_{i}|a_{i+1})\in\mathbb{Z}E^{\prime}$ to it. Notice
that if $a_{1}=a_{n}$, then
$\delta\left(\sum\limits_{i=1}^{n-1}(a_{i}|a_{i+1})\right)=0$. Let
$\mathcal{C}=\ker\delta$. We call $\mathcal{C}$ the _space of cycles_ of
$\Gamma$. The space of cycles is a $\mathbb{Z}$-submodule of the free
$\mathbb{Z}$-module
$\mathbb{Z}E^{\prime}/\\{(a\,|\,b)+(b\,|\,a)\,|\,\\{a,b\\}\in E\\}$ and thus
it is a free $\mathbb{Z}$-module. Consequently, it has a $\mathbb{Z}$-basis.

######  Lemma 4.5 ([8, Section 4.4]).

Let $\Gamma=(V,E)$ be a connected graph. Then $rank(\mathcal{C})=|E|-|V|+1$.

Let $T=(V_{T},E_{T})$ be a spanning tree of a graph $\Gamma=(V,E)$ and let
$e\in V\setminus V_{T}$. We shall denote by $T+e$ the subgraph of $\Gamma$
with vertex set $V$ and edge set $E_{T}\cup\\{e\\}$. Moreover, for any cycle
$C=(a_{1}\,|\,\ldots\,|\,a_{n}=a_{1})$, we say that an edge $(a\,|\,b)$ is in
$C$ if $(a\,|\,b)\in\\{(a_{i}\,|\,a_{i+1})\\}\,|\,1\leq i\leq n-1\\}$.

######  Lemma 4.6.

Let $\Gamma=(V,E)$ be a connected graph with $|{V}|=n$ and $|{E}|=m$. There
exists a basis of $\mathcal{C}$, $\mathcal{B}=\\{C_{1},\ldots,C_{m-n+1}\\}$,
such that, for all $1\leq i\leq m-n+1$, there exists $(b_{i}\,|\,c_{i})$ in
$C_{i}$ such that $(b_{i}\,|\,c_{i})$ is not in $C_{k}$ for $k\neq i$.

###### Proof.

Let $\Gamma=(V,E)$ be a connected graph with $\left|{V}\right|=n$ and
$\left|{E}\right|=m$. Pick a spanning tree $T=(V_{T},E_{T})$ of $\Gamma$. Let
$E\setminus E_{T}=\\{e_{1},\ldots,e_{m-n+1}\\}$. Notice that for each
$i\in\\{1,\ldots,m-n+1\\}$, $T+e_{i}$ contains one simple cycle. Thus, the
quiver $D(T+e_{i})$ contains two corresponding simple cycles, $C_{i}$ and
$-C_{i}$. Let

$\displaystyle\mathcal{B}=\\{C_{i}\,|\,i=1,\ldots,m-n+1\\}.$

Clearly the elements of $\mathcal{B}$ are linearly independent. Moreover,
$\left|{\mathcal{B}}\right|=m-n+1$. Thus, by Lemma 4.5, $\mathcal{B}$ is a
$\mathbb{Q}$-basis of $\mathcal{C}\otimes_{\mathbb{Z}}\mathbb{Q}$. But since
the coefficient of $e_{i}$ in $C_{i}$ is one, it follows that $\mathcal{B}$ is
a $\mathbb{Z}$-basis of $\mathcal{C}$. Finally, it is also clear that for all
$i=1,\ldots,m-n+1$ we have $(b_{i}\,|\,c_{i})$ in $C_{k}$ if and only if
$k=i$, where $(b_{i}\,|\,c_{i})$ is the corresponding directed edge of $e_{i}$
in $C_{i}$. ∎

Let $\Gamma=(V,E)$ be a connected graph and $D\Gamma=(V,E^{\prime})$ be its
double graph. The _graph cohomology_ of $\Gamma$ is defined to be the space of
group homomorphisms from $\mathcal{C}$ to $\Bbbk^{*}$:

$\displaystyle H^{1}(\Gamma,\Bbbk^{*})=\operatorname{Hom}_{\rm
group}(\mathcal{C},\Bbbk^{*}).$

Note that the operation in this group is pointwise multiplication.

Let $z\in\mathcal{C}$. Take a representative $\sum\limits_{e\in
E^{\prime}}\alpha_{e}e$ of $z$ in
$\mathbb{N}E^{\prime}=\Big{\\{}\sum\limits_{e\in
E^{\prime}}\alpha_{e}e\,|\,\alpha_{e}\in\mathbb{N}\text{ for all }e\in
E^{\prime}\Big{\\}}$. Since $z\in\mathcal{C}=\ker\delta$, for each $a\in V$ we
have $\sum\limits_{e\in S}\alpha_{e}=\sum\limits_{e\in T}\alpha_{e}$ where
$S=\\{e\in E^{\prime}\,|\,s(e)=a\\}$ and $T=\\{e\in E^{\prime}\,|\,t(e)=a\\}$.
Thus, we can choose $n_{a}\in\mathbb{N}$ and vertices
$b_{a,1},\ldots,b_{a,n_{a}},c_{a,1},\ldots,c_{a,n_{a}}$ such that

$\displaystyle\sum_{a\in
V}\sum_{i=1}^{n_{a}}\left((b_{a,i}\,|\,a)+(a\,|\,c_{a,i})\right)=2z.$

Now, for a collection of skew-zigzag coefficients $v$, define

$\displaystyle f_{v,a}(z)=\prod_{i=1}^{n_{a}}v^{a}_{b_{a,i},c_{a,i}}.$

######  Lemma 4.7.

Let $\Gamma=(V,E)$ be a connected graph, $v$ be a collection of skew-zigzag
coefficients, $a\in V$ and $z\in\mathcal{C}$. Set $b_{a,1},\ldots,b_{a,n_{a}}$
and $c_{a,1},\ldots,c_{a,n_{a}}$ as in the previous paragraph. Then,
$f_{v,a}(z)$ is independent of the order chosen for
$b_{a,1},\ldots,b_{a,n_{a}}$ and $c_{a,1},\ldots,c_{a,n_{a}}$ and $f_{v,a}(z)$
is independent of the representative of $z$ chosen in $\mathbb{N}E^{\prime}$.

###### Proof.

For the first claim, it suffices to show that $f_{v,a}(z)$ remains unchanged
when we interchange $c_{a,j}$ and $c_{a,j+1}$ for some
$j\in\\{1,\dots,n_{a}-1\\}$. Indeed, we have

$\displaystyle
v^{a}_{b_{a,j},c_{a,j+1}}v^{a}_{b_{a,j+1},c_{a,j}}\prod_{\begin{subarray}{c}i=1\\\
i\neq j,j+1\end{subarray}}^{n_{a}}v^{a}_{b_{a,i},c_{a,i}}$
$\displaystyle\qquad{}=v^{a}_{b_{a,j},c_{a,j}}v^{a}_{c_{a,j},c_{a,j+1}}v^{a}_{b_{a,j+1},c_{a,j+1}}v^{a}_{c_{a,j+1},c_{a,j}}\prod_{\begin{subarray}{c}i=1\\\
i\neq
j,j+1\end{subarray}}^{n_{a}}v^{a}_{b_{a,i},c_{a,i}}=\prod_{i=1}^{n_{a}}v^{a}_{b_{a,i},c_{a,i}},$

where the second equality follows from Lemma 3.4 and the third equality
follows from the second condition of Definition 3.1.

For the second part of the lemma, it suffices to show that $f_{v,a}(z)$
remains unchanged when you remove $b_{a,n}$ and $c_{a,n}$ when
$b_{a,n}=c_{a,n}$. But this is obvious as
$v^{a}_{b_{a,n},c_{a,n}}=v^{a}_{b_{a,n},b_{a,n}}=1$. ∎

Now, for any collection of skew-zigzag coefficients, $v$, define the map
$f_{v}$ to be

$\displaystyle f_{v}\colon\ \mathcal{C}\to\Bbbk^{*},\qquad z\mapsto\prod_{a\in
V}f_{v,a}(z).$

It is clear by the definition of $f_{v}$ that we have
$f_{v}(z_{1}+z_{2})=f_{v}(z_{1})f_{v}(z_{2})$ for any
$z_{1},z_{2}\in\mathcal{C}$. Thus, $f_{v}\in H^{1}(\Gamma,\Bbbk^{*})$. In
addition, if $C$ is a cycle in $D\Gamma$, then it is clear that
$f_{v}(C)=v^{\rm cycle}_{C}$.

Let ${\rm SZC}=\\{v\,|\,v\text{ is a collection of skew-zigzag
coefficients}\\}$. For $u,v\in{\rm SZC}$, define $u\cdot v$ to be the set of
skew-zigzag coefficients defined as follows:

$\displaystyle(u\cdot v)^{a}_{b,c}=u^{a}_{b,c}v^{a}_{b,c},$

for all $\\{a,b\\},\\{a,c\\}\in E$. It is straightforward to check that this
introduces a group structure on ${\rm SZC}$. We then define an equivalence
relation on ${\rm SZC}$ as follows:

$\displaystyle\begin{split}&v\equiv u\iff\text{there exists an isomorphism
}\phi\colon A_{v}(\Gamma)\to A_{u}(\Gamma)\\\ &\hphantom{v\equiv
u\iff{}}{}\text{such that }\phi([a])=[a]\text{ for all }a\in V.\end{split}$

By Lemma 4.3 and Proposition 4.4, we have

$\displaystyle v\equiv u\iff v_{P}^{\rm cycle}=u_{P}^{\text{cycle}}\text{ for
every cycle }P\text{ in }\Gamma.$

Now let

$\displaystyle\Sigma=\\{A_{v}(\Gamma)\,|\,v\in{\rm SZC}\\}.$

Let $\sim$ be the equivalence relation on $\Sigma$ defined by

$\displaystyle A_{v}(\Gamma)\sim A_{u}(\Gamma)\iff\text{there exists an
isomorphism }\phi\colon A_{v}(\Gamma)\to A_{u}(\Gamma)$
$\displaystyle\hphantom{A_{v}(\Gamma)\sim A_{u}(\Gamma)\iff{}}{}\text{such
that }\phi([a])=[a]\ \forall\,a\in V.$

Let $v\in{\rm SZC}$. From now on, we shall use $[v]$ to denote the equivalence
class of $v$ in ${\rm SZC}/_{\equiv}$. Furthermore, we shall use
$[A_{v}(\Gamma)]_{\sim}$ and $[A_{v}(\Gamma)]_{\cong}$ to denote the
equivalence classes of $A_{v}(\Gamma)$ in $\Sigma/_{\sim}$ and
$\Sigma/_{\cong}$ respectively, where $\cong$ denotes isomorphism of graded
algebras. Note that $\Sigma/_{\sim}$ and ${\rm SZC}/_{\equiv}$ are naturally
isomorphic sets via the map

$\displaystyle\phi\colon\ \Sigma/_{\sim}\to{\rm
SZC}/_{\equiv},\qquad[A_{v}(\Gamma)]_{\sim}\mapsto[v].$ (4.7)

######  Theorem 4.8.

Let $\Gamma=(V,E)$ be a connected graph. Then we have

$\displaystyle\Sigma/_{\sim}\cong{\rm SZC}/_{\equiv}\cong
H^{1}(\Gamma,\Bbbk^{*}),$

where the first isomorphism is an isomorphism of sets and the second
isomorphism is an isomorphism of groups.

###### Proof.

Define the map $\psi$ by

$\displaystyle\psi\colon\ {\rm SZC}\to H^{1}(\Gamma,\Bbbk^{*}),\qquad v\mapsto
f_{v}.$

It is clear that this map is a group homomorphism. Let $v,u\in{\rm SZC}$.
Then, we have

$\displaystyle\psi(v)=\psi(u)\iff f_{v}=f_{u}\iff f_{v}(P)=f_{u}(P)\text{ for
all }P\in\mathcal{C}$ $\displaystyle\hphantom{\psi(v)=\psi(u)}{}\iff
v_{P}^{\rm cycle}=u_{P}^{\rm cycle}\text{ for all }P\in\mathcal{C}\iff v\equiv
u.$

Hence, ${\rm SZC}/\ker\psi={\rm SZC}/_{\equiv}$.

Let us now prove that $\psi$ is surjective. By Lemma 4.6, there exists a basis
of cycles $\mathcal{B}=\\{C_{1},\ldots,C_{|E|-|V|+1}\\}$ of $\mathcal{C}$ such
that, for all $1\leq i\leq|E|-|V|+1$, there exists $(b_{i}\,|\,c_{i})\in
C_{i}$ such that $(b_{i}\,|\,c_{i})\in C_{k}$ if and only if $k=i$.

Let $f$ be a group homomorphism from $\mathcal{C}$ to $\Bbbk^{*}$. We define
coefficients as follows:

$\displaystyle v_{b,b}^{a}=1\text{ for all }\\{a,b\\}\in E,$ $\displaystyle
v_{b,c}^{a}=1\text{ for all }\\{a,b\\},\\{a,c\\}\in E,\qquad a\neq b_{i}\text{
for all }i=1,\ldots,|E|-|V|+1.$

Then, for all $i=1,\ldots,|E|-|V|+1$ we define

$\displaystyle 1/v_{c_{i},a}^{b_{i}}=v_{a,c_{i}}^{b_{i}}=f(C_{i})\text{ for
all }a\neq c_{k},\ 1\leq k\leq|E|-|V|+1,$ $\displaystyle
v_{c_{k},c_{i}}^{b_{i}}=f(C_{i})/f(C_{k})\text{ for all
}k\in\\{1,\ldots,|E|-|V|+1\\}\text{ such that }b_{i}=b_{k},$ $\displaystyle
v_{a,c}^{b_{i}}=1\text{ for all }\\{b_{i},a\\},\\{b_{i},c\\}\in E,a\neq
c_{i}\neq c.$

It is clear that the set $v=(v_{b,c}^{a}\,|\,\\{a,b\\},\\{a,c\\}\in E)$ is a
collection of skew-zigzag coefficients. Moreover, since $(b_{i}\,|\,c_{i})$ is
in $C_{k}$ if and only if $k=i$ for all $i=1,\ldots,|E|-|V|+1$, it is also
clear that $f_{v}(C_{i})=f(C_{i})$ for all $i=1,\ldots,|E|-|V|+1$. Thus, we
must have $\psi(v)=f_{v}=f$. Consequently, $\psi$ is surjective.

Since $\psi$ is surjective, the first isomorphism theorem implies ${\rm
SZC}/_{\equiv}\cong H^{1}(\Gamma,\Bbbk^{*})$. The result then follows using
(4.7). ∎

######  Remark 4.9.

In [5, Section 4], the authors state that “the moduli space of skew-zigzag
algebras is naturally isomorphic to $H^{1}(\Gamma,\mathbb{C}^{*})$”. In light
of Theorem 4.8, we assume that the moduli space they had in mind was
$\Sigma/\sim$.

Let $\Gamma=(V,E)$ be a graph. We say that a permutation $\sigma$ of the
elements of $V$ is a _graph automorphism_ of $\Gamma$ if, for all $a,b\in V$,
we have

$\displaystyle\\{a,b\\}\in E\iff\\{\sigma(a),\sigma(b)\\}\in E.$

The identity permutation is called the trivial graph automorphism. A graph is
said to be _asymmetric_ if it admits only the trivial graph automorphism,
otherwise it is said to be _symmetric_.

Suppose $\sigma$ is a graph automorphism of $\Gamma$. For any path
$P=(a_{1}\,|\,\dots\,|\,a_{n})$ we let
$\sigma(P)=(\sigma(a_{1})\,|\,\dots\,|\,\sigma(a_{n}))$. Notice that if $P$ is
a cycle, then $\sigma(P)$ is also a cycle. Moreover, if $P_{1}$ and $P_{2}$
are two paths, then it is clear that
$\sigma(P_{1}P_{2})=\sigma(P_{1})\sigma(P_{2})$. Thus, the map $\sigma$
induces an automorphism of the path algebra $\Bbbk D\Gamma$

$\displaystyle\phi_{\sigma}\colon\ \Bbbk D\Gamma\to\Bbbk D\Gamma,\qquad
P\mapsto\sigma(P).$

Fix a collection of skew-zigzag coefficients $v$. We define $(\sigma
v)_{b,c}^{a}$ for all $\\{a,b\\},\\{a,c\\}\in E$ as follows:

$\displaystyle(\sigma
v)_{b,c}^{a}=v_{\sigma^{-1}(b),\sigma^{-1}(c)}^{\sigma^{-1}(a)}.$

Then we define $\sigma v=((\sigma v)_{b,c}^{a}\,|\,\\{a,b\\},\\{a,c\\}\in E)$.
The map $(\sigma,v)\mapsto\sigma v$ defines an action of the group of graph
automorphisms on the set of skew-zigzag coefficients.

It is straightforward to verify that $\sigma(I_{v})=I_{\sigma v}$. Thus we
obtain an isomorphism

$\displaystyle\sigma\colon\ A_{v}(\Gamma)\to A_{\sigma
v}(\Gamma),\qquad[P]\mapsto[\sigma(P)].$ (4.8)

Let

$\displaystyle\psi\colon\ A_{v}(\Gamma)\to A_{u}(\Gamma)$

be an isomorphism of graded algebras where $v$ and $u$ are two collections of
skew-zigzag coefficients. Suppose that $V=\\{a_{1},\ldots,a_{n}\\}$. Although
it follows from more advance concepts, we will use an elementary approach to
prove that vertices must be mapped to vertices. For all $1\leq i\leq n$, let
$\psi([a_{i}])=\sum\limits_{j=1}^{n}\alpha_{ij}[a_{j}]$, where
$\alpha_{ij}\in\Bbbk$ for all $1\leq i,j\leq n$. For any two vertices
$a_{i},a_{k}\in V$, $1\leq i,k\leq n$, $i\neq k$, we have

$\displaystyle
0=\psi(0)=\psi([a_{i}][a_{k}])=\left(\sum_{j=1}^{n}\alpha_{ij}[a_{j}]\right)\left(\sum_{j=1}^{n}\alpha_{kj}[a_{j}]\right)=\sum_{j=1}^{n}\alpha_{ij}\alpha_{kj}[a_{j}].$

Thus, for any $j=1,\ldots,n$, at least one of $\alpha_{ij}$ or $\alpha_{kj}$
is 0. Hence, if $\alpha_{ij}\neq 0$ for some $1\leq i,j\leq n$, then
$\alpha_{kj}=0$ for all $1\leq k\leq n$, $k\neq i$. Moreover, since $\psi$ is
injective, $\psi([a_{i}])\neq 0$ for all $i=1,\ldots,n$, and so there exists
$j_{i}\in\\{1,\ldots,n\\}$ such that $\alpha_{ij_{i}}\neq 0$ and thus
$\alpha_{kj_{i}}=0$ for all $1\leq k\leq n$, $k\neq i$. Therefore, for any
$1\leq i,k\leq n$, $i\neq k$ we have $j_{i}\neq j_{k}$. As a result, we must
have $\alpha_{ij}=0$ for all $1\leq i,j\leq n$, $j\neq j_{i}$. Consequently,
we must have $\psi([a_{i}])=\alpha_{ij_{i}}[a_{j_{i}}]$ for some
$j_{i}\in\\{1,\ldots,n\\}$. Furthermore, notice that we have

$\displaystyle\phi([a_{i}])=\phi([a_{i}][a_{i}])=\phi([a_{i}])\phi([a_{i}])=\alpha_{ij_{i}}\alpha_{ij_{i}}[a_{j_{i}}]=\alpha_{ij_{i}}\phi([a_{i}]),$

for all $1\leq i\leq n$. Hence we must have $\alpha_{ij_{i}}=1$. Therefore,
any isomorphism $\psi\colon A_{v}(\Gamma)\to A_{u}(\Gamma)$ induces a graph
automorphism

$\displaystyle\sigma_{\psi}\colon\ V\to V,\qquad a_{i}\mapsto a_{j_{i}}.$

Then, we obtain the isomorphism

$\displaystyle\sigma_{\psi}^{-1}\circ\psi\colon\ A_{v}(\Gamma)\to
A_{\sigma^{-1}u}(\Gamma),$

Therefore

$\displaystyle\psi=\sigma_{\psi}\circ\big{(}\sigma_{\psi}^{-1}\circ\psi\big{)},\qquad\text{where
}\sigma_{\psi}^{-1}\circ\psi([a])=[a]\text{ for all }a\in V.$ (4.9)

######  Lemma 4.10.

Let $\Gamma$ be a connected asymmetric graph with at least three vertices, and
let $v$ and $u$ be two collections of skew-zigzag coefficients. Then
$A_{v}(\Gamma)\cong A_{u}(\Gamma)$ as graded algebras if and only if $v\equiv
u$.

###### Proof.

Let $\Gamma$ be a connected asymmetric graph with at least three vertices, and
$v$ and $u$ be two collections of skew-zigzag coefficients. Suppose that we
have an isomorphism $\phi\colon A_{v}(\Gamma)\to A_{u}(\Gamma)$. Then in
particular the map

$\displaystyle\sigma_{\phi}\colon\ V\to V,\qquad a\mapsto
s\left(\phi([a])\right),$

is a graph automorphism. Since $\Gamma$ is asymmetric, we must have
$\sigma_{\phi}(a)=a$ for all $a\in V$. Thus $\phi([a])=[a]$ for all $a\in V$.
Hence $v\equiv u$ by Lemma 4.3.

The reverse implication follows from the definition of the equivalence
relation. ∎

Let $\operatorname{Aut}(\Gamma)$ be the group of graph automorphisms of
$\Gamma$. We define a group action of $\operatorname{Aut}(\Gamma)$ on
$H^{1}(\Gamma,\Bbbk^{*})$ by

$\displaystyle(\sigma
f)(C)=f\big{(}\sigma^{-1}(C)\big{)},\qquad\sigma\in\operatorname{Aut}(\Gamma),\quad
f\in H^{1}(\Gamma,\Bbbk^{*}),\quad C\in\mathcal{C}.$

For any $\sigma\in\operatorname{Aut}(\Gamma)$ and any collection of skew-
zigzag coefficients $v$ we have an isomorphism given by (4.8) $\sigma\colon
A_{v}(\Gamma)\to A_{\sigma v}(\Gamma)$. It is clear that $A_{v}(\Gamma)\mapsto
A_{\sigma v}(\Gamma)$ defines a group action of $\operatorname{Aut}(\Gamma)$
on $\Sigma$ and that it preserves the equivalence relation $\sim$. Therefore,
we have an induced action of $\operatorname{Aut}(\Gamma)$ on $\Sigma/_{\sim}$,
given by $X\mapsto\sigma X$ for all $X\in\Sigma/_{\sim}$, where $\sigma
X=\\{\sigma x\,|\,x\in X\\}$.

######  Lemma 4.11.

For any graph $\Gamma$ we have the following isomorphism of sets

$\displaystyle(\Sigma/_{\sim})/\operatorname{Aut}(\Gamma)\cong\Sigma/_{\cong}.$

###### Proof.

Define

$\displaystyle\alpha\colon\
\Sigma/_{\sim}\to\Sigma/_{\cong},\qquad[A_{v}(\Gamma)]_{\sim}\mapsto[A_{v}(\Gamma)]_{\cong}.$

Clearly this is a well-defined surjective map. Moreover, notice that if $v$
and $u$ are two collections of skew-zigzag coefficients, then we have

$\displaystyle\alpha([A_{v}(\Gamma)]_{\sim})=\alpha([A_{u}(\Gamma)]_{\sim})\iff[A_{v}(\Gamma)]_{\cong}=[A_{u}(\Gamma)]_{\cong}$
$\displaystyle\hphantom{\alpha([A_{v}(\Gamma)]_{\sim})=\alpha([A_{u}(\Gamma)]_{\sim})}{}\iff\text{there
exists an isomorphism }\Phi\colon A_{v}(\Gamma)\to A_{u}(\Gamma).$

Recall from (4.9) that every isomorphism between skew-zigzag algebras can be
written as the composition of a graph automorphism and an isomorphism that
fixes the vertices. Thus, there exists $\sigma\in\operatorname{Aut}(\Gamma)$
and an isomorphism $\gamma$ that fixes the vertices such that

$\displaystyle\Phi=\sigma\circ\gamma\colon\
A_{v}(\Gamma)\xrightarrow{\gamma}A_{\sigma^{-1}u}(\Gamma)\xrightarrow{\sigma}A_{u}(\Gamma).$

Since $\gamma$ fixes the vertices, we have
$[A_{v}(\Gamma)]_{\sim}=[A_{\sigma^{-1}u}(\Gamma)]_{\sim}$ and so,

$\displaystyle\sigma[A_{v}(\Gamma)]_{\sim}=\sigma[A_{\sigma^{-1}u}(\Gamma)]_{\sim}=[A_{u}(\Gamma)]_{\sim}.$

Now, let $v$ and $u$ be two collections of skew-zigzag coefficients such that
there exists $\sigma\in\operatorname{Aut}(\Gamma)$ such that
$\sigma[A_{v}(\Gamma)]_{\sim}=[A_{u}(\Gamma)]_{\sim}$. Then, since
$A_{\sigma^{-1}u}(\Gamma)\in[A_{v}(\Gamma)]_{\sim}$, there exists an
isomorphism $\gamma$ that fixes the vertices such that we have the following

$\displaystyle
A_{v}(\Gamma)\xrightarrow{\gamma}A_{\sigma^{-1}u}(\Gamma)\xrightarrow{\sigma}A_{u}(\Gamma).$

Since $\sigma\circ\gamma$ is an isomorphism, we must have
$[A_{v}(\Gamma)]_{\cong}=[A_{u}(\Gamma)]_{\cong}$ and thus,
$\alpha([A_{v}(\Gamma)]_{\sim})=\alpha([A_{u}(\Gamma)]_{\sim})$.

Consequently, for any $X_{1},X_{2}\in\Sigma/_{\sim}$, we have

$\displaystyle\alpha(X_{1})=\alpha(X_{2})\iff\text{ there exists
}\sigma\in\operatorname{Aut}(\Gamma)\text{ such that }\sigma X_{1}=X_{2}.$

Hence, we finally obtain
$(\Sigma/_{\sim})/\operatorname{Aut}(\Gamma)\cong\Sigma/_{\cong}$. ∎

######  Theorem 4.12.

For any graph $\Gamma$ we have the following isomorphism of sets

$\displaystyle\Sigma/_{\cong}\cong
H^{1}(\Gamma,\Bbbk^{*})/\operatorname{Aut}(\Gamma).$

###### Proof.

By Theorem 4.8, we know that the map

$\displaystyle\psi\circ\phi\colon\ \Sigma/_{\sim}\to
H^{1}(\Gamma,\Bbbk^{*}),\qquad[A_{v}(\Gamma)]_{\sim}\mapsto f_{v},$

is an isomorphism, where the map $\phi$ is as in (4.7) and $\psi\colon{\rm
SZC}/_{\equiv}\to H^{1}(\Gamma,\Bbbk^{*}),$ $[v]\mapsto f_{v}$. Let us now
prove that it preserves the $\operatorname{Aut}(\Gamma)$-action, i.e., for all
$\sigma\in\operatorname{Aut}(\Gamma)$ and $v\in{\rm SZC}$, we have
$(\psi\circ\phi)(\sigma[A_{v}(\Gamma)]_{\sim})=\sigma(\psi\circ\phi)([A_{v}(\Gamma)]_{\sim})$.
We have

$\displaystyle(\psi\circ\phi)(\sigma[A_{v}(\Gamma)]_{\sim})=\psi([\sigma
v])=f_{\sigma v},$

and

$\displaystyle\sigma(\psi\circ\phi)([A_{v}(\Gamma)]_{\sim})=\sigma(\psi([v]))=\sigma(f_{v}).$

Let $C=(a_{1}\,|\,\ldots\,|\,a_{n})\in C$ be a cycle. Then,

$\displaystyle\sigma(f_{v})(C)=f_{v}\big{(}\sigma^{-1}(C)\big{)}=f_{v}\big{(}\big{(}\sigma^{-1}(a_{1})\,|\,\ldots\,|\,\sigma^{-1}(a_{n})\big{)}\big{)}$
$\displaystyle\hphantom{\sigma(f_{v})(C)=f_{v}\big{(}\sigma^{-1}(C)\big{)}}{}=v_{\sigma^{-1}(a_{n-1}),\sigma^{-1}(a_{2})}^{\sigma^{-1}(a_{1})}\cdots
v_{\sigma^{-1}(a_{n-2}),\sigma^{-1}(a_{n})}^{\sigma^{-1}(a_{n-1})}=f_{\sigma
v}(C).$

Hence $\sigma(f_{v})=f_{\sigma v}$. Consequently, we must have
$(\psi\circ\phi)(\sigma[A_{v}(\Gamma)]_{\sim})=\sigma(\psi\circ\phi)([A_{v}(\Gamma)]_{\sim})$.
So $\psi\circ\phi$ is an $\operatorname{Aut}(\Gamma)$-set isomorphism and
thus,

$\displaystyle(\Sigma/_{\sim})/\operatorname{Aut}(\Gamma)\cong
H^{1}(\Gamma,\Bbbk^{*})/\operatorname{Aut}(\Gamma).$

Lemma 4.11 then yields $\Sigma/_{\cong}\cong
H^{1}(\Gamma,\Bbbk^{*})/\operatorname{Aut}(\Gamma)$. ∎

In [5, Section 4] the authors state the following result without proof.

######  Corollary 4.13.

If $\Gamma$ is a tree then all of its skew-zigzag algebras are isomorphic.

###### Proof.

The follows immediately from Theorem 4.12 and the fact that trees have trivial
graph cohomology. ∎

######  Example 4.14.

Consider the graph $\Gamma=(V,E)$ given by $V=\\{a,b,c\\}$ and
$E=\\{\\{a,b\\},\\{b,c\\}$, $\\{a,c\\}\\}$. The associated double graph is

$a$$b$$c$

By Proposition 4.4, if $u,v\in{\rm SZC}$, then $A_{u}(\Gamma)\sim
A_{v}(\Gamma)$ if and only if $u^{\rm cycle}_{P}=v^{\rm cycle}_{P}$ for every
cycle $P$. Furthermore, recall that by (4.9), if $u,v\in{\rm SZC}$ are such
that $A_{v}(\Gamma)\cong A_{u}(\Gamma)$ via an isomorphism $\phi$, then $\phi$
can be written as the composition of a graph automorphism $\sigma$ with an
isomorphism $\gamma$ that fixes the vertices:

$\displaystyle\sigma\circ\gamma\colon\
A_{v}(\Gamma)\xrightarrow{\gamma}A_{\sigma^{-1}u}(\Gamma)\xrightarrow{\sigma}A_{u}(\Gamma).$

Thus, if $P$ is a cycle, then $v^{\rm cycle}_{P}=u^{\rm
cycle}_{\sigma^{-1}(P)}$. Conversely, if there exists
$\sigma\in\operatorname{Aut}(\Gamma)$, such that $v^{\rm cycle}_{P}=u^{\rm
cycle}_{\sigma^{-1}(P)}$ for every cycle $P$, then we have $A_{v}(\Gamma)\cong
A_{\sigma^{-1}u}(\Gamma)$ by Proposition 4.4 and
$A_{\sigma^{-1}u}(\Gamma)\cong A_{u}(\Gamma)$ by (4.8). Thus,
$A_{v}(\Gamma)\cong A_{u}(\Gamma)$. Consequently, $A_{v}(\Gamma)\cong
A_{u}(\Gamma)$ if and only if there exists some
$\sigma\in\operatorname{Aut}(\Gamma)$ such that
$v_{P}^{\text{cycle}}=u_{\sigma^{-1}(P)}^{\text{cycle}}$ for all cycles $P$.
Therefore, in order to compute $\Sigma/_{\sim}$ and $\Sigma/_{\cong}$, it
suffices to consider the cycle products of $u$ and $v$ ($u,v\in{\rm SZC}$)
along every cycle.

It is clear that the $C=(a\,|\,b\,|\,c\,|\,a)$ yields a basis for the cycle
space $\mathcal{C}$. So, if $u,v\in{\rm SZC}$, then $A_{u}(\Gamma)\sim
A_{v}(\Gamma)$ if and only if $u^{\rm cycle}_{C}=v^{\rm cycle}_{C}$ and
$A_{u}(\Gamma)\cong A_{v}(\Gamma)\iff u^{\rm cycle}_{C}=v^{\rm cycle}_{C}$ or
$u^{\rm cycle}_{C^{*}}=v^{\rm cycle}_{C}$.

Let $\Bbbk=\mathbb{C}$ and $xRy\iff x=y\text{ or }x^{-1}=y$
($x,y\in\mathbb{C}^{*}$). Consider the maps

$\displaystyle\phi\colon\
\Sigma/_{\sim}\to\mathbb{C}^{*},\qquad[A_{v}(\Gamma)]_{\sim}\mapsto v^{\rm
cycle}_{C},\qquad\psi\colon\
\Sigma/_{\cong}\to\mathbb{C}^{*}/R,\qquad[A_{v}(\Gamma)]_{\cong}\mapsto v^{\rm
cycle}_{C}.$

These maps are well-defined and injective by the above remark. Surjectivity
follows from the fact that if $x\in\mathbb{C}^{*}$, then the following is a
collection of skew-zigzag coefficients:

$\displaystyle
v_{b,b}^{a}=v_{c,c}^{a}=v_{a,a}^{b}=v_{c,c}^{b}=v_{a,a}^{c}=v_{b,b}^{c}=v_{b,c}^{a}=v_{c,b}^{a}=v_{a,c}^{b}=v_{c,a}^{b}=1,$
$\displaystyle v_{a,b}^{c}=x,\qquad v_{b,a}^{c}=x^{-1}.$

Thus, $\Sigma/_{\sim}\cong\mathbb{C}^{*}$, whereas
$\Sigma/_{\cong}\cong\mathbb{C}^{*}/R$. Note that,
$\mathbb{C}^{*}\ncong\mathbb{C}^{*}/R$ since, in $\mathbb{C}^{*}/R$, every
element is its own inverse which is not true in $\mathbb{C}^{*}$.

##  5 Other constructions of some skew-zigzag algebras

We conclude with a discussion of other constructions of certain skew-zigzag
algebras that have appeared in the literature [2, 3].

Let $\Gamma=(V,E)$ be a connected graph. We call
$\Omega=(\epsilon_{a,b}\in\Bbbk^{*}\,|\,a,b\in V)$ a collection of
_orientation coefficients_ if, for any pair of vertices $a,b\in V$, we have
$\epsilon_{a,b}=0$ if $\\{a,b\\}\notin E$ and $\epsilon_{a,b}=-\epsilon_{b,a}$
if $\\{a,b\\}\in E$. If $\Gamma$ has at least 3 vertices, we define the
algebra $B^{\Gamma}_{\Omega}$ to be the quotient algebra of the path algebra
of $D\Gamma$ by the two sided ideal $I_{\Omega}$ generated by elements of the
form

  * •

$(a\,|\,b\,|\,c)$ for $\\{a,b\\},\\{b,c\\}\in E$ and $a\neq c$, and

  * •

$\epsilon_{a,b}(a\,|\,b\,|\,a)-\epsilon_{a,c}(a\,|\,c\,|\,a)$ such that $a$ is
connected to both $b$ and $c$.

######  Lemma 5.1.

Let $\Gamma$ be a connected graph. For any collection of orientation
coefficients, $\Omega$, there exists a collection of skew-zigzag coefficients,
$v$, such that $B^{\Gamma}_{\Omega}=A_{v}(\Gamma)$.

###### Proof.

Let $\Omega=(\epsilon_{a,b})$ be a collection of orientation coefficients and
set $v_{b,c}^{a}=\epsilon_{a,c}/\epsilon_{a,b}$ for all
$\\{a,b\\},\\{a,c\\}\in E$. For any $\\{a,b\\},\\{a,c\\},\\{a,d\\}\in E$ we
have

$\displaystyle v_{b,b}^{a}=\frac{\epsilon_{a,b}}{\epsilon_{a,b}}=1,\qquad
v_{b,c}^{a}v_{c,b}^{a}=\frac{\epsilon_{a,c}}{\epsilon_{a,b}}\frac{\epsilon_{a,b}}{\epsilon_{a,c}}=1,\qquad\text{and}\qquad
v_{b,c}^{a}v_{c,d}^{a}v_{d,b}^{a}=\frac{\epsilon_{a,c}}{\epsilon_{a,b}}\frac{\epsilon_{a,d}}{\epsilon_{a,c}}\frac{\epsilon_{a,b}}{\epsilon_{a,d}}=1.$

Thus the set $v=(v_{b,c}^{a}\,|\,\\{a,b\\},\\{a,c\\}\in E)$ is a collection of
skew-zigzag coefficients. It is clear that $I_{\Omega}=I_{v}$. Thus, we have
$B^{\Gamma}_{\epsilon}=A_{v}(\Gamma)$. ∎

######  Definition 5.2 (orientation).

Let $\Gamma=(V,E)$ be a connected graph and let $D\Gamma=(V,E^{\prime})$ be
its double graph. A set $\epsilon\subseteq E^{\prime}$ is said to be an
_orientation_ of $D\Gamma$ if, for every $\\{a,b\\}\in E$, exactly one
directed edge in $D\Gamma$ between $a$ and $b$ is in $\epsilon$.

In [3, Section 2.1, p. 109] the authors fix an orientation $\epsilon$ of
$D\Gamma$. Then, they define orientation coefficients
$\Omega=(\epsilon_{a,b}\,|\,a,b\in V)$ as follows:

$\displaystyle\epsilon_{a,b}=\begin{cases}1&\text{if }(a\,|\,b)\in\epsilon,\\\
-1&\text{if }(b\,|\,a)\in\epsilon,\\\ 0&\text{if $a$ and $a$ are not
connected.}\end{cases}$ (5.1)

Notice that $\epsilon_{a,b}=-\epsilon_{b,a}$. Thus, by Lemma 5.1, we have
$B_{\Omega}^{\Gamma}=A_{v}(\Gamma)$, where

$\displaystyle
v=\left(v^{a}_{b,c}=\frac{\epsilon_{a,c}}{\epsilon_{a,b}}\,|\,\\{a,b\\},\\{a,c\\}\in
E\right).$

However, the following proposition shows that the converse of Lemma 5.1 is
false. Thus, the alternate definition of skew-zigzag algebras in terms of
orientation coefficients is more restrictive.

######  Proposition 5.3.

If $\Gamma$ is not a bipartite graph, then $A(\Gamma)$ is not isomorphic to
$B_{\Omega}^{\Gamma}$ for any collection of orientation coefficients $\Omega$.

###### Proof.

Suppose that $\Gamma$ is not a bipartite graph and that there exists an
isomorphism

$\displaystyle\phi\colon\ A(\Gamma)\to B^{\Gamma}_{\Omega}$

for some collection of orientation coefficients $\Omega$. For all $a\in V$ let
$\phi([a])=[x_{a}]$. So, for all $\\{a,b\\}\in E$ we have
$\phi([a\,|\,b])=\alpha_{a,b}[x_{a}\,|\,x_{b}]$ for some
$\alpha_{a,b}\in\Bbbk$. Consequently, for any $\\{a,b\\}$, $\\{a,c\\}\in E$ we
have

$\displaystyle\alpha_{a,b}\alpha_{b,a}[x_{a}\,|\,x_{b}\,|\,x_{a}]=\phi([a\,|\,b\,|\,a])=\phi([a\,|\,c\,|\,a])=\alpha_{a,c}\alpha_{c,a}[x_{a}\,|\,x_{c}\,|\,x_{a}].$

Therefore,

$\displaystyle\frac{\alpha_{a,c}\alpha_{c,a}}{\alpha_{a,b}\alpha_{b,a}}=\frac{\epsilon_{x_{a},x_{c}}}{\epsilon_{x_{a},x_{b}}}.$
(5.2)

By [4, Proposition 1.6.1], since $\Gamma$ is not bipartite, it contains a
cycle of odd length. Thus, there is a cycle $C=(a_{1}\,|\,\ldots\,|\,a_{n})$
in $D\Gamma$ with $n$ even. So, (5.2) yields

$\displaystyle
1=\frac{\alpha_{a_{1},a_{2}}\alpha_{a_{2},a_{1}}}{\alpha_{a_{1},a_{n-1}}\alpha_{a_{n-1},a_{1}}}\frac{\alpha_{a_{2},a_{3}}\alpha_{a_{3},a_{2}}}{\alpha_{a_{2},a_{1}}\alpha_{a_{1},a_{2}}}\cdots\frac{\alpha_{a_{n-1},a_{1}}\alpha_{a_{1},a_{n-1}}}{\alpha_{a_{n-1},a_{n-2}}\alpha_{a_{n-2},a_{n-1}}}$
$\displaystyle\hphantom{1}{}=\frac{\epsilon_{x_{a_{1}},x_{a_{2}}}}{\epsilon_{x_{a_{1}},x_{a_{n-1}}}}\frac{\epsilon_{x_{a_{2}},x_{a_{3}}}}{\epsilon_{x_{a_{2}},x_{a_{1}}}}\cdots\frac{\epsilon_{x_{a_{n-1}},x_{a_{1}}}}{\epsilon_{x_{a_{n-1}},x_{a_{n-2}}}}.$
(5.3)

Since $\\{a_{i},a_{i+1}\\}\in E$ for all $1\leq i\leq n-1$,
$\epsilon_{x_{a_{j}},x_{a_{k}}}\neq 0$ for $1\leq j,k,\leq n-1$, $j\neq k$.
Moreover, we know that $\epsilon_{a,b}=-\epsilon_{b,a}$ for all $a,b\in V$.
Consequently, (5.3) yields

$\displaystyle 1=(-1)^{n-1}=-1.$

This contradiction implies that $A_{u}(\Gamma)\ncong B^{\Gamma}_{\Omega}$ for
any $\Omega$. ∎

Suppose $(\epsilon_{a,b})$ is a collection of orientation coefficients. For
$x\in V$, define $V_{x}$ and $y_{x}$ as in Proposition 3.7. Notice that if we
modify the set $J$ in (3.2) by setting

$\displaystyle
J^{\prime}\coloneqq\\{[a],[b\,|\,c],\epsilon_{x,y_{x}}[x\,|\,y_{x}\,|\,x]\,|\,a,x\in
V,\text{ }b,c\in V\text{ such that }\\{b,c\\}\in E\\},$

then $J^{\prime}$ is independent of the choice of $y_{x}$ for every $x\in V$.
Indeed, for any $y,z\in V_{x}$, we have

$\displaystyle\epsilon_{x,y}[x\,|\,y\,|\,x]=\epsilon_{x,z}[x\,|\,z\,|\,x].$

In [2, Section 6.1, p. 2516], the authors fix an orientation $\epsilon$ of
$\Gamma$ and define coefficients $\epsilon_{a,b}$ $(a,b\in V)$ as in (5.1).
They then define a diagrammatic algebra using these coefficients. Their
algebra is in fact isomorphic to the algebra $B_{\Omega}^{\Gamma}$, for
$\Omega=(\epsilon_{a,b}\,|\,a,b\in V)$, via the map

$\displaystyle A\to A(\Gamma),\qquad\leavevmode\hbox to12.38pt{\vbox
to81.59pt{\pgfpicture\makeatletter\hbox{\hskip 6.19064pt\lower-11.37155pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ } {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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So, by Lemma 5.1, $A$ is isomorphic to $A_{v}(\Gamma)$ where
$v=\big{(}v^{a}_{b,c}=\frac{\epsilon_{a,c}}{\epsilon_{a,b}}\,|\,\\{a,b\\},\\{a,c\\}\in
E\big{)}$. However, if $\Gamma$ is not bipartite, then this is not isomorphic
to the zigzag algebra by Proposition 5.3.

Note on the LaTeXversion. For the interested reader, the tex file of this
paper includes hidden details of some straightforward computations and
arguments that are omitted in the pdf file. These details can be displayed by
switching the details toggle to true in the tex file and recompiling.

### Acknowledgements

This work was completed under the supervision of Professor Alistair Savage.
The author would like to thank Professor Savage immensely for his patience and
guidance throughout this paper as well as the opportunity to write this paper.
The author would also like to thank the University of Ottawa and the Work-
Study Program for their support. Finally, the author would like to thank the
referees for their useful comments and for providing a reference for
Proposition 3.9.

## References

  * [1] Assem I., Simson D., Skowroński A., Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, Vol. 65, Cambridge University Press, Cambridge, 2006.
  * [2] Cautis S., Licata A., Heisenberg categorification and Hilbert schemes, Duke Math. J. 161 (2012), 2469–2547, arXiv:1009.5147.
  * [3] Cautis S., Licata A., Sussan J., Braid group actions via categorified Heisenberg complexes, Compos. Math. 150 (2014), 105–142, arXiv:1207.5245.
  * [4] Diestel R., Graph theory, Graduate Texts in Mathematics, Vol. 173, 4th ed., Springer, Heidelberg, 2010, available at http://diestel-graph-theory.com/.
  * [5] Huerfano R.S., Khovanov M., A category for the adjoint representation, J. Algebra 246 (2001), 514–542, math.QA/0002060.
  * [6] Kock J., Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, Vol. 59, Cambridge University Press, Cambridge, 2004.
  * [7] Rosso D., Savage A., A general approach to Heisenberg categorification via wreath product algebras, arXiv:1507.06298.
  * [8] Sunada T., Topological crystallography. With a view towards discrete geometric analysis, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer, Tokyo, 2013.
  * [9] Zimmermann A., Representation theory. A homological algebra point of view, Algebra and Applications, Vol. 19, Springer, Cham, 2014.


# Thermal radiation and near-field thermal imaging of a plasmonic Su-
Schrieffer-Heeger chain

Florian Herz and Svend-Age Biehs∗ Institut für Physik, Carl von Ossietzky
Universität, D-26111 Oldenburg, Germany<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We investigate the direct thermal emission spectrum of a plasmonic Su-
Schrieffer-Heeger chain of InSb nanoparticles as well as its far-field
emission due to near-field scattering by a sharp tip as it is used in
scattering type thermal microscopes. We discuss the measurability of the
topological phase transition in such far-field and near-field experiments and
highlight the signatures of the topological edge modes. We further discuss the
impact of a substrate.

Nanoscale thermal imaging microscopes like the thermal radiative scanning
tunneling microscope (TRSTM) DeWilde ; Babuty , the thermal infrared near-
field spectroscope (TINS) Huth2011 ; Jones ; OCallahan , and the scanning
noise microscope (SNoiM) Lin ; WengEtAl2018 ; Komiyama  are collecting far-
field thermal radiation which is either scattered or directly emitted by a
thermal profiler. The sharp tip at the foremost part of the thermal profiler
allows to scan surfaces with a high lateral resolution. Interestingly,
microscopes like the SNoiM that scatter the evanescent near-field of the
thermally fluctuating sources inside the sample also allow to measure local
temperatures of these thermal sources like the electron temperature of a hot
electron gas in a quantum well structure WengEtAl2018 , for instance.
Recently, another experimental technique has been introduced to measure
thermal emission spectra of single silica spheres with sizes smaller than the
thermal wavelength without any thermal profiler ExperimentDeWilde .

First theoretical approaches have tried to model the signal obtained with the
scattering type thermal microscopes by a single dipole representing the
foremost part of the probe Joulain2 ; Jarzembski ; Herz2018 ; Herz2021 . A
more realistic modeling of the tip apex Edalatpour2  within the so-called
discrete dipole approximation (DDA) Yurkin  has been brought forward. These
methods are based on a many-body treatment of thermal radiation within the
framework of fluctuational electrodynamics RMP  and have also been used to
handle near-field radiation problems Edalatpour1 ; Ekeroth ; Villa . Recently,
the many-body theory for describing far-field emission of a collection of
nanoscale emitters in vacuum Ekeroth ; Centini2015 ; Centini2020  has been
generalized for arbitrary environments Herz2022  where the temperatures of the
environment and different emitters can be chosen separately. Within the DDA
this method is flexible enough to calculate the direct thermal emission of
macroscopic objects as measured in Ref. ExperimentDeWilde  as well as the
signal of different scattering based thermal microscopes for which the
temperatures for the tip, the sample, and the environment are different.

In this letter, we want to address theoretically the question whether it is
possible to find a signature of the topological edge modes in a plasmonic Su-
Schrieffer-Heeger (SSH) chain with the different kinds of scattering type
near-field microscopes or the direct measuring method described in Ref.
ExperimentDeWilde . That the SSH model can be realized with a relatively
simple chain of plasmonic nanoparticles (NPs) has been shown by several
theoretical works Weick1 ; Weick2 ; Weick3 ; OESSH ; ACSphotonSSH ; JAPSSH .
Very recently, an experimental proof could be provided using a photo-emission
electron microscopy ExpSSH  technique. Since the topological edge modes can
also be excited thermally and, therefore, contribute to the near-field
radiative heat flux OTTSSH ; OTTHoney  and the thermal near-field energy
density OTTSSH2D , we expect that also thermal radiation measurements will be
able to measure the topological transition and maybe even the breakdown of the
bulk-edge correspondence PocockEtAl2019 ; OTTHoney  in such simple plasmonic
SSH structures. To verify at least theoretically whether this expectation can
be fulfilled with a direct thermal emission measurement as in Ref.
ExperimentDeWilde  or one of the scattering type near-field thermal
microscopes DeWilde ; Babuty ; Huth2011 ; Jones ; OCallahan ; Lin ;
WengEtAl2018 ; Komiyama , we calculate the directly far-field emitted spectral
power $P_{\rm dir}$ of a collection of $N$ InSb NPs with temperatures
$T_{1}=\ldots=T_{N}\equiv T_{\rm ch}$ in a SSH chain configuration embedded in
a vacuum background filled with a photon gas with temperature $T_{b}<T_{\rm
ch}$ using the expression derived in Eq. (62) of our work Herz2022 . This
expression can be obtained by calculating the mean Poynting vector
$\langle\mathbf{S}\rangle$ due to thermal sources within the NPs only
employing the framework of fluctuational electrodynamics. Then integrating it
over an “observation plane” above the NP chain as depicted in Fig. 1

$P_{\rm tot}=\int\\!\\!\\!{\rm d}x\\!\\!\int\\!\\!\\!{\rm
d}y\,\langle\mathbf{S}\rangle\cdot\mathbf{e}_{z}$ (1)

with the unit vector $\mathbf{e}_{z}$ in z-direction. This quantity gives then
the emitted power by thermal radiation of the NP in the far-field $P_{\rm
tot}=P_{\rm dir}$. All the details of the calculation can be found in Ref.
Herz2022 . By adding a substrate there will also be thermal sources in the
substrate with temperature $T_{s}$ resulting in a contribution of thermal
radiation $P_{\rm s}$ of the substrate itself. Additionally, one has to
consider the part $P_{\rm abs}$ of the substrate emission absorbed within the
NPs which does not reach the detection plane, a scattering part $P_{\rm sc}$
due to scattering between particles and substrate, and the direct part $P_{\rm
dir}$ resulting in $P_{\rm tot}=P_{\rm s}+P_{\rm abs}+P_{\rm sc}+P_{\rm dir}$.
The thermal emission of the substrate itself as well as the part of this
thermal emission which is absorbed in the NPs will be omitted in the entire
manuscript because we focus on the thermal emission of the NP chain itself.
The explicit expression for the scattering part $P_{\rm sc}$ used below is
given in Eq. (68) of Ref. Herz2022 . Finally, we also calculate the far-field
scattering of a silicon tip as in the TINS experiments using the tip
configuration from Ref. Herz2022 . The results for $P_{\rm dir}$ and $P_{\rm
sc}$ will allow us to explore theoretically the thermal emission spectra of
SSH chains directly as it can be measured with direct methods like in Ref.
ExperimentDeWilde  or with the tip-based methods such as TRSTM, TINS, or
SNoiM.

Figure 1:  Sketch of a SSH NP chain with temperatures
$T_{1}=\ldots=T_{N}=T_{\rm ch}$ at distance $d$ to a substrate with
temperature $T_{s}$ emitting thermal radiation into a vacuum background at
temperature $T_{b}$. The emitted far-field power is determined by the
integration of the mean Poynting vector $\langle\mathbf{S}\rangle$ over the
observation plane indicated by the dashed line. The NPs of the two sub-
lattices with lattice constant $a$ are labeled by $A$ and $B$. The distance
between sub-lattices $A$ and $B$ is $t=\beta a/2$.

Before we start to carry out the numerical evaluations, we revisit some
properties of plasmonic SSH chains. The eigenmodes in a chain of $N$ identical
spherical NPs with radius $R$ described by a scalar polarizability $\alpha$
can be obtained following the procedure detailed in Ref. ford . The induced
dipole moment $\mathbf{p}_{i}$ for a given NP $i$ ($i=1,\ldots,N$) at a
position $\mathbf{r}_{i}$ due to the field of all other dipoles is
$\mathbf{p}_{i}=\epsilon_{0}\alpha\mathbf{E}(\mathbf{r}_{i})$ with the field
of all other dipoles given by

$\mathbf{E}(\mathbf{r})=\mu_{0}\omega^{2}\sum_{j\neq
i}\mathds{G}(\mathbf{r}_{i},\mathbf{r}_{j})\mathbf{p}_{j}.$ (2)

Here we make use of the Green’s tensor
$\mathds{G}(\mathbf{r}_{i},\mathbf{r}_{j})$. It can in general be written as
the sum of a vacuum contribution $\mathds{G}_{\rm vac}$ and a scattering part
$\mathds{G}_{\rm sc}$ which is important when considering a certain
environment like a substrate. Furthermore, we have introduced the permittivity
and permeability of vacuum $\epsilon_{0}$ and $\mu_{0}$ which are related to
the vacuum light velocity by $c^{2}=1/\epsilon_{0}\mu_{0}$. Combining both
equations into a block matrix equation gives the eigenvalue equation

$\tilde{\underline{\mathds{M}}}\underline{\mathbf{p}}=\frac{1}{\alpha}\underline{\mathbf{p}}$
(3)

where the $3N$-dimensional block vector is defined by
$\underline{\mathbf{p}}=(\mathbf{p}_{1},\ldots,\mathbf{p}_{N})^{t}$ and the
$3N\times 3N$ block matrix by ($i,j=1,\ldots,N$)

$\tilde{\underline{\mathds{M}}}_{ij}=(1-\delta_{ij})\frac{\omega^{2}}{c^{2}}\mathds{G}(\mathbf{r}_{i},\mathbf{r}_{j}).$
(4)

For any given set of dipoles, the complex eigenfrequencies of the modes of the
system can be determined by solving the eigenvalue equation (3) by determining
the non-trivial solutions from the condition
$\det(\tilde{\underline{M}}-\frac{1}{\alpha}\underline{\mathds{1}})=0$. In
this work, we use the so-called strong-scattering expression of the
polarizability which is given by

$\alpha_{\perp/\parallel}=\frac{V\chi_{E}}{1-\frac{\omega^{2}}{c^{2}}V\chi_{E}\langle
G_{\perp/\parallel}\rangle}$ (5)

where $\chi_{E}=\epsilon-1$ is the susceptibility of the NP with permittivity
$\epsilon$ and $V=4\pi R^{3}/3$ is its volume. The volume average of the
Green’s function $\langle\mathds{G}\rangle={\rm diag}(\langle
G_{\perp}\rangle,\langle G_{\perp}\rangle,\langle G_{\parallel}\rangle)$ is
performed over the particle volume taking the full Green’s function into
account. It regularizes the divergence typically encountered when taking the
self interaction into account. The explicit expressions for the case of a
vacuum environment or a planar substrate can be found in Ref. Herz2022 , for
instance.

Interestingly, when considering an infinite chain of NPs along an axis
parallel to the x-axis with two particles per unit cell labeled by $A$ and $B$
as depicted in Fig. 1 and introducing a Bloch ansatz for the dipole moments
the eigenvalue equation can be translated to a $6$ dimensional eigenvalue
equation

$\underline{\mathds{M}}\underline{\mathbf{p}}_{AB}=\frac{1}{\alpha}\underline{\mathbf{p}}_{AB}$
(6)

relating the eigenfrequencies to the Bloch vector $k$. Now, we have the block
vectors $\underline{\mathbf{p}}_{AB}=(\mathbf{p}_{A},\mathbf{p}_{B})^{t}$ for
the dipole moments of the two particles $A$ and $B$ in the unit cell and the
block matrix ($\gamma,\delta=A,B$)

$\underline{\mathds{M}}_{\gamma,\delta}=\frac{\omega^{2}}{c^{2}}\sum_{j\neq
0}\mathds{G}(\mathbf{r}_{\gamma_{0}},\mathbf{r}_{\delta_{j}}){\rm
e}^{\mathrm{i}kx_{\delta_{j}}}$ (7)

where the positions of the particles are defined by
$\mathbf{r}_{A_{j}}=(jd,0,z)^{t}$ and
$\mathbf{r}_{B_{j}}=((j+\beta/2)d,0,z)^{t}$ where $x_{\delta_{j}}$ is the
corresponding x-component of $\mathbf{r}_{A_{j}}$ or $\mathbf{r}_{B_{j}}$.
Note that we shifted the particle positions by $z$ from the x-axis because we
will later consider the chain at a distance $z=d+R$ above a substrate which
will be in the x-y plane as indicated in Fig. 1. If there is no substrate and
the chain is in vacuum, then the Green’s tensor is diagonal
$\mathds{G}=\mathds{G}_{\rm vac}={\rm
diag}(G_{\parallel},G_{\perp},G_{\perp})$ due to the high symmetry with the
explicit expressions for the components given in Ref. OTTSSH , for instance.
In this case, the $6$ dimensional eigenvalue equation splits into a two
dimensional eigenvalue equation for the longitudinal modes and two identical
eigenvalue equations for the transversal modes in the chain
($\nu=\parallel,\perp$)

$\underline{\mathds{M}}_{\nu}\underline{\mathbf{p}}_{AB,\nu}=\frac{1}{\alpha}\underline{\mathbf{p}}_{AB,\nu}$
(8)

where $\underline{\mathbf{p}}_{AB,\nu}=(p_{A,\nu},p_{B,\nu})^{t}$ contains
only the components of the dipoles parallel or perpendicular to the chain axis
(here, either $x$ or $y$/$z$ direction) and correspondingly
($\gamma,\delta=A,B$)

$\underline{M}_{\gamma,\delta,\nu}=\frac{\omega^{2}}{c^{2}}\sum_{j\neq
0}G_{\nu}(\mathbf{r}_{\gamma_{0}},\mathbf{r}_{\delta_{j}}){\rm
e}^{\mathrm{i}kx_{\delta_{j}}}.$ (9)

It has now been shown that exactly the eigenvalue equations (8) are in the
quasi-static limit equivalent to a two-band Schrödinger equation for the SSH
model when focusing to nearest-neighbor interaction Weick1 ; Weick2 ; Weick3 ;
OESSH ; ACSphotonSSH ; JAPSSH . As a consequence, in this limit it can be
shown that the Zak phase is either $0$ for $\beta\leq 1$ or $\pi$ for
$\beta>1$ so that there is a phase transition from a topological trivial phase
(TTP) for $\beta\leq 1$ to a non-trivial phase (TNTP) for $\beta>1$. This
transition is robust when including long-range interactions, retardation, and
dissipation Weick2 ; Weick3 ,

Now, the bulk edge correspondence guarantees that in a finite chain of NPs
there will be topologically protected edge modes in exactly that TNTP. These
edge modes have been studied for plasmonic SSH chains in great detail Weick1 ;
Weick2 ; Weick3 ; OESSH ; ACSphotonSSH ; JAPSSH . They can increase the near-
field energy density of the particles at the edge of the chain OTTSSH2D , but
also provide an important heat flux channel along the chain OTTSSH . In Fig.
2(a) and (b) we show the eigenmode frequencies for the longitudinal and
transversal modes in vacuum for different values of $\beta$ for a chain of
InSb NPs. It can first be seen that there is a symmetry for $\beta=1\pm x$ for
different values of $x$. This symmetry is only broken by the appearance of the
edge modes in the band gap for $\beta>1$. When looking at the far-field
spectrum of the direct thermal emission of such a chain of NPs in Fig. 2(c)
and (d) we can see that the thermal emission is given by a contribution of the
lower band of the longitudinal modes and a contribution of the upper band of
the transversal modes. This is because only the modes in these bands have a
dipole moment per unit cell and are, therefore, also called bright modes in
contrast to the dark modes Weick1  in the other bands which do not couple to
the radiation field. Apart from this feature, it can be observed that the
topological edge modes do contribute to the thermal emission even though these
modes are highly localized at the edges of the chains Weick1 ; OESSH ;
ACSphotonSSH ; JAPSSH . To visualize this feature, we plot the emission ration

$\eta=\frac{P^{\beta=1+x}}{P^{\beta=1-x}}$ (10)

for $x=0.1,0.2,0.3$. In Fig. 2(e) it can be seen that the far field emission
patterns for $\beta=1\pm x$ are similar but there is a noticeable difference
between the spectra $\beta>1$ and $\beta<1$ at the edge mode frequency
$\omega_{\rm em}=1.751\times 10^{14}\,{\rm rad/s}$ which approximately
coincides with the single particle resonance frequency $\omega=1.75\times
10^{14}\,{\rm rad/s}$. This feature indicates a stronger far-field emission at
the edge mode frequency in the TNTP. On the other hand, for $\beta>1$ the
emission of the two bands is slightly diminished which is in agreement with
the fact that one mode of each band for $\beta<1$ becomes an edge mode so that
for $\beta>1$ less band-modes can radiate into the far-field for $\beta>1$.

Figure 2:  Thermal emission of a free standing SSH chain of $N=10$ InSb NPs of
radius $R=100\,{\rm nm}$ and lattice constant $a=8R$ assuming a temperature of
$T_{\rm ch}=700\,{\rm K}$ of the NPs and a temperature of $T_{b}=300\,{\rm K}$
for the environment. (a) and (b) eigenmode frequencies for the longitudinal
and transversal modes as a function of $\beta$ derived from the condition
$\det(\tilde{\underline{M}}-\frac{1}{\alpha}\underline{\mathds{1}})=0$. (c)
Thermal emission spectrum $P_{\rm dir,\omega}$ for different values of
$\beta$. (d) Thermal emission spectrum $P_{\rm dir,\omega}$ for $\beta=0.7$
showing the contribution of the longitudinal and transversal modes,
separately. (e) Ratio $\eta=P^{\beta=1+x}/P^{\beta=1-x}$ for $x=0.1,0.2,0.3$.
The vertical dashed line indicates the single particle resonance frequency
$\omega=1.75\times 10^{14}\,{\rm rad/s}$.

Let us now turn to the impact of a substrate. As substrate we choose Ge which
basically has no dispersion or dissipation but a relatively large permittivity
of $\epsilon_{\rm Ge}=16$ in the infrared region. Now, we have not only a
direct thermal emission $P_{\rm dir}$ of the NP chain but, due to the presence
of the substrate, also a scattered part $P_{\rm sc}$, direct thermal emission
of the substrate $P_{\rm sub}$, and thermal emission of the substrate $P_{\rm
abs}$ which is absorbed by the particles. Here, we only focus on the
scattering part and the direct thermal emission of the NPs because these parts
contain the information about the thermal emission of the NPs. In Fig. 3 the
features are very similar to the SSH chain without substrate in Fig. 2. The
difference is that, due to the presence of the surface, we obtain two edge
modes; a transversal edge mode (y polarized) and a mixed transversal-
longitudinal mode (z-x polarized). In Fig. 3 we only see the enhanced $\eta$
due to the mixed edge mode. The edge mode resonance is more and more red-
shifted when approaching the substrate as could be expected from previous
observations Babuty ; Edalatpour2 ; Herz2022 . For example, for $d=100\,{\rm
nm}$ we find the resonance at $\omega=1.745\times 10^{14}\,{\rm rad/s}$ which
is slightly smaller than the edge mode’s eigenfrequency $\omega_{\rm
em,mix}=1.747\times 10^{14}\,{\rm rad/s}$. Such small differences between the
position of the resonance in $\eta$ and the exact edge mode frequency has
already been observed for the free chain in Fig. 2. Furthermore, the main
contribution stems from the directly emitted part of the NPs. We want to
emphasize that the curves for $50\,{\rm nm}$ have to be taken with some
caution because for such small distances multi-polar modes might play an
important role. Therefore, to obtain results for the case where the NPs touch
the interface, more elaborated DDA calculations for each NP of the chain have
to be carried out which are out of the scope of this letter. However, the
trend of the red-shift of the edge mode resonance seems reasonable, since it
approaches the single particle resonance frequency observed in Fig. 2(e)
without substrate. Hence, even by placing the NPs on a substrate, the edge
modes will lead to an increased thermal emission at the corresponding red-
shifted edge mode frequency. Therefore, far-field measurements like in Ref.
ExperimentDeWilde  could be carried out to observe the topological phase
transition in an SSH chain of NPs.

Figure 3:  Thermal emission of the SSH chain with parameters as in Fig 2 in
edge to edge distance $d=50\,{\rm nm},100\,{\rm nm},200\,{\rm nm},300\,{\rm
nm}$ to a Ge substrate. Ratio $\eta=P^{\beta=1.3}/P^{\beta=0.7}$ from Eq. (10)
for the sum $P=P_{\rm dir}+P_{\rm sc}$ of direct thermal emission of the chain
$P_{\rm dir}$ and the scattering part $P_{\rm sc}$. We choose $T_{s}=T_{\rm
ch}=500\,{\rm K}$ and $T_{b}=300\,{\rm K}$.

Finally, we want to examine the possibility to measure the topological phase
transition in the SSH chain with a thermal profiler scanning at an edge-to-
edge distance of 50 nm above the SSH chain as depicted in Fig. 4(a). To this
end, we consider a Si tip modeled by $5338$ voxels within the DDA as in Ref.
Herz2022  and as depicted in Fig. 4(b). As already seen in Fig. 3, the bands
and resonances shift due to presence of the substrate and, here, also due to
the tip so that the total emitted power $P=P_{\rm dir}+P_{\rm sc}$ is similar
to $P_{\rm dir}$ for the free chain in Fig. 2(c) and (d) but with a red-shift.
As might be expected, $\eta$ shown in Fig. 4(d) is quite similarly enhanced at
the edge mode frequency as in the case without tip shown in Fig. 3 when
placing the tip over the edge NP of the chain. Interestingly this feature also
exists if the tip is positioned over the center NP even though the enhancement
is slightly smaller for the center NP than for the edge NP. The most striking
feature is shown in Fig. 4(e) where the ratio of $P_{e}/P_{c}$ is plotted.
Here, $P_{e}$ is the emitted spectral power for the tip over the edge particle
and $P_{c}$ is the spectral emitted power when having the tip over the center
particle. First, it can be seen that in the regions of the lower and upper
bands around $1.73\times 10^{14}\,{\rm rad/s}$ and $1.755\times 10^{14}\,{\rm
rad/s}$ the ratio $P_{e}/P_{c}$ exhibits a peak for the TTP and TNTP. On the
other hand, at the frequency $1.743\times 10^{14}\,{\rm rad/s}$ we see that
$P_{e}/P_{c}$ is increased and shows a local maximum in the TNTP for
$\beta=1.3$ [as also seen in Fig. 3 and 4(d)], whereas for the TTP with
$\beta=0.7$ the ratio $P_{e}/P_{c}$ shows a local minimum. Furthermore, at the
frequency $1.75\times 10^{14}\,{\rm rad/s}$ we find the opposite behavior;
$P_{e}/P_{c}$ shows a local minimum for the TNTP and maximum for the TTP. We
associate these features with the mixed transversal-longitudinal and the
transversal edge modes at $\omega_{\rm em,mix}=1.747\times 10^{14}\,{\rm
rad/s}$ and $\omega_{\rm em,tr}=1.749\times 10^{14}\,{\rm rad/s}$,
respectively. Note that the transversal mode frequency coincides very well
with the frequency where $P_{e}/P_{c}$ shows a local minimum, whereas the
mixed edge mode frequency $1.747\times 10^{14}\,{\rm rad/s}$ is a little bit
higher than the frequency $1.743\times 10^{14}\,{\rm rad/s}$ for which we
observed the maximum in $P_{e}/P_{c}$ and for $\eta$ in Figs. 3. This shift of
resonance and edge mode frequency is in agreement with the observation of the
free SSH chain in Fig. 2(e) and the SSH chain above the Ge substrate in Fig.
3.

Figure 4:  Thermal emission and scattering of the SSH chain of $N=10$ InSb NPs
in edge to edge distance $d=100\,{\rm nm}$ to a Ge substrate due to the
presence of a scattering tip. (a) Sketch of the configuration. (b) DDA of the
scattering tip made of 5388 Si voxels. (c) We show the total emitted spectral
power $P=P_{\rm dir}+P_{\rm sc}$ for the tip over an edge and a center
particle for $\beta=0.7$ and $1.3$. (d) The ratio
$\eta=P^{\beta=1.3}/P^{\beta=0.7}$ from Eq. (10) for $P=P_{\rm dir}+P_{\rm
sc}$. (e) The ratio of $P$ above the edge particle and $P$ above the center
particle for $\beta=0.7$ and $1.3$. We choose $T_{s}=T_{\rm ch}=500\,{\rm K}$
and $T_{\rm tip}=T_{b}=300\,{\rm K}$. The shaded areas visualize the bright
bands of the mixed and transversal modes and the vertical dashed lines the
corresponding edge mode frequencies $\omega_{\rm em,mix}=1.747\times
10^{14}\,{\rm rad/s}$ and $\omega_{\rm em,tr}=1.749\times 10^{14}\,{\rm
rad/s}$.

In conclusion, we have first shown that the thermal far-field emission of a
single SSH chain of NPs has a distinct contribution of the edge modes in the
TNTP. Furthermore, the presence of a substrate leads to a red-shift of the
photonic bands in the NP chain and the edge mode frequency. The increased
thermal emission at the edge-mode frequency can still be observed so that the
topological phase transition should be observable in far-field experiments
like in Ref. ExperimentDeWilde . Furthermore, we have shown that with a
thermal near-field profiler, such as in TRSTM DeWilde ; Babuty , TINS Huth2011
; Jones ; OCallahan , and SNoiM Lin ; WengEtAl2018 ; Komiyama , similar
features can be seen as for the direct thermal emission of the SSH chain.
However, the advantage of the thermal profiler is that it can measure locally
and our results indicate that in the TNTP there is an enhancement of the
thermal emission when the tip is over the edge particle compared to the case
where the tip is over the center particle. This enhancement becomes striking
when the topological trivial and non-trivial cases are compared so that there
is a clear signature of the edge states measurable with a near field thermal
profiler.

S.-A. B. acknowledges support from Heisenberg Programme of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under the project No.
404073166 and F. H. acknowledges support from the Studienstiftung des
deutschen Volkes.

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