# On Leavitt path algebras of Hopf graphs

###### Abstract.

In this paper, we provide the structure of Hopf graphs associated to pairs
$(G,\mathfrak{r})$ consisting of groups $G$ together with ramification datas
$\mathfrak{r}$ and their Leavitt path algebras. Consequently, we characterize
the Gelfand-Kirillov dimension, the stable rank, the purely infinite
simplicity and the existence of a nonzero finite dimensional representation of
the Leavitt path algebra of a Hopf graph via properties of ramification data
$\mathfrak{r}$ and $G$.

Mathematics Subject Classifications 2020: 16S88, 16S99, 05C25

Key words: Hopf graph; Purely infinite simple; Finite dimensional
representation; Gelfand-Kirillov dimension; Leavitt path algebra.

T. G. Nam111Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Cau Giay,
Hanoi, Vietnam. E-mail address<EMAIL_ADDRESS>and N. T. Phuc222Faculty of
Mathematics - Informatics Teacher Education, Dong Thap University, Vietnam.
E-mail address<EMAIL_ADDRESS>Acknowledgements: The authors were
supported by the Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under Grant 101.04-2020.01. We also take an opportunity
to express their deep gratitude to the anonymous referee for extremely careful
reading, highly professional working with our manuscript, and quite valuable
suggestions.

##  1\. Introduction

Given a (row-finite) directed graph $E$ and a field $K$, Abrams and Aranda
Pino in [3], and independently Ara, Moreno, and Pardo in [14], introduced the
_Leavitt path algebra_ $L_{K}(E)$. Abrams and Aranda Pino later extended the
definition in [5] to all countable directed graphs. Goodearl in [17] extended
the notion of Leavitt path algebras to all (possibly uncountable) directed
graphs $E$. In [28], Tomforde generalized the construction of Leavitt path
algebras by replacing the field with a commutative ring. Katsov, Nam and
Zumbragel in [19] considered the concept of Leavitt path algebras with
coefficients in a commutative semiring. Leavitt path algebras generalize the
Leavitt algebras $L_{K}(1,n)$ of [23], and also contain many other interesting
classes of algebras. In addition, Leavitt path algebras are intimately related
to graph $C^{*}$-algebras (see [27]). During the past seventeen years, Leavitt
path algebras have become a topic of intense investigation by mathematicians
from across the mathematical spectrum. For a detailed history and overview of
Leavitt path algebras we refer the reader to the survey article [1].

Cibils and Rosso [16] have introduced the notion of the Hopf graph
$\Gamma_{G,\mathfrak{r}}$ of a group $G$ with a ramification data which is a
function from $\mathcal{C}$ to $\mathbb{N}$, denoted by
$\mathfrak{r}=\sum\limits_{C\in\mathcal{C}}\mathfrak{r}_{C}C$, where
$\mathcal{C}$ is the set of conjugacy classes of $G$. They then classified all
graded Hopf algebras structures over path coalgebras using Hopf graphs. It
turns out that the path coalgebra $KE$ of a graph $E$ over a field $K$ admits
a structure of a Hopf algebra if and only if $E$ is a Hopf graph. We should
note that Hopf graphs are similar to Cayley graphs, which have the set of
vertices given by the elements of a group and arrows corresponding to
multiplication by elements of a chosen system of generators. In recent years,
there have been several works around Leavitt path algebras of Cayley graphs of
finite groups and in particular computing their Grothendieck group $K_{0}$
(see, e.g., [6, 8, 11, 24]) and regarding their simplicity and Invariant Basis
Number property (see, e.g., [24, 26]).

Motivated by the above results, in this article we investigate Leavitt path
algebras of Hopf graphs. More namely, we describe connected and strongly
connected components of Hopf graphs $\Gamma_{G,\mathfrak{r}}$, and obtain that
the Leavitt path algebra $L_{K}(\Gamma_{G,\mathfrak{r}})$ is isomorphic to the
direct sum of $|G/\Delta^{0}_{G,\mathfrak{r}}|$-copies of the Leavitt path
algebra $L_{K}(\Delta_{G,\mathfrak{r}})$, where $\Delta^{0}_{G,\mathfrak{r}}$
is the subgroup of $G$ generated by the set $\\{c\in C\mid
C\in\mathcal{C},\mathfrak{r}_{C}>0\\}$ and $\Delta_{G,\mathfrak{r}}$ is the
subgraph of $\Gamma_{G,\mathfrak{r}}$ having set of vertices
$\Delta^{0}_{G,\mathfrak{r}}$ (Theorem 2.10). Consequently, we characterize
the Gelfand-Kirillov dimension (Theorem 3.2), the purely infinite simplicity
(Theorem 3.5), the stable rank (Theorem 3.7), as well as classify all finite
dimensional representations (Theorem 3.9) of the Leavitt path algebra
$L_{K}(\Gamma_{G,\mathfrak{r}})$ in terms of both ramification data
$\mathfrak{r}$ and the subsemigroup $S_{G,\mathfrak{r}}$ of $G$ generated by
the set $\\{c\in C\mid C\in\mathcal{C},\mathfrak{r}_{C}>0\\}$. In particular,
we extend the criteria for the simplicity and Invariant Basis Number property
of Leavitt path algebras of Cayley graphs, introduced in [24, 26], to Hopf
graphs. We should mention that graph-theoretic characterizations on graphs $E$
of these properties for Leavitt path algebras $L_{K}(E)$ have been established
in literatures (see, e.g, [25], [7], [15], and [20], respectively); while
based on these criteria, our characterizations are established completely on
properties of both ramification data $\mathfrak{r}$ and $G$.

Throughout this article, the set of nonnegative integers is denoted by
$\mathbb{N}$, the integers by $\mathbb{Z}$.

##  2\. The structure of Hopf graph Leavitt path algebras

The main aim of this section is to provide fundamental properties of Hopf
graphs (Proposition 2.6), the structure of Hopf graph and their Leavitt path
algebras (Theorem 2.10) via properties of ramification data.

We begin this section by recalling some general notions of graph theory.

A (directed) graph $E=(E^{0},E^{1},r,s)$ (or shortly $E=(E^{0},E^{1})$)
consists of two disjoint sets $E^{0}$ and $E^{1}$, called _vertices_ and
_edges_ respectively, together with two maps $r,s:E^{1}\longrightarrow E^{0}$.
The vertices $r(e)$ and $s(e)$ are referred to as the _range_ and the _source_
of the edge $e$, respectively. The graph is called _row-finite_ if
$|s^{-1}(v)|<\infty$ for all $v\in E^{0}$. A graph $E$ is _finite_ if both
sets $E^{0}$ and $E^{1}$ are finite. We say that $E$ is a _trivial graph_ if
$E$ has only one vertex and no edges. A vertex $v$ for which $s^{-1}(v)$ is
empty is called a _sink_ ; a vertex $v$ for which $r^{-1}(v)$ is empty is
called a _source_ ; a vertex $v$ is called an _isolated vertex_ if it is both
a source and a sink; and a vertex $v$ is _regular_ if $0<|s^{-1}(v)|<\infty$.

A _finite path_ $p=e_{1}\cdots e_{n}$ in a graph $E$ is a sequence of edges
$e_{1},\cdots,e_{n}$ such that $r(e_{i})=s(e_{i+1})$ for $i=1,\cdots,n-1$. In
this case, we say that the path $p$ starts at the vertex $s(p):=s(e_{1})$ and
ends at the vertex $r(p):=r(e_{n})$, and has _length_ $|p|:=n$. We consider
the vertices in $E^{0}$ to be paths of length $0$. We denote by
$\text{Path}(E)$ the set of all finite paths of $E$. We denote by $p^{0}$ the
set of its vertices, that is, $p^{0}=\\{s(e_{i}),r(e_{i})\ |\
i=1,\cdots,n\\}$. A finite path $p$ of positive length is closed if
$s(p)=r(p)$, in which case $p$ is said to be based at the vertex $s(p)$. The
closed path $p$ is called a _cycle_ if $p$ does not pass through any of its
vertices twice. A cycle $c$ is called a _single cycle_ if
$|r^{-1}(v)|=|s^{-1}(v)|=1$ for all $v\in c^{0}$. A graph $E$ is _acyclic_ if
it has no cycles. An edge $f$ is an _exit_ for a path $p=e_{1}\cdots e_{n}$ if
$s(f)=s(e_{i})$ but $f\neq e_{i}$ for some $1\leq i\leq n$. For vertices
$v,w\in E^{0}$, we write $v\geq w$ if there exists a path in $E$ from $v$ to
$w$, i.e., a finite path $p$ with $s(p)=v$ and $r(p)=w$. By an _infinite path_
in $E$ we mean a sequence $q=e_{1}\cdots e_{i}\cdots$ for which
$r(e_{i})=s(e_{i+1})$ for all positive integer $i$. We denote by $E^{\infty}$
the set of all infinite paths of $E$, and by $E^{\geq\infty}$ the set
$E^{\infty}$ together with the elements of $\text{Path}(E)$ whose range vertex
is a singular vertex.

Let $E=(E^{0},E^{1},r,s)$ be a graph. We define the extended graph of $E$ as
the new graph
$\widehat{E}=(E^{0},E^{1}\cup(E^{1})^{*},\widehat{r},\widehat{s})$, where
$(E^{1})^{*}=\\{e^{*}\mid e\in E^{1}\\}$, and the functions $\widehat{r}$ and
$\widehat{s}$ are defined as

$\widehat{r}(e)=r(e),~{}\widehat{s}(e)=s(e),~{}\widehat{r}(e^{*})=s(e)\text{
and }\widehat{s}(e^{*})=r(e)\text{ for any }e\in E^{1}.$

We sometimes refer to the edges in the graph $E$ as real edges, the path in
the graph $E$ as real path and the additional edges in $\widehat{E}$ (i.e. the
elements of $(E^{1})^{*}$) as ghost edges.

We say that a graph $E$ is _connected_ if given any two vertices $u,v\in
E^{0}$ there exists a path $h$ in $\widehat{E}$ for which $\widehat{s}(h)=u$
and $\widehat{r}(h)=v$. The _connected components_ of a graph $E$ are the
graphs $\\{E_{i}\\}_{i\in\Lambda}$ such that $E$ is the disjoint union
$E=\sqcup_{i\in\Lambda}E_{i}$, where every $E_{i}$ is connected.

A graph $E$ is called _strongly connected_ if given any pair of vertices
$(u,v)\in E^{0}\times E^{0}$ there exists a path $h$ in $E$ such that $s(h)=u$
and $r(h)=v$. We denote by $\mathcal{S}_{E}$ the set of all strongly connected
components of $E$. It is obvious that $\mathcal{S}_{E}$ is partially ordered
by the relation $\geq$ defined by: for all
$\Pi_{1},\Pi_{2}\in\mathcal{S}_{E}$, $\Pi_{1}\geq\Pi_{2}$ if $v\geq w$ for
some $v\in\Pi_{1}^{0}$ and $w\in\Pi_{2}^{0}$.

For an arbitrary graph $E=(E^{0},E^{1},s,r)$ and an arbitrary field $K$, the
_Leavitt path algebra_ $L_{K}(E)$ of the graph $E$ _with coefficients in_ $K$
is the $K$-algebra generated by the sets $E^{0}$ and $E^{1}\cup(E^{1})^{*}$,
satisfying the following relations for all $v,w\in E^{0}$ and $e,f\in E^{1}$:

  * (1)

$vw=\delta_{v,w}w$;

  * (2)

$s(e)e=e=er(e)$ and $r(e)e^{*}=e^{*}=e^{*}s(e)$;

  * (3)

$e^{*}f=\delta_{e,f}r(e)$;

  * (4)

$v=\sum_{e\in s^{-1}(v)}ee^{*}$ for any regular vertex $v$;

where $\delta$ is the Kronecker delta.

It is worth mentioning the following simple fact.

######  Remark 2.1 (cf. [9, Proposition 1.4] and [7, Proposition 1.2.14]).

Let $K$ be a field and $E$ a graph with its connected components
$\\{E_{i}\\}_{i\in\Lambda}$. Then
$L_{K}(E)\cong\bigoplus_{i\in\Lambda}L_{K}(E_{i})$.

We next recall the notion of Hopf graphs introduced by Cibils and Rosso in
[16]. Let $G$ be an arbitrary group and $\mathcal{C}$ the set of all conjugacy
classes of $G$. We call a _ramification data_ of $G$ is a function
$\mathfrak{r}:\mathcal{C}\longrightarrow\mathbb{N}$, denoted by
$\mathfrak{r}=\sum\limits_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ (where
$\mathfrak{r}_{C}:=\mathfrak{r}(C)$). The support of $\mathfrak{r}$ is the set

$\rm{supp}(\mathfrak{r})=\\{C\in\mathcal{C}\mid\mathfrak{r}_{C}>0\\}.$

We say that $\mathfrak{r}$ has _finite support_ if
$\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ is a finite set. We denote by
$S_{G,\mathfrak{r}}$ the _subsemigroup_ of $G$ generated by
$\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$.

######  Definition 2.2 ([16, Definition 3.1]).

Let $G$ be an arbitrary group with a ramification data
$\mathfrak{r}=\sum\limits_{C\in\mathcal{C}}\mathfrak{r}_{C}C$. The Hopf graph
associated to the pair $(G,\mathfrak{r})$, denoted by
$\Gamma_{G,\mathfrak{r}}$, has set of vertices $\Gamma_{G,\mathfrak{r}}^{0}=G$
and has $\mathfrak{r}_{C}$ edges from $x$ to $xc$ for each $x\in G$ and $c\in
C$.

It is worth mentioning that in [16, Theorem 3.3] Cibils and Rosso showed that
Hopf graphs are precisely the graphs such that the path algebra can be endowed
with a graded Hopf algebra structure.

For clarification, we illustrate the notion of Hopf graphs by presenting the
following examples.

######  Example 2.3.

Consider the trivial group $G=\\{1_{G}\\}$ and the ramification data
$\mathfrak{r}=n[1_{G}]$. Then $\Gamma_{\mathbb{Z},\mathfrak{r}}$ is the rose
with $n$ petals graph:

$\textstyle{R_{n}=}$$\textstyle{\bullet_{1_{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Example 2.4.

Let $G=S_{3}$ be the symmetric group of order $6$, and write

$G=\\{id,(12),(13),(23),(123),(132)\\}.$

We then have $\mathcal{C}=\\{C_{1}:=[id],C_{2}:=[(12)],C_{3}:=[(123)]\\}$,
where

$C_{1}=\\{id\\},C_{2}=\\{(12),(13),(23)\\},C_{3}=\\{(123),(132)\\}.$

(1) Consider the ramification data $\mathfrak{r}=C_{3}$. We then have that
$\Gamma_{G,\mathfrak{r}}$ is the following graph:

$\textstyle{\bullet_{id}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(123)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(132)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\bullet_{(12)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(13)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(23)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

(2) Consider the ramification data $\mathfrak{r}=C_{2}$. Then
$\Gamma_{G,\mathfrak{r}}$ is the following graph:

$\textstyle{\bullet_{(123)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(132)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(12)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(13)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(23)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{id}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Example 2.5.

Consider the group of all integers $\mathbb{Z}$. We then have
$\mathcal{C}=\\{[n]\mid n\in\mathbb{Z}\\}$, where $[n]=\\{n\\}$ for all
$n\in\mathbb{Z}$.

(1) Consider the ramification data $\mathfrak{r}=[0]+[2]$. Then
$\Gamma_{\mathbb{Z},\mathfrak{r}}$ is the following graph:

$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...}$

(2) Consider the ramification data $\mathfrak{r}=[2]+[3]$. Then
$\Gamma_{\mathbb{Z},\mathfrak{r}}$ is the following graph:

$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...}$

The following proposition provides us with fundamental properties of Hopf
graphs, which are useful to prove the main results of this article.

######  Proposition 2.6.

Let $G$ be an arbitrary group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$. Then the following
statements hold:

$(1)$ There exists a path in $\widehat{\Gamma_{G,\mathfrak{r}}}$
$(\Gamma_{G,\mathfrak{r}}$, respectively$)$ from a vertex $g$ to a vertex $h$
if and only if $h=gw$ for some $w\in\langle S_{G,\mathfrak{r}}\rangle\ (w\in
S_{G,\mathfrak{r}}$, respectively$)$, where $\langle
S_{G,\mathfrak{r}}\rangle$ is the subgroup of $G$ generated by
$S_{G,\mathfrak{r}}$;

$(2)$ $|s^{-1}(g)|=|r^{-1}(g)|$ for all $g\in G$;

$(3)$ $\Gamma_{G,\mathfrak{r}}$ is a row-finite graph if and only if
$\mathfrak{r}$ has finite support;

$(4)$ If $\mathfrak{r}=0$, then $\Gamma_{G,\mathfrak{r}}$ is a disjoint union
of isolated vertices;

$(5)$ If $\mathfrak{r}\neq 0$, then $\Gamma_{G,\mathfrak{r}}$ has neither
sinks nor sources;

$(6)$ $\Gamma_{G,\mathfrak{r}}$ has a cycle if and only if
$S_{G,\mathfrak{r}}$ is a submonoid of $G$;

$(7)$ If $G$ is a finite group, then $g$ is the base of at least
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|$ cycles for all $g\in G$.

###### Proof.

(1) Let $(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)^{-1}:=\\{c^{-1}\mid
c\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C\\}$. There exists a path
$p=e_{1}e_{2}\cdots e_{n}$ in $\widehat{\Gamma_{G,\mathfrak{r}}}$ from $g$ to
$h$ if and only if $g=\widehat{s}(e_{1})$, $h=\widehat{r}(e_{n})$, and for any
$1\leq i\leq n$, if $e_{i}$ is a real edge, then
$\widehat{r}(e_{i})=\widehat{s}(e_{i})c_{i}$ for some
$c_{i}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$, else
$\widehat{r}(e_{i})=\widehat{s}(e_{i})c_{i}$ for some
$c_{i}\in(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)^{-1}$. We then have
$h=\widehat{r}(e_{n})=\widehat{s}(e_{1})c_{1}\cdots c_{n}=gc_{1}\cdots c_{n}$,
where $c_{i}$’s are in
$\bigcup_{C\in\rm{supp}(\mathfrak{r})}C\cup(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)^{-1}$.
Let $w:=c_{1}\cdots c_{n}$. We then have $h=gw$ and $w\in\langle
S_{G,\mathfrak{r}}\rangle$. If $p$ is a real path, then
$c_{i}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ for all $1\leq i\leq n$, and
so $w\in S_{G,\mathfrak{r}}$, proving (1).

(2) It is obvious that
$|s^{-1}(g)|=|r^{-1}(g)|=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|$ for all
$g\in G$.

(3) By item (2), $|s^{-1}(g)|=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|$ for
all $g\in G$. Since $\mathfrak{r}_{C}$ is a nonnegative integer for all
$C\in\mathcal{C}$, we must have $|s^{-1}(g)|<\infty$ if and only if
$\sum_{C\in\mathcal{C}}|C|<\infty$. Equivalently, $\Gamma_{G,\mathfrak{r}}$ is
row-finite if and only if $\mathfrak{r}$ has finite support.

(4) If $\mathfrak{r}=0$, then $|s^{-1}(g)|=|r^{-1}(g)|=0$ for all $g\in G$,
and so, $\Gamma_{G,\mathfrak{r}}$ is a disjoint union of isolated vertices.

(5) If $\mathfrak{r}\neq 0$, then $|s^{-1}(g)|=|r^{-1}(g)|\neq 0$ for all
$g\in G$, and so, $\Gamma_{G,\mathfrak{r}}$ has neither sinks nor sources.

(6) $(\Longrightarrow)$. Assume that $\Gamma_{G,\mathfrak{r}}$ has a cycle
$\alpha=e_{1}e_{2}\cdots e_{n}$. Then, there exist elements
$\\{c_{i}\\}_{i=1}^{n}\subseteq\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ such
that $r(e_{i})=s(e_{i})c_{i}$ for all $i=1,\cdots,n$. Since $\alpha$ is a
cycle, we have $s(e_{1})=r(e_{n})=s(e_{1})c_{1}c_{2}\cdots c_{n}$, and so,
$1_{G}=c_{1}\cdots c_{n}\in S_{G,\mathfrak{r}}$. This shows that
$S_{G,\mathfrak{r}}$ is a submonoid of $G$.

$(\Longleftarrow)$. Assume that $S_{G,\mathfrak{r}}$ is a submonoid of $G$.
This implies that there exist elements
$\\{c_{i}\\}_{i=1}^{n}\subseteq\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ such
that $c_{1}\cdots c_{n}=1_{G}$. Let $g$ be an arbitrary element of $G$. We
then have that in $\Gamma_{G,\mathfrak{r}}$ there exist edges
$\\{e_{i}\\}^{n}_{i=1}$ such that $s(e_{1})=g$, $s(e_{i})=gc_{1}\cdots
c_{i-1}$ for all $i\geq 2$, and $r(e_{i})=gc_{1}\cdots c_{i}$ for all $1\leq
i\leq n$. Since $c_{1}\cdots c_{n}=1_{G}$, we obtain that $s(e_{1})=r(e_{n})$,
and so, $\Gamma_{G,\mathfrak{r}}$ has a cycle $e_{1}\cdots e_{n}$.

(7) Let $g\in G$ and $c\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$. Since $G$
is a finite group, there exists a positive integer $n$ such that
$c^{n}=1_{G}$. By item (6), there exists a cycle $\alpha$ in
$\Gamma_{G,\mathfrak{r}}$ based at $g$. This implies that $g$ is the base of
at least $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|$ cycles, thus finishing
the proof. ∎

In the remainder of this section, we investigate the structure of Hopf graphs
and their Leavitt path algebras. To do so, we need some useful notions and
facts.

######  Definition 2.7.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$.

(1) We denote by $\Delta_{G,\mathfrak{r}}$ the following subgraph of
$\Gamma_{G,\mathfrak{r}}$: if $\mathfrak{r}=0$, then $\Delta_{G,\mathfrak{r}}$
is the trivial graph, else

$\Delta_{G,\mathfrak{r}}=(\Delta_{G,\mathfrak{r}}^{0},\Delta_{G,\mathfrak{r}}^{1},r|_{\Delta_{G,\mathfrak{r}}^{1}},s|_{\Delta_{G,\mathfrak{r}}^{1}}),$

where $\Delta_{G,\mathfrak{r}}^{0}:=\langle S_{G,\mathfrak{r}}\rangle$ and
$\Delta_{G,\mathfrak{r}}^{1}:=r^{-1}(\Delta_{G,\mathfrak{r}}^{0})$.

(2) We denote by $\Lambda_{G,\mathfrak{r}}$ the following subgraph of
$\Gamma_{G,\mathfrak{r}}$: if $S_{G,\mathfrak{r}}$ is not a submonoid of $G$,
then $\Lambda_{G,\mathfrak{r}}$ is the trivial graph, else

$\Lambda_{G,\mathfrak{r}}=(\Lambda_{G,\mathfrak{r}}^{0},\Lambda_{G,\mathfrak{r}}^{1},r|_{\Lambda_{G,\mathfrak{r}}^{1}},s|_{\Lambda_{G,\mathfrak{r}}^{1}}),$

where $\Lambda_{G,\mathfrak{r}}^{0}:=G(S_{G,\mathfrak{r}})$ is the maximal
subgroup of $G$ contained in $S_{G,\mathfrak{r}}$ and
$\Lambda_{G,\mathfrak{r}}^{1}:=r^{-1}(\Lambda_{G,\mathfrak{r}}^{0})\cap
s^{-1}(\Lambda_{G,\mathfrak{r}}^{0})$.

For clarification, we illustrate the graphs $\Delta_{G,\mathfrak{r}}$ and
$\Lambda_{G,\mathfrak{r}}$ by presenting the following examples.

######  Example 2.8.

(1) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.4 (1). We
then have that $\Lambda_{G,\mathfrak{r}}=\Delta_{G,\mathfrak{r}}$ is the
following graph:

$\textstyle{\bullet_{id}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(123)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{(132)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

(2) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.4 (2). We
then have
$\Lambda_{G,\mathfrak{r}}=\Delta_{G,\mathfrak{r}}=\Gamma_{G,\mathfrak{r}}$.

(3) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (1). We
then have that $\Lambda_{G,\mathfrak{r}}$ is the following graph:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and $\Delta_{G,\mathfrak{r}}$ is the following graph:

$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{-2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{...}$

(4) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (2). We
then have that $\Lambda_{G,\mathfrak{r}}$ is the trivial graph and
$\Delta_{G,\mathfrak{r}}=\Gamma_{G,\mathfrak{r}}$.

It is worth mentioning the following fact.

######  Lemma 2.9.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$. Then
$\Lambda_{G,\mathfrak{r}}^{0}$ and $\Delta_{G,\mathfrak{r}}^{0}$ are normal
subgroups of $G$.

###### Proof.

We first note that for all $c\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ and
$g\in G$, we have $gcg^{-1}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ and
$gc^{-1}g^{-1}\in(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)^{-1}$. We now prove
that $\Lambda_{G,\mathfrak{r}}^{0}$ is a normal subgroup of $G$. Indeed, let
$g\in G$ and $h\in\Lambda_{G,\mathfrak{r}}^{0}$. Write $h=c_{1}c_{2}\cdots
c_{n}$, where $c_{i}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ for all $i$. We
then have

$ghg^{-1}=gc_{1}g^{-1}gc_{2}g^{-1}\cdots gc_{n}g^{-1}\text{ and
}gc_{i}g^{-1}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$

for all $i$, and so, $ghg^{-1}\in S_{G,\mathfrak{r}}$. Moreover, since
$h\in\Lambda_{G,\mathfrak{r}}^{0}$, $h^{-1}\in S_{G,\mathfrak{r}}$. We then
have $(ghg^{-1})^{-1}=gh^{-1}g^{-1}\in S_{G,\mathfrak{r}}$. So
$ghg^{-1}\in\Lambda_{G,\mathfrak{r}}^{0}$. This shows that
$\Lambda_{G,\mathfrak{r}}^{0}$ is a normal subgroup of $G$.

Similarly, we obtain that $\Delta_{G,\mathfrak{r}}^{0}$ is a normal subgroup
of $G$, thus finishing the proof. ∎

We are now in a position to give the main result of this section, which
provides us with the structure of Hopf graphs and their Leavitt path algebras.

######  Theorem 2.10.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$, and $K$ an arbitrary
field. Then the following statements hold:

$(1)$ $\Delta_{G,\mathfrak{r}}$ is a connected component of
$\Gamma_{G,\mathfrak{r}}$ and $\Gamma_{G,\mathfrak{r}}=\underset{i\in
G/\Delta_{G,\mathfrak{r}}^{0}}{\sqcup}\Omega_{i}$ where $\Omega_{i}$ is
isomorphic to $\Delta_{G,\mathfrak{r}}$ as graphs;

$(2)$ $\Lambda_{G,\mathfrak{r}}$ is a strongly connected component of
$\Delta_{G,\mathfrak{r}}$ and $\Pi\cong\Lambda_{G,\mathfrak{r}}$ as graphs for
all strongly connected component $\Pi$ of $\Delta_{G,\mathfrak{r}}$.
Furthermore,

$\displaystyle|\mathcal{S}_{\Delta_{G,\mathfrak{r}}}|=\begin{cases}|\Delta_{G,\mathfrak{r}}^{0}/\Lambda_{G,\mathfrak{r}}^{0}|&\textnormal{if
}S_{G,\mathfrak{r}}\textnormal{ is a submonoid of $G$;}\\\
|\Delta_{G,\mathfrak{r}}^{0}|&\textnormal{otherwise};\end{cases}$

$(3)$ $\Delta_{G,\mathfrak{r}}$ is a strongly connected component of
$\Gamma_{G,\mathfrak{r}}$ if and only if $\mathfrak{r}=0$ or
$S_{G,\mathfrak{r}}$ is a subgroup of $G$. Consequently, if
$S_{G,\mathfrak{r}}$ is finite, then $\Delta_{G,\mathfrak{r}}$ is a strongly
connected component of $\Gamma_{G,\mathfrak{r}}$;

$(4)$ If, in addition, $G$ is commutative, then
$(\mathcal{S}_{\Delta_{G,\mathfrak{r}}},\geq)$ is downward directed;

$(5)$ $L_{K}(\Gamma_{G,\mathfrak{r}})\cong
L_{K}(\Delta_{G,\mathfrak{r}})^{(G/\Delta_{G,\mathfrak{r}}^{0})}$.

###### Proof.

(1) Let $x$ and $y$ be elements of $\Delta_{G,\mathfrak{r}}^{0}$. Then, there
exist elements
$\\{c_{i}\\}^{n}_{i=1}\subseteq(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)\cup(\bigcup_{C\in\rm{supp}(\mathfrak{r})}C)^{-1}$
such that $x^{-1}y=c_{1}\cdots c_{n}$ and $y=xc_{1}\cdots c_{n}$. By
Proposition 2.6 (1), there exists a path in
$\widehat{\Delta_{G,\mathfrak{r}}}$ from $x$ to $y$. Let $x$ be an element of
$\Delta_{G,\mathfrak{r}}^{0}$ and let $z$ be an element of $G$. If there
exists a path in $\widehat{\Delta_{G,\mathfrak{r}}}$ from $z$ to $x$, then by
Proposition 2.6 (1), there exists an element $w\in\langle
S_{G,\mathfrak{r}}\rangle=\Delta_{G,\mathfrak{r}}^{0}$ such that $x=zw$, and
so $z=xw^{-1}\in\Delta_{G,\mathfrak{r}}^{0}$. This implies that
$\Delta_{G,\mathfrak{r}}$ is a connected component of
$\Gamma_{G,\mathfrak{r}}$.

Let $\Omega$ be a connected component of $\Gamma_{G,\mathfrak{r}}$. We claim
that $\Omega$ is isomorphic to $\Delta_{G,\mathfrak{r}}$ as graphs. Indeed,
let $g$ be an arbitrary vertex of $\Omega$, and let
$\varphi^{0}:\Delta_{G,\mathfrak{r}}^{0}\longrightarrow\Omega^{0}$ be the map
defined by: $\varphi^{0}(w)=gw$ for all $w\in\Delta_{G,\mathfrak{r}}^{0}$.
Then, if $\varphi^{0}(w)=gw=gw^{\prime}=\varphi^{0}(w^{\prime})$, then
$w=w^{\prime}$, and so, $\varphi^{0}$ is an injection. Let $h$ be an element
of $\Omega^{0}$. Since $\Omega$ is a connected graph, there exists a path in
$\widehat{\Omega}$ from $g$ to $h$. By Proposition 2.6 (1), $h=gw$ for some
$w\in\langle S_{G,\mathfrak{r}}\rangle=\Delta_{G,\mathfrak{r}}^{0}$, that
means, $h=\varphi^{0}(w)$. This shows that $\varphi^{0}$ is surjective, and
hence $\varphi^{0}$ is a bijection.

For each edge $e$ in $\Delta_{G,\mathfrak{r}}$ from $v$ to $w$, we have $w=vc$
for some $c\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$, and $\varphi^{0}(v)=gv$
and $\varphi^{0}(w)=gw=gvc$. Therefore, there is a unique edge in $\Omega$
from $\varphi^{0}(v)$ to $\varphi^{0}(w)$ with respect to $e$. This shows that
there is a bijection
$\varphi^{1}:\Delta_{G,\mathfrak{r}}^{1}\longrightarrow\Omega^{1}$ from the
set of edges of $\Delta_{G,\mathfrak{r}}$ to the set of edges of $\Omega$ such
that $\varphi^{0}$ and $\varphi^{1}$ commute with the source and range maps,
and so $\Delta_{G,\mathfrak{r}}$ is isomorphic to $\Omega$ as graphs, proving
the claim.

We note that
$\Omega^{0}=\varphi^{0}(\Delta_{G,\mathfrak{r}}^{0})=g\Delta_{G,\mathfrak{r}}^{0}$,
and so, the cardinality of all connected components of
$\Gamma_{G,\mathfrak{r}}$ is equal to the cardinality of the quotient group
$G/\Delta_{G,\mathfrak{r}}^{0}$, as desired.

(2) Let $g$ and $h$ be elements of $\Lambda_{G,\mathfrak{r}}^{0}$. Since
$\Lambda_{G,\mathfrak{r}}^{0}$ is a subgroup of $G$,
$g^{-1}h,h^{-1}g\in\Lambda_{G,\mathfrak{r}}^{0}$, and so, $g^{-1}h,h^{-1}g\in
S_{G,\mathfrak{r}}$. Since $h=g(g^{-1}h),g=h(h^{-1}g)$ and by Proposition 2.6
(1), we obtain that there exist two paths $p$ and $q$ in
$\Gamma_{G,\mathfrak{r}}$ such that $s(p)=g,r(p)=h$ and $s(q)=h,r(q)=g$. Let
$g\in\Lambda_{G,\mathfrak{r}}^{0}$ and $h\in G$ such that there exist two
paths $p$ and $q$ in $\Gamma_{G,\mathfrak{r}}$ for which $s(p)=g,r(p)=h$ and
$s(q)=h,r(q)=g$. We claim that $h\in\Lambda_{G,\mathfrak{r}}^{0}$. Indeed, by
Proposition 2.6 (1), $h=gw$ and $g=hw^{\prime}$ for some $w,w^{\prime}\in
S_{G,\mathfrak{r}}$. Then $h=gw=hw^{\prime}w$, and so, $w^{\prime}w=1_{G}$. It
implies that $w^{-1}=w^{\prime}\in S_{G,\mathfrak{r}}$, and so,
$h^{-1}=w^{-1}g^{-1}\in S_{G,\mathfrak{r}}$. It is obvious that $h=gw\in
S_{G,\mathfrak{r}}$, and hence, $h\in\Lambda_{G,\mathfrak{r}}^{0}$, proving
the claim. Therefore, $\Lambda_{G,\mathfrak{r}}$ is a strongly connected
component of $\Delta_{G,\mathfrak{r}}$.

We next prove that $\Pi\cong\Lambda_{G,\mathfrak{r}}$ as graphs for all
strongly connected component $\Pi$ of $\Delta_{G,\mathfrak{r}}$. By repeating
the approach described in the proof of (1), we consider the map
$\varphi^{0}:\Lambda_{G,\mathfrak{r}}^{0}\longrightarrow\Pi^{0}$ defined by:
$\varphi^{0}(w)=gw$ for all $w\in\Lambda_{G,\mathfrak{r}}^{0}$, where $g$ is
an arbitrary vertex of $\Pi$. It is sufficient to show that $\varphi^{0}$ is
surjective. Let $h$ be an element of $\Pi^{0}$. Since $\Pi$ is a strongly
connected graph, there exist two paths in $\Gamma_{G,\mathfrak{r}}$ from $g$
to $h$ and from $h$ to $g$. By using the above argument again, we have $h=gw$
and $g=hw^{-1}$ for some $w,w^{-1}\in S_{G,\mathfrak{r}}$. It implies that
$w\in\Lambda_{G,\mathfrak{r}}^{0}$, and so, $h=gw=\varphi^{0}(w)$, as desired.

If $S_{G,\mathfrak{r}}$ is not a submonoid of $G$, then
$\Lambda_{G,\mathfrak{r}}$ is the trivial graph, and so,
$|\mathcal{S}_{\Delta_{G,\mathfrak{r}}}|=|\Delta_{G,\mathfrak{r}}^{0}|$.
Otherwise, if $S_{G,\mathfrak{r}}$ is a submonoid of $G$, then, similarly to
the proof of (1), we obtain that the cardinality of all strongly connected
components of $\Delta_{G,\mathfrak{r}}$ is equal to the cardinality of the
quotient group $\Delta_{G,\mathfrak{r}}^{0}/\Lambda_{G,\mathfrak{r}}^{0}$,
thus showing (2).

(3) $(\Longleftarrow)$ If $\mathfrak{r}=0$, then $\Delta_{G,\mathfrak{r}}$ is
a trivial graph, and so, it is a strongly connected component of
$\Gamma_{G,\mathfrak{r}}$. Suppose that $\mathfrak{r}\neq 0$ and
$S_{G,\mathfrak{r}}$ is a subgroup of $G$. We then have
$\Delta_{G,\mathfrak{r}}^{0}=\Lambda_{G,\mathfrak{r}}^{0}=S_{G,\mathfrak{r}}$.
Therefore, $\Delta_{G,\mathfrak{r}}=\Lambda_{G,\mathfrak{r}}$ is a strongly
connected component of $\Gamma_{G,\mathfrak{r}}$ by item (2).

$(\Longrightarrow)$ Suppose that $\Delta_{G,\mathfrak{r}}$ is a strongly
connected component of $\Gamma_{G,\mathfrak{r}}$. If $\Delta_{G,\mathfrak{r}}$
is the trivial graph, then $\mathfrak{r}=0$. Assume that
$\Delta_{G,\mathfrak{r}}$ has an edge. We claim that $S_{G,\mathfrak{r}}$ is a
subgroup of $G$. Indeed, let $c\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ and
$e$ an edge in $\Delta_{G,\mathfrak{r}}$ such that $r(e)=s(e)c$. Since
$\Delta_{G,\mathfrak{r}}$ is a strongly connected component of
$\Gamma_{G,\mathfrak{r}}$, there exists a path $p_{e}$ in
$\Delta_{G,\mathfrak{r}}$ such that $s(p_{e})=r(e)$ and $r(p_{e})=s(e)$, and
so, $ep_{e}$ is a cycle in $\Delta_{G,\mathfrak{r}}$. By Proposition 2.6 (6),
$S_{G,\mathfrak{r}}$ is a submonoid of $G$. We write $p_{e}=f_{1}\cdots f_{n}$
where $r(f_{i})=s(f_{i})w_{i}$ for some
$w_{i}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$, $1\leq i\leq n$. Since
$ep_{e}$ is a cycle in $\Delta_{G,\mathfrak{r}}$,
$s(e)=r(p_{e})=r(f_{n})=s(f_{n})w_{n}=\cdots=s(e)cw_{1}\cdots w_{n}$, and so,
$cw_{1}\cdots w_{n}=1_{G}$. It shows that $S_{G,\mathfrak{r}}$ is a subgroup
of $G$, proving the claim.

Assume that $S_{G,\mathfrak{r}}$ is a finite subsemigroup of $G$. If
$S_{G,\mathfrak{r}}=\varnothing$, then $\Delta_{G,\mathfrak{r}}$ is the
trivial graph, therefore, $\Delta_{G,\mathfrak{r}}$ is a strongly connected
component of $\Gamma_{G,\mathfrak{r}}$. Otherwise, $S_{G,\mathfrak{r}}$ is a
nonempty finite subsemigroup of $G$, then $S_{G,\mathfrak{r}}$ is also a
subgroup of $G$, and so, $\Delta_{G,\mathfrak{r}}$ is a strongly connected
component of $\Gamma_{G,\mathfrak{r}}$.

(4) Assume that $G$ is a commutative group. Let
$x,y\in\Delta_{G,\mathfrak{r}}^{0}$. We claim that there exists an element
$z\in\Delta_{G,\mathfrak{r}}^{0}$ such that $x\geq z$ and $y\geq z$. Since $G$
is a commutative, we can write $x=c_{1}c_{2}\cdots
c_{n}d_{1}^{-1}d_{2}^{-1}\cdots d_{m}^{-1}$ and $y=k_{1}k_{2}\cdots
k_{n^{\prime}}l_{1}^{-1}l_{2}^{-1}\cdots l_{m^{\prime}}^{-1}$, where
$c_{i},d_{j},k_{i^{\prime}},l_{j^{\prime}}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$
for all $1\leq i\leq n$, $1\leq j\leq m$, $1\leq i^{\prime}\leq n^{\prime}$,
$1\leq j^{\prime}\leq m^{\prime}$. Consider the path $p=e_{1}\cdots e_{m}$ and
$q=f_{1}\cdots f_{n^{\prime}}$ such that $s(e_{1})=x,r(e_{i})=s(e_{i})d_{i}$,
$1\leq i\leq m$ and $s(f_{1})=r(p),r(f_{i})=s(f_{i})k_{i}$, $1\leq i\leq
n^{\prime}$. We then have $s(pq)=x$ and $r(pq)=c_{1}c_{2}\cdots
c_{n}k_{1}k_{2}\cdots k_{n^{\prime}}=:z\in\Delta_{G,\mathfrak{r}}^{0}$, and
so, $x\geq z$. Similarly, there exist paths $p^{\prime}$, $q^{\prime}$ in
$\Delta_{G,\mathfrak{r}}$ such that $s(p^{\prime}q^{\prime})=y$,
$r(p^{\prime}q^{\prime})=k_{1}k_{2}\cdots k_{n^{\prime}}c_{1}c_{2}\cdots
c_{n}=z$, and so, $y\geq z$, proving the claim. This implies that
$(\mathcal{S}_{\Delta_{G,\mathfrak{r}}},\geq)$ is downward directed.

(5) It follows from item (1) and Remark 2.1, thus finishing the proof. ∎

It is worth mentioning the following note.

######  Remark 2.11.

There exist two graphs $E$ and $F$ for which $E$ is a Hopf graph and $F$ is
not a Hopf graph, but $L_{K}(E)\cong L_{K}(F)$, where $K$ is an arbitrary
field. For example, let $E=\Gamma_{G,\mathfrak{r}}$ be the Hopf graph of
Example 2.3 with $n=3$ and $F$ the following graph:

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Then, since $F$ has two strongly connected components which are not isomorphic
to each other, and by Theorem 2.10 (2), $F$ is not a Hopf graph. It is well
known that $L_{K}(E)\cong L_{K}(1,3)$ and $L_{K}(F)\cong M_{3}(L_{K}(1,3))$
(by [26, Theorem 2.11 (2)]). By [2, Theorem 4.4], we have $L_{K}(1,3)\cong
M_{3}(L_{K}(1,3))$, and so $L_{K}(E)\cong L_{K}(F)$.

We close this section with the following useful corollary.

######  Corollary 2.12.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$. Then the following
statements hold:

$(1)$ If $\mathfrak{r}=0$, then $\Gamma_{G,\mathfrak{r}}$ is a disjoint union
of isolated vertices and

$L_{K}(\Gamma_{G,\mathfrak{r}})=K^{(G)}.$

$(2)$ If $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and
$S_{G,\mathfrak{r}}$ is a submonoid of $G$, then $\Gamma_{G,\mathfrak{r}}$ is
a disjoint union of single cycles, $S_{G,\mathfrak{r}}$ is a finite normal
subgroup of $G$ and

$L_{K}(\Gamma_{G,\mathfrak{r}})=M_{|S_{G,\mathfrak{r}}|}(K[x,x^{-1}])^{(G/S_{G,\mathfrak{r}})}.$

###### Proof.

(1) It immediately follows from Proposition 2.6 (4) and Theorem 2.10.

(2) Assume that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and
$S_{G,\mathfrak{r}}$ is a submonoid of $G$. We then have
$\bigcup_{C\in\rm{supp}(\mathfrak{r})}C=\\{c\\}$ for some $c\in G$. Since
$S_{G,\mathfrak{r}}$ is a submonoid of $G$, there exists a positive integer
$n$ such that $c^{n}=1_{G}$, and so,
$S_{G,\mathfrak{r}}=\Delta_{G,\mathfrak{r}}^{0}$ is the cyclic subgroup of $G$
generated by $c$, and $S_{G,\mathfrak{r}}$ is also a normal subgroup of $G$
(by Lemma 2.9). By [26, Theorem 2.11],
$L_{K}(\Delta_{G,\mathfrak{r}})=M_{{|S_{G,\mathfrak{r}}|}}(K[x,x^{-1}])$.
Then, by Theorem 2.10, $\Gamma_{G,\mathfrak{r}}$ is a disjoint union of single
cycles and
$L_{K}(\Gamma_{G,\mathfrak{r}})=M_{|S_{G,\mathfrak{r}}|}(K[x,x^{-1}])^{(G/S_{G,\mathfrak{r}})}$,
thus finishing the proof. ∎

##  3\. Applications

In this section, based on Theorem 2.10, we characterize the Gelfand-Kirillov
dimension (Theorem 3.2), the stable rank (Theorem 3.7), the purely infinite
simplicity (Theorem 3.5) and the existence of a nonzero finite dimensional
representation (Theorem 3.9) of Leavitt path algebras of Hopf graphs via
ramification datas.

###  3.1. Gelfand-Kirillov dimension

We begin this subsection by recalling some general notions on the Gelfand-
Kirillov dimension of algebras. Given a field $K$ and a finitely generated
$K$-algebra $A$. The Gelfand-Kirillov dimension of $A$ (GKdim(A) for short) is
defined to be

$\text{GKdim}(A):=\limsup\limits_{n\rightarrow\infty}\log_{n}(\dim(V^{n})),$

where $V$ is a finite dimensional subspace of $A$ that generates $A$ as an
algebra over $K$. This definition is independent of the choice of $V$. If $A$
does not happen to be finitely generated over $K$, the Gelfand-Kirillov
dimension of $A$ is defined to be

$\text{GKdim}(A)=\sup\\{\text{GKdim}(B)\mid B\text{ is a finitely generated
subalgebra of }A\\}.$

In [12] Alahmadi, Alsulami, Jain and Zelmanov determined the Gelfand-Kirillov
dimension of Leavitt path algebras of finite graphs. In [25] Moreno-Fernández
and Siles Molina extended this to arbitrary graphs. We should mention this
result here. To do so, we need to recall useful notions of graph theory.

Let $E$ be an arbitrary graph. A cycle $c$ in $E$ is said to be an exclusive
cycle if it is disjoint with every other cycle; equivalently, no vertex on $c$
is the base of a different cycle other than a cyclic permutation of $c$. We
say that $E$ satisfies Condition (EXC) if every cycle of $E$ is an exclusive
cycle.

For two cycles $c$ and $c^{\prime}$, we write $c\Rightarrow c^{\prime}$ if
there exists a path that starts in $c$ and ends in $c^{\prime}$. A sequence of
cycles $c_{1},\cdots,c_{k}$ is a chain of length $k$ if
$c_{1}\Rightarrow\cdots\Rightarrow c_{k}$. We say that such a chain has an
exit if the cycle $c_{k}$ has an exit. Let $d_{1}$ be the maximal length of a
chain of cycles in $E$, and let $d_{2}$ be the maximal length of a chain of
cycles with an exit in $E$. For every field $K$, by [25, Theorem 3.21],
$\text{GKdim}(L_{K}(E))$ is finite if and only if $E$ satisfies Condition
(EXC) and the maximal length of chains of cycles in $E$ is finite. In this
case, $\text{GKdim}(L_{K}(E))=\max\\{2d_{1}-1,2d_{2}\\}$. Consequently, every
positive integer can arise as the Gelfand-Kirillov dimension of a Leavitt path
algebra.

######  Lemma 3.1.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ such that
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$. Then, for every cycle
$\alpha$ in $\Gamma_{G,\mathfrak{r}}$, there exists a cycle $\beta$ in
$\Gamma_{G,\mathfrak{r}}$ such that $\beta$ is not a cyclic permutation of
$\alpha$ and $\beta\Rightarrow\alpha$.

###### Proof.

Let $\alpha=e_{1}\cdots e_{n}$ be a cycle in $\Gamma_{G,\mathfrak{r}}$. We
claim that $\beta\Rightarrow\alpha$ for some cycle $\beta$ in
$\Gamma_{G,\mathfrak{r}}$. Indeed, by Proposition 2.6 (6),
$S_{G,\mathfrak{r}}$ is a submonoid of $G$, and so, there exists elements
$\\{c_{i}\\}_{i=1}^{n}\subseteq\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ such
that $c_{1}\cdots c_{n}=1_{G}$, $s(\alpha)=s(e_{1})$ and
$s(e_{i})=s(\alpha)c_{1}\cdots c_{i-1}$ for all $2\leq i\leq n$, and
$r(e_{i})=s(\alpha)c_{1}\cdots c_{i}$ for all $1\leq i\leq n$. We consider the
following two cases:

_Case_ 1: $|\bigcup_{C\in\rm{supp}(\mathfrak{r})}C|=1$. We then have
$\rm{supp}(\mathfrak{r})=\\{C\\}$ and $C=\\{c\\}$ for some $c\in G$, and
$\mathfrak{r}_{C}\geq 2$. Therefore, by the definition of
$\Gamma_{G,\mathfrak{r}}$, there exists $\mathfrak{r}_{C}$ edges from
$s(e_{i})$ to $r(e_{i})$ for all $1\leq i\leq n$, and so, $s(\alpha)$ is base
of at least $\mathfrak{r}_{C}$ cycles, that means, there exists a cycle
$\beta$ in $\Gamma_{G,\mathfrak{r}}$ such that $\beta$ is not a cyclic
permutation of $\alpha$ and $\beta\Rightarrow\alpha$, as desired.

_Case_ 2: $|\bigcup_{C\in\rm{supp}(\mathfrak{r})}C|\geq 2$. We consider the
following subcases:

_Case 2.1_ : $x^{m}=1_{G}$ for some
$x\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C\setminus\\{c_{n}\\}$ and positive
integer $m$. Then, $s(\alpha)$ is the base of a cycle $\beta=f_{1}\cdots
f_{|x|}$ which is different from $\alpha$, where $|x|$ is the order of $x$,
$s(f_{1})=s(\alpha)$, $s(f_{i})=s(\alpha)x^{i-1}$ for all $2\leq i\leq n$, and
$r(f_{i})=s(\alpha)x^{i}$ for all $1\leq i\leq n$. This implies that
$\beta\Rightarrow\alpha$, as desired.

_Case 2.2_ : $x^{m}\neq 1_{G}$ for all
$x\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C\setminus\\{c_{n}\\}$ and positive
integer $m$. Let $x$ be an arbitrary element of
$\bigcup_{C\in\rm{supp}(\mathfrak{r})}C\setminus\\{c_{n}\\}$. We then have
$s(\alpha)x^{-1}\neq s(\alpha)$, $s(\alpha)x^{-1}\neq s(\alpha)c_{1}\cdots
c_{n-1}$ and there exists an edge in $\Gamma_{G,\mathfrak{r}}$ from
$s(\alpha)x^{-1}$ to $s(\alpha)$. Moreover, we receive that $s(\alpha)x^{-1}$
is the base of the cycle $\beta=f_{1}\cdots f_{n}$, where
$s(\alpha)x^{-1}=s(f_{1})$ and $s(f_{i})=s(\alpha)x^{-1}c_{1}\cdots c_{i-1}$
for all $2\leq i\leq n$, and $r(f_{i})=s(\alpha)x^{-1}c_{1}\cdots c_{i}$ for
all $1\leq i\leq n$. These observations show that there exists a cycle $\beta$
in $\Gamma_{G,\mathfrak{r}}$ such that $\beta$ is not a cyclic permutation of
$\alpha$ and $\beta\Rightarrow\alpha$, as desired.

Therefore, in any case, we arrive at that there exists a cycle $\beta$ in
$\Gamma_{G,\mathfrak{r}}$ such that $\beta$ is not a cyclic permutation
$\alpha$ and $\beta\Rightarrow\alpha$, thus finishing the proof. ∎

In the following theorem, we compute the Gelfand-Kirillov dimension of Leavitt
path algebras of Hopf graphs via ramification datas, showing that for Hopf
graphs, there are only very few possible Gelfand-Kirillov dimensions.

######  Theorem 3.2.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ and $K$ an arbitrary
field. Then

$\displaystyle\rm{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\begin{cases}\infty&\textnormal{if
}\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2\textnormal{ and
}S_{G,\mathfrak{r}}\textnormal{ is a submonoid of $G$;}\\\ 1&\textnormal{if
}\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1\textnormal{ and
}S_{G,\mathfrak{r}}\textnormal{ is a submonoid of $G$;}\\\
0&\textnormal{otherwise}.\end{cases}$

###### Proof.

If $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and $S_{G,\mathfrak{r}}$ is a
submonoid of $G$, then by Corollary 2.12 (2), $\Gamma_{G,\mathfrak{r}}$ is a
disjoint union of single cycles, and so,
$\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=1$, by [25, Theorem 3.21]. If
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$ and $S_{G,\mathfrak{r}}$ is
a submonoid of $G$, then by Proposition 2.6 (6), $\Gamma_{G,\mathfrak{r}}$ has
a cycle. Then, by Lemma 3.1, we obtain that either $\Gamma_{G,\mathfrak{r}}$
does not satisfy Condition (EXC) or $\Gamma_{G,\mathfrak{r}}$ satisfies
Condition (EXC) and the maximal length of chains of cycles in
$\Gamma_{G,\mathfrak{r}}$ is infinite. By [25, Theorem 3.21] again, we
immediately get that $\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\infty$.
Otherwise, $S_{G,\mathfrak{r}}$ is not a submonoid of $G$, and so,
$\Gamma_{G,\mathfrak{r}}$ has no cycles, by Proposition 2.6 (6). Then, by [25,
Theorem 3.21], $\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=0$, thus
finishing the proof. ∎

For clarification, we illustrate the statement of Theorem 3.2 by presenting
the following examples.

######  Example 3.3.

(1) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.4 (1). We
then have that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=2$ and
$S_{G,\mathfrak{r}}=\\{id,(123),(132)\\}$ is a submonoid of $G=S_{3}$, and so
$\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\infty$, by Theorem 3.2.

(2) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (1). We
then have that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=2$ and
$S_{G,\mathfrak{r}}=\\{0,2n\mid n\geq 1\\}$ is a submonoid of $G=\mathbb{Z}$,
and so $\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\infty$, by Theorem 3.2.

(3) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.3 with
$n=1$. We have that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=n=1$ and
$S_{G,\mathfrak{r}}=\\{1_{G}\\}$ is a submonoid of $G$, and so
$\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=1$, by Theorem 3.2.

(4) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (2). We
have that $S_{G,\mathfrak{r}}=\\{2m+3n\mid m,n\geq 1\\}$ is not a submonoid of
$G=\mathbb{Z}$, and so $\text{GKdim}(L_{K}(\Gamma_{G,\mathfrak{r}}))=0$, by
Theorem 3.2.

###  3.2. Purely infinite simplicity

An idempotent $e$ in a ring $R$ is called _infinite_ if $eR$ is isomorphic as
a right $R$-module to a proper direct summand of itself. $R$ is called _purely
infinite_ in case every right ideal of $R$ contains an infinite idempotent. In
[4, 5] Abrams and Aranda Pino provided criteria for Leavitt path algebras of
countable graphs to be purely infinite simple. This result was extended to
arbitrary graphs in [7, Theorem 3.1.10]. We should mention the result in
Theorem 3.4 below. To do so, we need to recall some notions.

Let $E$ be a graph and $H$ a subset of $E^{0}$. We say $H$ is _hereditary_ if
for all $v\in H$ and $w\in E^{0}$, $v\geq w$ implies $w\in H$. We say $H$ is
_saturated_ if whenever $v\in E^{0}$ has the property that $v$ is regular,
$\\{r(e),e\in s^{-1}(v)\\}\subseteq H$, then $v\in H$. We say that a vertex
$v\in E^{0}$ is _cofinal_ if for every $\gamma\in E^{\geq\infty}$ there is a
vertex $w$ in the path $\gamma$ such that $v\geq w$. We say that a graph $E$
is _cofinal_ if every vertex in $E$ is cofinal. We should note that a graph
$E$ is cofinal if and only if the only hereditary and saturated subset of
$E^{0}$ are $\varnothing$ and $E^{0}$ (see, e.g. [7, Lemma 2.9.6]).

The following theorem provides us with criteria for Leavitt path algebras of
graphs to be purely infinite simple, which is very useful to prove the main
result of this subsection.

######  Theorem 3.4 ([7, Theorem 3.1.10]).

The Leavitt path algebra $L_{K}(E)$ of a graph $E$ with coefficients in a
field $K$ is purely infinite simple if and only if the following conditions
are satisfied:

$(1)$ The only hereditary and saturated subsets of $E^{0}$ are $\varnothing$
and $E^{0}$;

$(2)$ Every cycle in $E$ has an exit;

$(3)$ $E$ has a cycle.

We are now in a position to give the main result of this subsection, which
provides criteria for Leavitt path algebras of Hopf graphs to be purely
infinite simple via ramification datas which plays an important role in the
proof of Theorem 3.7 below.

######  Theorem 3.5.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ and $K$ an arbitrary
field. Then $L_{K}(\Gamma_{G,\mathfrak{r}})$ is purely infinite simple if and
only if $S_{G,\mathfrak{r}}=G$ and
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$.

###### Proof.

$(\Longrightarrow)$. Assume that $L_{K}(\Gamma_{G,\mathfrak{r}})$ is purely
infinite simple. By Theorem 3.4, $L_{K}(\Gamma_{G,\mathfrak{r}})$ has a cycle,
and so, $S_{G,\mathfrak{r}}$ is a submonoid of $G$, by Proposition 2.6 (6). If
$\mathfrak{r}=0$, then by Proposition 2.6 (4), $\Gamma_{G,\mathfrak{r}}$ is a
disjoint union of isolated vertices, a contradiction. Consider the case
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$. Since $S_{G,\mathfrak{r}}$ is a
submonoid of $G$ and by Corollary 2.12 (2),
$L_{K}(\Gamma_{G,\mathfrak{r}})=M_{|S_{G,\mathfrak{r}}|}(K[x,x^{-1}])^{(G/\Delta_{G,\mathfrak{r}}^{0})}$,
and so, $L_{K}(\Gamma_{G,\mathfrak{r}})$ is not simple, a contradiction.
Therefore, we have $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2.$

If $\langle S_{G,\mathfrak{r}}\rangle\neq G$, then by Theorem 2.10, we have
$L_{K}(\Gamma_{G,\mathfrak{r}})=L_{K}(\Gamma_{G,\mathfrak{r}})^{(G/\langle
S_{G,\mathfrak{r}}\rangle)}$, and so, $L_{K}(\Gamma_{G,\mathfrak{r}})$ is not
simple, a contradiction. This implies that $\langle
S_{G,\mathfrak{r}}\rangle=G$.

We claim that $S_{G,\mathfrak{r}}$ is a group. Indeed, if
$S_{G,\mathfrak{r}}=\\{1_{G}\\}$, then the claim is obvious. Consider the case
when $S_{G,\mathfrak{r}}\neq\\{1_{G}\\}$. Let
$d\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ and $d\neq 1_{G}$. Then, there
exists an edge $e$ in $\Gamma_{G,\mathfrak{r}}$ such that $s(e)=d$ and
$r(e)=d^{2}$. Since $S_{G,\mathfrak{r}}$ is a submonoid of $G$, there exists
elements
$\\{w_{i}\\}_{i=1}^{n}\subseteq\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$ such
that $w_{1}\cdots w_{n}=1_{G}$, and so, there exists a cycle
$\alpha=e_{1}e_{2}\cdots e_{n}$ in $\Gamma_{G,\mathfrak{r}}$ such that
$d=s(e_{1})$ and $r(e_{i})=s(e_{i})w_{i}$ for all $1\leq i\leq n$. Since
$L_{K}(\Gamma_{G,\mathfrak{r}})$ is simple and by [7, Theorem 2.9.7],
$\Gamma_{G,\mathfrak{r}}$ is cofinal. Then, there exists a path $p=f_{1}\cdots
f_{k}$ in $\Gamma_{G,\mathfrak{r}}$ which starts at $d^{2}$ and ends in
$\alpha$. Write $r(f_{i})=s(f_{i})c_{i}$ for all $1\leq i\leq k$, where
$c_{i}\in\bigcup_{C\in\rm{supp}(\mathfrak{r})}C$. Since $p$ ends in $\alpha$,
$r(p)=s(e_{i})$ for some $1\leq i\leq n$. Let $\beta:=ef_{1}\cdots
f_{k}e_{i}e_{i+1}\cdots e_{n}$. We then have that $\beta$ is a closed path in
$\Gamma_{G,\mathfrak{r}}$ based at $d^{2}$, and so,
$s(e)=d=r(e_{n})=s(e)dc_{1}\cdots c_{k}w_{i}\cdots w_{n}$ and $dc_{1}\cdots
c_{k}w_{i}\cdots w_{n}=1_{G}$. This implies that $d$ is invertible in
$S_{G,\mathfrak{r}}$, and hence $S_{G,\mathfrak{r}}$ is a subgroup of $G$,
proving the claim. Then, since $\langle S_{G,\mathfrak{r}}\rangle=G$, we
obtain that $S_{G,\mathfrak{r}}=G$.

$(\Longleftarrow)$. Since $S_{G,\mathfrak{r}}=G$ and by Theorem 2.10 and
Proposition 2.6 (6), $\Gamma_{G,\mathfrak{r}}$ is both strongly connected and
has a cycle. It implies that $\Gamma_{G,\mathfrak{r}}$ is cofinal, and so, the
only hereditary and saturated subset of $\Gamma^{0}_{G,\mathfrak{r}}$ are
$\varnothing$ and $\Gamma^{0}_{G,\mathfrak{r}}$. Since
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$, every cycle in
$\Gamma_{G,\mathfrak{r}}$ has an exit. Then, by Theorem 3.4,
$L_{K}(\Gamma_{G,\mathfrak{r}})$ is purely infinite simple, thus finishing the
proof. ∎

Consequently, we obtain the following corollary which extends [26, Proposition
4.1] and [24, Theorem 3.1] to Hopf graphs.

######  Corollary 3.6.

Let $G$ be a finite group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C\neq 0$ and $K$ an
arbitrary field. Then the following statements are equivalent:

$(1)$ $L_{K}(\Gamma_{G,\mathfrak{r}})$ is purely infinite simple;

$(2)$ $L_{K}(\Gamma_{G,\mathfrak{r}})$ is simple;

$(3)$ $S_{G,\mathfrak{r}}=G$ and
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$.

###### Proof.

The equivalence of (1) and (3) immediately follows from Theorem 3.5. We note
that by Theorem 3.4 and [7, Theorem 2.9.1], the Leavitt path algebra
$L_{K}(E)$ of an arbitrary graph $E$ is purely infinite simple if and only if
$L_{K}(E)$ is simple and $E$ has a cycle. Since $G$ is finite,
$S_{G,\mathfrak{r}}$ is a subgroup of $G$, and so, $\Gamma_{G,\mathfrak{r}}$
has a cycle by Proposition 2.6 (6). These observations show the equivalence of
(1) and (2), thus finishing the proof. ∎

###  3.3. Stable rank

Let $S$ be a unital ring containing an associate ring $R$ as a two-sided
ideal. Following [29], a vector $(a_{i})_{i=1}^{n}$ in $S$ is called
$R$-_unimodular_ if $a_{1}-1;a_{i}\in R$ for $i>1$ and there exists
$b_{1}-1;b_{i}\in R\,(i>1)$ such that $\sum_{i=1}^{n}a_{i}b_{i}=1$. We denote
by $\text{sr}(R)$ the _stable rank_ of $R$, which is the least number $m$ for
which for any $R$-unimodular vector $(a_{i})_{i=1}^{m}+1$ there exists
$r_{i}\in R$ such that the vector $(a_{i}+r_{i}a_{m+1})_{i=1}^{m}$ is
$R$-unimodular. If such an $m$ does not exist, the stable rank of $R$ is
defined to be infinite.

In [15, Theorem 2.8] Ara and Pardo showed that the only possible values for
the stable rank of the Leavitt path algebra of a row-finite graphs are $1$,
$2$ and $\infty$. In [22, Theorem 4.7] Larki and Riazi extended this to an
arbitrary graph. In the following theorem, by using [22, Theorem 4.7] and
Theorems 2.10 and 3.5, we compute the stable rank of Leavitt path algebras of
Hopf graphs via ramification datas.

######  Theorem 3.7.

Let $G$ be a group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ and $K$ an arbitrary
field. Then

$\displaystyle\rm{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\begin{cases}1&\textnormal{if
}S_{G,\mathfrak{r}}\textnormal{ is not a submonoid of }G\\\
\infty&\textnormal{if }\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq
2\textnormal{ and }S_{G,\mathfrak{r}}\textnormal{ is a finite subgroup of
}G\\\ 2&\textnormal{otherwise}.\end{cases}$

###### Proof.

If $S_{G,\mathfrak{r}}$ is not a submonoid of $G$, then
$\Gamma_{G,\mathfrak{r}}$ is acyclic, by Proposition 2.6 (6). By [22, Theorem
4.7] (1), $\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=1$. If
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$ and $S_{G,\mathfrak{r}}$ is
a finite subgroup of $G$, then by Theorem 3.5,
$L_{K}(\Delta_{G,\mathfrak{r}})$ is both a unital purely infinite simple ring
and a quotient of $L_{K}(\Gamma_{G,\mathfrak{r}})$. By [22, Theorem 4.7] (2),
$\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\infty$. Otherwise, we have the
following cases:

_Case_ 1: $S_{G,\mathfrak{r}}$ is a submonoid of $G$ and
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\leq 1$. We then have that
$\Gamma_{G,\mathfrak{r}}$ has a cycle, by Proposition 2.6 (6). By Corollary
2.12, $\Gamma_{G,\mathfrak{r}}$ is a disjoint union of single cycles, and so,
$\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=2$, by [22, Theorem 4.7] (3).

_Case_ 2: $S_{G,\mathfrak{r}}$ is an infinite subgroup of $G$ and
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$. Then, by Theorem 3.5,
$L_{K}(\Delta_{G,\mathfrak{r}})$ is non-unital purely infinite simple. This
implies that $L_{K}(\Gamma_{G,\mathfrak{r}})$ has no unital purely infinite
simple quotients, and so, $\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=2$, by
[22, Theorem 4.7] (3).

_Case_ 3: $S_{G,\mathfrak{r}}$ is an infinite submonoid of $G$, but not a
group and $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$. Then, since
$S_{G,\mathfrak{r}}$ is a submonoid of $G$ and by Proposition 2.6 (6),
$\Gamma_{G,\mathfrak{r}}$ contains a cycle. We next claim that
$L_{K}(\Gamma_{G,\mathfrak{r}})$ has no unital purely infinite simple
quotients. Indeed, by Theorem 2.10, it is enough to show that
$L_{K}(\Delta_{G,\mathfrak{r}})$ has no unital purely infinite simple
quotients. Assume that $L_{K}(\Delta_{G,\mathfrak{r}})$ has a unital purely
infinite simple quotient. By [22, Proposition 3.3], there exists a hereditary
and saturated subset $H$ of $\Delta_{G,\mathfrak{r}}^{0}$ such that
$L_{K}(\Delta_{G,\mathfrak{r}}\setminus H)$ is unital purely infinite simple,
where $\Delta_{G,\mathfrak{r}}\setminus H$ is the quotient graph defined by:

$(\Delta_{G,\mathfrak{r}}\setminus H)^{0}=\Delta_{G,\mathfrak{r}}^{0}\setminus
H$ and $(\Delta_{G,\mathfrak{r}}\setminus
H)^{1}=\\{e\in\Delta_{G,\mathfrak{r}}^{1}\mid r(e)\notin H\\}$.

In particular, we receive that $\Delta_{G,\mathfrak{r}}^{0}\setminus H$ is a
nonempty finite set. Let $v\in\Delta_{G,\mathfrak{r}}^{0}\setminus H$. Since
$S_{G,\mathfrak{r}}$ is not a group, there exists an element $c\in
S_{G,\mathfrak{r}}$ such that $wc\neq 1_{G}$ for all $w\in
S_{G,\mathfrak{r}}$. We then have that $\\{vc^{-k}\\}^{\infty}_{k=0}$ are
distinct vertices in $\Delta_{G,\mathfrak{r}}$, where $c^{0}:=1_{G}$. Since
$vc^{-k}=(vc^{-k-1})c$, for each $k\geq 0$, there exists an edge $e_{k}$ in
$\Delta_{G,\mathfrak{r}}$ such that $s(e_{k})=vc^{-k-1}$ and
$r(e_{k})=vc^{-k}$, and so, there exists a path in $\Delta_{G,\mathfrak{r}}$
from $vc^{-k}$ to $v$ for all $k\geq 1$. Since $H$ is hereditary and $v\notin
H$, $vc^{-k}\notin H$ for all $k$. This shows that
$\Delta_{G,\mathfrak{r}}^{0}\setminus H$ is an infinite set, a contradiction,
proving the claim. From these observations and [22, Theorem 4.7 (3)], we
immediately obtain that $\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=2$, thus
finishing the proof. ∎

We end this subsection by presenting the following example.

######  Example 3.8.

(1) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.4 (1). We
then have that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=2$ and
$S_{G,\mathfrak{r}}=\\{id,(123),(132)\\}$ is a finite subgroup of $G=S_{3}$,
and so $\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=\infty$, by Theorem 3.7.

(2) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (1). We
then have that $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=2$ and
$S_{G,\mathfrak{r}}=\\{0,2n\mid n\geq 1\\}$ is an infinite submonoid of
$G=\mathbb{Z}$, and so $\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=2$, by
Theorem 3.7.

We note that Leavitt path algebras of these graphs have infinite Gelfand-
Kirillov dimension (Example 3.3).

(3) Consider the Hopf graph $\Gamma_{G,\mathfrak{r}}$ of Example 2.5 (2). We
then have that $S_{G,\mathfrak{r}}=\\{2m+3n\mid m,n\geq 1\\}$ is not a
submonoid of $G=\mathbb{Z}$, and so
$\text{sr}(L_{K}(\Gamma_{G,\mathfrak{r}}))=1$, by Theorem 3.7.

###  3.4. Finite dimensional representations

We begin this subsection by recalling notions of graph theory introduced in
[20, 21]. Let $E$ be a graph. For a cycle $c$ and a sink $v$ in $E$, we write
$c\Rightarrow v$ if there exists a path in $E$ which starts in $c$ and ends at
$v$. A sink $v$ in $E$ is called maximal if there is not any cycle $c$ in $E$
such that $c\Rightarrow v$. A cycle $c$ in $E$ is called maximal if there is
not any cycle $d$ in $E$ which is different from a cyclic permutation of $c$
such that $d\Rightarrow c$. The predecessors of a vertex $v$ in $E$ is the set
$E_{\geq v}:=\\{w\in E^{0}\mid w\geq v\\}$ and the predecessors of a cycle $c$
in $E$ is the set $E_{\geq v}$, where $v$ is an arbitrary vertex on $c$.

In [20, Theorem 6.5] Koç and Özaydin proved that the Leavitt path algebra
$L_{K}(E)$ of a row-finite graphs $E$ with coefficients over a field $K$ has a
nonzero finite dimensional representations if and only if $E$ has a maximal
sink or cycle with finitely many predecessors. Moreover, they have classified
all finite dimensional representations of Leavitt path algebras of row-finite
graphs (see [21, Theorem 4.7]). In the following theorem, based essentially on
these results, we classify all finite dimensional representations of Leavitt
path algebras of Hopf graphs via ramification datas.

######  Theorem 3.9.

Let $G$ be a group with a finite support ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ and $K$ an arbitrary
field. Then $L_{K}(\Gamma_{G,\mathfrak{r}})$ has a nonzero finite dimensional
representation if and only if one of the following conditions holds:

$(1)$ $\mathfrak{r}=0$;

$(2)$ $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and $S_{G,\mathfrak{r}}$
is a submonoid of $G$.

Furthermore, if the above conditions are satisfied, then

$\displaystyle\mathcal{M}_{L_{K}(\Gamma_{G,\mathfrak{r}})}^{fd}\backsimeq\begin{cases}(\mathcal{M}_{K}^{fd})^{(G)}&\textnormal{
if }\mathfrak{r}=0\\\
(\mathcal{M}_{K[x,x^{-1}]}^{fd})^{(G/S_{G,\mathfrak{r}})}&\textnormal{
otherwise,}\end{cases}$

where $\mathcal{M}_{A}^{fd}$ is the category of finite dimensional
$A$-modules, $\mathcal{N}^{(X)}$ is the $X$-indexed direct sum of copies of
the category $\mathcal{N}$, and $\backsimeq$ denotes equivalence of
categories.

###### Proof.

$(\Longrightarrow)$. Assume that $L_{K}(\Gamma_{G,\mathfrak{r}})$ has a
nonzero finite dimensional representation. By [20, Theorem 6.5],
$\Gamma_{G,\mathfrak{r}}$ has a maximal sink or cycle with finitely many
predecessors. We consider the following cases:

_Case_ 1: $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq 2$. Then, by
Proposition 2.6 (5) and Lemma 3.1 respectively, $\Gamma_{G,\mathfrak{r}}$ has
neither maximal sinks nor maximal cycles, a contradiction.

_Case_ 2: $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and
$S_{G,\mathfrak{r}}$ is not a submonoid of $G$. Then, by Propositions 2.6 (5)
and (6) respectively, $\Gamma_{G,\mathfrak{r}}$ has neither sinks nor cycles.
So, $\Gamma_{G,\mathfrak{r}}$ has neither maximal sinks nor maximal cycles, a
contradiction.

Therefore, in any case, we arrive at a contradiction, thus proving the
statement.

$(\Longleftarrow)$. If $\mathfrak{r}=0$, we then have
$L_{K}(\Gamma_{G,\mathfrak{r}})=K^{(G)}$, by Corollary 2.12 (1). This implies
that
$\mathcal{M}_{L_{K}(\Gamma_{G,\mathfrak{r}})}^{fd}\backsimeq(\mathcal{M}_{K}^{fd})^{(G)}$.
Consider the case when $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|=1$ and
$S_{G,\mathfrak{r}}$ is a submonoid of $G$. By Corollary 2.12 (2), we
immediately obtain that $S_{G,\mathfrak{r}}$ is a finite subgroup of $G$ and
$L_{K}(\Gamma_{G,\mathfrak{r}})=(M_{|S_{G,\mathfrak{r}}|}(K[x,x^{-1}]))^{(G/S_{G,\mathfrak{r}})}$,
and so,
$\mathcal{M}_{L_{K}(\Gamma_{G,\mathfrak{r}})}^{fd}\backsimeq(\mathcal{M}_{K[x,x^{-1}]}^{fd})^{(G/S_{G,\mathfrak{r}})}$,
thus finishing the proof. ∎

A unital ring $R$ is said to have _Invariant Basis Number_ if, for any pair of
positive integers $m\text{ and }n$, $R^{m}\cong R^{n}$ (as right modules)
implies that $m=n$. A unital ring $R$ is said to have _Unbounded Generating
Number_ if for all $m,n\in\mathbb{N}$ and any right $R$-module $K$,
$R^{n}\cong R^{m}\oplus K$ (as right $R$-modules) implies that $n\geq m$. A
straightforward computation immediately establishes that if $R$ has Unbounded
Generating Number, then $R$ has Invariant Basis Number.

Criteria for Leavitt path algebras of finite graphs to have Invariant Basis
Number have been established in [18, Theorem 3.1] and [26, Theorem 3.5]. In
[10, Theorem 3.16] Abrams and the authors completely classified finite graphs
$E$ for which the Leavitt path algebra $L_{K}(E)$ of $E$ with coefficients
over a field $K$ have Unbounded Generating Number. In [10, Example 3.19]
Abrams and the authors established that, within the class of Leavitt path
algebras, the Invariant Basis Number property is strictly weaker than the
Unbounded Generating Number property. However, these properties are equivalent
to each other within the class of Leavitt path algebras of Cayley graphs ([26,
Corollary 4.3]). In the following corollary (Corollary 3.10), we prove that
these properties are equivalent to each other for Leavitt path algebras of
Hopf graphs. Before doing so, we need some useful notions and facts.

Following [14], for any directed graph $E=(E^{0},E^{1},s,r)$ we define the
monoid $M_{E}$ as follows. We denote by $T$ the free abelian monoid (written
additively) with generators $E^{0}$ and define relations on $T$ by setting

$v=\sum_{e\in s^{-1}(v)}r(e)$

for every regular vertex $v\in E^{0}$. Let $\sim_{E}$ be the congruence
relation on $T$ generated by these relations. Then $M_{E}=T/_{\sim_{E}}$, and
we also denote an element of $M_{E}$ by $[x]$, where $x\in T$.

Following [26, Corollary 3.4], the Leavitt path algebra $L_{K}(E)$ of a finite
graph $E$ with coefficients in a field $K$ has Invariant Basis Number if and
only if for any pair of positive integers $m$ and $n$,

if $\ m[\sum_{v\in E^{0}}v]=n[\sum_{v\in E^{0}}v]$ in $M_{E}$, then $m=n$.

We end this article with the following fact which extends [26, Theorem 4.2] to
Hopf graphs.

######  Corollary 3.10.

Let $G$ be a finite group with a ramification data
$\mathfrak{r}=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}C$ and $K$ an arbitrary
field. Then, the following statements are equivalent:

$(1)$ $L_{K}(\Gamma_{G,\mathfrak{r}})$ has a nonzero finite dimensional
module;

$(2)$ $L_{K}(\Gamma_{G,\mathfrak{r}})$ has Unbounded Generating Number;

$(3)$ $L_{K}(\Gamma_{G,\mathfrak{r}})$ has Invariant Basis Number;

$(4)$ $\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\leq 1$.

###### Proof.

(1) $\Longleftrightarrow$ (2). It follows from [10, Theorem 3.16] and [20,
Theorem 6.5].

(2) $\Longrightarrow$ (3). It is obvious.

(3) $\Longrightarrow$ (4). Assume that $L_{K}(\Gamma_{G,\mathfrak{r}})$ has
Invariant Basis Number, and $m:=\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\geq
2$. Write

$\text{supp}(\mathfrak{r})=\\{C_{1},C_{2},\cdots,C_{k}\\}$ and
$C_{i}=\\{c_{i1},c_{i2},\cdots,c_{in_{i}}\\}$.

We then have $G=Gc_{ij}$ for all $1\leq i\leq k$ and $1\leq j\leq n_{i}$
(since $G$ is a finite group) and every vertex $g$ in
$\Gamma_{G,\mathfrak{r}}$ emits exactly $\mathfrak{r}_{C_{i}}$ edges to the
vertex $gc_{ij}$ for all $1\leq i\leq k$ and $1\leq j\leq n_{i}$, and $g$
emits only to these vertices $gc_{ij}$. This implies that

$[g]=\sum^{n_{1}}_{j=1}\mathfrak{r}_{C_{1}}[gc_{1j}]+\sum^{n_{2}}_{j=1}\mathfrak{r}_{C_{2}}[gc_{2j}]+\cdots+\sum^{n_{k}}_{j=1}\mathfrak{r}_{C_{k}}[gc_{kj}]$
in $M_{\Gamma_{G,\mathfrak{r}}}$

and

$[\sum_{g\in G}g]=\sum^{n_{1}}_{j=1}\mathfrak{r}_{C_{1}}[\sum_{g\in
G}gc_{1j}]+\cdots+\sum^{n_{k}}_{j=1}\mathfrak{r}_{C_{k}}[\sum_{g\in
G}gc_{kj}]$ in $M_{\Gamma_{G,\mathfrak{r}}}.$

Since $G=Gc_{ij}$ for all $1\leq i\leq k$ and $1\leq j\leq n_{i}$, we obtain
that

$\sum_{g\in G}g=\sum_{g\in G}gc_{ij}$ for all $1\leq i\leq k$ and $1\leq j\leq
n_{i},$

so

$[\sum_{g\in G}g]=\sum^{k}_{i=1}n_{i}\mathfrak{r}_{C_{i}}[\sum_{g\in
G}g]=m[\sum_{g\in G}g]$ in $M_{\Gamma_{G,\mathfrak{r}}},$

showing that $L_{K}(\Gamma_{G,\mathfrak{r}})$ has no Invariant Basis Number by
[26, Corollary 3.4], a contradiction. Therefore, we must have
$\sum_{C\in\mathcal{C}}\mathfrak{r}_{C}|C|\leq 1$.

(4) $\Longrightarrow$ (1). It follows from Theorem 3.9 and the fact that if
$G$ is a finite group, then $S_{G,\mathfrak{r}}$ is a subgroup of $G$, thus
finishing the proof. ∎

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