# The Commuting Algebra

Edward L. Green  Edward L. Green,
Department of Mathematics
Virginia Tech
Blacksburg, VA 24061
USA<EMAIL_ADDRESS>and  Sibylle Schroll  Insitut für Mathematik,
Universität zu Köln, Weyertal 86-90, Köln, Germany and Institutt for
matematiske fag, NTNU, N-7491 Trondheim, Norway<EMAIL_ADDRESS>
###### Abstract.

Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field.
We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all
differences of parallel paths in $Q$. We show that $KQ/C$ is always finite
dimensional and its global dimension is finite. Furthermore, we prove that
$KQ/C$ is Morita equivalent to an incidence algebra.
The paper starts with a more general setting, where $KQ$ is replaced by $KQ/I$
with $I$ a two-sided ideal in $KQ$.

_2020 Mathematics Subject Classification._ 16P10 16P20 16S50 05E10

This work was partially supported through the DFG through the project SFB/TRR
191 Symplectic Structures in Geometry, Algebra and Dynamics (Projektnummer
281071066-TRR 191).

##  1\. Introduction

In the study of the representation theory of finite dimensional $K$-algebras
with $K$ a field, the algebras one encounters often are of the form $KQ/I$,
where $I$ is an admissible ideal; that is $J^{n}\subseteq I\subseteq J^{2}$,
for some positive integer $n$, and $J$ is the ideal in $KQ$ generated by the
arrows of $Q$. It is not unreasonable to say that $J$ plays a ‘special’ role
in the theory.

Our overall goal is to show that there is another ‘special’ ideal in a path
algebra that connects any (not necessarily finite dimensional) path algebra of
a finite quiver $Q$ with a subring of a matrix ring and with an incidence
algebra. It is well known that incidence algebras, partially ordered sets and
Hasse diagrams are all interrelated and that their representation theory is
well-studied and well-understood. For some examples of classical as well as
recent work in that direction, see [B, FI, IZ, IM, K, NR]. If $\Lambda=KQ/I$
is a not necessarily finite dimensional algebra, we consider a different type
of ’special’ ideal, $C$, in $\Lambda$ which has the property that $(KQ/I)/C$
is always finite dimensional and contains information about $\Lambda$. There
are no restrictions on $Q$ nor $I$ other than $Q$ is a finite quiver. In
particular, $I$ need not be finitely generated and $KQ/I$ need not be left nor
right Noetherian. For the special case $I=0$ and $\Lambda=KQ$, we prove that
$KQ/C$ has finite global dimension. Moreover, in this case, we show that a
basic finite dimensional algebra, Morita equivalent to $KQ/C$, is an incidence
algebra. This paper provides a detailed analysis of $\Lambda/C$,with special
attention given to the case $\Lambda=KQ$.

We summarize the results of the paper. In Section 2, we define a quasi-
commuting ideal and quasi-commuting algebras. Theorem 3.1 proves that all
quasi-commuting algebras are finite dimensional $K$-algebras. Section 3
further studies properties of quasi-commuting algebras. In Section 4, we turn
our attention to the relation on the vertex set of $Q$ given by $v\sim w$ if
there are paths in $Q$ from $v$ to $w$ and from $w$ to $v$. This leads to
showing that if $I=0$ and all the coefficients in the commuting relations
defining the ideal $C$ are equal to 1, the commuting algebra is in block
matrix form (see Definition 4.3 and Theorem 4.4) In particular, we see that a
commuting algebra is a subring of a full matrix ring. In Section 5, we apply
Morita equivalence theory to find a basic algebra Morita equivalent to the
commuting algebra. We call such an algebra a skeleton of $Q$. We show that the
vertex set of a skeleton can be partially ordered (Theorem 6.2) and hence
skeletons have finite global dimension (Theorem 6.4). It follows that
commuting algebras have finite global dimension. Thus starting with a finite
quiver $Q$, a skeleton of $KQ$ is always an incidence algebra.

##  2\. Quasi-commuting algebras

We begin by recalling the definition of _parallel paths_ in a quiver. Let $p$
and $q$ be (finite) paths in $Q$. Then we say $p$ is _parallel_ to $q$,
denoted $p\|q$, if there exist vertices $v,w$ in $Q$ such that $vp=p,vq=q$,
$pw=p$, and $qw=q$. It is convenient to let ${\mathcal{B}}$ denote the set of
finite paths in $Q$. Note that ${\mathcal{B}}$ includes the paths of length
zero; namely, the vertices of $Q$, and that ${\mathcal{B}}$ is an infinite set
if and only there is an oriented cycle of length $\geq 1$ in $Q$.

We fix an ideal $I$ in $KQ$ that is contained in the ideal in $KQ$ generated
by paths of length 2. We let $KQ/I$ be denoted by $\Lambda$, the canonical
surjection $KQ\to\Lambda$ by $\pi$ and let $f:{\mathcal{B}}\to K^{*}$ be a set
map.

######  Definition 2.1.

Let _the quasi-commuting ideal of $\Lambda$ and $f$_, denoted by $C^{f}$, be
the ideal in $\Lambda$ generated by all $f(p)\pi(p)-f(q)\pi(q)$, where
$p,q\in{\mathcal{B}}$ and $p\|q$. We call $\Lambda/C^{f}$ _the quasi-commuting
algebra of $\Lambda$ and $f$_. In the special case where $f$ is the constant
map equal to $1$, we call $\Lambda/C^{f}$, _the commuting algebra of
$\Lambda$_ and $C=C^{f}$ the commuting ideal in $\Lambda$.

Denote the canonical surjection $\Lambda\to\Lambda/C^{f}$ by $\rho$.

##  3\. Properties of quasi-commuting algebras

The following result is fundamental to the study of the structure of quasi-
commuting algebras.

######  Theorem 3.1.

Let $Q$ be a quiver with $n$ vertices. Then keeping the notation above, we
have

  1. (1)

if $v_{i}$ and $v_{j}$ are vertices in $Q$, then
$\operatorname{dim}\nolimits_{K}(v_{i}\Lambda/C^{f}v_{j})\leq 1$, for all
$1\leq i,j\leq n$.

  2. (2)

Every quasi-commuting algebra of $\Lambda$ is finite dimensional, with
dimension over $K$ no greater than $n^{2}$, where $n$ is the number of
vertices of $Q$.

###### Proof.

We begin with a proof of (1). Let $B$ be the $K$-basis of paths in
$v_{i}KQv_{j}$, for vertices $v_{i}$ and $v_{j}$ of $Q$. In particular,
$B=v_{i}{\mathcal{B}}v_{j}$. Let $\rho\pi(B)$ denote the set
$\\{\rho\pi(p)|p\in B\\}$. Clearly $\rho\pi(B)$ generates
$v_{i}(\Lambda/C^{f})v_{j}$. Suppose that
$v_{i}(\Lambda/C^{f}_{\Lambda})v_{j}\neq 0$. It follows that there is some
$p^{*}\in B$ such that $\rho\pi(p^{*})\neq 0$. We show that $\rho\pi(p^{*})$
generates $v_{i}(\Lambda/C^{f})v_{j}$ as a $K$-vector space; thus showing
$\operatorname{dim}\nolimits_{K}(v_{i}(\Lambda/C^{f})v_{j})=1$.

We see that $\rho\pi(B)$ generates $v_{i}\Lambda/C^{f}v_{j}$. Let $p$ be a
path from $v_{i}$ to $v_{j}$ in $Q$. Then $\pi(f(p^{*})p^{*}-f(p)p)\in C^{f}$.
Hence $[f(p^{*})/f(p)]\rho\pi(p^{*})=\rho\pi(p)$ and we conclude that
$\rho\pi(p^{*})$ generates $v_{i}\Lambda/C^{f}v_{j}$.

To prove (2), note that
$\Lambda/C^{f}=\oplus_{i=1}^{n}\oplus_{j=1}^{n}v_{i}(\Lambda/C^{f})v_{j}$. The
result follows from (1).

∎

The next result will be used frequently.

######  Corollary 3.2.

Let $Q$ be a quiver and $I$ an ideal in $KQ$ contained in the ideal generated
by paths of length $2$. Suppose $v,w$ are vertices in $Q$ (not necessarily
distinct). Let $f\colon{\mathcal{B}}\to K^{*}$. The following statements are
equivalent

  1. (1)

There is a path $p$ from $v$ to $w$ in $Q$ such that $\rho\pi(p)\neq 0$

  2. (2)

$\operatorname{dim}\nolimits_{K}v(\Lambda/C^{f})w\neq 0$

  3. (3)

$\operatorname{dim}\nolimits_{K}v(\Lambda/C^{f})w=1$.

###### Proof.

Parts (2) and (3) are seen to be equivalent by Theorem 3.1.

It is obvious that (1) implies part (2).

Now assume part (2) holds. Then, since $v_{i}\Lambda/C^{f}v_{j}\neq 0$, there
is a path $p$ from $v_{i}$ to $v_{j}$ such that $\rho\pi(p)\neq 0$ and we are
done. ∎

######  Remark 3.3.

Note that if $I=0$, this implies that $\rho$ is the identity map and thus
there is a non-zero path from $v$ to $w$ in $Q$ if and only if
$v(\Lambda/C^{f})w=v(KQ/C^{f})w\neq 0$.

Combining Theorem 3.1 and Corollary 3.2, we have the following.

######  Corollary 3.4.

Let $Q$ be a quiver and $I$ an ideal in $KQ$ contained in the ideal generated
by paths of length $2$. Suppose $v,w$ are vertices in $Q$ (not necessarily
distinct). The following statements are equivalent

  1. (1)

$\rho\pi(p)=0$, for all paths $p$ from $v$ to $w$ in $Q$

  2. (2)

For all paths $q$ from $v$ to $w$ in $Q$, $q\in(vC^{f}w)+I$

  3. (3)

$\operatorname{dim}\nolimits_{K}(v(\Lambda/C^{f})w)=0$.

######  Corollary 3.5.

Let $v$ and $w$ be two, not necessarily distinct, vertices in $Q$. If $p$ is a
path from $v$ to $w$ such that $\rho\pi(p)\neq 0$, then $\rho\pi(p)$ is a
$K$-basis for $v(KQ/C^{f})w$. If $q$ is another path from $v$ to $w$ such that
$\rho\pi(q)\neq 0$, then $f(q)\rho\pi(q)=f(p)\rho\pi(p)$.

###### Proof.

Using Theorem 3.1 the result follows. ∎

The following result is easy to prove and we include a proof for completeness.

######  Lemma 3.6.

Let $V$ be a $K$-vector space with $K$-basis ${\mathcal{B}}^{*}=\\{b_{i}\mid
i\in{\mathcal{I}}\\}$. Let $b\in{\mathcal{B}}^{*}$ and let
$f:{\mathcal{B}}^{*}\to K^{*}$. Define $C$ to be the $K$-subspace of $V$
generated by the set $\\{f(b_{i})b_{i}-f(b)b\mid i\in{\mathcal{I}}\\}$. Then
$b\not\in C$.

###### Proof.

If $b\in C$ then $b$ is a finite linear combination of elements of the form
$f(b_{i})b_{i}-f(b)b$; say $b=\sum_{b_{i}\neq
b}\alpha_{i}(f(b_{i})b_{i}-f(b)b)$.
Thus we obtain

$b+(\sum_{b_{i}\neq b}\alpha_{i})f(b)b=\sum_{b_{i}\neq
b}(\alpha_{i}f(b_{i})b_{i}).$

This contradicts ${\mathcal{B}}^{*}$ is a linearly independent set.∎

As noted in the proof of Theorem 3.1, the quasi-commuting algebra,
$\Lambda/C^{f}$, has a direct sum decomposition

$(*)\quad\quad\Lambda/C^{f}=\oplus_{i=1}^{n}\oplus_{j=1}^{n}v_{i}(\Lambda/C^{f})v_{j}.$

We use this observation in the next section, and we end this section, with the
following result:

######  Lemma 3.7.

Let $L$ and $I$ be ideals in $KQ$ such that $L\subseteq I\subseteq J^{2}$, the
ideal of $KQ$ generated by paths of length 2. Then for any $f:{\mathcal{B}}\to
K^{*}$, the quasi-commuting algebra of $KQ/L$ and $f$ maps onto the quasi-
commuting algebra of $KQ/I$ and $f$.

###### Proof.

Let $\pi_{I}\colon KQ\to KQ/I$, $\pi_{L}\colon KQ\to KQ/L$, $\rho_{I}\colon
KQ/I\to(KQ/I)/C_{I}^{f}$, and $\rho_{L}\colon KQ/L\to(KQ/L)/C_{L}^{f}$ be the
canonical surjections. Then $C_{L}^{f}$ is generated by
$f(p)\pi_{L}(p)-f(q)\pi_{L}(q)$ for parallel paths $p$ and $q$ and
$\pi_{L}(f(p)p-f(q)q)\in C_{L}^{f}$ and $\pi_{I}(f(p)p-f(q)q)\in C_{I}^{f}$.
Thus $C_{L}^{f}\to C_{I}^{f}$ is a surjection. The result follows from the
exact commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{(KQ/L)/C_{L}^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(KQ/I)/C_{I}^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{KQ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{KQ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{L}^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{I}^{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

∎

Letting $L=0$ in the above lemma, we see that if $I$ is an ideal contained in
$J^{2}$, the quasi-commuting algebra of $KQ$ for some function $f$ maps onto
the commuting algebra of $KQ/I$ for the same $f$.

##  4\. Path connected components

For the remainder of the paper, we will restrict our attention to the
‘hereditary’ case; that is, we assume $I=0$, and we study the quasi-commuting
algebras of a path algebra $KQ$.

Next, consider the relation $\sim$ on the vertex set of $Q$ given by $v\sim w$
if and only if there are paths from $v$ to $w$ and from $w$ to $v$. Then
$\sim$ is easily seen to be an equivalence relation on the vertex set of $Q$.
The equivalence class of a vertex $v$, denoted $\wr v\wr$, is called the _path
connected component of $v$_ . It is well-known that different path connected
components are disjoint and that the disjoint union of the path connected
components is the set of vertices. Note that $v\in\wr v\wr$ since $v$ is a
path of length $0$ with $v$ both the start and end vertex.

We will see in Theorem 4.4 that the path connected components are related to
the ring structure of the commuting algebra of $KQ$. Before that the next few
results show that in studying the structure of quasi-commuting algebras, we
may assume that $Q$ has no loops or multiple arrows.

Our next goal is to describe the ring structure of a quasi-commuting algebra
of $KQ$ and $f$. In the hereditary case ($I=0$), we have $\pi\colon KQ\to
KQ/I$ is the identity map. Thus by Corollary 3.2, we have that $vKQ/C^{f}w\neq
0$ if and only if there is a path $p$ in $Q$ from $v$ to $w$.

We fix the following notation: $Q$ is a quiver with $n$ vertices
$v_{1},\dots,v_{n}$ and $f:{\mathcal{B}}\to K^{*}$ is a set map. Let
${\mathcal{D}}=\\{D_{1},\dots,D_{m}\\}$ be the path connected components of
$Q$. For $i=1,\dots,m$, set $d_{i}=|D_{i}|$ and hence,
$\sum_{i=1}^{m}d_{i}=n$.

######  Definition 4.1.

If $a$ and $b$ are positive integers, $M_{a\times b}(K)$ denotes the $a\times
b$ matrix with each entry $K$. We also define $M_{a\times b}(0)$ to be the
$a\times b$ matrix, all of whose entries are $0$.

The next result will be applied in the Structure Theorem below.

######  Proposition 4.2.

Let $v$ and $w$ be vertices in $Q$ with $v$ in $D_{i}$ and $w$ in $D_{j}$.

  1. (1)

If $i\neq j$, and there is a path in $Q$ from $v$ to $w$, then there is no
path in $Q$ from any vertex in $D_{j}$ to any vertex in $D_{i}$.

  2. (2)

There is a path in $Q$ from $v$ to $w$ if and only if there are paths in $Q$
from each vertex in $D_{i}$ to each vertex in $D_{j}$.

  3. (3)

the $d_{i}\times d_{i}$-matrix with entries $v_{i}(KQ/C^{f})v_{i}$ is
$M_{i,i}(K)$, the $d_{i}\times d_{i}$-matrix ring with entries $K$.

  4. (4)

if $i\neq j$ and there are no paths from $D_{i}$ to $D_{j}$, then the
$d_{i}\times d_{j}$-matrix with entries $v_{i}(KQ/C^{f})v_{j}$ is
$M_{i,j}(0)$.

  5. (5)

if $i\neq j$, and there is a path from a vertex in $D_{i}$ to a vertex in
$D_{j}$ the $d_{i}\times d_{j}$-matrix with entries $v_{i}(KQ/C^{f})v_{j}$ is
$M_{i,j}(K)$ and the $d_{j}\times d_{i}$-matrix with entries
$v_{j}(KQ/C^{f})v_{i}$ is $M_{j,i}(0)$.

###### Proof.

  1. (1)

Since $i\neq j$, $D_{i}\cap D_{j}=\emptyset$. Suppose there are paths from a
vertex $v\in D_{i}$ to $w\in D_{j}$ and from a vertex $w^{\prime}\in D_{j}$ to
$v^{\prime}\in D_{i}$. But since there are paths from $w$ to $w^{\prime}$ and
from $v^{\prime}$ to $v$, concatenating the paths, we obtain a cycle at $v$
containing $w$. Thus $v$ and $w$ are in the same connected component - a
contradiction.

  2. (2)

If $v,v^{\prime}\in D_{i}$ and $w,w^{\prime}\in D_{j}$, and
$v\xrightarrow{p}w$ is a path in $Q$, then there is a path $v^{\prime}\to
v\xrightarrow{q}w\to w^{\prime}$.

  3. (3)

Use (2), that $v(KQ/C^{f})v$ is 1-dimensional, and that $D_{i}$ is path
connected.

  4. (4)

Clear.

  5. (5)

Follows from (1), (2) and (4).

∎

######  Definition 4.3.

Given positive integers $d_{1},\dots,d_{m}$ and $n=\sum_{i=1}^{m}d_{i}$. An
$n\times n$ display of $0$s and $K$s is said to be in _$(d_{1},\dots,d_{m})$
block form_, if

$A=\begin{pmatrix}B_{1,1}&B_{2,2}&\cdots&B_{2,m}\\\ \vdots&\vdots&&\vdots\\\
B_{m,1}&B_{m,2}&\cdots&B_{m,m}\end{pmatrix}$

where each $B_{i,i}$ is the $d_{i}\times d_{i}$ matrix ring with entries $K$,
and each $B_{i,j}$ is a $d_{i}\times d_{j}$-matrix with entries either 0 or
$K$.

For the rest of the paper, we assume that $f(p)=1$ for all paths $p$ in
${\mathcal{B}}$. For this choice of $f$, we let $C=C^{f}$ and recall that we
say that $KQ/C$ is the commuting algebra of $KQ$.

The next result shows that the commuting algebra of $KQ$ is isomorphic to a
subring of $n\times n$ matrix ring with entries $K$.

######  Theorem 4.4 (Structure Theorem).

Let $Q$ be a finite quiver with $n$ vertices, $\\{v_{1},\dots,v_{n}\\}$. Let
$D_{1},\dots,D_{m}$ be the path connected components of $Q$ with
$|D_{i}|=d_{i}$, for $i=1,\dots,m$. Reorder the vertices so that the first
$d_{1}$ vertices are the vertices in $D_{1}$, the next $d_{2}$ vertices are
the vertices in $D_{2},\dots$, the last $d_{m}$ vertices are the vertices in
$D_{m}$. The commuting algebra of $KQ$ is in $(d_{1},\dots,d_{m})$ block form
with $B_{i,j}=M_{i,j}(K)$ if there is a path in $Q$ from some vertex in
$D_{i}$ to a vertex in $D_{j}$ and $B_{i,j}=M_{i,j}(0)$ if there is no path in
$Q$ from any vertex in $D_{i}$ to a vertex in $D_{j}$.

###### Proof.

The proof directly follows from Proposition 4.2. ∎

######  Definition 4.5.

Keeping the notation of the theorem, if the vertices are ordered in such a
fashion that the first $d_{1}$ vertices are the vertices in $D_{1}$, the next
$d_{2}$ vertices are the vertices in $D_{2}$, …, the last $d_{m}$ vertices are
the vertices in $D_{m}$, we say the ordering of the vertices is _consistent_
with ${\mathcal{D}}$.

##  5\. Morita equivalence

Using Morita equivalence, we find a basic algebra in the Morita class of a
commuting algebra. Recall that a finite dimensional $K$-algebra, $\Lambda$, is
called _basic_ if every simple right $\Lambda$-module is 1-dimensional. Note
that a finite dimensional basic $K$-algebra is unique, up to ring isomorphism,
in its Morita class. We choose a an ordering of the vertices of $Q$ consistent
with ${\mathcal{D}}$.

Select one vertex, $w_{i}\in D_{i}$, from each $D_{i}$. Let
$P=\oplus_{i=1}^{m}\rho(w_{i})KQ/C$. Note that each $\rho(w_{i})$ is a nonzero
idempotent in $KQ/C$ since $\rho$ is a ring homorphism and $w_{i}\not\in C$ by
Lemma 3.6.

######  Lemma 5.1.

The right $KQ/C$-module $P$ is a right projective generator for
$\operatorname{\mathrm{mod}}\nolimits(KQ/C)$.

###### Proof.

Clearly $P$ is a projective $KQ/C$-module. Let $v$ be a vertex in $Q$. To show
that $P$ is a generator we need to show that every indecomposable projective
module, $\rho(v)KQ/C$, is isomorphic to one of the $\rho(w_{i})KQ/C$. We have
that $v\in D_{i}$, for some $i$. Thus there is a path $p$ from $v$ to $w_{i}$
and $q$ from $w_{i}$ to $v$ since both $v$ and $w_{i}$ are in the same path
connected component. Consider $\rho(pq)+C\in\rho(w_{i})KQ/C$ and
$\rho(qp)+C\in\rho(v)KQ/C$. Then $qp-w_{i}\in C$ and $pq-v\in C$. Hence
$\rho(pq)=\rho(v)$ and $\rho(qp)=\rho(w_{i})$. It follows that multiplication
on the right by the elements $\rho(p)+C$ and $\rho(q)+C$ induce the desired
inverse isomorphisms. ∎

######  Theorem 5.2.

Let $Q$ be a quiver with $n$ vertices $\\{v_{1},\dots v_{n}\\}$ and let
$D_{1},\dots D_{m}$ be the path connected components of $Q$. Set
$P=\oplus_{j=1}^{m}\rho(w_{j})KQ/C$, where, for each $j=1,\dots,m$, $w_{j}$ is
a vertex in $D_{j}$. Then the $K$-algebra
$\operatorname{End}\nolimits_{KQ/C}(P)$ is a basic algebra in the Morita class
of $KQ/C$.

###### Proof.

By Lemma 5.1, we have that $P$ is a finitely generated projective generator
for the category of finitely generated right $KQ/C$-modules. By Morita
equivalence, the category of finitely generated right $KQ/C$-modules is
equivalent to the category of finitely generated right
$\operatorname{End}\nolimits_{KQ/C}(P)$-modules. Since
$\\{\rho(w_{1}),\dots,\rho(w_{m})\\}$ is a full set of orthogonal non-
isomorphic idempotents in $KQ/C$, $\operatorname{End}\nolimits_{KQ/C}(P)$ is a
basic algebra Morita equivalent to $KQ/C$. ∎

We immediately have the following.

######  Proposition 5.3.

The following results hold:

  1. (1)

For $1\leq i\leq m$,
$\operatorname{End}\nolimits_{KQ/C}\rho(w_{i})(KQ/C)\rho(w_{i})$ is isomorphic
to $K$.

  2. (2)

For $1\leq i,j\leq m$ with $i\neq j$,
$\operatorname{End}\nolimits_{KQ/C}\rho(w_{i})(KQ/C)\rho(w_{j})$ is isomorphic
to $K$ if and only if there is a path $p$ in $Q$ from $w_{j}$ to $w_{i}$.
Otherwise, $\operatorname{End}\nolimits_{KQ/C}\rho(w_{i})(KQ/C)\rho(w_{j})=0$.

  3. (3)

For $1\leq i,j\leq m$ with $i\neq j$, if
$\operatorname{End}\nolimits_{KQ/C}\rho(w_{i})(KQ/C)\rho(w_{j})$ is isomorphic
to $K$, then
$\operatorname{End}\nolimits_{KQ/C}\rho(w_{j})(KQ/C)\rho(w_{i})=0$.

###### Proof.

Since, for each $1\leq i\leq m$, $w_{i}\in D_{i}$, the result follows from
applying Proposition 4.2

∎

######  Definition 5.4.

We call $\operatorname{End}\nolimits_{KQ/C}(P)$ the _skeleton of $KQ$_ and
denote it by $Sk(Q)$.

In the next section we investigate the structure of $Sk(Q)$.

##  6\. The skeleton of an algebra

For every finite dimensional algebra, $\Lambda=KQ/I$, the _skeleton_ of
$\Lambda$ is a basic algebra in the Morita equivalence class of the commuting
algebra of $\Lambda$. We believe that, in general, the skelton of an algebra
contains basic structural information about the algebra. In this section, we
only deal with path algebras; that is, $I=0$, unless otherwise stated.

In the construction of the commuting algebra of $KQ$, we see that there are
$m$ paths connected components of $Q$, each corresponding to a vertex of the
skeleton, $Sk(Q)$, of $KQ$. Let $\\{x_{1},\dots,x_{m}\\}$ be the vertex set of
$Sk(Q)$ where each $x_{i}$ corresponds to a path connected component $D_{i}$
of $Q$.

######  Proposition 6.1.

Let $KQ/C$ be the commuting algebra of $KQ$. Then the commuting algebra of
$KQ/C$ is isomorphic to $KQ/C$. Moreover, the commuting algebra of $Sk(Q)$ is
isomorphic to $Sk(Q)$.

###### Proof.

We show that the commuting ideal of $KQ/C$ is $(0)$. Let $p\|q$ be parallel
paths in $Q$ and $\rho\colon KQ\to KQ/C$ be the canonical surjection. Then the
commuting ideal of $KQ/C$ is generated by $\\{\rho(p)-\rho(q)$ with $p\|q\\}$.
But $\rho(p)-\rho(q)=\rho(p-q)=0$, since $p-q\in C$. ∎

######  Proposition 6.2.

Setting $x_{i}\leq x_{j}$ if and only if there is a path from $w_{i}\in D_{i}$
to $w_{j}\in D_{j}$ in $Q$, is a partial ordering of the vertex set of
$Sk(Q)$.

###### Proof.

Let $1\leq i,j\leq m$ with $i\neq j$. Suppose that $x_{i}\leq x_{j}$. We need
to show that ${x_{j}\not\leq x_{i}}$. But it follows directly from Proposition
4.2 that there is no non-zero path from $x_{j}$ to $x_{i}$ and hence
${x_{j}\not\leq x_{i}}$. Note that if $i=j$ then $x_{i}=x_{j}$. ∎

######  Definition 6.3.

If $S$ is a finite partially ordered set, then
$s_{i_{1}}<s_{i_{2}}<\cdots<s_{i_{t}}$, for $s_{i_{j}}\in S$, is called a
_chain of length $t$_.

######  Theorem 6.4.

Let $Q$ be a finite quiver. Then the commuting algebra of $KQ$ and $Sk(Q)$ are
finite dimensional $K$-algebras with finite global dimension $\leq$ the length
of the longest chain of vertices of $Sk(Q)$.

###### Proof.

Let $\\{x_{1}\dots,x_{m}\\}$ be the vertices of $Sk(Q)$. By a proof similar to
the proof of Theorem 3.1, the skeleton of $Q$ is finite dimensional. By
Proposition 6.2 $\\{x_{1},\dots,x_{n}\\}$ are partially ordered by $x_{i}\leq
x_{j}$ if either $i=j$ or there is a path in $Q$ from a vertex in $D_{i}$ to a
vertex in $D_{j}$. By standard arguments it follows that the global dimension
of the skeleton of $KQ$ is less than or equal to the longest chain of
vertices. The result follows since the commuting algebra of $Q$ is Morita
equivalent to the skeleton of $Q$ and global dimension is preserved by Morita
equivalence. ∎

Let $(S,\preceq)$ be a finite partially ordered set. We say that, for $s$ and
$t$ in $S$, that $s$ is an _immediate pedecesor_ of $t$ or $t$ is an
_immediate successor of $s$ if $s\prec t$_, and there is no $u\in S$ such that
$s\prec u\prec t$. The _incidence algebra of the partially ordered set
$(S,\preceq)$_ is the commuting algebra, denoted $Incid(S)$, of the quiver
$Q(S)$, where $S$ is the vertex set of $(Q(S),\preceq)$ and there is an arrow
from vertex $s$ to vertex $t$ if $t$ is an immediate successor of $s$.

Note that the incidence algebra of a partially ordered set is a uniquely
defined algebra.

######  Theorem 6.5.

Let $Sk(Q)$ be the skeleton of $KQ$, for a finite quiver $Q$. Let $V$ be the
vertex set of $Sk(Q)$. Viewing $V$ as a partially ordered set, the incidence
algebra $Incid(V)$ is isomorphic to $Sk(Q)$.

###### Proof.

Since $Sk(Q)$ is finite dimensional, with one dimensional simple modules, it
is isomorphic to the quotient of a path algebra $KQ^{*}/I^{*}$. Let $V$ be the
vertex set of $Q^{*}$. By Proposition 6.2, $(V,\preceq)$ is a partially
ordered set, and we let $Incid(V)$ be its incidence algebra. The vertices of
both $Incid(V)$ and $Sk(Q)$ are the same. The algebra $Sk(Q)$ has one vertex
for each path connected component of $Q$. There is an arrow from $v$ to $w$ if
there is a path from $v$ to $w$ in $Sk(Q)$. On the other hand, there is an
arrow from $v$ to $w$ in $Incid(V)$ if $v\prec w$ that is if $v$ is a
immediate predecessor of $w$. It is an easy exercise check that there is a
$K$-algebra isomorphism from $Sk(Q)$ to $Incid(V)$. ∎

As is well-known, incidence algebras and algebras of partially ordered finite
sets are isomorphic, see, for example [IM, Section 1.2] or more generally,
also see [S, SO]. Recall that the incidence algebra of a partially ordered set
$P$ is the $K$-algebra with basis given by elements $p^{x}_{y}$ whenever two
elements $x,y\in P$ are such that $x<y$. The multiplication of basis elements
is given by $p^{x}_{y}p^{w}_{z}=p^{x}_{z}$ if $y=w$ and the product is zero
otherwise. On the other hand the algebra of a partially ordered set is the
quotient $KQ/I$ of the path algebra of a quiver $Q$ where the vertices of the
quiver are the elements of the partially ordered set $P$ and there is an arrow
from vertex $v$ to vertex $w$ if $v<w$ and if there is no other element $s$ in
$P$ such that $v<s<w$ unless $s=v$ or $s=w$. The ideal $I$ is generated by
$p-q$ for any parallel paths $p$ and $q$ in $Q$ of length at least two. These
can be viewed as the commuting algebras of path algebras whose quiver is a
Hasse diagram. Since the skeleton of an algebra is Morita equivalent to the
commuting algebra of $KQ$, for any path algebra, the category of finitely
generated modules over the skeleton of the commuting algebra of $KQ$ embeds
into the the category of $KQ$-modules. This observation allows one to apply
the rich theory of representations of partially ordered sets to study of a
‘piece’ of the (usually wild) category of $KQ$-modules.

##  7\. Examples

######  Example 7.1.

Let $Q$ be the quiver:

---
$\textstyle{v_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

There are two path connected components

$D_{1}=\\{v_{1},v_{2},v_{3},v_{4}\\}\mbox{ and }D_{2}=\\{v_{5},x_{6}\\}.$

The commuting algebra for $KQ$ in block form is:

$\begin{pmatrix}K&K&K&K&K&K\\\ K&K&K&K&K&K\\\ K&K&K&K&K&K\\\ K&K&K&K&K&K\\\
0&0&0&0&K&K\\\ 0&0&0&0&K&K\end{pmatrix}$

and the skeleton of $KQ$ has 2 vertices corresponding to $D_{1}$ and $D_{2}$
and an arrow from “$D_{1}$” to “$D_{2}$”

The skeleton is the incidence algebra of $w_{1}\to w_{2}$. The incidence
algebra is isomorphic to

$\begin{pmatrix}K&K\\\ 0&K\end{pmatrix}$

######  Example 7.2.

Consider the quivers

$Q_{1}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.18057pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.64995pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.99792pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{c}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 70.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{6\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.87032pt\raise-40.95137pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
5.50002pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.6783pt\raise-41.87497pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.50002pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\mbox{
and }Q_{2}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.18057pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.64995pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 35.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.99792pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{c}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 70.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{6\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.87032pt\raise-40.95137pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
5.50002pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.6783pt\raise-41.87497pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.50002pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces$

Both $Q_{1}$ and $Q_{2}$ have the same commuting algebra, namely, the $6\times
6$ matrix ring and their skeleton is given by a single vertex. But $KQ_{1}$
and $KQ_{2}$ are not isomorphic. The skeleton of both $KQ_{1}$ and $KQ_{2}$ is
a single vertex.

######  Example 7.3.

Let $Q=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.18057pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.64995pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
35.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 35.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.99792pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
70.0pt\raise-18.22221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{c}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 70.0pt\raise-26.99998pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.87032pt\raise-40.95137pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
5.50002pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.6783pt\raise-31.01387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise-36.44443pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{6}$}}}}}}}\ignorespaces}}}}\ignorespaces$
The path connected components are $D_{1}=\\{1,2,3,4\\},D_{2}=\\{5\\}$, and
$D_{3}=\\{6\\}$ and the commuting algebra $KQ/C$ is the $6\times 6$ matrix
ring

$KQ/C=\begin{pmatrix}K&K&K&K&K&K\\\ K&K&K&K&K&K\\\ K&K&K&K&K&K\\\
K&K&K&K&K&K\\\ 0&0&0&0&K&K\\\ 0&0&0&0&0&K\end{pmatrix}$

The skeleton of $KQ$ is $w_{1}\to w_{2}\to w_{3}$.

The next example shows that if $f(p)=1$ but $I\neq 0$, then the global
dimension of the commuting algebra can be infinite.

######  Example 7.4.

First take $I=0$. and $Q=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.003pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 31.003pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{v_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
56.76274pt\raise-13.54805pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{a}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
69.00902pt\raise-29.46262pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{v_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.19417pt\raise-13.54805pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{c}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
32.94821pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
35.00601pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
69.00902pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{v_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.50394pt\raise-41.54053pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{b}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 7.003pt\raise-36.10999pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces$
The commuting algebra of $Q$ is the $3\times 3$ matrix ring with entries in
$K$. We know that the skeleton of the $3\times 3$ matrix ring with entries in
$K$ is just $K$.

Continuing, we now consider $KQ/I$ where $I=\langle ab,bc,ca\rangle$. In this
case, $KQ/I$ is a monomial algebra and has no paths of length 2. It is well-
known that this algebra has infinite global dimension. Then the global
dimension of $((KQ/I)/C_{KQ/I})$ is also infinite since $C_{KQ/I}=0$.

We give some further examples of commuting algebras and skeletons.

######  Example 7.5.

Let $Q$ be the quiver that has two vertices $v$ and $w$ and $n$ arrows from
$v$ to $w$. Then the commuting algebra of $Q$ isomorphic to the skeleton of
$KQ$ and consists of two vertices and one arrow from $v$ to $w$.

######  Example 7.6.

If the underlying graph of a quiver $Q$ is a tree then the commuting algebra
of $Q$ is isomorphic to the skeleton of $KQ$ and is the algebra is itself.
This follows from Proposition 6.2.

######  Example 7.7.

Let $Q$ be an oriented cycle with $n$ vertices and $n$ arrows. Then the
commuting algebra of $Q$ is isomorphic to the $n\times n$-matrix ring whereas
the skeleton of $Q$ corresponds to a vertex with no arrows. So in this case
the commuting algebra of $KQ$ is not isomorphic to the skeleton of $KQ$.

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