# Two vertex geometrically irreducible algebras

Grzegorz Bobiński  Grzegorz Bobiński
Faculty of Mathematics and Computer Science,
Nicolaus Copernicus University
ul. Chopina 12/18
87-100 Toruń
Poland<EMAIL_ADDRESS>and  Grzegorz Zwara  Grzegorz Zwara
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
ul. Chopina 12/18
87-100 Toruń
Poland<EMAIL_ADDRESS>
##  1\. Introduction and main result

Throughout the paper $\Bbbk$ denotes an algebraically closed field of
arbitrary characteristic.

Given a finite dimension $\Bbbk$-algebra $A$ and a nonnegative integer $d$ we
denote by $\operatorname{mod}_{A}(d)$ the variety of $d$-dimensional
$A$-modules, which consists of the $\Bbbk$-algebra homomorphisms
$A\to\mathbb{M}_{d\times d}(\Bbbk)$ and parameterizes $A$-modules of dimension
$d$. The algebra $A$ is called geometrically irreducible if the connected
components of $\operatorname{mod}_{A}(d)$ are irreducible, for all $d$.
Geometrically irreducible algebras have been studied in [BobSch2019a,
BobSch2019b, Bob2021a]. In particular, it is conjectured in [BobSch2019b] that
every geometrically irreducible algebra is a gluing (in a sense explained in
[BobSch2019b]) of algebras with at most two simples. This conjecture has been
confirmed for the algebras without shortcuts ([BobSch2019b]*Theorem 1.2).
Consequently, the geometrically irreducible algebras with at most two simples
seem to be of particular interest. The aim of this paper is to complete the
classification of this class of algebras initiated in [BobSch2019b, Bob2021a].

We introduce some families of algebras. For $n\geq 0$, let $Q(n)$ be the
quiver

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For $l\geq 1$ (and $n\geq 1$), let

$\rho^{(l)}:=\sum_{i=0}^{l}\varepsilon_{0}^{l-i}\alpha_{1}\varepsilon_{1}^{i}.$

For $m\geq 2$ and $n,l\geq 1$, let $A(n,m,l)$ be the path algebra of the
quiver $Q(n)$ bounded by $\varepsilon_{0}^{m}$, $\varepsilon_{1}^{m}$ and
$\rho^{(l)}$. Similarly, for $m_{0},m_{1}\geq 1$ and $n\geq 0$, let
$A^{\prime}(n,m_{0},m_{1})$ be the path algebra of the quiver $Q(n)$ bounded
by $\varepsilon_{0}^{m_{0}}$ and $\varepsilon_{1}^{m_{1}}$.

The following theorem is the first main result of the paper.

######  Theorem 1.1.

Let $A$ be an algebra which has exactly two simples. Then $A$ is geometrically
irreducible if and only if $A$ is Morita equivalent to one of the following
algebras:

  1. (1)

$A(n,m,1)$, for some $m\geq 2$ and $n\geq 1$, or

  2. (2)

$A(n,m,m-1)$, for some $m\geq 2$ and $n\geq 1$, or

  3. (3)

$A^{\prime}(n,m_{0},m_{1})$, for some $m_{0},m_{1}\geq 1$ and $n\geq 0$.

The main result of [BobSch2019b] (Theorem 1.1) states that if $A$ is a
geometrically irreducible algebra with exactly two simples, then (up to Morita
equivalence) $A$ is one of the algebras listed in Theorem 1.1. On the other
hand, it is relatively easy to see that the algebras
$A^{\prime}(n,m_{0},m_{1})$, $m_{0},m_{1}\geq 1$ and $n\geq 0$, are
geometrically irreducible (see for example [BobSch2019b]*Proposition 3.3).
Moreover, [Bob2021a]*Corollary 2.1 implies that the algebras $A(n,m,1)$,
$m\geq 2$ and $n\geq 1$, are also geometrically irreducible. Thus in order to
prove Theorem 1.1 it suffices to prove the following.

######  Theorem 1.2.

If $m\geq 2$ and $n\geq 1$, then the algebra $A(n,m,m-1)$ is geometrically
irreducible.

The paper is organized as follows. In Section 2 we recall basic definitions,
while in Section 3 we introduce classes of hom- and ext-irreducible algebras,
which play crucial role in our proof of the main result presented in Section
4. The final Section 5 contains a classification of the hom- and ext-
irreducible algebras.

The both authors gratefully acknowledge the support of the National Science
Centre grant no. 2020/37/B/ST1/00127.

##  2\. Preliminaries

Throughout the paper, by $\mathbb{Z}$, $\mathbb{N}$, and $\mathbb{N}_{+}$ we
denote the sets of integers, nonnegative integers, and positive integers,
respectively. If $i,j\in\mathbb{Z}$, then $[i,j]$ denotes the set of all
$l\in\mathbb{Z}$ such that $i\leq l\leq j$.

A quiver $Q$ consists of a finite set $Q_{0}$ of vertices, a finite set
$Q_{1}$ of arrows, and two maps $s,t\colon Q_{1}\to Q_{0}$, which assign to an
arrow $\alpha\in Q_{1}$ its starting and terminating vertex, respectively. An
arrow $\alpha\in Q_{1}$ is called a loop if $s\alpha=t\alpha$. If $\alpha$ is
an arrow in a quiver $Q$, then we define the degree $\deg\alpha$ of $\alpha$
by $\deg\alpha:=0$, if $\alpha$ is a loop, and $\deg\alpha:=1$, otherwise. We
denote by $Q_{1}^{(i)}$ the set of arrows of degree $i$, $i=0,1$ (in
particular, $Q_{1}^{(0)}$ is the set of loops in $Q$). By a path of length
$l\in\mathbb{N}_{+}$ in $Q$ we mean a sequence
$\sigma=\alpha_{1}\cdots\alpha_{l}$ of arrows such that
$s\alpha_{i}=t\alpha_{i+1}$, for all $i\in[1,l-1]$. We put
$s\sigma:=s\alpha_{l}$ and $t\sigma:=t\alpha_{1}$, and call them the starting
and the terminating vertex of $\sigma$, respectively. By the degree
$\deg\sigma$ of $\sigma$ we mean $\sum_{i=1}^{l}\deg\alpha_{i}$. By an
oriented cycle we mean a path $\sigma$ of positive length such that
$s\sigma=t\sigma$. For each vertex $x$ of $Q$ we also consider the trivial
path $\mathbf{1}_{x}$ at vertex $x$ (i.e.,
$s\mathbf{1}_{x}:=x=:t\mathbf{1}_{x})$ of length (and degree) $0$. If
$X\subseteq Q_{0}$, then we put $\mathbf{1}_{X}:=\sum_{x\in X}\mathbf{1}_{x}$.

With a quiver $Q$ we associate its path algebra $\Bbbk Q$, which as a
$\Bbbk$-vector space has a basis formed by all paths in $Q$ and whose
multiplication is induced by the composition of paths. By a relation $\rho$ in
$Q$ we mean a linear combination of paths of length at least $2$ with the same
starting and terminating vertex (denoted by $s\rho$ and $t\rho$,
respectively). If $\rho=\sum_{i=1}^{n}\lambda_{i}\sigma_{i}$ is a relation,
where $\lambda_{1}$, …, $\lambda_{n}$ are nonzero scalars and $\sigma_{1}$, …,
$\sigma_{n}$ pairwise different paths, the by the degree $\deg\rho$ of $\rho$
we mean $\min\\{\deg\sigma_{i}:\text{$i\in[1,n]$}\\}$. By a bound quiver we
mean a pair $(Q,I)$ consisting of a quiver $Q$ and an ideal $I$ of $\Bbbk Q$
generated by relations, such that there exists $n\in\mathbb{N}_{+}$ with
$\sigma\in I$, for each path $\sigma$ of length (at least) $n$. If $(Q,I)$ is
a bound quiver, then we call $\Bbbk Q/I$ the path algebra of $(Q,I)$. If $A$
is the path algebra of a bound quiver $(Q,I)$ and the ideal $I$ is generated
by relations $\rho_{1}$, …, $\rho_{n}$, then we also say that $A$ is the path
algebra of $Q$ bounded by $\rho_{1}$, …, $\rho_{n}$.

If $A$ is an algebra, then there exists a bound quiver $(Q,I)$ such that $A$
and $\Bbbk Q/I$ are Morita equivalent. Moreover, it follows from [Bon1991]
that $A$ is geometrically irreducible if and only if $\Bbbk Q/I$ is
geometrically irreducible. Thus from now on we concentrate on path algebras of
bound quivers. The quiver $Q$ is uniquely determined by $A$ and is called the
Gabriel quiver of $A$. The algebra $A$ is called weakly triangular, if there
are no oriented cycles of positive degree in $Q$.

Let $(Q,I)$ be a bound quiver. By a representation of $(Q,I)$ we mean a
collection $V$ of finite dimensional $\Bbbk$-vector spaces $V_{x}$, $x\in
Q_{0}$, together with $\Bbbk$-linear maps $V_{\alpha}\colon V_{s\alpha}\to
V_{t\alpha}$, $\alpha\in Q_{1}$, such that $V_{\rho}$ (defined in a natural
way) vanishes for all relations $\rho\in I$. The collection
$(\dim_{k}V_{x})\in\mathbb{N}^{Q_{0}}$ is called the dimension vector of $V$
(and the elements of $\mathbb{N}^{Q_{0}}$ are called dimension vectors). If
$U$ and $V$ are representations of $(Q,I)$, then a homomorphism $f\colon V\to
U$ is a collection of $\Bbbk$-linear maps $f_{x}\colon V_{x}\to U_{x}$, $x\in
Q_{0}$, such that $U_{\alpha}\circ f_{s\alpha}=f_{t\alpha}\circ V_{\alpha}$,
for each $\alpha\in Q_{1}$. If $A$ is the path algebra of $(Q,I)$ then the
category $\operatorname{mod}A$ of (finite dimensional left) $A$-modules is
equivalent to the category $\operatorname{rep}(Q,I)$ of representations of
$(Q,I)$. By abuse of language we call objects of $\operatorname{rep}(Q,I)$
representations of $A$. If $x\in Q_{0}$ then we denote by $S_{A}(x)$ the
simple representation at vertex $x$, i.e., $S_{A}(x)_{x}:=\Bbbk$,
$S_{A}(x)_{y}:=0$, for $y\neq x$, and $S_{A}(x)_{\alpha}:=0$, for $\alpha\in
Q_{1}$.

According to [Bon1991] the above equivalence has its geometric counterpart.
More precisely, let $A$ be the path algebra of a bound quiver $(Q,I)$. For a
dimension vector $\mathbf{d}$ we denote by
$\operatorname{rep}_{A}(\mathbf{d})$ the set of representations $V$ of $A$
such that $V_{x}=\Bbbk^{d_{x}}$, for each $x\in Q_{0}$. One easily observes
that $\operatorname{rep}_{A}(\mathbf{d})$ can be identified with a closed
subset of $\prod_{\alpha\in Q_{1}}\mathbb{M}_{d_{t\alpha}\times
d_{s\alpha}}(\Bbbk)$, thus carries a structure of an affine variety. Then
results of [Bon1991] imply that $A$ is geometrically irreducible if and only
if $\operatorname{rep}_{A}(\mathbf{d})$ is irreducible, for each
$\mathbf{d}\in\mathbb{N}^{Q_{0}}$.

Let $A$ be the path algebra of a bound quiver $(Q,I)$ and $\mathbf{d}$ a
dimension vector. Put $\operatorname{GL}(\mathbf{d}):=\prod_{x\in
Q_{0}}\operatorname{GL}(d_{x})$. The group $\operatorname{GL}(\mathbf{d})$
acts on $\operatorname{rep}_{A}(\mathbf{d})$ by conjugation, i.e., if
$g\in\operatorname{GL}(\mathbf{d})$ and
$V\in\operatorname{rep}_{A}(\mathbf{d})$, then $(g\ast
V)_{\alpha}:=g_{t\alpha}V_{\alpha}g_{s\alpha}^{-1}$, for $\alpha\in Q_{1}$.
Note that $g\in\operatorname{GL}(\mathbf{d})$ can be viewed as an isomorphism
$g\colon V\to g\ast V$.

We finish this subsection by listing basic properties of geometrically
irreducible algebras proved in [BobSch2019a]*Section 3.

######  Lemma 2.1.

The following hold for a geometrically irreducible algebra $A$ with the
Gabriel quiver $Q$.

  1. (1)

$A$ is weakly triangular.

  2. (2)

For each vertex $x\in Q_{0}$ there exists at most one loop $\alpha\in Q_{1}$
with $s\alpha=x$.

  3. (3)

If $|Q_{0}|=1$, then $A\simeq k[x]/(x^{m})$. ∎

##  3\. Hom-, mono- and ext-irreducible algebras

Let $\Lambda$ be an algebra. For dimension vectors $\mathbf{d}$ and
$\mathbf{e}$ we denote by $\mathcal{H}_{\Lambda}(\mathbf{e},\mathbf{d})$ the
set of triples $(V,W,f)$, such that
$V\in\operatorname{rep}_{\Lambda}(\mathbf{e})$,
$W\in\operatorname{rep}_{\Lambda}(\mathbf{d})$, and
$f\in\operatorname{Hom}_{\Lambda}(V,W)$. Then
$\mathcal{H}_{\Lambda}(\mathbf{e},\mathbf{d})$ has a natural structure of an
affine variety. We denote by $\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$
the subset of $\mathcal{H}_{\Lambda}(\mathbf{e},\mathbf{d})$ consisting of the
triples $(V,W,f)$ such that $f$ is a monomorphism. Note that
$\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$ is an open subset of
$\mathcal{H}_{\Lambda}(\mathbf{e},\mathbf{d})$. Moreover,
$\mathcal{M}(\mathbf{e},\mathbf{d})$ is nonempty if and only if
$\mathbf{e}\leq\mathbf{d}$ (i.e., $e_{x}\leq d_{x}$, for each $x\in Q_{0}$).
We say that $\Lambda$ is hom-irreducible (mono-irreducible) if
$\mathcal{H}_{\Lambda}(\mathbf{e},\mathbf{d})$
($\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$, respectively) is irreducible,
for all dimension vectors $\mathbf{d}$ and $\mathbf{e}$ (such that
$\mathbf{e}\leq\mathbf{d}$, respectively). We have the following obvious
observation.

######  Proposition 3.1.

If an algebra $\Lambda$ is hom-irreducible, then $\Lambda$ is mono-
irreducible. ∎

In order to define ext-irreducible algebras we need to recall an
interpretation of extensions of $\Lambda$-modules in terms of cocycles. Fix
the path algebra $\Lambda$ of a bound quiver $(Q,I)$, dimension vectors
$\mathbf{d}$ and $\mathbf{e}$, and
$V\in\operatorname{rep}_{\Lambda}(\mathbf{d})$ and
$U\in\operatorname{rep}_{\Lambda}(\mathbf{e})$. Let
$\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}:=\prod_{\alpha\in
Q_{1}}\mathbb{M}_{d_{t\alpha}\times e_{s\alpha}}(\Bbbk)$. For each
$Z\in\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$ we have a representation
$W^{V,Z,U}$ of $Q$ defined by:
$W^{V,Z,U}_{x}:=\Bbbk^{d_{x}+e_{x}}(=\Bbbk^{d_{x}}\oplus\Bbbk^{e_{x}})$, for
$x\in Q$, and
$W^{V,Z,U}_{\alpha}:=\left[\begin{smallmatrix}V_{\alpha}&Z_{\alpha}\\\
0&U_{\alpha}\end{smallmatrix}\right]$, for $\alpha\in Q_{1}$. Given an exact
sequence

$0\to V\xrightarrow{f}W\xrightarrow{h}U\to 0$

of $\Lambda$-modules, there exists
$Z\in\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$ and an isomorphism $g\colon
W^{V,Z,U}\to W$, such that the following diagram is commutative

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{W^{V,Z,U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{g}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\mu$ and $\pi$ are the natural maps. Commutativity of the above diagram
means in particular that $f=g\circ\mu$. Moreover, if
$W\in\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})$, then
$g\in\operatorname{GL}(\mathbf{d}+\mathbf{e})$ and $W=g\ast W^{V,Z,U}$. Let
$\mathbb{Z}^{U,V}$ be the set of $Z\in\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$
such that $W^{V,Z,U}\in\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})$.
Then $Z\in\mathbb{Z}^{U,V}$ if and only if $Z_{\rho}^{U,V}=0$, for each
relation $\rho\in I$, where

$Z_{\rho}^{U,V}=\sum_{i=1}^{n}\sum_{j=1}^{l_{i}}\lambda_{i}V_{\alpha_{i,1}}\cdots
V_{\alpha_{i,j-1}}Z_{\alpha_{i,j}}U_{\alpha_{i,j+1}}\cdots
U_{\alpha_{i,l_{i}}},$

provided $\rho=\sum_{i=1}^{n}\lambda_{i}\alpha_{i,1}\cdots\alpha_{i,l_{i}}$.

Given an algebra $\Lambda$ and dimension vectors $\mathbf{d}$ and $\mathbf{e}$
we denote by $\mathcal{E}_{\Lambda}(\mathbf{e},\mathbf{d})$ the set of triples
$(U,V,Z)$ such that $U\in\operatorname{rep}_{\Lambda}(\mathbf{e})$,
$V\in\operatorname{rep}_{\Lambda}(\mathbf{d})$, and $Z\in\mathbb{Z}^{U,V}$.
Obviously $\mathcal{E}_{\Lambda}(\mathbf{e},\mathbf{d})$ has a structure of an
affine variety and we call $\Lambda$ ext-irreducible if the varieties
$\mathcal{E}_{\Lambda}(\mathbf{e},\mathbf{d})$ are irreducible, for all
dimension vectors $\mathbf{d}$ and $\mathbf{e}$.

The following proposition, whose proof uses a construction from [Bon1994], is
the main result of this section.

######  Proposition 3.2.

An algebra $\Lambda$ is ext-irreducible if and only if $\Lambda$ is mono-
irreducible.

###### Proof.

Assume first that $\Lambda$ is ext-irreducible and fix dimension vectors
$\mathbf{d}$ and $\mathbf{e}$ such that $\mathbf{e}\leq\mathbf{d}$. Consider
the map
$\Phi\colon\operatorname{GL}(\mathbf{d})\times\mathcal{E}_{\Lambda}(\mathbf{d}-\mathbf{e},\mathbf{e})\to\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$
given by:

$\Phi(g,(U,V,Z)):=(V,g\ast W^{V,Z,U},g\circ\mu),$

for $g\in\operatorname{GL}(\mathbf{d})$,
$V\in\operatorname{rep}_{\Lambda}(\mathbf{e})$,
$U\in\operatorname{rep}_{\Lambda}(\mathbf{d}-\mathbf{e})$, and
$Z\in\mathbb{Z}^{U,V}$, where $\mu\colon V\to W^{V,Z,U}$ is the natural
inclusion. One easily checks that $\Phi$ is a well-defined morphism of
varieties. We show that $\Phi$ is onto and this will imply that
$\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$ is irreducible, since
$\operatorname{GL}(\mathbf{d})$ is irreducible and by assumption
$\mathcal{E}_{\Lambda}(\mathbf{d}-\mathbf{e},\mathbf{e})$ is irreducible.

Let $(V,W,f)\in\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{d})$ and
$U:=\operatorname{Coker}f$. Without loss of generality we may assume
$U\in\operatorname{rep}_{\Lambda}(\mathbf{d}-\mathbf{e})$. We have an exact
sequence

$0\to V\xrightarrow{f}W\to U\to 0.$

By remarks preceding the proposition, there exists $Z\in\mathbb{Z}^{U,V}$ and
$g\in\operatorname{GL}(\mathbf{d})$ such that $W=g\ast W^{V,Z,U}$ and
$f=g\circ\mu$. In other words, $(V,W,f)=\Phi(g,(U,V,Z))$.

Now assume that $\Lambda$ is mono-irreducible and fix dimension vectors
$\mathbf{d}$ and $\mathbf{e}$. Let $\mathcal{Y}$ be the set of tuples
$(V,W,f,h)$ such that
$(V,W,f)\in\mathcal{M}_{\Lambda}(\mathbf{d},\mathbf{d}+\mathbf{e})$ and
$h\in\prod_{x\in Q_{0}}\mathbb{M}_{(d_{x}+e_{x})\times e_{x}}(\Bbbk)$, and
$\mathcal{X}$ be the subset of $\mathcal{Y}$ consisting of the tuples
$(V,W,f,h)$ such that the collection $[f,h]:=([f_{x},h_{x}])_{x\in
Q_{0}}\in\prod_{x\in
Q_{0}}\mathbb{M}_{(d_{x}+e_{x})\times(d_{x}+e_{x})}(\Bbbk)$ belongs to
$\operatorname{GL}(\mathbf{e}+\mathbf{d})$. Then $\mathcal{Y}$ is a vector
bundle over $\mathcal{M}_{\Lambda}(\mathbf{e},\mathbf{e}+\mathbf{d})$ and
$\mathcal{X}$ is a nonempty open subset of $\mathcal{Y}$, hence irreducible.

We have obvious maps
$\pi_{1,1}\colon\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})\to\mathbb{A}_{Q}^{\mathbf{d},\mathbf{d}}$,
$\pi_{1,2}\colon\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})\to\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$,
$\pi_{2,1}\colon\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})\to\mathbb{A}_{Q}^{\mathbf{d},\mathbf{e}}$,
and
$\pi_{2,2}\colon\operatorname{rep}_{\Lambda}(\mathbf{d}+\mathbf{e})\to\mathbb{A}_{Q}^{\mathbf{e},\mathbf{e}}$,
where $Q$ is the Gabriel quiver of $\Lambda$. If $(V,W,f,h)\in\mathcal{X}$,
$g:=[f,h]$, and $W^{\prime}:=g^{-1}\ast W$, then $\pi_{1,1}(W^{\prime})=V$,
$\pi_{2,1}(W^{\prime})=0$, and $\pi_{1,2}(W^{\prime})\in\mathbb{Z}^{U,V}$,
where $U:=\pi_{2,2}(W)$. Consequently, we may define
$\Psi\colon\mathcal{X}\to\mathcal{E}_{\Lambda}(\mathbf{e},\mathbf{d})$ by

$\Psi(V,W,f,h):=(\pi_{2,2}([f,h]^{-1}\ast W),V,\pi_{1,2}([f,h]^{-1}\ast W)),$

for $(V,W,f,h)\in\mathcal{X}$. Obviously $\Psi$ is a regular map, which is
easily seen to be onto, hence $\mathcal{E}_{\Lambda}(\mathbf{e},\mathbf{d})$
is irreducible. ∎

We formulate the following obvious consequence of Propositions 3.1 and 3.2.

######  Corolary 3.3.

If an algebra $\Lambda$ is hom-irreducible, then $\Lambda$ is ext-irreducible.
∎

In Section 5 we classify the hom- and ext-irreducible algebras. In particular,
we show that the hereditary algebras, which are not semisimple, are ext-
irreducible, but not hom-irreducible.

##  4\. Proof of the main result

Throughout this section we fix $m\geq 2$. In order to simplify notation we put
$A:=A(1,m,1)$ and $B_{n}:=A(n,m,m-1)$, for $n\in\mathbb{N}_{+}$. Moreover, by
$A^{\prime}$ we denote the path algebra of the quiver

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{0}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{1}}$$\scriptstyle{\alpha}$

bounded by $\varepsilon_{0}^{m}$, $\varepsilon_{1}^{m}$, and
$\varepsilon_{0}\alpha-\alpha\varepsilon_{1}$. Finally, we put
$\Lambda:=k[x]/(x^{m})$. Note that the dimension vectors for $\Lambda$ are the
elements of $\mathbb{N}$, while the dimension vectors for $A$, $A^{\prime}$
and $B_{n}$ are the pairs $(d,e)$ with $d,e\in\mathbb{N}$. Furthermore, the
correspondence given by

$\varepsilon_{0}\mapsto\varepsilon_{0},\alpha\mapsto\alpha_{1},\varepsilon_{1}\mapsto-\varepsilon_{1},$

induces an isomorphism $A^{\prime}\simeq A$. Since the algebra $A$ is
geometrically irreducible by [Bob2021a]*Corollary 1.2, $A^{\prime}$ is
geometrically irreducible as well.

The crucial point in the proof is the following.

######  Lemma 4.1.

Let $d,e\in\mathbb{N}$. Then the following hold.

  1. (1)

$\operatorname{rep}_{A^{\prime}}((d,e))\cong\mathcal{H}_{\Lambda}(e,d)$.

  2. (2)

$\operatorname{rep}_{B_{1}}((d,e))\cong\mathcal{E}_{\Lambda}(e,d)$.

###### Proof.

(1) Note that $\Lambda$ is the path algebra of the quiver $Q$ of the form

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$

bounded by $\varepsilon^{m}$. Consequently, $\operatorname{rep}_{\Lambda}(d)$
consists of the matrices $V\in\mathbb{M}_{d\times d}(\Bbbk)$ such that
$V^{m}=0$. Similarly, $\operatorname{rep}_{\Lambda}(e)$ consists of the
matrices $U\in\mathbb{M}_{e\times e}(\Bbbk)$ such that $U^{m}=0$. Finally, if
$V\in\operatorname{rep}_{\Lambda}(d)$ and
$U\in\operatorname{rep}_{\Lambda}(e)$, then
$\operatorname{Hom}_{\Lambda}(U,V)$ consists of the matrices
$f\in\mathbb{M}_{d\times e}(\Bbbk)$ such that $fU=Vf$. Consequently, we have
an isomorphism
$\operatorname{rep}_{A^{\prime}}((d,e))\cong\mathcal{H}_{\Lambda}(e,d)$ given
by

$\operatorname{rep}_{A^{\prime}}((d,e))\ni
M\to(M_{\varepsilon_{1}},M_{\varepsilon_{0}},M_{\alpha_{1}})\in\mathcal{H}_{\Lambda}(e,d).$

(2) Note that $\mathbb{A}_{Q}^{e,d}=\mathbb{M}_{d\times e}(\Bbbk)$, while
$\mathcal{E}_{\Lambda}(e,d)$ consists of the triples $(V,U,Z)$ with
$V\in\operatorname{rep}_{\Lambda}(d)$, $U\in\operatorname{rep}_{\Lambda}(e)$,
and $Z\in\mathbb{A}_{Q}^{e,d}$, such that $Z_{\varepsilon^{m}}^{U,V}=0$. By
easy induction one shows that
$Z^{U,V}_{\varepsilon^{l}}=\sum_{i=0}^{l-1}V^{l-1-i}ZU^{i}$, for
$l\in\mathbb{N}_{+}$. Consequently the map

$\operatorname{rep}_{B_{1}}((d,e))\ni
M\to(M_{\varepsilon_{1}},M_{\varepsilon_{0}},M_{\alpha_{1}})\in\mathcal{E}_{\Lambda}(e,d)$

is easily seen to be an isomorphism. ∎

We have the following consequences.

######  Corolary 4.2.

The algebra $\Lambda$ is hom- and ext-irreducible.

###### Proof.

If $d,e\in\mathbb{N}$, then
$\mathcal{H}_{\Lambda}(e,d)\cong\operatorname{rep}_{A^{\prime}}((d,e))$ by
Lemma 4.1(1). Since $A^{\prime}$ is geometrically irreducible,
$\mathcal{H}_{\Lambda}(e,d)$ is irreducible, thus $\Lambda$ is hom-
irreducible. Ext-irreducibility follows from Corollary 3.3. ∎

######  Corolary 4.3.

The algebra $B_{1}$ is geometrically irreducible.

###### Proof.

Fix a dimension vector $(d,e)$. Since $\Lambda$ is ext-irreducible by
Corollary 4.2 and
$\operatorname{rep}_{B_{1}}((d,e))\cong\mathcal{E}_{\Lambda}(e,d)$ by Lemma
4.1(2), $\operatorname{rep}_{B_{1}}((d,e))$ is irreducible and the claim
follows. ∎

###### Proof of Theorem 1.2.

Fix $n\in\mathbb{N}_{+}$. If $(d,e)$ is a dimension vector, then

$\operatorname{rep}_{B_{n}}((d,e))\cong\operatorname{rep}_{B_{1}}((d,e))\times\prod_{i=2}^{n}\mathbb{M}_{d\times
e}(\Bbbk).$

Since $B_{1}$ is geometrically irreducible (Corollary 4.3),
$\operatorname{rep}_{B_{n}}((d,e))$ is irreducible, thus $B_{n}$ is
geometrically irreducible. ∎

##  5\. Classification results

It is natural to ask about a classification of the hom- and ext-irreducible
algebras. We solve this problem in this section.

We start with the following remark. One could define hom- and ext-
irreducibility for not necessarily basic algebras, using module varieties
instead of representation varieties (compare constructions in
[Bon1994]*subsection 2.1). A comparision of these constructions (which we
omit), based on results of [Bon1991], shows that notions of hom- and ext-
irreducibility defined in this way coincides with these defined in Section 3.
Consequently, we will work further with the path algebras of bound quivers,
but formulate the results for arbitrary algebras.

For $m\in\mathbb{N}_{+}$, let $\Lambda_{m}:=k[x]/(x^{m})$. We have the
following classification of the hom-irreducible algebras.

######  Theorem 5.1.

Up to Morita equivalence, the algebras $\Lambda_{m}$, $m\in\mathbb{N}_{+}$,
are precisely the hom-irreducible connected algebras.

###### Proof.

We already know that the algebras $\Lambda_{m}$, $m\in\mathbb{N}_{+}$, are
hom-irreducible (Corollary 4.2). It remains to show that they are the only
hom-irreducible connected algebras.

Let $(Q,I)$ be a bound quiver such that $A:=\Bbbk Q/I$ is a hom-irreducible
connected algebra. It is obvious that $A$ is geometrically irreducible
($\operatorname{rep}_{A}(\mathbf{d})=\mathcal{H}_{A}(\mathbf{0},\mathbf{d})$).
Suppose there exists an arrow $\alpha_{1}\colon x\to y$ in $Q$ with $x\neq y$.
Since $A$ is weakly triangular by Lemma 2.1(1), there are no arrows $y\to x$
in $Q$. Let $\alpha_{2},\ldots,\alpha_{n}\colon x\to y$ be the remaining
arrows from $x$ to $y$ in $Q$. Let $\mathbf{d}$ and $\mathbf{e}$ be the
dimension vectors such that

$d_{z}:=\delta_{x,z}+\delta_{y,z}\qquad\text{and}\qquad e_{z}:=\delta_{x,z},$

for $z\in Q_{0}$, where $\delta_{u,v}$ is the Kronecker delta.

Fix a loop $\alpha$ in $Q$. Recall that there exists $n\in\mathbb{N}_{+}$ such
that $\alpha^{n}\in I$. Consequently, if
$W\in\operatorname{rep}_{A}(\mathbf{d})$, then $W_{\alpha}^{n}=0$. Since
$d_{z}\leq 1$, for each $z\in Q_{0}$, this implies $W_{\alpha}=0$. Similarly,
$V_{\alpha}=0$, for each $V\in\operatorname{rep}_{A}(\mathbf{e})$.

Thus $\mathcal{H}_{A}(\mathbf{e},\mathbf{d})$ is isomorphic to the variety

$\\{(b,a_{1},\ldots,a_{n})\in\Bbbk^{n+1}:\text{$a_{1}b=\cdots=a_{n}b=0$}\\},$

which is easily seen to be reducible.

The above implies that $|Q_{0}|=1$ (due to connectedness of $A$). Using again
geometrical irreducibility of $A$ we get $A\simeq\Lambda_{m}$, for some
$m\in\mathbb{N}_{+}$, by Lemma 2.1(3). ∎

In order to classify the ext-irreducible algebras, we introduce the
corresponding class of algebras. We call an algebra $A$ a loop extension of a
hereditary algebra, if $A$ is the path algebra of a quiver $Q$ bound by
relations of degree $0$. Note that $A/\langle Q_{1}^{(0)}\rangle$ is
isomorphic to the path algebra of the quiver $(Q_{0},Q_{1}^{(1)})$ (hence
hereditary). In other words, $A$ is obtained from a hereditary algebra (i.e.,
the path algebra of a quiver with no relations) by adding loops and relations
involving only these loops. In particular $A$ is weakly triangular. If
additionally, for each vertex $x$ of $Q$ there is at most one loop $\alpha$
such that $s\alpha=x$, then we call $A$ a simple loop extension of a
hereditary algebra. We show that the ext-irreducible algebras are exactly the
simple loop extensions of hereditary algebras.

We start with the following.

######  Lemma 5.2.

Every hereditary algebra is ext-irreducible.

###### Proof.

Let $A$ be a hereditary algebra. Then $A=\Bbbk Q$, for some quiver $Q$.
Consequently, if we fix dimension vectors $\mathbf{d}$ and $\mathbf{e}$, then
$\operatorname{rep}_{A}(\mathbf{d})$ and $\operatorname{rep}_{A}(\mathbf{e})$
are affine spaces, and
$\mathbb{Z}^{U,V}=\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$, for all
representations $V\in\operatorname{rep}_{A}(\mathbf{d})$ and
$U\in\operatorname{rep}_{A}(\mathbf{e})$. Consequently,
$\mathcal{E}_{A}(\mathbf{e},\mathbf{d})=\operatorname{rep}_{A}(\mathbf{e})\times\operatorname{rep}_{A}(\mathbf{d})\times\mathbb{A}_{Q}^{\mathbf{e},\mathbf{d}}$
is an affine space, thus irreducible. ∎

As a consequence we obtain the first part of our classification result.

######  Lemma 5.3.

Every simple loop extension of a hereditary algebra is ext-irreducible.

###### Proof.

Let $A=\Bbbk Q/I$ be a simple loop extension of a hereditary algebra. Let $H$
be the path algebra of the quiver $(Q_{0},Q_{1}^{(1)})$. Next, for each loop
$\varepsilon$, let $m_{\varepsilon}$ be the minimal $m$ such that
$\varepsilon^{m}\in I$, and
$\Lambda_{\varepsilon}:=k[x]/(x^{m_{\varepsilon}})$. Note that
$I=\langle\varepsilon^{m_{\varepsilon}}:\text{$\varepsilon\in
Q_{1}^{(0)}$}\rangle$.

Let $\mathbf{d}$ and $\mathbf{e}$ be dimension vectors. Then

$\mathcal{E}_{A}(\mathbf{e},\mathbf{d})=\mathcal{E}_{H}(\mathbf{e},\mathbf{d})\times\prod_{\varepsilon\in
Q_{1}^{(0)}}\mathcal{E}_{\Lambda_{\varepsilon}}(e_{t\varepsilon},d_{s\varepsilon}).$

Since $H$ is ext-irreducible by Lemma 5.2, and the algebras
$\Lambda_{\varepsilon}$ are ext-irreducible by Corollary 4.2,
$\mathcal{E}_{A}(\mathbf{e},\mathbf{d})$ is irreducible, and the claim
follows. ∎

In order to prove that the simple loop extensions of hereditary algebras are
the only ext-irreducible algebras, we need additional facts on relations in
(geometrically irreducible) algebras. Let $Q$ be a quiver. Recall that a set
$R$ of relations is called minimal, if $\langle R\rangle\neq\langle
R\setminus\\{\rho\\}\rangle$, for each $\rho\in R$. If $R$ is a set of
relations, then for $x,y\in R$ we denote by $R_{x,y}$ the set of $\rho\in R$
such that $s\rho=x$ and $t\rho=y$. The first fact is the following
generalization of [Bon1983]*Proposition 1.2.

######  Proposition 5.4.

Let $A=\Bbbk Q/I$ be a weakly triangular algebra. If $R$ is a minimal set of
relations generating $I$, then the cardinality $\\#R_{x,y}$ of $R_{x,y}$ is
independent of $R$, for all $x,y\in Q_{0}$, $x\neq y$. More precisely,

$\\#R_{x,y}=\dim_{\Bbbk}\operatorname{Ext}_{A}^{2}(S_{A}(x),S_{A}(y)).$

###### Proof.

Let $J$ be the ideal in $\Bbbk Q$ generated by the arrows. Then according to
[Bon1983]*Corollary 1.2

$\dim_{\Bbbk}\operatorname{Ext}_{A}^{2}(S_{A}(x),S_{A}(y))=\dim_{\Bbbk}\mathbf{1}_{y}(I/(IJ+JI))\mathbf{1}_{x}.$

Next, let $R^{(0)}$ be the set of relations of degree $0$ in $R$,
$A^{\prime}=\Bbbk Q/\langle R^{(0)}\rangle$, $I^{\prime}=I/\langle
R^{(0)}\rangle$, and $J=J^{\prime}/\langle R^{(0)}\rangle$. Since $x\neq y$,

$\dim_{\Bbbk}\mathbf{1}_{y}(I/(IJ+JI))\mathbf{1}_{x}=\dim_{\Bbbk}\mathbf{1}_{y}(I^{\prime}/(I^{\prime}J^{\prime}+J^{\prime}I^{\prime}))\mathbf{1}_{x}.$

Thus in order to finish the proof it suffices to show

$\\#R_{x,y}=\dim_{\Bbbk}\mathbf{1}_{y}(I^{\prime}/(I^{\prime}J^{\prime}+J^{\prime}I^{\prime}))\mathbf{1}_{x}.$

In other words, it is enough to show that the residue classes of the elements
of $R_{x,y}$ form a basis of
$\mathbf{1}_{y}(I^{\prime}/(I^{\prime}J^{\prime}+J^{\prime}I^{\prime}))\mathbf{1}_{x}$.

One easily sees that the residue classes of the elements of $R_{x,y}$ span
$\mathbf{1}_{x}(I^{\prime}/(I^{\prime}J^{\prime}+J^{\prime}I^{\prime}))\mathbf{1}_{y}$
as a vector space (see for example the proof of the corresponding statement in
the proof of [MarSalSte2021]*Theorem 4.1). Thus assume the residue classes of
the elements of $R_{x,y}$ are linearly dependent. This implies that exists
$\rho\in R_{x,y}$ such that $\rho\in
K^{\prime}+I^{\prime}J^{\prime}+J^{\prime}I^{\prime}$, where
$K^{\prime}:=K/\langle R^{(0)}\rangle$ and $K:=\langle
R\setminus\\{\rho\\}\rangle$. Consequently,
$K^{\prime}+I^{\prime}J^{\prime}+J^{\prime}I^{\prime}=I^{\prime}$. Note that
$I^{\prime}J^{\prime}+J^{\prime}I^{\prime}$ is the radical of $A^{\prime}$ as
an $A^{\prime}$-$A^{\prime}$-bimodule (here we use that $A$ is weakly
triangular, hence there exists $n\in\mathbb{N}_{+}$ with $\sigma\in\langle
R^{(0)}\rangle$, for each path $\sigma$ in $Q$ of length $n$). Thus the
Nakayama lemma (see for example [AndFul1992]*Corollary 15.13) implies
$I^{\prime}=K^{\prime}$, i.e., $I=K$, which contradicts the minimality of $R$.
∎

Before we continue we formulate an easy generalization of the results of
[BobSch2019a]*Subsection 5.5.

######  Lemma 5.5.

Let $Q$ be the quiver

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117.16211pt\raise 9.50693pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8125pt\hbox{$\scriptstyle{\alpha_{k1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.0pt\raise 4.30554pt\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
116.07831pt\raise-10.06248pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.25694pt\hbox{$\scriptstyle{\alpha_{kn_{k}}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
118.88022pt\raise 1.84445pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cdots}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.0pt\raise-4.30554pt\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,$

where $k,n_{1},\ldots,n_{k}\in\mathbb{N}_{+}$. If $A^{\prime}$ is a
geometrically irreducible algebra, whose Gabriel quiver $Q^{\prime}$ is a
subquiver of $Q$, and $R^{\prime}$ is a minimal set of relations such that
$A^{\prime}\simeq\Bbbk Q^{\prime}/\langle R^{\prime}\rangle$, then
$\deg\rho\in\\{0,1\\}$, for each $\rho\in R^{\prime}$.

###### Proof.

We adapt arguments from the proof of [BobSch2019a]*Corollary 5.5: Let
$R^{\prime(0)}$ be the set of relations of degree $0$ in $R^{\prime}$. Without
loss of generality, we may assume
$R^{\prime(0)}:=\\{\varepsilon_{i}^{m_{i}}:\text{$i\in
Q_{0}^{\prime\prime}$}\\}$, for some $m_{i}\geq 2$, $i\in
Q_{0}^{\prime\prime}$, where $Q_{0}^{\prime\prime}$ is the set of $i\in[0,k]$
such that $\varepsilon_{i}\in Q_{1}^{\prime}$. Choose $m_{i}\geq 2$, for each
$i\in Q_{0}\setminus Q_{0}^{\prime\prime}$, and let
$R:=R^{\prime}\cup\\{\varepsilon_{i}^{m_{i}}:\text{$i\in Q_{0}\setminus
Q_{0}^{\prime\prime}$}\\}$. If $A:=\Bbbk Q/\langle R\rangle$, then
$A^{\prime}$ being geometrically irreducible implies $A$ is geometrically
irreducible. Consequently, the claim follows from [BobSch2019a]*Proposition
5.4. ∎

The proof of the following fact uses ideas from the proof of
[BobSch2019b]*Lemma 5.1.

######  Proposition 5.6.

Let $A$ be a geometrically irreducible algebra with Gabriel quiver $Q$, and
let $R$ be a minimal set of relations such that $A\simeq\Bbbk Q/\langle
R\rangle$. If $\deg\rho\neq 1$, for each $\rho\in R$, then $\deg\rho=0$, for
each $\rho\in R$.

###### Proof.

Let $R^{(0)}$ be the set of relations of degree $0$ in $R$ and $I:=\langle
R^{(0)}\rangle$. Our aim is to show that $\rho\in I$, for each $\rho\in
R\setminus R^{(0)}$.

For a path $\sigma=\alpha_{1}\cdots\alpha_{l}$ in $Q$, let
$\operatorname{supp}\sigma:=\\{t(\alpha_{1}),s(\alpha_{1}),\ldots,s(\alpha_{l})\\}$.
Note that $\deg\sigma=\\#\operatorname{supp}\sigma-1$, since $A$ is weakly
triangular by Lemma 2.1(1). If $\rho=\sum_{i=1}^{k}\lambda_{i}\sigma_{i}$ is a
relation in $R$, we write $\rho=\sum_{X\subseteq Q_{0}}\rho_{X}$, where
$\rho_{X}:=\sum_{i:\operatorname{supp}\sigma_{i}=X}\lambda_{i}\sigma_{i}$, for
$X\subseteq Q_{0}$. By induction on $\\#X$ we show $\rho_{X}\in I$, for each
$\rho\in R\setminus R^{(0)}$. The claim is obvious if $\\#X=1$ (indeed, weak
triangularity implies that $\rho_{X}=\rho\in R^{(0)}$, if $\\#X=1$ and
$\rho_{X}\neq 0$) and there is nothing to prove if $\\#X=2$ (if $\\#X=2$ and
$\rho_{X}\neq 0$, then $\deg\rho=1$ by weak triangularity).

Now fix $X\subseteq Q_{0}$ with $\\#X>2$, and assume $\rho_{Y}\in I$, for each
$\rho\in R\setminus R^{(0)}$ and $Y$ with $\\#Y<\\#X$. By replacing every
$\rho\in R\setminus R^{(0)}$ by $\rho-\sum_{Y:\\#Y<\\#X}\rho_{Y}$ (note that
$(\rho-\sum_{Y:\\#Y<\\#X}\rho_{Y})_{X}=\rho_{X}$) we may assume

($*$) $\text{$\rho_{Y}=0$, for each $\rho\in R\setminus R^{(0)}$ and $Y$ with
$\\#Y<\\#X$}.$

We number the elements $x_{0}$, …, $x_{l}$ of $X$ in such a way that there are
no paths from $x_{i}$ to $x_{j}$ in $Q$, for $i<j$ (here we use again that $A$
is weakly triangular). Let $Q_{1}^{\prime}$ be the union of the set of all
loops at $x_{0}$, …, $x_{l}$ and the set of arrows $\alpha\in Q_{1}$ such that
$s\alpha=x_{i}$ and $t\alpha=x_{i-1}$, for some $i\in[1,l]$. If
$Q^{\prime}:=(X,Q_{1}^{\prime})$, then $\rho_{X}$ is a linear combination of
paths in $Q^{\prime}$, for each $\rho\in R\setminus R^{(0)}$. Let
$A^{\prime}:=A/\langle(\mathbf{1}_{Q_{0}\setminus X}\rangle+\langle
Q_{1}\setminus Q_{1}^{\prime}\rangle)$. We show that $A^{\prime}$ is
geometrically irreducible.

First observe that $A^{\prime}\simeq\Bbbk Q^{\prime}/I^{\prime}$, where
$I^{\prime}$ is the ideal in $\Bbbk Q^{\prime}$ generated by the restrictions
$\rho|_{Q^{\prime}}$ of $\rho\in R$ to $Q^{\prime}$. If $\rho\in R^{(0)}$,
then either $\rho|_{Q^{\prime}}=0$ (if $s\rho\not\in X$) or
$\rho|_{Q^{\prime}}=\rho$ (if $s\rho\in X$). On the other hand, if $\rho\in
R\setminus R^{(0)}$, then assumption ($*$ ‣ 5) implies
$\rho|_{Q^{\prime}}=\rho_{X}$. Consequently, $I^{\prime}=\langle
R^{\prime}\rangle$, where

$R^{\prime}:=\\{\rho\in R^{(0)}:\text{$s\rho\in
X$}\\}\cup\\{\rho_{X}:\text{$\rho\in R\setminus R^{(0)}$}\\}.$

Now let $A^{\prime\prime}:=A/\langle\mathbf{1}_{Q_{0}\setminus X}\rangle$.
Then obviously $A^{\prime\prime}$ is geometrically irreducible. Moreover,
$A^{\prime\prime}\simeq\Bbbk Q^{\prime\prime}/I^{\prime\prime}$, where
$Q^{\prime\prime}$ is the full subquiver of $Q$ with the vertex set $X$ and
$I^{\prime\prime}$ is the ideal in $\Bbbk Q^{\prime\prime}$ generated by the
restrictions $\rho|_{Q^{\prime\prime}}$ of $\rho\in R$ to $Q^{\prime\prime}$.
Obviously, $\rho|_{Q^{\prime\prime}}=\rho|_{Q^{\prime}}$, if $\rho\in
R^{(0)}$. Furthermore, if $\rho\in R\setminus R^{(0)}$, then
$\rho|_{Q^{\prime\prime}}=\rho_{X}$, again by assumption ($*$ ‣ 5). In other
words, $I^{\prime\prime}=\langle R^{\prime}\rangle$. Consequently,
$\operatorname{rep}_{A^{\prime\prime}}(\mathbf{d})$ is the product of
$\operatorname{rep}_{A^{\prime}}(\mathbf{d})$ and an affine space, for each
dimension vector $\mathbf{d}$, hence $A^{\prime\prime}$ being geometrically
irreducible implies $A^{\prime}$ is geometrically irreducible as well.

Now we apply Lemma 5.5 to $A^{\prime}$ (note that $A^{\prime}$ being
geometrically irreducible implies that there is at most one loop at each
vertex of $Q^{\prime}$ by Lemma 2.1(2)) and conclude that $\rho_{X}$ belongs
to the ideal generated by the relations in $R^{(0)}\cap R^{\prime}$, for each
$\rho\in R\setminus R^{(0)}$. Consequently, $\rho_{X}\in I$, for each $\rho\in
R\setminus R^{(0)}$, and this finishes the proof. ∎

Now we are able to conclude a classification of the ext-irreducible algebras.

######  Theorem 5.7.

Up to Morita equivalence, an algebra $A$ is ext-irreducible if and only if $A$
is a simple loop extension of a hereditary algebra.

Before giving the proof we need some preparations. Let $A$ be the path algebra
of a bound quiver $(Q,I)$. For each loop $\alpha$, let $m_{\alpha}$ be the
minimal $m\in\mathbb{N}_{+}$ such that $\alpha^{m}\in I$. A minimal set $R$ of
relations generating $I$ is called normalized if, for each loop $\alpha$,
$\alpha^{m_{\alpha}}$ is the unique element of $R$ which has a summand
containing $\alpha^{m_{\alpha}}$ as a subpath (in particular,
$\alpha^{m_{\alpha}}\in R$). In general a normalized minimal set of relations
generating $I$ may not exists. However, it obviously exists if $A$ is a
quotient of a simple loop extension of a hereditary algebra.

Now let $A=\Bbbk Q/I$ be a geometrically irreducible algebras. According to
Lemma 2.1(2), for each vertex $x$ there is at most one loop $\alpha$ such that
$s\alpha=x$. Moreover, $A$ is weakly triangular by Lemma 2.1(1). Consequently,
$A$ is a quotient of a simple loop extension of a hereditary algebras. In
particular, there exists a normalized minimal set of relations generating $I$.

###### Proof.

We already know from Lemma 5.3 that the simple loop extensions of hereditary
algebras are ext-irreducible. Thus we need to show that if $A=\Bbbk Q/I$ is
ext-irreducible, then $A$ is a simple loop extension of a hereditary algebra.
Recall from Proposition 3.2 that $A$ is mono-irreducible. Moreover, $A$ is
geometrically irreducible
($\operatorname{rep}_{A}(\mathbf{d})=\mathcal{E}_{A}(\mathbf{0},\mathbf{d})$).

Let $R$ be a normalized minimal set of generators of $I$. According to
Proposition 5.6 it sufficies to show $\deg\rho\neq 1$, for all $\rho\in R$.
Suppose this is not the case and fix $\rho\in R$ with $\deg\rho=1$. Put
$x:=t\rho$ and $y:=s\rho$. If
$B:=A/\langle\mathbf{1}_{Q_{0}\setminus\\{x,y\\}}\rangle$, then $B$ is also
mono- and geometrically irreducible. Moreover, $B$ has exactly two simples,
hence by Theorem 1.1 (or [BobSch2019b]*Theorem 1.1) we may assume $B=A(n,m,1)$
or $B=A(n,m,m-1)$, for some $n\in\mathbb{N}_{+}$ and $m\geq 2$ (the cases of
$n=0$ and $B=A^{\prime}(n,m_{0},m_{1})$ are excluded by Proposition 5.4, since
from $R$ being normalized it follows that $\rho$ induces a relation of degree
$1$ in $B$, which cannot be generated by relations of degree $0$). In order to
treat the both above cases simultaneously, let $l:=2$ in the former case, and
$l:=m$ in the latter one.

Let $\mathbf{d}:=(1,l)$ and $\mathbf{e}:=(1,1)$. Then
$\mathcal{M}_{B}(\mathbf{e},\mathbf{d})$ consist of the tuples

$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[0]}$$\scriptstyle{[\lambda]}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[0]}$$\scriptstyle{[\mu_{1}]}$$\scriptstyle{[\mu_{n}]}$$\scriptstyle{\vdots}$$\scriptstyle{W}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[0]}$$\textstyle{k^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{V_{1}}$$\scriptstyle{V_{n}}$$\scriptstyle{\vdots}$

where $\mu_{1},\ldots,\mu_{n},\lambda\in\Bbbk$,
$V_{1},\ldots,V_{n}\in\mathbb{M}_{1\times l}(\Bbbk)$, $U\in\mathbb{M}_{l\times
l}(\Bbbk)$, $W\in\mathbb{M}_{l\times 1}(\Bbbk)$, such that

$V_{1}U^{l-1}=0,\;U^{l}=0,\;\lambda\mu_{1}=V_{1}W,\;\ldots,\lambda\mu_{n}=V_{n}W,\;UW=0,$

and $\lambda$ and $W$ are nonzero.

Let $\mathcal{U}_{1}$ be the set of the tuples as above such that
$\operatorname{rk}U=l-1$ and $\mathcal{U}_{2}$ be set of the tuples such that
$\mu_{1}\neq 0$. They are easily seen to be nonempty open subsets of
$\mathcal{M}_{B}(\mathbf{e},\mathbf{d})$. If
$\mathcal{M}_{B}(\mathbf{e},\mathbf{d})$ were irreducible, $\mathcal{U}_{1}$
and $\mathcal{U}_{2}$ would have a nonempty intersection. However,
$V_{1}U^{l-1}=0$ and $UW=0$, imply
$\operatorname{Im}U^{l-1}\subseteq\operatorname{Ker}V_{1}$ and
$\operatorname{Im}W\subseteq\operatorname{Ker}U$. Moreover, if
$\operatorname{rk}U=l-1$, then $U^{l}=0$ infer
$\operatorname{Ker}U=\operatorname{Im}U^{l-1}$, hence
$\operatorname{Im}W\subseteq\operatorname{Ker}V_{1}$. Consequently,
$\lambda\mu_{1}=V_{1}W=0$. Since $\lambda\neq 0$, $\mu_{1}=0$, i.e.,
$\mathcal{U}_{1}\cap\mathcal{U}_{2}=\varnothing$. Thus
$\mathcal{M}_{B}(\mathbf{e},\mathbf{d})$ is reducible, a contradiction. ∎

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# Constraining nuclear Parton Density Functions with forward photon production
at the LHC

Marco van Leeuwen
Nikhef, National Institute for Subatomic Physics, P.O. Box 41882, 1009 DB
Amsterdam
and Utrecht University, P.O. Box 80000, 3508 TA Utrecht, The Netherlands

##  1 Introduction

In this paper we explore the use of results on forward particle production at
the LHC to constrain nuclear Parton Density Functions (nPDFs). The case study
here is based on a possible future measurement of forward photon production
with the Forward Calorimeter that is currently under discussion as an upgrade
of the ALICE experiment. As a starting point, we use the recent nNNPDF 1.0
nuclear PDFs, which have been determined using fixed target neutral-current
DIS data, which constrain the gluon density at $x\gtrsim 10^{-2}$ [1]. The
Bayesian reweighting technique is used to include the constraints from the
future measurement.

##  2 Method

As a starting point, we use the recent nNNPDF 1.0 nuclear PDFs [1], which have
been determined using fixed target neutral-current DIS data, which constrain
the gluon density at $x\gtrsim 10^{-2}$. The nNNPDF 1.0 nuclear PDFs provide a
set of replicas which sample the allowed parton distributions in the nucleus.
The reweighting process starts by calculating the photon production cross
section in the rapidity range covered by the proposed measurement,
$3.5<\eta<4.5$ in the centre-of-mass system. The cross section is calculated
using the INCNLO code [2], with a patch to improve the numerical stability at
forward rapidity, provided by Ilkka Helenius [10] and interface code to read
the PDFs using the LHAPDF library [6]. The calculation is carried out using
the nNNPDF 1.0 PDFs for the Pb nucleus and the NNPDF 3.1 [5] PDF for the
proton.

The reweighting itself follows the Bayesian approach outlined in [4, 8]. For
each replica $R_{i}$ the probability $P_{i}(\mathrm{data}|R_{i})$ to observe
the pseudo data points as provided by the ALICE-FOCAL collaboration is
calculated and the reweighted (posterior) probability $P^{\prime}_{i}$ for
each replica is calculated using Bayes’ theorem:

$P^{\prime}_{i}=1/N_{rep}\frac{P_{i}(\mathrm{data}|R_{i})}{\sum_{i}P_{i}(\mathrm{data}|R_{i})}$
(1)

where the prefactor $1/N_{rep}$ is the (prior) probability for each replica
before reweighting. The probabilities $P_{i}(\mathrm{data}|R_{i})$ are given
by the $\chi^{2}$ probability distribution

$P(\chi^{2}|N)=\frac{1}{2^{\frac{N}{2}}\Gamma\left(\frac{N}{2}\right)}\left(\chi^{2}\right)^{\frac{N}{2}-1}e^{-\frac{\chi^{2}}{2}}$

with as input the $\chi^{2}$ between the measured data points and the cross
sections calculated with each replica $R_{i}$. The calculation of the
$\chi^{2}$ which takes into account the correlated systematic uncertainties is
discussed below around Eq. 2.

Confidence intervals for the final results are calculated by sorting the
replicas (or cross sections) by value and summing the reweighted
probabilities. For example, for the 90% CL, the probabilities are summed until
a cumulative probability of 0.05 is reached to find the lower bound of the
confidence interval; further summing until 0.95 is reached determines the
upper bound.

It has also been suggested to use an ’unweighting’ procedure to generate a new
set of equal-probability replicas [3]. This is useful if the results are to be
distributed for further fits or theoretical predictions. However, for the
present purpose, this step is not necessary.

###  2.1 Treatment of the systematic uncertainties

A key point in the calculation of the probabilities for the data is taking
into account correlated systematic uncertainties. For simplicity, we have
decided in this case to add the systematic and statistical uncertainties on
the FOCAL pseudo data in quadrature for each data point, which amounts to
assuming that the systematic uncertainties are uncorrelated from point to
point, and to add a fully correlated uncertainty of $\sigma_{N}=5\%$, to
represent normalisation uncertainties from the total cross section
determination in the experiment, the $T_{pA}$ scaling factor for nuclei, and
the energy scale uncertainty from the experiment. The effect of the correlated
uncertainty is taken into account by introducing a parameter $\epsilon$ which
quantifies the systematic deviation. The $\chi^{2}$ deviation of the data from
the curves is then calculated as follows:

$\chi^{2}(\epsilon)=\sum_{i}{\left(\frac{(1-\epsilon\sigma_{N})y_{i}-f(x_{i})}{\tilde{\sigma}_{i}}\right)^{2}}+\epsilon^{2}$
(2)

where the measured data points $(x_{i},y_{i})$ are shifted by a relative
amount $\epsilon\sigma_{N}$. The function $f(x_{i})$ is the differential cross
section calculated for $p_{\mathrm{T}}$ bin $i$ of the pseudo data. The
uncertainties on the data points $\sigma_{i}$ are rescaled, assuming that the
relative uncertainty is constant
$\tilde{\sigma}_{i}=(1-\epsilon\sigma_{N})\sigma_{i}$.

A minimization algorithm is used to find the value of $\epsilon$ for which the
$\chi^{2}$ is the smallest. This value of the $\chi^{2}$ is then used to
calculate the unnormalized posterior probability $P(\mathrm{data}|R_{j})$ from
the $\chi^{2}$ probability distribution with number of degrees of freedom
equal to the number of data points minus one.

##  3 Results

Figure 1: Nuclear modification factor $R_{\mathrm{p-Pb}}$ for direct photons
at $\eta=4.0$. The grey bands indicate the 68 and 90% CL intervals from the
nNNPDF 1.0. The points are pseudo data reflecting the projected uncertainties
on the direct photon measurement using FoCal and the colored bands show the 68
and 90 % CL intervals of the $R_{\mathrm{p-Pb}}$ calculations that are
compatible with the pseudo data.

Figure 1 shows the nuclear modification factor of photons at forward rapidity
$3.5<\eta<4.5$ calculated using the current NNPDF 1.0 and NNPDF 3.1 for the
proton PDFs. The figure shows the theoretical calculations, with 68% and 90%
CL intervals, the pseudo data from the ALICE-FOCAL collaboration and the
posterior uncertainties, also depicted as a 68% and 90% CL intervals (in red).
The uncertainties on the NNPDF are large, because this observable probes
$x<10^{-4}$ and the DIS measurements that constrain the NNPDFs are in the
range $x>10^{-2}$. Note also that he probability distribution is very non-
Gaussian, as can be seen from comparing the 68% and 90% CL intervals. A
measurement of forward photon production with the precision foreseen for the
FoCal clearly provides experimental input to constrain the nuclear PDFs.

Figure 2: Nuclear modification factor $R_{g}(x)$ for the gluon distribution in
the Pb nucleus at a scale $Q^{2}=10\;\mathrm{GeV}^{2}$. The grey bands
indicate the 68 and 90 % CL intervals from nNNPDF 1.0. The red colored bands
show the 68 and 90 % CL intervals of the posterior PDFs constrained by the
foreseen FoCal measurement.

Figure 2 shows the resulting posterior distributions for the nuclear
modification of the gluon PDF at a scale $Q^{2}=10\mathrm{GeV}^{2}$. For
comparison, also the constraints provided by possible future measurements at
the high energy EIC, as presented in the nNNPDF paper [1], are shown. It can
be seen that the forward photon measurement at the LHC constrains the gluon
nPDF down to $x\approx 5\cdot 10^{-5}$.

In addition to direct photon measurements, other forward particle production
measurements are also sensitive to the gluon density at small-$x$. In
particular, forward open charm measurements are already available and are
being used to constrain both proton and nuclear PDFs [9, 7]. The different
processes cover different kinematic ranges and have different theoretical
uncertainties, and thus provide complementary tests of the small-$x$ structure
of the nucleus.

## Acknowledgments

The author would like to thank the nNNPDF team (Jake Ethier, Juan Rojo and
Rabah Abdul Khalek) for useful discussion, including help with the reweighting
procedure, and for providing the 1000-replica version of the nNNPDF1.0. Thanks
also to ALICE-FoCal collaboration for providing the pseudo data.

## References

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