A note on species realizations and nondegeneracy of potentials

TL;DR

This paper develops a mutation theory for species with potential, demonstrating that certain skew-symmetrizable matrices can have realizations with non-degenerate potentials, partially answering open questions in the field.

Contribution

It introduces a mutation framework for species with potential and provides examples of matrices with realizations admitting non-degenerate potentials, even when not globally unfoldable.

Findings

A mutation theory for species with potential is established.

Certain skew-symmetrizable matrices admit non-degenerate potentials.

Examples include matrices not globally unfoldable but still having non-degenerate potentials.

Abstract

In this note we show that a mutation theory of species with potential can be defined so that a certain class of skew-symmetrizable integer matrices have a species realization admitting a non-degenerate potential. This gives a partial affirmative answer to a question raised by Jan Geuenich and Daniel Labardini-Fragoso. We also provide an example of a class of skew-symmetrizable $4 \times 4$ integer matrices, which are not globally unfoldable nor strongly primitive, and that have a species realization admitting a non-degenerate potential.

Equations

$$\mathfrak{h}(f_1{f}_2)(f)=\mathfrak{h}(f_1)(f_{2}f)+\mathfrak{h}(f_2)(f{f}_1)$$

$$\delta (f_1{f}_2)=\mathfrak{h}(f_1)(f_2)+\mathfrak{h}(f_2)(f_1)$$

$$\mathfrak{h}(f_1{f}_2)(f)$$

$$=u(\Delta (f_1)f_{2}f)+u(f_1\Delta (f_2)f)$$

$$=u(\Delta (f_1)f_{2}f)+u(\Delta (f_2)f{f}_1)$$

$$=\mathfrak{h}(f_1)(f_{2}f)+\mathfrak{h}(f_2)(f{f}_1).$$

$$\mathfrak{h}(f_1{f}_2)(f)(l)$$

$$=\sum _{i_1+i_2+j=l}\mathfrak{h}(f_1(i_1)f_2(i_2))(f(j))$$

$$=\sum _{i_1+i_2+j=l}\mathfrak{h}(f_1(i_1))(f_2(i_2)f(j))+\sum _{i_1+i_2+j=l}\mathfrak{h}(f_2(i_2))(f(j)f_1(i_1))$$

$$=\sum _{i_1+t=l}\mathfrak{h}(f_1(i_1))((f_{2}f)(t))+\sum _{i_2+r=l}\mathfrak{h}(f_2(i_2))((f{f}_1)(r))$$

$$=\mathfrak{h}(f_1)(f_{2}f)(l)+\mathfrak{h}(f_2)(f{f}_1)(l)$$

$$=\left(\mathfrak{h}(f_1)(f_{2}f)+\mathfrak{h}(f_2)(f{f}_1)\right)(l).$$

$${}_k\hat{T}$$

$${\tilde{T}}_k$$

$$B=\left[\begin{array}{cccc}0& -a& 0& b\\ 1& 0& -1& 0\\ 0& a& 0& -b\\ -1& 0& 1& 0\end{array}\right]$$

Full text

1. Introduction

In [6, p.14], motivated by the seminal paper [5] of Derksen-Weyman-Zelevinsky, J. Geuenich and D. Labardini-Fragoso raise the following question:

Question [6, Question 2.23] Can a mutation theory of species with potential be defined so that every skew-symmetrizable matrix $B$ have a species realization which admit a non-degenerate potential?

In [1], we show that for every skew-symmetrizable matrix $B=(b_{ij})_{i,j}\in\mathbb{Z}^{n\times n}$ that admits a skew-symmetrizer $D=\operatorname{diag}(d_{1},\ldots,d_{n})$, with the property that each $d_{j}$ divides each and every element $b_{ij}$, then for every finite sequence $k_{1},\ldots,k_{l}$ of elements of $\{1,\ldots,n\}$, there exists a species realization of $B$ and a potential $P$ on this species such that all the iterated mutations $\bar{\mu}_{k_{1}}P$, $\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}P,\ldots,\bar{\mu}_{k_{l}}\cdots\bar{\mu}_{k_{1}}P$ are $2$-acyclic.

In [7], D. Labardini-Fragoso and A. Zelevinsky give a partially positive answer to Question $2.23$ provided that the skew-symmetrizer has pairwise coprime diagonal entries. We remark that this is a stronger condition than the one we impose in [1]. Indeed, since $DB$ is skew-symmetric then $-d_{j}b_{ji}=d_{i}b_{ij}$. Using the fact that $b_{ji}$ is an integer and that $(d_{i},d_{j})=1$, it follows that $d_{j}$ divides $b_{ij}$, as claimed.

In Theorem 3.5 and Corollary 3.6 we give a partially affirmative answer to Question $2.23$ by proving the following: let $B=(b_{ij})\in\mathbb{Z}^{n\times n}$ be a skew-symmetrizable matrix with skew-symmetrizer $D=\operatorname{diag}(d_{1},\ldots,d_{n})$. If $d_{j}$ divides $b_{ij}$ for every $j$ and every $i$, then the matrix $B$ can be realized by a species that admits a non-degenerate potential.

Finally, in Section 5 we give an example of a class of skew-symmetrizable $4\times 4$ integer matrices which are not globally unfoldable nor strongly primitive (in the sense of [7, Definition 14.1]), and that have a species realization admitting a non-degenerate potential. This gives an example of a class of skew-symmetrizable $4\times 4$ integer matrices which are not covered by [7].

2. Preliminaries

The following material is taken from [1].

Definition 2.1**.**

Let $F$ be a field and let $D_{1},\ldots,D_{n}$ be division rings, each containing $F$ in its center and of finite dimension over $F$. Let $S=\displaystyle\prod_{i=1}^{n}D_{i}$ and let $M$ be an $S$-bimodule of finite dimension over $F$. Define the algebra of formal power series over $M$ as the set:

$\mathcal{F}_{S}(M):=\left\{\displaystyle\sum_{i=0}^{\infty}a(i):a(i)\in M^{\otimes i}\right\}$

where $M^{0}=S$. Note that $\mathcal{F}_{S}(M)$ is an associative unital $F$-algebra where the product is the one obtained by extending the product of the tensor algebra $T_{S}(M)=\displaystyle\bigoplus_{i=0}^{\infty}M^{\otimes i}$.

Let $\{e_{1},\ldots,e_{n}\}$ be a complete set of primitive orthogonal idempotents of $S$.

Definition 2.2**.**

An element $m\in M$ is legible if $m=e_{i}me_{j}$ for some idempotents $e_{i},e_{j}$ of $S$.

Definition 2.3**.**

Let $Z=\displaystyle\sum_{i=1}^{n}Fe_{i}\subseteq S$. We say that $M$ is $Z$-freely generated by a $Z$-subbimodule $M_{0}$ of $M$ if the map $\mu_{M}:S\otimes_{Z}M_{0}\otimes_{Z}S\rightarrow M$ given by $\mu_{M}(s_{1}\otimes m\otimes s_{2})=s_{1}ms_{2}$ is an isomorphism of $S$-bimodules. In this case we say that $M$ is an $S$-bimodule which is $Z$-freely generated.

Throughout this paper we will assume that $M$ is $Z$-freely generated by $M_{0}$.

Definition 2.4**.**

Let $A$ be an associative unital $F$-algebra. A cyclic derivation, in the sense of Rota-Sagan-Stein [9], is an $F$-linear function $\mathfrak{h}:A\rightarrow\mathrm{End}_{F}(A)$ such that:

[TABLE]

for all $f,f_{1},f_{2}\in A$.

Definition 2.5**.**

Let $A$ be an associative unital $F$-algebra. Given a cyclic derivation $\mathfrak{h}:A\rightarrow\mathrm{End}_{F}(A)$, we define the associated cyclic derivative $\delta:A\rightarrow A$ as $\delta(f)=\mathfrak{h}(f)(1)$.

From (2.1) one obtains:

[TABLE]

for all $f_{1},f_{2}\in A$. In particular, $\delta(f_{1}f_{2})=\delta(f_{2}f_{1})$.

We now construct a cyclic derivative on $\mathcal{F}_{S}(M)$. First, we define a cyclic derivation on the tensor algebra $A=T_{S}(M)$ as follows. Consider the map

$\hat{u}:A\times A\rightarrow A$

given by $\hat{u}(f,g)=\displaystyle\sum_{i=1}^{n}e_{i}gfe_{i}$ for every $f,g\in A$; this is an $F$-bilinear map which is $Z$-balanced. Therefore, there exists $u:A\otimes_{Z}A\rightarrow A$ such that $u(a\otimes b)=\hat{u}(a,b)$. Now we define an $F$-derivation $\Delta:A\rightarrow A\otimes_{Z}A$ as follows: for $s\in S$, we define $\Delta(s)=1\otimes s-s\otimes 1$; for $m\in M_{0}$, we let $\Delta(m)=1\otimes m$. Then we define $\Delta:M\rightarrow T_{S}(M)$ such that for $s_{1},s_{2}\in S$ and $m\in M_{0}$, we have

$\Delta(s_{1}ms_{2})=\Delta(s_{1})ms_{2}+s_{1}\Delta(m)s_{2}+s_{1}m\Delta(s_{2}).$

The above map is well defined because $M\cong S\otimes_{Z}M_{0}\otimes_{Z}S$ via the multiplication map $\mu_{M}$. Now $\Delta$ can be extended to an $F$-derivation on $A$.

We define $\mathfrak{h}:A\rightarrow\mathrm{End}_{F}(A)$ as follows

$\mathfrak{h}(f)(g)=u(\Delta(f)g).$

We have:

[TABLE]

Therefore $\mathfrak{h}$ is a cyclic derivation on $T_{S}(M)$. We now extend $\mathfrak{h}$ to $\mathcal{F}_{S}(M)$: take $f,g\in\mathcal{F}_{S}(M)$, then we have $\mathfrak{h}(f(i))(g(j))\in M^{\otimes i+j}$; thus we define $\mathfrak{h}(f)(g)(l)=\displaystyle\sum_{i+j=l}\mathfrak{h}(f(i))(g(j))$ for every non-negative integer $l$.

Proposition 2.6**.**

The $F$-linear map $\mathfrak{h}:\mathcal{F}_{S}(M)\rightarrow\mathrm{End}_{F}(\mathcal{F}_{S}(M))$ is a cyclic derivation.

Proof.

Let $f,f_{1},f_{2}\in\mathcal{F}_{S}(M)$. Then for any non-negative integer $l$, we have:

[TABLE]

The result follows. ∎

From the above we obtain a cyclic derivative $\delta$ on $\mathcal{F}_{S}(M)$ defined as

$\delta(f):=\mathfrak{h}(f)(1)$

for every $f\in\mathcal{F}_{S}(M)$.

Definition 2.7**.**

Let $\mathcal{C}$ be a subset of $M$. We say that $\mathcal{C}$ is a right $S$-local basis of $M$ if every element of $\mathcal{C}$ is legible and if for each pair of idempotents $e_{i},e_{j}$ of $S$, we have that $\mathcal{C}\cap e_{i}Me_{j}$ is a $D_{j}$-basis for $e_{i}Me_{j}$.

Note that a right $S$-local basis $\mathcal{C}$ induces a dual basis $\{u,u^{\ast}\}_{u\in\mathcal{C}}$, where $u^{\ast}:M_{S}\rightarrow S_{S}$ is the morphism of right $S$-modules defined by $u^{\ast}(v)=0$ if $v\in\mathcal{C}\setminus\{u\}$; and $u^{\ast}(u)=e_{j}$ if $u=e_{i}ue_{j}$.

Let $T$ be a $Z$-local basis of $M_{0}$ and $L$ be a $Z$-local basis of $S$. The former means that for each pair of idempotents $e_{i},e_{j}$ of $S$, $T\cap e_{i}Me_{j}$ is an $F$-basis of $e_{i}M_{0}e_{j}$, and the latter means that $L(i)=L\cap e_{i}S$ is an $F$-basis of the division algebra $e_{i}S=D_{i}$. It follows that the non-zero elements of the set $\{sa:s\in L,a\in T\}$ is a right $S$-local basis of $M$. Therefore, for every $s\in L$ and $a\in T$, we have the map $(sa)^{\ast}\in\operatorname{Hom}_{S}(M_{S},S_{S})$ induced by the dual basis.

Definition 2.8**.**

Let $\mathcal{D}$ be a subset of $M$. We say that $\mathcal{D}$ is a left $S$-local basis of $M$ if every element of $\mathcal{D}$ is legible and if for each pair of idempotents $e_{i},e_{j}$ of $S$, we have that $\mathcal{D}\cap e_{i}Me_{j}$ is a $D_{i}$-basis for $e_{i}Me_{j}$.

As before, a left $S$-local basis $\mathcal{D}$ induces a dual basis $\{u,^{\ast}u\}_{u\in\mathcal{D}}$ where ${}^{\ast}u:_{S}M\rightarrow_{S}S$ is the morphism of left $S$-modules defined by ${}^{\ast}u(v)=0$ if $v\in\mathcal{D}\setminus\{u\}$; and ${}^{\ast}u(u)=e_{i}$ if $u=e_{i}ue_{j}$.

Let $\psi$ be any element of $\mathrm{Hom}_{S}(M_{S},S_{S})$. We will extend $\psi$ to an $F$-linear endomorphism of $\mathcal{F}_{S}(M)$, which we will denote by $\psi_{\ast}$.

First, we define $\psi_{\ast}(s)=0$ for $s\in S$; and for $M^{\otimes l}$, where $l\geq 1$, we define $\psi_{\ast}(m_{1}\otimes\dotsm\otimes m_{l})=\psi(m_{1})m_{2}\otimes\cdots\otimes m_{l}\in M^{\otimes(l-1)}$ for $m_{1},\dots,m_{l}\in M$. Finally, for $f\in\mathcal{F}_{S}(M)$ we define $\psi_{\ast}(f)\in\mathcal{F}_{S}(M)$ by setting $\psi_{\ast}(f)(l-1)=\psi_{\ast}(f(l))$ for each integer $l>1$. Then we set

$\psi_{\ast}(f)=\displaystyle\sum_{l=0}^{\infty}\psi_{\ast}(f(l)).$

Definition 2.9**.**

Let $\psi\in M^{*}=\mathrm{Hom}_{S}(M_{S},S_{S})$ and $f\in\mathcal{F}_{S}(M)$. We define $\delta_{\psi}:\mathcal{F}_{S}(M)\rightarrow\mathcal{F}_{S}(M)$ as

$\delta_{\psi}(f)=\psi_{\ast}(\delta(f))=\displaystyle\sum_{l=0}^{\infty}\psi_{\ast}(\delta(f(l))).$

Definition 2.10**.**

Given an $S$-bimodule $N$ we define the cyclic part of $N$ as $N_{cyc}:=\displaystyle\sum_{j=1}^{n}e_{j}Ne_{j}$.

Definition 2.11**.**

A potential $P$ is an element of $\mathcal{F}_{S}(M)_{cyc}$.

For each legible element $a$ of $e_{i}Me_{j}$, we let $\sigma(a)=i$ and $\tau(a)=j$.

Definition 2.12**.**

Let $P$ be a potential in $\mathcal{F}_{S}(M)$, we define a two-sided ideal $R(P)$ as the closure of the two-sided ideal of $\mathcal{F}_{S}(M)$ generated by all the elements $X_{a^{\ast}}(P)=\displaystyle\sum_{s\in L(\sigma(a))}\delta_{(sa)^{\ast}}(P)s$, $a\in T$.

Definition 2.13**.**

An algebra with potential is a pair $(\mathcal{F}_{S}(M),P)$ where $P$ is a potential in $\mathcal{F}_{S}(M)$ and $M_{cyc}=0$.

The following construction follows the one given in [5, p.20]. Let $k$ be an integer in $[1,n]$. Using the $S$-bimodule $M$, we define a new $S$-bimodule $\mu_{k}M=\widetilde{M}$ as:

$\widetilde{M}:=\bar{e}_{k}M\bar{e}_{k}\oplus Me_{k}M\oplus(e_{k}M)^{\ast}\oplus^{\ast}(Me_{k})$

where $\bar{e}_{k}=1-e_{k}$, $(e_{k}M)^{\ast}=\operatorname{Hom}_{S}((e_{k}M)_{S},S_{S})$, and ${}^{\ast}(Me_{k})=\operatorname{Hom}_{S}(_{S}(Me_{k}),_{S}S)$. One can show (see [1, Lemma 8.7]) that $\mu_{k}M$ is $Z$-freely generated.

Definition 2.14**.**

Let $P$ be a potential in $\mathcal{F}_{S}(M)$ such that $e_{k}Pe_{k}=0$. Following [5], we define:

$\mu_{k}P:=[P]+\displaystyle\sum_{sa\in_{k}\hat{T},bt\in\tilde{T}_{k}}[btsa]((sa)^{\ast})(^{\ast}(bt))$

where:

[TABLE]

3. Existence of non-degenerate potentials

For every integer $m\geq 2$, let $B(T)_{m}$ be the $F$-basis of $(M^{\otimes m})_{cyc}$ consisting of all elements of the form $x=t_{1}(x)a_{1}(x)t_{2}(x)\cdots t_{m}(x)a_{m}(x)t_{m+1}(x)$ where $t_{i}(x)\in L(\sigma(a_{i}(x)))$, $t_{m+1}(x)\in L(\tau(a_{m}(x)))$, $a_{i}(x)\in T$, and let $B(T)=\displaystyle\bigcup_{m=2}^{\infty}B(T)_{m}$. If $B(T)$ is non-empty, then $B(T)$ is clearly countable. Note than an enumeration of the elements of $B(T)$ gives rise to an algebra isomorphism between $F[Z_{x}]_{x\in B(T)}$, the free commutative $F$-algebra on the set $B(T)$, and the polynomial ring in countably many variables.

Definition 3.1**.**

Let $P$ be a potential in $\mathcal{F}_{S}(M)$. We say that $P$ is $2$-acyclic if no element of $(M^{\otimes 2})_{cyc}$ appears in the expansion of $P$.

The following definition is motivated by [5, Definition 7.2].

Definition 3.2**.**

Let $k_{1},\ldots,k_{l}$ be a finite sequence of elements of $\{1,\ldots,n\}$ such that $k_{p}\neq k_{p+1}$ for $p=1,\ldots,l-1$. We say that an algebra with potential $(\mathcal{F}_{S}(M),P)$ is $(k_{l},\ldots,k_{1})$-non-degenerate if all the iterated mutations $\bar{\mu}_{k_{1}}P$, $\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}P,\ldots,\bar{\mu}_{k_{l}}\cdots\bar{\mu}_{k_{1}}P$ are $2$-acyclic. We say that $(\mathcal{F}_{S}(M),P)$ is non-degenerate if it is $(k_{l},\ldots,k_{1})$-non-degenerate for every sequence of integers as above.

Definition 3.3**.**

If $W\in F[Z_{x}]_{x\in B(T)}$ we define $Z(W):=\{f\in F^{B(T)}:W(f(x))_{x\in B(T)}\neq 0\}$ where $F^{B(T)}$ is the $F$-vector space of all functions $B(T)\rightarrow F$.

We now recall the definition of species realization of a skew-symmetrizable integer matrix, in the sense of [6] (Definition 2.22).

Definition 3.4**.**

Let $B=(b_{ij})\in\mathbb{Z}^{n\times n}$ be a skew-symmetrizable matrix, and let $I=\{1,\ldots,n\}$. A species realization of $B$ is a pair $(\mathbf{S},\mathbf{M})$ such that:

1. (1)

$\mathbf{S}=(F_{i})_{i\in I}$ is a tuple of division rings; 2. (2)

$\mathbf{M}$ is a tuple consisting of an $F_{i}-F_{j}$ bimodule $M_{ij}$ for each pair $(i,j)\in I^{2}$ such that $b_{ij}>0$; 3. (3)

for every pair $(i,j)\in I^{2}$ such that $b_{ij}>0$, there are $F_{j}-F_{i}$-bimodule isomorphisms

$\operatorname{Hom}_{F_{i}}(M_{ij},F_{i})\cong\operatorname{Hom}_{F_{j}}(M_{ij},F_{j})$; 4. (4)

for every pair $(i,j)\in I^{2}$ such that $b_{ij}>0$ we have $\operatorname{dim}_{F_{i}}(M_{ij})=b_{ij}$ and $\operatorname{dim}_{F_{j}}(M_{ij})=-b_{ji}$.

Theorem 3.5**.**

Suppose that the underlying field $F$ is uncountable, then $\mathcal{F}_{S}(M)$ admits a non-degenerate potential.

Proof.

We follow the guidelines of [5, Corollary 7.4]. Let $k_{1},\ldots,k_{l}$ be a sequence as in Definition 3.2. By [1, Proposition 12.5] there exists a potential $P\in\mathcal{F}_{S}(M)$ such that $P$ is $(k_{l},\ldots,k_{1})$-non-degenerate. Now using [1, Proposition 12.3] we can find a nonzero polynomial $W_{(k_{1},\ldots,k_{l})}\in F[Z_{x}]_{x\in B(T)}$ such that every potential belonging to $Z(W_{(k_{1},\ldots,k_{l})})$, is $(k_{l},\ldots,k_{1})$-non-degenerate. This collection $\mathcal{A}$ is a subset of $F[Z_{x}]_{x\in B(T)}$. Moreover, it is a countable family since it is indexed by a subset of all finite sequences of $\mathbb{N}$, and the latter is clearly countable. It remains to show that $\displaystyle\bigcap_{A\in\mathcal{A}}Z(A)\neq\emptyset$. We may realize the polynomial ring $F[Z_{x}]_{x\in B(T)}$ as the polynomial ring $F[t_{1},t_{2},\ldots]$. As in [5, Corollary 7.4], since $F$ is uncountable we can find $\lambda_{1}\in F$ such that $G(\lambda_{1})\neq 0$ for all $G\in\mathcal{A}\cap F[t_{1}]$. Then, we can find $\lambda_{2}\in F$ such that $G(\lambda_{1},\lambda_{2})\neq 0$ for all $G\in\mathcal{A}\cap F[t_{1},t_{2}]$. Repeating this process, we can find a sequence $(\lambda_{i})_{i\geq 1}$ of elements of $F$ such that $G(\lambda_{1},\lambda_{2},\ldots)\neq 0$ for all $G\in\mathcal{A}$. Note that we can guarantee this by the fact that $\mathcal{A}$ can be written as the union of the factors that appear in the following ascending chain:

$\mathcal{A}\cap F[t_{1}]\subseteq\mathcal{A}\cap F[t_{1},t_{2}]\subseteq\mathcal{A}\cap F[t_{1},t_{2},t_{3},\ldots]\subseteq\ldots$

This completes the proof. ∎

Corollary 3.6**.**

Let $B=(b_{ij})\in\mathbb{Z}^{n\times n}$ be a skew-symmetrizable matrix with skew-symmetrizer $D=\operatorname{diag}(d_{1},\ldots,d_{n})$. If $d_{j}$ divides $b_{i,j}$ for every $j$ and every $i$, then the matrix $B$ can be realized by a species admitting a non-degenerate potential.

Proof.

Note that there exists a Galois extension $U/V$ such that $\operatorname{Gal}(U/V)$ is isomorphic to $S_{m}$, where $m:=d_{1}\cdots d_{n}$. Indeed, let $F=\mathbb{C}$ denote the set of all complex numbers (or any uncountable field), $U=F(x_{1},\ldots,x_{m})$, the field of rational functions in the indeterminates $x_{1},\ldots,x_{m}$; and $V=F(s_{1},\ldots,s_{m})$, the subfield generated by the elementary symmetric polynomials $s_{1},\ldots,s_{m}$. Then $\operatorname{Gal}(U/V)\cong S_{m}$ and by Cayley’s Theorem, $G:=\displaystyle\bigoplus_{i=1}^{n}\mathbb{Z}_{d_{i}}$ may be realized a subgroup of $S_{m}$. Now, let $L$ be the fixed field of $G$ in $U$, then by the Fundamental Theorem of Galois Theory we have that $U/L$ is Galois and $\operatorname{Gal}(U/L)\cong G$. Applying [1, Proposition 11.2] yields that the matrix $B$ can be realized by a species whose underlying field is precisely $L$. Note that $L$ is uncountable since it contains the uncountable field $V$. Applying Theorem 3.5 yields that such species admits a non-degenerate potential. This completes the proof. ∎

4. Rigidity and nondegeneracy

The following definition is taken from [1, Definition 43] and it is motivated by [5, Definition 6.7].

Definition 4.1**.**

Let $(\mathcal{F}_{S}(M),P)$ be an algebra with potential. The deformation space $\operatorname{Def}(M,P)$ is the quotient $\mathcal{F}_{S}(M)^{\geq 1}/(R(P)+[\mathcal{F}_{S}(M),\mathcal{F}_{S}(M)])$ where $[\mathcal{F}_{S}(M),\mathcal{F}_{S}(M)]$ denotes the commutator of $\mathcal{F}_{S}(M)$.

The following definition is also based on [5, Definition 6.10].

Definition 4.2**.**

An algebra with potential $(\mathcal{F}_{S}(M),P)$ is rigid if the deformation space $\operatorname{Def}(M,P)$ is zero.

Lemma 4.3**.**

Let $(\mathcal{F}_{S}(M),P)$ and $(\mathcal{F}_{S}(M^{\prime}),Q)$ be right-equivalent algebras with potentials. Then $(\mathcal{F}_{S}(M),P)$ is rigid if and only if $(\mathcal{F}_{S}(M^{\prime}),Q)$ is rigid.

Proof.

Let $\varphi:\mathcal{F}_{S}(M)\rightarrow\mathcal{F}_{S}(M^{\prime})$ be an algebra with $\varphi|_{S}=id_{S}$. By [1, Theorem 5.3] we have $\varphi(R(P))=R(\varphi(P))$, and by continuity of $\varphi$, it follows that $\varphi([\mathcal{F}_{S}(M),\mathcal{F}_{S}(M)])\subseteq[\mathcal{F}_{S}(M^{\prime}),\mathcal{F}_{S}(M^{\prime})]$. This implies that $\varphi$ induces a surjection: $\psi:\mathcal{F}_{S}(M)^{\geq 1}/(R(P)+[\mathcal{F}_{S}(M),\mathcal{F}_{S}(M)])\twoheadrightarrow\mathcal{F}_{S}(M^{\prime})^{\geq 1}/(R(Q)+[\mathcal{F}_{S}(M^{\prime}),\mathcal{F}_{S}(M^{\prime})])$, which in fact is an isomorphism because $\varphi$ is. ∎

Lemma 4.4**.**

An algebra with potential $(\mathcal{F}_{S}(M),P)$ is rigid if and only if the mutated algebra $(\mathcal{F}_{S}(\bar{\mu}_{k}M),\bar{\mu}_{k}P)$ is rigid.

Proof.

This is the statement of [1, Proposition 10.1]. ∎

Lemma 4.5**.**

Let $P$ be a reduced potential in $\mathcal{F}_{S}(M)$ and let $k_{1},k_{2}$ be distinct integers in $[1,n]$ such that $\mathcal{F}_{S}(\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}M)_{cyc}=0$. Then $P$ is a non-degenerate potential in $\mathcal{F}_{S}(M)$.

Proof.

Due to the fact that there are no potentials in $\mathcal{F}_{S}(\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}M)$ it follows that $\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}P=0$. Since mutation at $k_{2}$ preserves rigidity and mutation is an involution (see [1, Theorem 8.12]), Lemma 4.3 yields that $(\mathcal{F}_{S}(\bar{\mu}_{k_{1}}M),\bar{\mu}_{k_{1}}P)$ is also rigid. Applying the same argument with $k_{1}$ one gets that $(\mathcal{F}_{S}(M),P)$ is a rigid and reduced algebra; therefore $P$ is non-degenerate. ∎

An induction gives the following

Proposition 4.6**.**

Let $k_{1},\ldots,k_{l}$ be a finite sequence of elements of $\{1,\ldots,n\}$ such that $k_{i}\neq k_{i+1}$ for every $i=1,\ldots,l-1$. Let $P$ be a reduced potential in $\mathcal{F}_{S}(M)$ such that $\mathcal{F}_{S}(\bar{\mu}_{k_{l}}\cdots\bar{\mu}_{k_{2}}\bar{\mu}_{k_{1}}M)_{cyc}=0$. Then $P$ is a non-degenerate potential in $\mathcal{F}_{S}(M)$.

5. A species realization for a certain class of $4\times 4$ skew symmetrizable matrices

In this section we provide an example of a class of skew-symmetrizable $4\times 4$ integer matrices, which are not globally unfoldable nor strongly primitive (in the sense of [7, Definition 14.1]), and that have a species realization admitting a non-degenerate potential.

In what follows, let

[TABLE]

where $a,b$ are positive integers such that $a<b$, $a$ does not divide $b$ and $(a,b)\neq 1$.

Note that there are infinitely many such pairs $(a,b)$. For example, let $p$ and $q$ be primes such that $p<q$. For any $n\geq 2$, define $a=p^{n}$ and $b=p^{n-1}q$. Then $a<b$, $a$ does not divide $b$ and $(a,b)=p^{n-1}\neq 1$. Note that $B$ is skew-symmetrizable since it admits $D=\operatorname{diag}(1,a,1,b)$ as a skew-symmetrizer.

Remark*.*

By [7, Example 14.4] we know that the class of all matrices given by (5.1) does not admit a global unfolding. Moreover, since we are not assuming that $a$ and $b$ are coprime, then such matrices are not strongly primitive; hence they are not covered by [7].

We have the following

Proposition 5.1**.**

The class of all matrices given by (5.1) are not globally unfoldable nor strongly primitive, yet they can be realized by a species admitting a non-degenerate potential.

Proof.

The fact that they are not globally unfoldable follows at once by [7, Example 14.4], and by construction, they are not strongly primitive. Let $(d_{1},d_{2},d_{3},d_{4})=(1,a,1,b)$ be the skew-symmetrizer tuple, then $d_{j}$ divides $b_{ij}$ for every $j$ and every $i$. Applying Corollary 3.6 yields that $B$ can be realized by a species admitting a non-degenerate potential. This completes the proof. ∎

Acknowledgements

I thank Raymundo Bautista and Daniel Labardini-Fragoso for helpful conversations. I also thank the referee for his valuable suggestions and remarks that improved the quality of this paper.