TITLE: A question about the quivers with potentials
QUESTION [8 upvotes]: Let $Q=(Q_0,Q_1,h,t)$ be a quiver consisted of a pair of finite sets $Q_0$(vectors),and $Q_1$ (arrows) supplied with two maps $h : Q_1 → Q_0$ (head) and $t : Q_1 → Q_0$ (tail ). This definition allows the underlying graph to have multiple edges and (multiple) loops.
Then the path algebra of $Q$ is the graded tensor algebra
$R\langle Q \rangle=\bigoplus_{d=0}^{\infty}A^d$, where $R=k^{Q_0},A=k^{Q_1}$, $k$ is a field of characteristic zero. Any path algebra has a maximal ideal generated by elements of positive degree, we define the completed path algebra $R\langle\langle Q \rangle\rangle=\prod_{d=0}^{\infty}A^d$ as the completion of $R\langle Q \rangle$ with respect to powers of the maximal ideal.
Let $R\langle\langle Q \rangle\rangle_{cyc}$ be the linear subspace of the completed path algebra generated by cyclic paths, i.e. products of the form $\prod_{i=1}^{n}a_i$ such that $t(a_1)=s(a_n)$. A potential on $Q$ is an element of $R\langle\langle Q \rangle\rangle_{cyc}$ considered up to cyclic shift: $a_1a_2...a_n\leftrightarrow a_na_1...a_{n-1}$. A quiver with potential is a pair $(Q,W)$ where $Q$ is a quiver and $W$ is an element of $R\langle\langle Q \rangle\rangle_{cyc}$ considered up to cyclic shifts.
You can see https://arxiv.org/pdf/0704.0649.pdf and https://arxiv.org/pdf/1701.00672.pdf for more details.
I want to know whether the definition of potentials for quivers with weighted arrows or weighted vertices coincide with the above definition? For example,
Let $B = (b_{ij})$ is a sign-skew-symmetric real square matrix. $B$ is 2-finite if it has integer entries, and any matrix $B'=\mu_{k}(B)=(b_{ij}^{'})$ mutation equivalent to $B$ is sign-skew-symmetric and satisfies $\mid b_{ij}^{'}b_{ji}^{'}\mid \leq 3$ for all $i$ and $j$. In the paper https://arxiv.org/pdf/math/0208229.pdf. On page 26, Definition 7.3. S. Fomin and A. Zelevinsky define a 2-finite diagrams. The diagram of a sign-skew-symmetric matrix $B = (b_{ij}),i,j\in I$ is the weighted directed graph $\Gamma (B)$ with the vertex set $I$ such that there is a directed 
edge from $i$ to $j$ if and only if $b_{ij} > 0$, and this edge is assigned the weight $|b_{ij}b_{ji}|$. On page 28, Proposition 8.1, gives a mutation rule for diagrams.  
Whether the definition of potentials for these diagrams with weighted edges have to be done?
Any help will be appreciated.

REPLY [6 votes]: Let $B$ be an $n\times n$ skew-symmetrizable matrix with integer coefficients. The first question seems to be "how to 'realize' $B$ through a 'path algebra'?". There are at least three different approaches to this question (Demonet,Dlab-Ringel,Geiss-Leclerc-Schröer). The three approaches seem to fit in the following general idea:
A 'modulation' or 'species realization' of $B$ is a pair $(\mathbf{F},\mathbf{A})$ consisting of an $n$-tuple $\mathbf{F}=(F_k)_{1\leq k\leq n}$ of rings and a tuple $\mathbf{A}=(A_{jk})_{b_{jk}\geq 0}$ of bimodules (A_{jk} being an $F_j$-$F_j$-bimodule) satisfying certain properties (one of which is that each $A_{jk}$ is free when separately considered as left and as right module, with its right and left dimensions being prescribed by the entries of the matrix; I can list the conditions explicitly if you want, but at the moment I'd rather keep it short).
With the pair $(\mathbf{F},\mathbf{A})$ at hand, you can define a ring $R=\times_{1\leq k\leq n}F_k$ and an $R$-$R$-bimodule $A=\oplus_{b_{jk}\geq 0}A_{jk}$. The tensor algebra $T_R(A)$ can be thought of as a 'path algebra' associated to $B$ (when $B$ is skew-symmetric and $Q$ is the corresponding quiver, $T_R(A)$ is the usual path algebra of $Q$ if you take $F_k$ to be $\mathbb{C}$ for all $k\in\{1,\ldots,n\}$ and $A_{jk}$ to be  $\mathbb{C}$-vector space with basis the set of all arrows that go from $k$ to $j$). Then you can define a 'potential' to be an element of $T_R(A)/[T_R(A),T_R(A)]$.
Depending on which $\mathbf{F}$ you take, a tuple $\mathbf{A}$ of bimodules giving a species realization of $B$ may or may not exist. Wether one can define cyclic derivatives (and hence Jacobian algebras), perform mutations in this setting, or show that non-degenerate potentials exist, is a different matter, and things can become rather technical.
To finish, the difference between the approaches of Demonet, Dlab-Ringel and Geiss-Leclerc-Schröer lies mainly in which rings $F_k$ they take:

Demonet takes each $F_k$ to be a group algebra of certain cyclic group;
Dlab-Ringel take $F_k$ to be a division ring;
Geiss-Leclerc-Schröer take $F_k$ to be a truncated polynomial ring $\mathbb{C}[X]/X^{d_k}$.


Edit (this is the first time I post in MO, I hope to be doing it the right way...):
Just as a follow-up, it is probably a good idea to say a few words on how you can actually construct an explicit 'modulation' or 'species realization' for a given skew-symmetrizable matrix $B\in\mathbb{Z}^{n\times n}$.
Fix a matrix $D=\operatorname{diag}(d_1,\ldots,d_n)$, with $d_1,\ldots,d_n\in\mathbb{Z}_{>0}$, such that $DB$ is skew-symmetric. The matrix $C=(c_{ij})_{1\leq i,j\leq n}$ defined by
$$
c_{ij}=\frac{b_{ij}\operatorname{gcd}(d_i,d_j)}{d_j}
$$
has integer entries and is skew-symmetric. As such, it has an associated quiver $Q_C$ with $c_{ij}$ arrows from $j$ to $i$ whenever $c_{ij}\geq 0$.
Fix a degree-$d$ cyclic Galois extension $E/F$, where $d=\operatorname{lcm}(d_1,\ldots,d_n)$. For each $k\in\{1,\ldots,n\}$, set $F_k$ to be the unique subfield of $E$ such that $[F_k:F]=d_k$, and for every pair $(j,k)$ such that $b_{jk}\geq 0$, set $A_{jk}:=\oplus_{\ell=1}^{c_{jk}}F_j\otimes_{F_j\cap F_k}F_k$. The pair $((F_k)_{1\leq k\leq n},(A_{jk})_{b_{jk}\geq 0})$ then constitutes a species realization of $B$.
If you are interested in mutating the above species realization (with some potential), then you are forced to consider some 'twisted' version of the bimodules $F_j\otimes_{F_j\cap F_k}F_k$.