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.



A question about the quivers with potentials
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:


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




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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 $F\_j\otimes\_{F\_j\cap F\_k}F\_k$.


