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.

notmaximal in general, because the quotient is $k^{Q_0}$ which is not simple if $|Q_0|>1$. $\endgroup$ – Johannes Hahn Aug 1 '17 at 17:47