A question about the quivers with potentials

Question

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.

Answer 1

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 $F_j\otimes_{F_j\cap F_k}F_k$.

Answer 2

It seems an answer to your question may be given by the theory of species with potential; a species is a kind of generalisation of the path algebra of a quiver, designed so that the representation-finite species correspond to the Dynkin diagrams (the representation-finite path algebras corresponding only to the simply-laced Dynkin diagrams).

Since I do not know the details of this theory, I cannot give a very elaborate answer, but you might want to look at Geuenich–Labardini-Fragoso's work (e.g. https://arxiv.org/abs/1507.04304) if you have not already seen it. The authors discuss a mutation theory for species with potential, and describe examples coming from orbifolds. There are lots of references to earlier work as well, which could be interesting to explore. The earliest attempt to generalise Derksen–Weyman–Zelevinsky to skew-symmetrizable cases seems to be by Demonet (https://arxiv.org/abs/1003.5078) but I may have overlooked something.