Strongly primitive species with potentials: aims and limitations

TL;DR

This paper discusses the aims and limitations of strongly primitive species with potentials within the context of cluster algebras, based on joint work with Zelevinsky.

Contribution

It clarifies the goals and constraints of strongly primitive species with potentials in cluster algebra theory, extending previous joint research.

Findings

Identifies key aims of strongly primitive species with potentials

Highlights limitations in current approaches

Provides insights for future research directions

Abstract

This is an extended abstract of my talk at the Oberwolfach Workshop ''Cluster Algebras and Related Topics'' (December 8 - 14, 2013). It is based on a joint work with A. Zelevinsky (arXiv:1306.3495).

Equations

$$\frac{\gcd (d_i,d_j)b_{ij}}{d_j}$$

$$A_{ij}=\underset{a:j\to i}{⨁}{F}_i{\otimes }_{F_i\cap F_j}{F}_j.$$

$$R=\underset{i\in Q_0}{⨁}{F}_i\text{and}A=\underset{i,j\in Q_0}{⨁}{A}_{ij}.$$

$$R⟨⟨A⟩⟩=\prod _{\ell =0}^{\infty }{A}^{\ell }$$

$${\tilde{\mu }}_k(S)=[S]+\sum _{\stackrel{a}{\to }k\stackrel{b}{\to }}\sum _{\omega \in {\mathcal{B}}_k}{\omega }^{-1}{b}^{\ast }[b\omega a]a^{\ast }.$$

Full text

This talk is based on [4]. We define mutations of strongly primitive species with potentials, and show that if $B$ is any $n\times n$ integer matrix admitting a skew-symmetrizer with pairwise coprime diagonal entries, then for any given sequence of mutations which one wants to perform there exists a species realization $A$ of $B$ which admits a potential $S$ such that the mutations of $(A,S)$ along the given sequence are compatible with matrix mutation.

An $n\times n$ integer matrix $B$ is said to be skew-symmetrizable if there exist positive integers $d_{1},\ldots,d_{n}$ such that $DB$ is skew-symmetric, where $D=\operatorname{diag}(d_{1},\ldots,d_{n})$. Such $D$ is said to be a skew-symmetrizer of $B$.

A weighted quiver is a pair $(Q,\mathbf{d})$, where $Q=(Q_{0},Q_{1},t,h)$ is a loop-free quiver and $\mathbf{d}=(d_{i})_{i\in Q_{0}}$ is a tuple of positive integers attached to the vertices of $Q$. We call $\mathbf{d}$ the weight tuple of $(Q,\mathbf{d})$.

If $B$ is a skew-symmetrizable matrix with skew-symmetrizer $D$, we define a weighted quiver $(Q_{B},\mathbf{d})$ as follows: The vertex set of $Q_{B}$ is $\{1,\ldots,n\}$, and $Q_{B}$ has exactly

[TABLE]

arrows from $j$ to $i$ whenever $b_{ij}\geq 0$. The tuple $\mathbf{d}$ is defined to be the tuple of diagonal entries of $D$.

If $D$ (and hence $\mathbf{d}$) is kept fixed, the assignment $B\mapsto(Q_{B},\mathbf{d})$ is easily seen to be a bijection between the set of $n\times n$ integer matrices that are skew-symmetrized by $D$ and the set of 2-acyclic weighted quivers with weight tuple $\mathbf{d}$. This means that there is a weighted quiver counterpart of the notion of matrix mutation:

• (Step 1)

For each pair of arrows $a:j\to k$ and $b:k\to i$ of $Q$, introduce $\frac{\gcd(d_{i},d_{j})d_{k}}{\gcd(d_{i},d_{k})\gcd(d_{k},d_{j})}$ “composite” arrows from $j$ to $i$;

• (Step 2)

replace each $c\in Q_{1}$ incident to $k$ with an arrow $c^{*}$ going in the opposite direction;

• (Step 3)

choose a maximal collection of disjoint 2-cycles and remove them.

Let $d$ be the least common multiple of the tuple $\mathbf{d}=(d_{i})_{i\in Q_{0}}$, $F$ be a finite field, and $E$ be the unique degree-$d$ field extension of $F$. For $i\in Q_{0}$, set $F_{i}$ to be the unique degree-$d_{i}$ field subextension of $E/F$, and for every pair of vertices $i,j\in Q_{0}$, set

[TABLE]

Define

[TABLE]

Then $R$ is a semisimple ring and $A$ is an $R$-$R$-bimodule. We say that $A$ is the species of $(Q,\mathbf{d})$ over $E/F$.

The complete tensor algebra of $A$ over $R$ is called the complete path algebra of $A$ and denoted $R\langle\langle A\rangle\rangle$. Thus we have

[TABLE]

as an $R$-$R$-bimodule, where $A^{\ell}$ denotes the $\ell$-fold tensor product $A\otimes_{R}\ldots\otimes_{R}A$. If an element $S\in\prod_{\ell=1}^{\infty}{A^{\ell}}$ satisfies $S=\sum_{i\in Q_{0}}e_{i}Se_{i}$, where $e_{i}$ is the idempotent sitting in the $i^{\operatorname{th}}$ component of $R$, we say that $S$ is a potential on $A$ and that $(A,S)$ is a species with potential.

From now on we assume that $(Q,\mathbf{d})$ is strongly primitive, that is, we suppose that $\operatorname{gcd}(d_{i},d_{j})=1$ for all $i\neq j$. We shall say that $A$, the species of $(Q,\mathbf{d})$ over $E/F$, is strongly primitive as well, and that $(A,S)$ is a strongly primitive species with potential whenever $S$ is a potential on $A$. We also assume that the characteristic of $F$ is congruent to $1$ modulo $d$. This implies that there exists an element $v\in E$ such that the set $\mathcal{B}=\{1,v,v^{2},\ldots,v^{d-1}\}$ is an eigenbasis of $E/F$, that is, an $F$-basis of $E$ consisting of eigenvectors of all elements of the Galois group $\operatorname{Gal}(E/F)$. This eigenbasis is quite useful to obtain an explicit description of the elements of $R\langle\langle A\rangle\rangle$; indeed, setting $\mathcal{B}_{i}=\mathcal{B}\cap F_{i}$ for $i\in Q_{0}$ we have:

Given a 2-acyclic strongly primitive weighted quiver $(Q,\mathbf{d})$ and a vertex $k\in Q_{0}$, for each pair of arrows $a:j\to k$ and $b:k\to i$ of $Q$, the composite arrows introduced in the first step of the weighted-quiver mutation with respect to $k$ are denoted with the symbols $[b\omega a]$, where $\omega$ runs in the set $\mathcal{B}_{k}$.

We can state now the mutation rule for strongly primitive species with potentials.

[TABLE]

The species with potential which is the reduced part of $(\widetilde{\mu}_{k}(A),\widetilde{\mu}_{k}(S))$ will be called the mutation of $(A,S)$ with respect to $k$ and denoted $\mu_{k}(A,S)$.

We refer the reader to [4] for the definition of reduced parts as well as for a proof of their existence. Note that the underlying species of $\mu_{k}(A,S)$ is again strongly primitive, although its underlying weighted quiver may have oriented 2-cycles.

The following is the main result of the talk.

Besides Theorem 6, mutations of strongly primitive species with potentials share many other properties with the mutations of quivers with potentials of Derksen-Weyman-Zelevinsky. For example, they are well-defined up to right-equivalence and involutive up to right-equivalence, and they preserve Jacobi-finiteness. Following the spirit of [2], the notion of mutation is further lifted in [4] to the representation-theoretic level.

The class of skew-symmetrizable matrices whose associated weighted quivers are strongly primitive includes several examples of matrices that do not admit global unfoldings whatsoever (a global unfolding is an unfolding which is compatible with all possible sequences of mutations). So, the species framework in [4] provides a representation-theoretic approach to (the skew-symmetrizable matrices of) several cluster algebras where approaches via unfoldings do not work.

Now, how about matrices that admit global unfoldings?, and how about species with potentials for weighted quivers that are not strongly primitive? The answer to the first question is provided by the work [1] of L. Demonet, who developed an approach to mutations via group species with potentials. The framework of group species differs from the one we have presented above in that a group species attaches group algebras to the vertices of $Q$ rather than fields; the bimodules attached to the arrows are hence also different in nature from the bimodules $A_{ij}$ above. Group species with potentials provide a representation-theoretic approach that can be successfully applied to those matrices that admit global unfoldings through group actions. Unfortunately, for matrices that do not admit global unfoldings one needs a framework different from the group species framework.

Regarding the second question: if $\operatorname{gcd}(d_{i},d_{j})$ is not assumed to be equal to 1 for all $i\neq j$, one can easily construct an example of a weighted quiver $(Q,\mathbf{d})$ that has a 2-cycle $ab$ which is cyclically equivalent to [math] in $R\langle\langle A\rangle\rangle$, and this implies that none of the arrows $a,b$ belongs to the Jacobian ideal of any potential on the species $A$ of $(Q,\mathbf{d})$ over $E/F$. This means that none of $a,b$ can be deleted from $Q$. Ultimately, this yields an example of a 2-acyclic species (not strongly primitive) such that, no matter which potential on it we take, when we try to perform the three steps of weighted-quiver mutation at the level of species with potentials, the species framework presented above fails to delete 2-cycles from the underlying weighted quiver.