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Your argument looks correct to me. Note that the corresponding question for the derived category has been answered positively, and a parameterisation of total stability conditions given by QiuYu and Z …

You may be interested in the classification of quivers of finite mutation-type: http://www.emis.ams.org/journals/EJC/Volume_15/PDF/v15i1r139.pdf This extends the classification of mutation classes of …

This statement is false. Take $R=k$ a field, and $\mathcal{X}$ any finite-dimensional representation of $\mathcal{Q}$ over $k$. Then for $i>0$ and $v\in\mathcal{Q}_0$, there is some $n\geq0$ such that …

Yes. Using left modules, the indecomposable projective $eAe$-modules are $eP$ for $P$ an indecomposable projective $A$-module such that $e\operatorname{top}(P)=\operatorname{top}(P)$, and in this case …

This is true when $R$ is a field (or semisimple, so a product of fields), for the following reason. Since $v_1$ is a source, there is a monomorphism $\mathcal{X}'\to\mathcal{X}$, which fits into a sh …

For $n>0$, let $A_n=KQ_n/I_n$ as follows: $Q_n$ is the quiver with vertex set $\{1,2\}$ and arrows $\alpha_i\colon 1\to 2$ and $\beta_i\colon 2\to 1$ for $1\leq i\leq n$, and $I_n$ is generated by $\b …

It is not particularly productive to think about finite-dimensional Calabi–Yau algebras, at least if you are grading everything in degree $0$ (and possibly even if you have a more interesting grading— …

This is an application of the Yoneda lemma. There is an exact sequence $$0\to\Omega^2_\Lambda(N)\to P_1\to P_0\to N\to0$$ with $P_i$ projective. By definition of $\Lambda$, it follows that $P_i=\ope …

Up to possibly learning some new technology, you can get a good answer to this question using the cluster category of type $D_n$ (or indeed of any Dynkin type). Given a Dynkin quiver $Q$, its associat …

Keller–Reiten's main theorem in Acyclic Calabi–Yau categories implies that if $\mathcal{C}$ is a $2$-Calabi–Yau (algebraic) triangulated category admitting a cluster-tilting object $T$ such that the q …

Let $R$ be a ring (associative with unit, but not necessarily commutative, and definitely not necessarily Noetherian.) Then the category $\operatorname{GP}(R)$ consists of those $R$-modules having a c …

This is a case in which various heuristics about the 'niceness' of ADE quivers can interfere with each other. One result in this area that might conform to your expectations is that the algebra $\math …

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-fin …

For $H$ the path algebra of a 'star-shaped' quiver having three legs with lengths $r$, $s$, $t$, an answer seems to be implicit in Lamberti's combinatorial model for the cluster category: https://arx …

This is only a partial answer: For Q1, you might be interested in the examples in the papers arXiv:math/0609138 and arXiv:1309.7301, which both give several examples of AR quivers of categories of Go …