8 problems

Let $C$ be a cluster, let $(\mathcal{Q}_C,I_C)$ be its associated quiver with relations, and let $T$ be the corresponding tilting object in the cluster category $\mathcal{C}$. Writ…

Let $C$ be a cluster in a cluster algebra of simply-laced Dynkin type, and let the algebra associated combinatorially to $C$ be the algebra constructed by Caldero, Chapoton and Sch…

Igusa–Todorov's conjecture. There exists a tilted algebra $B$ such that $\Lambda\cong B\ltimes \operatorname{Ext}^2_B(DB,B)$ and the number of isomorphism classes of indecomposable…

Igusa–Todorov's conjecture. (a) The ordered $B$-modules form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maxima…

Parsons's strongness conjecture. Every companion basis of every cluster-tilted algebra of simply-laced Dynkin type is strong.

Derived-class characterization conjecture. The algebras satisfying the following definition are precisely the algebras that are derived equivalent to connected $m$-cluster tilted a…

Decomposition conjecture. Any $m$-cluster tilted algebra is a direct product of $m_i$-relation extensions of iterated tilted algebras.

Let $\Gamma$ be a cluster-tilted algebra of simply-laced Dynkin type, and let $\Psi$ be a companion basis giving rise to a positive quasi-Cartan companion of the exchange matrix as…