Describing the derived category: from Ikeda-Qiu to Happel

In 2018, in their study of deformations of spaces of stability conditions, Ikeda-Qiu introduced X-Calabi-Yau completions (=bigraded CY-completions) and proved that the cluster category associated with the X-CY-completion of an acyclic quiver is equivalent to the bounded derived category of its category of representations. We will report on joint work with Fan Li and Yu Qiu where we generalize this theorem to suitable differential graded algebras. We will then compare this result with Hanihara’s recent generalization to dg algebras of Happel’s description of the derived category as the singularity category of the repetitive algebra.