# Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?

Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?

(I moved this question from another thread Derived equivalences of Dyck paths , where the question was originally aksed when I thought the answer was elementary/well known but the question now doesnt fit so good anymore and I started a new thread. For Nakayama algebras the answer should be that acyclic ones can not be derived equivalent to non-acyclic ones as noted by Jeremy Rickard in that thread)

• If by cycle you mean oriented cycle, the answer is yes. Take for example $\Lambda(1,2,1)$ and $\Delta(1,2,1)$ from 10.2478/BF02475948. – Jabby Sep 30 at 9:32
• @Jabby Yes, that counts (although it would be also interested to see whether one could even have an acyclic quiver as an undirected graph derived equivalent to a quiver algebra with a oriented cycle). If you want you can make this an answer. – Mare Sep 30 at 11:58