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This question is perhaps a little stupid as I have barely any evidence for it, but another thread just reminded me about it so maybe someone has an idea or counterexample for this.

Let $A$ be a finite dimensional algebra that has the property that an algebra $B$ is derived invariant to $A$ iff it is Morita equivalent to $A$. Lets call such an algebra $A$ derived invariant. Assume $A$ has a cluster-tilting object $M$.

Question:

Is the algebra $End_A(M)$ quasi-hereditary?

Examples of such situations are $A=K[x]/(x^n)$ or $A$ being a preprojective algebra of Dynkin type and there the endomorphism algebras seem to be indeed quasi-hereditary. Are there other example of derived invariant algebras having a cluster-tilting object?

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  • $\begingroup$ Do you have any nice conditions which imply derived invariance? (How is it proved in your examples, for instance?) By 'cluster-tilting', do you mean '$d$-cluster-tilting for some $d$'? (I would guess yes because of $K[x]/(x^n)$, but better to be sure.) $\endgroup$
    – Matthew Pressland
    Commented Oct 2, 2018 at 16:14

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