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Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top boundary of its Auslander-Reiten quiver, so we can identify Nakayama algebras with a linear quiver with Dyck paths). Derived categories of Nakayama algebras appeared in the literature (see for example https://www.sciencedirect.com/science/article/pii/S0001870813000182 ) but it seems to be wide open to classify Dyck paths with respect to derived equivalences (or?).

Call a Nakayama algebra (or the corresponding Dyck path) bouncing in case the Kupisch series is of the form $[a_1+1,a_1,...,3,2,a_2+1,a_2,...,2,...,a_r+1,a_r,...,3,2,1]$.

The truth of the conjecture in What are the periodic Dyck paths? would imply the following: In case a Dyck path $D_1$ is derived equivalent to a bouncing Dyck path $D_2$, then also $D_1$ is bouncing. Thus, it would give that being bouncing is a derived invariant.

Question 1:

Do we know whether it is true that being bouncing is a derived invariant? That is: In case a Nakayama algebra $D_1$ is derived equivalent to a bouncing Nakayama algebra $D_2$, then also $D_1$ is bouncing?

Question 2:

Is it known which bouncing Nakayama algebras with a fixed number of simple modules are derived equivalent to eachother? How many equivalence classes (up to derived equivalence) exist of bouncing Nakayama algebras?

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    $\begingroup$ In answer to the second part of the parenthesized Question 3, I think it's the case that the first Hochschild cohomology $HH^1(A,A)$ (which is an invariant of the derived category) is zero if $A$ is a Nakayama algebra with acyclic quiver, but non-zero if $A$ is a Nakayama algebra with non-acyclic quiver. $\endgroup$
    – Jeremy Rickard
    Commented Sep 25, 2018 at 10:05
  • $\begingroup$ @JeremyRickard That is a good idea. That the Hochschild cohomology is zero for the linear quiver case is by a result of Happel. I cant remembers seeing the first Hochschild cohomology calculated for Nakayama algebras with a cyclic quiver but I guess you are right. $\endgroup$
    – Mare
    Commented Sep 25, 2018 at 12:53
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    $\begingroup$ @JeremyRickard Here what my computer suggests: Let $A$ be a Nakayama algebra with a cyclic quiver and Kupisch series $[c_1,...,c_n]$ with minimal entry $c=s(n+1)+k \geq 2$ for $k \leq n$ and $s \geq 0$. Then the vector space dimension of the first Hochschild cohomology is equal to $s+1$. So the first Hochschild cohomology counts in a way how often you walk around the circle completely +1 it seems. $\endgroup$
    – Mare
    Commented Sep 25, 2018 at 12:55
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    $\begingroup$ I think Happel showed, by producing an explicit non-inner derivation, that the path algebra of a non-acyclic quiver by an admissible and homogeneous (in terms of lengths of paths) ideal has non-zero $HH^1$. $\endgroup$
    – Jeremy Rickard
    Commented Sep 25, 2018 at 13:01
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    $\begingroup$ I think, but am not sure, that I once knew of an example of derived equivalent algebras with acyclic/non-acyclic quivers. I’ll try to remember what it is or where I saw it. $\endgroup$
    – Jeremy Rickard
    Commented Sep 25, 2018 at 13:04

1 Answer 1

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Question 1 has a positive answer by the answer of Gjergji Zaimi in this thread: What are the periodic Dyck paths? .

Here a positive answer to question 1 and 2 using a complicated classification result that uses several other deep results from the literature (so more elementary proofs of Question 2 are welcome):

Question 1 and 2 have a positive answer by the classification of iterated tilted algebras of Dynkin type $\mathcal{A}$. See the main theorem of https://www.tandfonline.com/doi/abs/10.1080/00927878108822697 , which implies here that the bouncing Dyck paths correspond to Nakayama algebras with a linear quiver having only relations of length two.

(edit: I think I found an elementary answer to Question 2 using tilting modules. I might post it here soon)

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