2 moar questions

Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).

1. Does there exist an isomorphism of additive categories $\mathrm{Rep}Q\to\mathrm{Rep}R$ ?

At the moment, I am not even 100% sure about the weaker statement that there exists an equivalence of additive categories $\mathrm{Rep}Q\to\mathrm{Rep}R$. However, if I am not mistaken, it follows from the isomorphy of the representation rings of $Q$ and $R$, which follows from Gabriel's theorem.(My intuition for this is pretty much zero.)

In before reflection functors - they are not isomorphisms. (But yes, it's a nice exercise to see that we can obtain $R$ from $Q$ by a sequence of admissible reflections, where an "admissible reflection" means choosing some sink in $Q$ and switching all the arrows to $Q$. Unfortunately, the corresponding reflection functors may turn some nonzero representations of $Q$ to zero.)

2. Are the representation rings of $Q$ and $R$ isomorphic? The isomorphy of their additive groups follows from Gabriel's theorem, but it is not clear to me what this does to tensor products.

1+2. Does there exist an isomorphism of tensor categories $\mathrm{Rep}Q\to\mathrm{Rep}R$ ?

3. If some of these answers are No, what if we restrict ourselves to Dynkin quivers?

I am new to quivers, so I'm sorry if this has been already talked over a dozen of times here.

1

# Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?

Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same). Does there exist an isomorphism of categories $\mathrm{Rep}Q\to\mathrm{Rep}R$ ?

At the moment, I am not even 100% sure about the weaker statement that there exists an equivalence of categories $\mathrm{Rep}Q\to\mathrm{Rep}R$. However, if I am not mistaken, it follows from the isomorphy of the representation rings of $Q$ and $R$, which follows from Gabriel's theorem.

In before reflection functors - they are not isomorphisms. (But yes, it's a nice exercise to see that we can obtain $R$ from $Q$ by a sequence of admissible reflections, where an "admissible reflection" means choosing some sink in $Q$ and switching all the arrows to $Q$. Unfortunately, the corresponding reflection functors may turn some nonzero representations of $Q$ to zero.)

I am new to quivers, so I'm sorry if this has been already talked over a dozen of times here.