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 $\mathrm{Rep}Q\to\mathrm{Rep}R$. (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.

2.Representations of a quiver do not come with a tensor product. – Bruce Westbury Mar 12 '10 at 13:02