This is a very simple question, but for some reason I'm having trouble with it. Here is the set up:
Let $k$ be an alg. closed field, let $Q$ be a quiver, and let $A = kQ$ be the path algebra of $Q$ over $k$. Then we have two functors, the $k$-dual (also called 'standard duality'), which is the contravariant Hom functor $D = Hom_k(-,k)$, and the $A$-dual, which is the contravariant functor $(-)^* = Hom_A(-,A)$.
My question is: how are these two functors related? How are they the same, and how are they different? I'm specifically thinking of quiver representations.
But, I'd prefer an answer in general: how does a/the "vector space duality" differ from a/the "algebra duality"? (Is that not the right way to pose the question?) Perhaps, how does the k-linear duality differ from the A-linear duality, conceptually?
[Edit, re: my comment, it was pointed out to me I originally messed up this example, so I am rewriting it to get the arrows correct] For concreteness: Let $Q = 1 \rightarrow 2 \rightarrow 3$ be a quiver, let $A=kQ$ be its path algebra. Take a representation, say, $M = k \rightarrow k \rightarrow 0$. Then $DM = Hom(k,k)\leftarrow Hom(k,k)\leftarrow Hom(0,k)$ and this is isomorphic to, say, $N = k \leftarrow k \leftarrow 0$. Then $N^*= Hom_A(A,k)\rightarrow Hom_A(A,k)\rightarrow Hom_A(A,0)$. Does this "just" end up being isomorphic to $M = k \rightarrow k \rightarrow 0$ again?
That being said, was my example too simple as to be trivial and not shed light on the situation? Or was I too fast and loose with my calculations, and perhaps said something very silly?
For the very generous: Can someone hold my hand through a more enlightening calculation, or otherwise point out where my lack of insight might be coming from?
Otherwise, just the answers to the highlighted questions would be appreciated!
For the record, my references for these questions are:
Auslander, Reiten, Smalø - Representations Theory of Artin Algebras
Assem, Simson, Skowronski - Elements of the Representation Theory of Associative Algebras
Schiffler - Quiver Representations