Given a connected finite dimensional path algebra $A=KQ$ of Dynkin type with enveloping algebra $A^e= A^{op} \otimes_K A$.
I can prove that there is a canonical exact sequence connecting the bimodules $A$ and $D(A)=Hom_K(A,K)$, that is one has either
($dim(Ext_{A^e}^{1}(D(A),A))=1$ and $dim(Ext_{A^e}^{i}(D(A),A))=0$ for $i \neq 1$)
or
($dim(Ext_{A^e}^{2}(D(A),A))=1$ and $dim(Ext_{A^e}^{i}(D(A),A))=0$ for $i \neq 2$).
One has the case $dim(Ext_{A^e}^{1}(D(A),A))=1$ iff $A$ is of Dynin type $\mathcal{A}$.
So for such algebas there should be a canonical non-split bimodule exact sequence of the form $0 \rightarrow A \rightarrow M \rightarrow D(A) \rightarrow 0$ or $ 0 \rightarrow A \rightarrow M_0 \rightarrow M_1 \rightarrow D(A) \rightarrow 0$.
However, I was not able to explicitly write down such a canonical exact sequence for general $KQ$ of Dynkin type. Note that $D(A)$ has projective dimension at most 2.
So my question is:
How do these canonical exact sequences look like?
I should remark that similar things are not true for non-Dynkin type path algebras and I had to assume that the field is algebraically closed (at least when looking more general at finite dimensional herditary algebras), but I would guess that this is true for general fieds.
edit: I would expect that for any hereditary algebra one has $dim(Ext_{A^e}^{1}(D(A),A))+dim(Ext_{A^e}^{2}(D(A),A)) \neq 0$ and maybe one can write down a "canonical" exact sequence between A and $D(A)$ and there are just some other too for some algebras. For example for the Kroenecker algebra one has $dim(Ext_{A^e}^{1}(D(A),A))=3$ and $dim(Ext_{A^e}^{2}(D(A),A))=5$ so there are many nonsplit exact sequences between A and $D(A)$ but maybe one is especially pretty?
