TITLE: Convolution algebra associated to a finite dimensional algebra
QUESTION [5 upvotes]: Given a finite dimensional $k$-algebra $A$ (we can assume it is given by a connected quiver with relations). One can form its trivial extension $T(A)$ (see for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra ), which is a Frobenius algebra and thus it has a coalgebra structure, see for example the book by Kock on Frobenius algebras and 2D topological quantum field theories.
Now given any $k$-coalgebra $C$ and $k$-algebra $A$, we can form the convolution algebra $W_{C,A}={\rm Hom}_k(C,A)$.
This gives rise to (at least) two new algebras from a given algebra $A$, namely $W_{T(A),A}$ and $W_{T(A),T(A)}$ using that $T(A)$ has a coalgebra structure as a Frobenius algebra. I have no real experience with this topic, so sorry in case my questions are stupid.

Question 1: Have either of the algebras $W_{T(A),A}$ or $W_{T(A),T(A)}$ been considered/described before? Are they quiver algebras and if so, can their quivers be described?
Question 2: What do those two algebras look like in the concrete example where $A=kQ$ is a path algebra (maybe of Dynkin type)?
Question 3: In case those questions are non-trivial, can those algebras be calculated by a computer algebra system like GAP/QPA?

REPLY [3 votes]: Question 3: The algebra constructions $W_{C,A}$ or even $W_{T(A),A}$ are not available in QPA.  There are no structures made for co-algebras in QPA.