Finding automorphisms and cyclic modules via QPA

Question

Given a symmetric finite dimensional algebra $A$ over a finite field with enveloping algebra $A$.

Assume we know that $\Omega_{A^e}^i(A) \cong A_{f}$, where $f$ is some automorphism of the algebra $A$. (so $A_{f}$ is the bimodule $A$ twisted by this automorphism)

Question 1: Is there a way to obtain $f$ using QPA? Note that we can calculate $\Omega_{A^e}^i(A)$ using QPA.

Question 2: Is there a good theoretical way how to obtain $f$ in a quick way? Does $f$ have special properties?

Quesiton 3: We know that $A_{f}$ is a cyclic $A^e$-module, how can we find an element $x$ in the enveloping algebra $A^e$ with QPA so that we have $A_f = x A^e$?

Question 4: Assume we know that $A$ as a bimodule admits a projective resolution (not necessarily minimal) of the form: $... \rightarrow A^e \rightarrow ... A^e \rightarrow A^e \rightarrow A \rightarrow 0$,so that every term can be choosen to be the regular module of the enveloping algebra. This means that $\Omega_{A^e}^i(A)=x_i A^e$ for some elements $x_i \in A^e$. Is there a canonical choice of the $x_i$ or some nice behavior? Can one obtain the $x_i$ in a nice form via QPA?

More generally, can we obtain a minimal free resolution of a module in QPA instead of a minimal projective resolution?

This is possible by filtering through all elements but in practise this takes too long.

Answer 1

Assume that we start with an admissible quotient $A = kQ/I$ for a path algebra over a field $k$ and a finite quiver $Q$. In addition assume that $\Omega^n_A(A/rad) \simeq A/rad$ as right $A$-modules, and if necessary $\Omega^n_A(S)$ is simple for all simple right $A$-modules $S$. This is all you need of assumptions. It will follow that the algebra is selfinjective.

Question 1:

1. Find $n$ such that $\Omega^n_A(S)$ is simple for all simple right $A$-modules $S$.
2. Find $B = \Omega^n_{A^e}(A)$.
3. Construct the natural algebra homomorphism $A \to A^e$ via the command g := TensorAlgebrasInclusion( A^e, 2 ).
4. Find $B_A$ via RestrictionViaAlgebraHomomorphism( g, B ).
5. Find isomorphism $\varphi\colon A_A\to B_A$ using IsomorphismOfModule( A, B ).
6. Find $\varphi( 1 )$. Call it $b$.
7. Define $f\colon A\to A$ by letting $f(a) = \varphi^{-1}(ab)$.

Then the automorphism $f$ is found.

Question 2: Don't know any better than the above. All taken from Theorem 1.4 in

Green, Edward L.; Snashall, Nicole; Solberg, Øyvind, The Hochschild cohomology ring of a selfinjective algebra of finite representation type., Proc. Am. Math. Soc. 131, No. 11, 3387-3393 (2003). ZBL1061.16017.

Question 3: The element $x$ can be taken as the element $\varphi(1) = b$ given in Question 1.

Question 4: Here already for $n = 2$ the assumptions needed for the above should be satisfied just using dimension arguments. Using the above, or maybe just taking the minimal generator of $\Omega^i_{A^e}(A)$ would give you the $x_i$ that you want.