5
$\begingroup$

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules. Recall that the Auslander algebra of $A$ is $End_A(M)$ and the stable Auslander algebra of $A$ is the stable endomorphism ring of $M$.

Questions:

Is the stable Auslander algebra of $A$ the quiver algebra having the stable Auslander-Reiten quiver with mesh relations and some additional relations (what are those additional relations? Probably the mesh relations where the deleted arrows are set to zero)?

Are two algebras $A_1$ and $A_2$ stable equivalent if and only if their stable Auslander algebras are isomorphic as algebras?

Is there a computational way using QPA to check whether two (representation-finite)algebras $A$ and $B$ are stable equivalent?

$\endgroup$
1
  • $\begingroup$ The second question should have a positive answer by the work of Auslander and Reiten in their article "Stable equivalence of artin algebras". $\endgroup$
    – Mare
    Jun 21, 2019 at 7:07

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.