Given a connected quiver algebra $A$ that is representation finite, the Auslander algebra $B_A$ of $A$ is defined as the endomorphism ring of the direct sum of each indecomposable $A$-module.
It is stated on page 232 in the book by Auslander-Reiten-Smalo that the quiver algebra $KQ/I$ of $B_A$ is given by $Q$ being the opposite Auslander-Reiten quiver of $A$ and $I$ are the mesh relations in case the field is algebraically closed of characteristic different from 2.
Question 1: Does this description still hold over finite fields of characteristic different from 2? (is there a reference in case the answer is yes?) Are there examples where this fails in case the field is not algebraically closed? Are then the relations still quadratic?
Question 2: Is there an existing method in the GAP-package QPA to obtain the Auslander algebra of an algebra $A$ quickly by quiver and relations in case one has a list of all indecomposable $A$-modules (which are easily obtainable for example for hereditary algebras of Dynkin type)?
Question 3: Is there a method to check in QPA whether a given quiver algebra is quadratic , that is isomorphic to $KQ/I$ with quadratic $I$ and obtain such $I$? For example calculating the Auslander algebra of the Nakayama algebra with Kupisch series $[3,3]$ as the endomorphism ring of all indecomposable modules gives a presentation $KQ/I$ with non-quadratic $I$ but it there should be one with quadratic $I$ and Im not sure how to obain it via QPA(although I know how to obtain it theoretically in this easy example).