# On Auslander algebras

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).

• "Crawley-Boevey: Matrix reductions for artinian rings and an application to rings of finite representation type" seems related to Question 1 (but is more general). – Julian Kuelshammer Apr 26 '19 at 10:52

Question 3: QPA can for a given admissible ideal $$I$$ in a path algebra $$kQ$$ find a minimal generating set and also check if these generators are quadratic. For the requested example it would look like this:

gap> A := NakayamaAlgebra(GF(3),[3,3]);;
gap> S1 := SimpleModules(A)[1];;
gap> pred := PredecessorsOfModule(S1,7)[1];;
gap> pred := Unique(Flat(pred));
[ <[ 1, 0 ]>, <[ 1, 1 ]>, <[ 0, 1 ]>, <[ 2, 1 ]>, <[ 1, 1 ]>, <[ 1, 2 ]> ]
gap> M := DirectSumOfQPAModules(pred);;
gap> B := EndOfModuleAsQuiverAlgebra(M)[3];;
gap> kQ := OriginalPathAlgebra(B);;
gap> I := Ideal(kQ, RelationsOfAlgebra(B));;
true
gap> mingens := MinimalGeneratingSetOfIdeal(I);
[ (Z(3)^0)*a1*a7, (Z(3)^0)*a3*a1+(Z(3))*a4*a5, (Z(3)^0)*a6*a2+(Z(3))*a7*a8, (Z(3)^0)*a8*a3 ]