Return to Question

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).

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).

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 (or even Koszul), 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).

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).

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 or of characteristic 2? 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 (or even Koszul), 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).

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 (or even Koszul), 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).