# Preprojective algebra of finite dimensional algebras

The preprojective algebra of a module $$M$$ over a finite dimensional algebra $$A$$ is defined as $$P_M:= \bigoplus\limits_{n=0}^{\infty}{Hom_A(M, \tau^{-n}(M))}$$ with the canonical multiplication.

Question 1: Is there an easy way to obtain quiver and relations of $$P_M$$ in case it is finite dimensional? Are explicit quiver and relations known when $$M=A$$ and $$A$$ is an acyclic Nakayama algebra?

Question 2: Can $$P_M$$ be obtained in qpa in some way in case it is finite dimensional, at least in some special cases like Nakayama algebras?

For Question 2: In QPA one can do the following using the latest uploaded extensions of QPA:

gap> A := NakayamaAlgebra( GF(2), [ 3, 2, 1 ] );
<GF(2)[<quiver with 3 vertices and 2 arrows>]>
gap> M := DirectSumOfQPAModules(IndecProjectiveModules(A));
<[ 1, 2, 3 ]>
gap> B := PreprojectiveAlgebra( M, 3 );;
gap> C := B[1];
<GF(2)[<quiver with 3 vertices and 4 arrows>]/
<two-sided ideal in <GF(2)[<quiver with 3 vertices and 4 arrows>]>, (5 generators)>>
gap> Display( AdjacencyMatrixOfQuiver( QuiverOfPathAlgebra( C ) ) );
[ [  0,  1,  0 ],
[  1,  0,  1 ],
[  0,  1,  0 ] ]


The command PreprojectiveAlgebra computes the preprojective algebra of the module $$M$$ if it is finite dimensional and degree $$n$$ is zero and it is given as a quotient of a path algebra.

The QPA-team.

• Thank you very much. So degree $n$ means that $\tau^{-n}(M)=0$ or does it mean that $Hom_A(M,\tau^{-n}(M)=0$? Also I noted that I put $\tau^n$ instead of $\tau^{-n}$ in the question, I will edit that. – Mare May 13 at 14:03
• It means $\operatorname{Hom}_A(M, \tau^{-n}(M)) = 0$. Should be zero from degree $n$ on. – Oeyvind Solberg May 13 at 14:05