Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension. Let $B$ be the Ext algebra of $M$, that is $B:=\bigoplus_{k=0}^{\infty}{Ext_A^i(M,M)}$. Note that $B$ is finite dimensional because of the assumptions on $M$.

Questions:

1.Is there a computer software that can check whether $B$ is again a quiver algebra and can calculate the quiver and relations of $M$ in case it is a quiver algebra?

I am just aware of the GAP-package QPA, which is able to calculate minimal generators of $B$.

  1. Is there a precise condition when $B$ is a quiver algebra?
  • 1
    Is your base field algebraically closed? Let's assume this. Aren't all finite dim basic algebras quiver algebras? So this is true as long as your module M is has no summands occuring with the same multiplicity (so that $$\text{End}(M)/\text{rad} (\text{End|(M))$$ is a bunch of fields). – Geordie Williamson Nov 5 at 21:55
  • @GeordieWilliamson I would allow arbitrary fields, since most computers cant handle algebraically closed fields (and yes, for alg. closed fields all basic algebras are quiver algebras). Your condition means exactly that $End_A(M)$ is a quiver algebra, which would also have been my guess that this condition is needed. For basic modules $M$, End_A(M) is always a quiver algebra in case $A$ is representation-finite and Im mostly interested in representation-finite $A$ for the start. – Mare Nov 5 at 22:10
  • @GeordieWilliamson For calculating the degree zero part $End_A(M)$ of $B$ (at least as an algebra in case this is needed), one needs to decompose $M$ into indecomposables and I think for this one needs even to assume that the field is finite since as far as I know there a no very effective algorithm to decompose a module into indecomposables over infinite fields (not sure whether this works for QPA for example for infinite fields). – Mare Nov 5 at 22:18

Q2: By Auslander-Reiten-Smalø, Theorem 1.9 page 65, says:

If $A$ is a finite dimensional elementary algebra over a field $k$ (that is, $A/\operatorname{rad}(A) \simeq k^n$ for some $n$), then $A\simeq kQ/I$ for some finite quiver $Q$ and some admissible ideal $I$ in $kQ$.

If $A\simeq kQ/I$ for a finite quiver $Q$ and an admissible ideal $I$ in $kQ$, the radical of $A$ is $\langle\textrm{arrows}\rangle/I$ and $A/\operatorname{rad}(A) \simeq kQ/\langle\textrm{arrows}\rangle \simeq k^{|Q_0|}$. Hence $A$ is an elementary algebra, and the above characterizes admissible quotients of path algebras.

This applied to $B=\operatorname{Ext}^*_A(M,M)$ gives that $M$ must be a basic module, that is, $M \simeq \oplus_{i=1}^n M_i$ with $M_i$ indecomposable and $M_i\not\simeq M_j$ for $i\neq j$. Furthermore, for $B$ to elementary $\operatorname{End}_A(M_i)/\operatorname{rad}\operatorname{End}_A(M_i)$ must be isomorphic to a fixed field $K$ for all $i$. This ensures that $B$ is a quiver algebras.

For Q1: Given the above, the software needs to be able compute the radical of $\operatorname{End}_A(M)$ and determine the structure of $\operatorname{End}_A(M)/\operatorname{rad}\operatorname{End}_A(M)$. For QPA this is to my knowledge only possible over finite fields. The command $\textrm{IsElementaryAlgebra}$ checks if a finite dimensional algebra over a finite field is elementary. To understand the structure of $\operatorname{End}_A(M)/\operatorname{rad}\operatorname{End}_A(M)$ is basically to decompose $M$ into indecomposable modules. The command $\textrm{DecomposeModule}(M)$ does this over finite fields. This gives a complete set of primitive idempotents of $\operatorname{End}_A(M)$, which correspond to the vertices in the quiver one wants to construct.

The arrows of degree $0$ is given by basis of $\operatorname{rad}\operatorname{End}_A(M)/\operatorname{rad}^2\operatorname{End}_A(M)$. The command $\textrm{EndOfModuleAsQuiverAlgebra( M )}$ would do this if the field $K$ is the same field $A$ is an algebra over.

Given that $\operatorname{rad}B = \operatorname{rad}(A) \oplus \operatorname{Ext}^{\geq 1}_A(M,M)$, we need to compute

$\operatorname{Ext}^n_A(M,M)/(\operatorname{rad}\operatorname{End}_A(M)\operatorname{Ext}^n_A(M,M) + \operatorname{Ext}^{n-1}_A(M,M)\operatorname{Ext}^1_A(M,M) + \cdots + \operatorname{Ext}^1_A(M,M)\operatorname{Ext}^{n-1}_A(M,M) + \operatorname{Ext}^n_A(M,M)\operatorname{rad}\operatorname{End}(M))$

to find the arrows of degree $n$. The command $\textrm{ExtAlgebraGenerators}(M, n)$ would find $\operatorname{Ext}^n_A(M,M)$ modulo everything (among other things) except for the two outer terms (so the documentation on this command is misleading as it is only correct if the degree zero part is a semisimple ring). However all of this is possible to do in QPA, but there is no readily available commands to preform these tasks. These necessary commands are something we want to implement in QPA version 1 or version 2.

I hope that these are useful comments.

Best regards, The QPA-team.

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