Example to periodic symmetric algebras

Question

In case $A$ is a symmetric finite dimensional algebra and $e$ an idempotent, $eAe$ is again symmetric.

Is there an easy counterexample for the following:

In case $A$ is additionally a periodic algebra, $eAe$ is also periodic?

Is there a criterion when $eAe$ is still periodic depending on $A$ and $e$?

In case $A$ is representation-finite the claim is true, but I am not very experienced with representation-infinite periodic algebras.

I did some examples with the computer and in those examples the following thing happened: Let $A$ be a symmetric algebra and $W$ the direct sum of all indecomposable projective $A$-modules $P$ such that $soc(P)$ is periodic. Then $End_A(W)$ was a periodic algebra.

Answer 1

The simplest counterexample I know is when A is the mesh algebra of generalized Dynkin type $C_3$ (see the paper "Periodic algebras" by Erdmann and Skowronski). Its quiver is the same as the preprojective algebra of Dynkin type $D_4$, but the relations are slightly different. It is symmetric and periodic of period $6$. If the central vertex is labeled 2, and the three other vertices are 0, 1, and 3, the indecomposable projective $e_2A$ has structure: $$\begin{array}{ccc} & 2 \\ 0 & 1 & 3 \\ 2 & \ 2 \\ 0 & 1 & 3 \\ & 2\end{array}$$ Thus $e_2Ae_2 \cong k[x,y]/(xy, x^2+y^2)$, which is not periodic.

Your other observation at the end is interesting. While I don't expect that to hold in general, at the moment I don't know a counterexample.