Question on admissible ideal

Question

I look at the quiver kQ with one point and two loops $x,y$ and relations $I:= <xy-yx, x^5+x^2 y^4, x^6+y^7 >$. Let J be the jacobsonradical of kQ. Then clearly $I \subseteq J^2.$ Now $I$ should be admissible (meaning $J^n \subseteq I$ for some n) iff $J^n+I/I$ is zero iff all path of length at least n are zero in $kQ/I$ iff $kQ/I$ is finite dimensional. I get kQ/I has dimension 38 with qpa but it also shows me that the ideal is not admissible. Where is my thinking error or is qpa wrong?

Answer 1

For a quiver with loops (or directed cycles), the Jacobson radical is not the ideal generated by arrows. In fact, for your quiver $Q$, the Jacobson radical of $kQ$ is zero.

But if by $J$ you actually mean the ideal generated by arrows, then it is not true that $kQ/I$ finite dimensional implies $J^n\subseteq I$ for some $n$.

Consider the simpler example of the quiver with a single loop $x$ and $I=\langle x^3-x^2\rangle$. Then $kQ/I=k[x]/(x^3-x^2)$ is $3$-dimensional, spanned by $1,x,x^2$, and (in $kQ/I$) $0\neq x^2=x^3=x^4=\dots$, so no power of $x$ is contained in $I$.