Given a finite dimensional quiver algebra A=KQ/I. It can be (possibly) written as $A= B_1 \otimes_k B_2 ... \otimes_k B_r$ and the $B_i$ can not be decomposed into smaller algebras. Is this factorisation unique? (all quivers are assumed to be connected and have at least one arrow)
In Nüsken, M. "Unique tensor factorization of algebras", Math Ann. (1999) 315-341 this is proved for $K$ of characteristic zero. As far as I know, it's still open in positive characteristic, although the same author proved some partial results in a later paper.