Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension. Is it true that the dimension of $A/[A,A]$ is equal to the number of simples of $A$? Here $[A,A]$ is the vector space generated by all elements of the form $ab-ba$.
Note that it is known that in general for such $A$ that the dimension of $A/([A,A]+rad(A))$ is equal to the number of simple $A$-modules. Thus the question should be equivalent to asking whether we have $rad(A) \subseteq [A,A]$ in case $A$ has finite global dimension.