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.



Commutator of finite global dimension algebras
Yes. See the result of Section 2.5 of a wonderful paper of Bernhard Keller :


<https://webusers.imj-prg.fr/~bernhard.keller/publ/ilc.pdf>


(and the references therein).


