Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension.
I want to ask the following questionss:
(1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring $M_n(Z)$?
(2) what are the relations between the Cartan matrix ( or Coxeter matrix) of a finite dimensional algebra $B$ and the global dimension of $B$? ( Here $B$ may have infinite global dimension)