# Cartan or Coxeter matrix of an algebra of infinite global dimension

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)

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(1) No. Example: $$\begin{array}{ccc} & \alpha & \\ 1 & \rightleftarrows & 2 \\ & \beta & \end{array}$$ with all paths of length 3 = 0. This is a symmetric algebra wigh Loewy structure $$\begin{array}{ccc} 1 & & 2\\ 2 & \oplus & 1 \\ 1 & & 2 \end{array}$$ and has infinite global dimension. Its Cartan matrix is $$\left(\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right)$$ whose determinant is not 1, hence not inveritble in $\mathrm{Mat}_2(\mathbb{Z})$.
Thank you very much, Aaron Is there a finite dimensional algebra $A$ satisfying the following conditions: (1) The Cartan matrix of $A$ is not invertible (2) $A$ has finite globle dimension. –  Aimin Xu Nov 28 '12 at 7:31