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Let $A$ be a finite dimensional algebra with Cartan matrix $C_A$.$C_A$ being a symmetric matrix is equivalent to the Coxeter matrix being minus the identity matrix in case $A$ has finite global dimension and thus for finite global dimension having a symmetric Cartan matrix is a derived invariant.

Question 1: Is having a symmetric Cartan matrix a derived invariant for general algebras $A$?

edit: This is true by Jeremy Rickard's comment.

Question 2: Is it true for a general finite dimensional algebra with symmetric Cartan matrix that the dominant and Gorenstein dimensions are even in case they are finite?

Probably not, but maybe it is true when adding an extra condition. Is it true when the algebra additionally has finite global dimension?

(I edited question 2: Before it was about the finitistic dimension instead of the Gorenstein dimension, but there is an algebra with symmetric Cartan matrix and finitistic dimension 1).

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    $\begingroup$ Symmetry of the Cartan matrix is a derived invariant. If $C$ and $C'$ are the Cartan matrices of two derived equivalent finite dimensional algebras, then $C'=X^TCX$ for some invertible integer matrix $X$. $\endgroup$
    – Jeremy Rickard
    Commented Jul 28, 2020 at 8:44

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