Symmetrizability of generalised Cartan matrix

Question

How to prove that a generalized Cartan matrix whose diagram contains no cycles is symmetrizable?

Any hint would be sufficient.

Thanks in Advance.

Answer 1

Hint

A size $n$ generalized Cartan Matrix $A = (A_{ij})_{1\leq i,j\leq n}$ is symmetrizable precisely if for all sequences $i_1,i_2,...,i_k$ of indices in $\{1,2,...,n\}$ you have $A_{i_1 i_2} A_{i_2 i_3} ... A_{i_k i_1} = A_{i_2 i_1} A_{i_3 i_2} ... A_{i_1 i_k}$.

Since $A_{ij}=0$ if there is no edge between $i$ and $j$ in the Dynkin diagram $\Gamma$ of $A$, we may assume that $i_1\to i_2\to ...\to i_k\to i_1$ constitutes a path in $\Gamma$. Then, since $\Gamma$ has no cycles by definition, any edge must be traversed in one direction precisely as often as it is traversed in the other direction.