Cartan matrices in the books of (1) Humphreys (page 59) and of (2) Carter (page 82 -- 83) are different. Moreover, they are not transpose of each other. Which one is correct? Thank you very much.
Edit: Now I know the differences. In (1), $c_{ij}=2(a_i, a_j)/(a_j, a_j)$. But in (2), $c_{ij} = 2(a_i, a_j)/(a_i, a_i)$. Which one is standard? In particular, which Cartan matrices are used in the paper [V. G. Drinfeld, A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212-216]? Thank you very much.
standardconvention than the user/author. Well, software should come with a document :-) – 2011-08-19