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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.

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Well, $(a_i,a_j)=(a_j,a_i)$. This implies that the definitions in Humpreys and Carter's book are indeed transposed to each other. Isn't it? I guess you have in mind that the Cartan numbers $\langle a_i, a_j\rangle$ and $\langle a_i, a_j\rangle$ are not always the same and forgot that $(,)$ is a inner product. – Júlio César Aug 17 '11 at 23:07
Glad that you figured it out! If the information content is the same, why would we need to elevate one of the two alternatives to a status of a standard? In some cases that may be advisable or even necessary, but this is a relatively esoteric case, and simply calling it a Cartan matrix conveys the message clearly those who know. The only situation that I can think of, where it would matter, would be that if you are writing/using a piece of software, and are expecting a different <strike> standard </strike> convention than the user/author. Well, software should come with a document :-) – Jyrki Lahtonen Aug 19 '11 at 6:33

Here is a similar question with a great answer by Humphreys!

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