Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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!

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.