1

I have been studying Lie Algebras from chapter 13 of Conformal Field Theory by Di Francesco et al., I understand that the entire structure of a Lie algebra is contained in its Cartan Matrix. What I do not understand is how one calculates it in the general case. Some examples are given on page 504-507 of the book, but they start with the Cartan matrix as the definition of the Lie algebra. How does one, for example, start with the Lie algebra sp(4) of the symplectic group SP(4), and find that the Cartan matrix is given as equation 13.102 of the book?

Meer Ashwinkumar
  • 555
  • 1
    If you have it available, you might want to consult Humphreys' book Introduction to Lie Algebras and Representation Theory, which explains this in detail. Note that the whole Cartan matrix thing is only for semisimple Lie algebra, not arbitrary ones. – Tobias Kildetoft Aug 25 '15 at 08:34
  • Thank you for your comment. I could not find it in Humphreys' book, the language is a little more abstract for me, could you point out the relevant section in the book? – Meer Ashwinkumar Aug 26 '15 at 00:28
  • This is done in chapter 8, though it is done for $\mathfrak{sl}_2$ in chapter 7, as a motivating example. Note, however, that he does not introduce root systems until chapter 9, which can make things a little confusing. So he associates a root system to the Lie algebra in chapter 8, he just does not call it a root system. – Tobias Kildetoft Aug 26 '15 at 09:36

0 Answers0