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I was taking an introductory course in Lie algebras and I just learned about how we associate a Cartan matrix to a semisimple Lie algebra. So, for the A-series, the determinant of this matrix goes to infinity while for other series it is constant within a series. I was curious if this determinant measures or implies anything about the corresponding Lie algebra.

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    I will just leave this as a comment as it is not directly about the Lie algebra (though I suppose it could say something about the representations via looking at highest weights). The determinant of the Cartan matrix gives the index of the root lattice inside the lattice of integral weights. – Tobias Kildetoft Jan 28 '15 at 07:56
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    Actually, I just found something more directly related to the representations (which is aparently well-known, I just had not seen it before). The quotient of the weight lattice by the root lattice indexes the minuscule representations of the Lie algebra, so the determinant of the Cartan matrix gives the number of such. If I get the time, I might leave a more detailed answer, but I thought this was interesting enough to put as a comment until then. – Tobias Kildetoft Feb 09 '15 at 08:47
  • Related: https://math.stackexchange.com/q/1305467/96384 – Torsten Schoeneberg Feb 15 '22 at 06:22

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