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Suppose we know the root system/Cartan matrix of a semisimple Lie algebra. Is there a formula that determines the dimension of the Lie algebra?

Thanks in advance.

Torsten Schoeneberg
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Daven
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    Compare https://math.stackexchange.com/q/279516/96384. – Torsten Schoeneberg Jun 15 '19 at 01:29
  • So the dimension of the root space gives the dimension of the cartan subalgebra. The number of roots gives the number of one dimensional root spaces corresponding to roots. Given a Cartan matrix, can we calculate the number of roots? – Daven Jun 15 '19 at 06:59
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    Actually this answer: https://math.stackexchange.com/questions/1467330/computing-information-about-a-lie-algebra-from-cartan-matrix?rq=1 helps. – Daven Jun 15 '19 at 07:02

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I assume you are talking about a split semisimple Lie algebra. This is always the case if you work over an algebraically closed field of characteristic $0$, like $\Bbb C$.

Then it's part of the standard structure theory that such a Lie algebra $L$ decomposes as

$$ L \simeq H \oplus \bigoplus_{\alpha \in R} L_\alpha$$

where $H$ is a Cartan subalgebra, $R$ is the corresponding root system, and $L_\alpha$ is the root space to the root $\alpha$. It's also a standard fact that in this case, all $L_\alpha$ are one-dimensional, whereas the dimension of $H$ equals the rank of $R$. Which implies

$$dim(L) = rank(R) + \lvert R \rvert.$$

Torsten Schoeneberg
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