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Suppose that $R$ is a root system in a complex $n$-dimensional vector space $V$ with a base $S = \{\alpha_1,...,\alpha_n\}$, and that $$(n(\alpha_i,\alpha_j))_{\alpha_i,\alpha_j \in S}$$ is the Cartan matrix of $R$ with respect to $S$.

Serre's theorem is the following:

Let $\mathfrak{g}$ be the Lie algebra defined by the $3n$ generators $X_i,Y_i,H_i$ ($i = 1...n$), and the relations

  1. $[H_i,H_j] = 0$,
  2. $[X_i,Y_i] = H_i$, and $[X_i,Y_j] = 0$ if $i \neq j$,
  3. $[H_i,X_j] = n(\alpha_i,\alpha_j)X_j$, and $[H_i,Y_j] = -n(\alpha_i,\alpha_j)Y_j$,
  4. $Ad_{X_i}^{\:(-n(\alpha_i,\alpha_j)+1)}(X_j) = 0$ if $i \neq j$,
  5. $Ad_{Y_i}^{\:(-n(\alpha_i,\alpha_j)+1)}(Y_j) = 0$ if $i \neq j$.

Then $\mathfrak{g}$ is a semisimple Lie algebra, where the subalgebra generated by $H_i$ is a Cartan subalgebra, and its root system is $R$.

I would like to clarify two things. Firstly, I think that if $R$ were an irreducible root system, then it follows that $\mathfrak{g}$ is simple. Is this true?

Secondly, if it is true that $\mathfrak{g}$ is simple in the case that $R$ is irreducible, does this follow directly from Serre's theorem, or other results? How do we deduce that $\mathfrak{g}$ is simple if $R$ is irreducible, as opposed to just semisimple?

Matt
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  • First question: yes, then the Lie algebra is simple. Second question: via Dynkin diagram (connected=simple). – Dietrich Burde Mar 19 '18 at 17:54
  • @Dietrich, how do we conclude that a connected Dynkin diagram gives rise to a simple Lie algebra? I understand that the classification tells us that, but I'm under the impression that to go from a connected Dynkin diagram to a semisimple (simple) Lie algebra, I first construct the Cartan matrix, and then use the above. At what point am I able to prove that my Lie algebra is simple? – Matt Mar 19 '18 at 17:58
  • You are able to prove it once you go back. If the Lie algebra decomposes, then this gives a corresponding decomposition of the root system (and hence of the Dynkin diagram). – Tobias Kildetoft Mar 19 '18 at 18:59
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    For an overview of the classification result, see my note http://pure.au.dk/portal/da/publications/the-classification-of-semisimple-lie-algebras(04c29758-17a5-4a7a-8f8e-df9bc6ee805b).html – Tobias Kildetoft Mar 19 '18 at 19:01

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