Understanding presentation of simple Lie algebras by means of Chevalley-Serre relations.

Asked Feb 14 '23 at 21:16

Active Feb 14 '23 at 21:28

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I came across the Chevalley-Serre Relations while studying semisimple Lie algebras. My question is how do the last two conditions regarding adjoint operator make sense? For $i = j$ we have $A_{ii} = 2,$ being the $i$-th diagonal entry of a Cartan matrix. Then the last two conditions are stating to invert the adjoint operator. Is it always possible?

I am eagerly looking forward for an answer which drives away this confusion. Any help would be warmly welcomed.

Thanks a bunch!

asked Feb 14 '23 at 21:16

Anacardium

2,716

The case $i=j$ should be excluded in the last two relations. See https://en.wikipedia.org/wiki/Serre%27s_theorem_on_a_semisimple_Lie_algebra – Torsten Schoeneberg Feb 14 '23 at 21:20
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Well, what sign do non zero off-diagonal elements of a Cartan matrix have? – Mariano Suárez-Álvarez Feb 14 '23 at 21:59
Also, related: https://math.stackexchange.com/q/2509230/96384. They also exclude $i=j$ in the last relation. Maybe write an e-mail to Wolfram, they should do the same. – Torsten Schoeneberg Feb 14 '23 at 23:12

Understanding presentation of simple Lie algebras by means of Chevalley-Serre relations.

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