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I assumed that the 1-dimensional Lie algebra was simple since I cannot think of any proper non-trivial ideal it could have (you either have no elements, or once you have one element you span the space). However every classification of simple Lie algebras I have seen does not include a 1-dimensional Lie algebras. Is there a reason for this or is the 1-dimensional Lie algebra somehow simple?

Thank you!

AXidenT
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1 Answers1

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A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals. So we want to exclude the $1$-dimensional Lie algebra from the simple Lie algebras. In dimension $2$ there are only two non-isomorphic Lie algebras over any field, the abelian one and the non-abelian solvable one. So the first simple Lie algebra we have has already dimension $3$, for example $\mathbb{sl}_2(K)$ for a field of characteristic not $2$.

Dietrich Burde
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  • Is there any compelling mathematical reason that you know of for why it is somehow "better" (e.g., the theory becomes simpler or more elegant) to demand that simple Lie algebras be non-abelian? Or is it just a matter of conventional fiat? – user43208 Dec 11 '23 at 15:19
  • @user43208 Abelian Lie algebras are just vector spaces with zero Lie bracket. For many results it is just convenient to exclude them, but indeed, we could also consider them as (semi)simple. I find it nicer to say that $L$ is semisimple if and only if the Killing form is non-degenerate (over characteristic zero). – Dietrich Burde Dec 11 '23 at 15:46