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This Wikipedia article talks about Lie algebra extension by a Lie-algebra, while this other artilce talks about extension by a module. This nLab article mensions central extensions by a ground field.

Are they all some special cases of something, or are they different concepts?

mma
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    At least the ground field would be a special case of a module. The circle group makes no sense to me, I assume you or your source mixed up Lie algebras and Lie groups there. – Torsten Schoeneberg Jan 10 '20 at 07:29
  • @TorstenSchoeneberg Oops.. you are right, $\mathbb T$ is for group extensions. – mma Jan 10 '20 at 07:40

1 Answers1

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Certainly one can extend Lie algebras by Lie algebras. Here the ground field is considered as a $1$-dimensional abelian Lie algebra. For example, we have the short exact sequence of Lie algebras $$ 0 \rightarrow \mathfrak{sl}_n(K)\rightarrow \mathfrak{gl}_n(K)\rightarrow K \rightarrow 0, $$ where $\mathfrak{gl}_n(K)$ is an extension of $K$ by $\mathfrak{sl}_n(K)$.

In the second link, $M$ is a Lie algebra module, but it is in fact considered as an abelian Lie algebra in the short exact sequence. So this is not different.

The second Lie algebra cohomology $H^2(L,M)$ classifies equivalence classes of abelian extenions $$ 0\rightarrow M \rightarrow \mathfrak{g}\rightarrow L \rightarrow 0, $$ i.e., where $M$ is an abelian Lie algebra. See for example this post:

Classification of Lie Algebra extensions in Weibel's book

and other related posts.

Dietrich Burde
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