Can each central (simple) K'-Lie algebra be represented as a scalar extension of a K-Lie algebra for any field extension K'/K

Asked Nov 27 '19 at 11:52

Active Dec 01 '19 at 15:50

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Let K'/K be a field extension and L' a central (simple?) K'-Lie algebra. Can we find a K-Lie algebra, such that $L'=K'\otimes_K L$.

in other words:

Let K'/K be a field extension and L' a central (simple?) K'-Lie algebra. Can we find a K'-base B of L', which satisfies $[B,B]\subseteq <B>_K$?

edited Dec 01 '19 at 15:50

asked Nov 27 '19 at 11:52

That's a good question (please improve formatting though). First of all, do you have any example of any field extension $K'\vert K$ and any Lie algebra over $K'$ which is not the scalar extension of one over $K$? I think those examples exist, but are already a bit non-trivial, so it would be interesting to look at them. – Torsten Schoeneberg Nov 28 '19 at 19:02
I would not ask, if I had – Isidor Konrad Maier Dec 01 '19 at 15:57
Dear Torsten Schoeneberg, I got the following representation theorem as a result of my master thesis. https://math.stackexchange.com/questions/3461566/representation-for-finite-dimensional-central-lie-algebras-over-a-a-field-of-cha My teacher is not sure, if it is true and if it can be found anywhere else. It would be a great help, if you could give a feedback till thursday or friday. Do you know/believe it? Or do you even know a source for it? – Isidor Konrad Maier Dec 03 '19 at 17:04
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As for examples for what I alluded to in my first comment, see https://math.stackexchange.com/a/95544/96384. – Torsten Schoeneberg Dec 08 '19 at 04:56

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