perfect Lie algebra but not semisimple

Question

Lie algebra $\mathfrak{g}$ is perfect if and only if $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$.

And we know semisimple Lie algebra must satisfy the $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$ then it's perfect.

What is the example of perfect Lie algebra but not semisimple? And what's the sufficient and necessary condition for a perfect Lie algebra to be semisimple Lie algebra?

See this MO-question, and also this MSE-question. – Dietrich Burde Sep 11 '17 at 16:54 — Dietrich Burde, Sep 11 '17 at 16:54

score 5 · Answer 1 · answered Sep 11 '17 at 06:14

5

I think the set of $4$-by-$4$ matrices of the form $$\pmatrix{*&*&*&*\\*&*&*&*\\0&0&*&*\\0&0&*&*}$$ will be an example.

answered Sep 11 '17 at 06:14

Angina Seng

161,540

5

you need to prescribe both diagonal blocks to have trace zero. – YCor Sep 11 '17 at 20:50

score 4 · Answer 2 · answered Sep 11 '17 at 16:59

Take any semisimple Lie algebra $L$ of dimension $n$ and an irreducible representation $V$ of $L$, of dimension $m \ge 2$ and define a bracket on $L \times V$ by $$ [(X,v),(Y,u)] := ([X,Y],Xu-Yv). $$ This turns $L \times V$ into a perfect Lie algebra with $\text{Rad}(L \times V) = V$. The dimension is $n+m$. It is not semisimple.

A necessary and sufficient condition for a perfect Lie algebra to be semisimple is that its solvable radical is trivial.

perfect Lie algebra but not semisimple

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