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The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of the sort $X=\begin{pmatrix}A & B\\ C & -A^{t}\end{pmatrix}$ when $B=B^{t},\, C=C^{t}$.

This far I was able to get, but how can I prove that it is simple?

Thanks!

  • 3
    Can you prove it is semisimple?2012-02-01
  • 2
    I can't. Although I'm pretty sure you'll have to show that the killing form is non-degenerate. Can't see exactly how.2012-02-01

2 Answers 2