## Symplectic Lie Algebras

Hi guys!!

I'm having a few issues understanding the concepts presented by my current mathematics course, and I was wondering whether I could get some help!!

I've been presented with the following question;

Consider the polynomial algebra $C [x_{1}, x_{2}, x_{3}]$ , with $x_{1}, x_{2}, x_{3}$ three independent, real variables. Show that the differential operators $x_{i} \frac{\partial}{\partial x_{j}}$ satisfy the gl(3) commutation relations. Use Ado's theorem to find expressions for the o(3) generators, $L_{1} , L_{2} , L_{3}$, in terms of (that partial fraction equation).

Show that $L^2 = x^2 \bigtriangledown^2 - x.(x.\bigtriangledown + 2)\bigtriangledown$.

You may find it helpful to use the result $\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta{km} - \delta_{jm}\delta_{kl}$.

And I'm a bit lost, at multiple steps.

In satisfying the commutation relations, I've done the following;

$[a_{ij} , a_{kl}] = \delta_{kj} a_{il} - \delta_{il} a_{kj} = \delta_{kj} x_{i} \frac{\partial}{\partial x_{l}} - \delta_{il} x_{k} \frac{\partial}{\partial x_{j}} = x_{i} \frac{\partial}{\partial x_{l}} - x_{k} \frac{\partial}{\partial x_{j}}$

However, I'm not sure where to go from here, and I'm not even sure what I'm meant to be showing.

I also have how to use Ado's theorem to express the generators in those terms, and I'm even more lost on the last part.

Any help (with some explanations) would be greatly appreciated!!