Define the following quaternionic matrices $1=\pmatrix{1&0\\0&1}, i=\pmatrix{0&-1\\1&0}, j=\pmatrix{0&-i\\-i&0}, k=\pmatrix{i&0\\0&-i}$
I am given that the symplectic group is the group of linear transformations preserving the inner product in $\mathbb{H}=\mathbb{R^4}$. I want to know if $\frac{1}{\sqrt{2}}\pmatrix{j&k\\k&j}$ and $\pmatrix{1&0\\0&i}$ are in the symplectic lie group $Sp(2)$?
My problem is that the definition is not clear enough to me, so this is difficult to think about. Any ideas out there?
You actually need to check if this two matrices preserve inner product of $\mathbb{H}^2=\mathbb{C}^4=\mathbb{R}^8$. So the simplest way is using those quaternionic matrices to get 4*4 matrices over $\mathbb{C}$ and check whether they are unitary.
Here is a detailed process of calculating inner product: the first matrix send $(u,v)\in\mathbb{H}^2$ to $\frac{1}{\sqrt{2}}(ju+kv,ku+jv)$ and:
$||(ju+kv,ku+jv)||^2=(ju+kv)^{*}(ju+kv)+(ku+jv)^{*}(ku+jv)=-(u^{*}j+v^{*}k)(ju+kv)-(u^{*}k+v^{*}j)(ku+jv)=u^{*}u+v^{*}v-u^{*}iv+v^{*}iu+u^{*}u+v^{*}v+u^{*}iv-v^{*}iu=2(u^{*}u+v^{*}v)$
put back the factor $\frac{1}{2}$ we get correct result.