Exercise 12-14 in John Lee's book, An introduction to Smooth Manifolds, reads as follows:
The real symplectic group is the subgroup $Sp(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$ consisting of $2n\times 2n$ matrices leaving the standard symplectic form $\omega = \sum_1^n dx^i \wedge dy^i$ invariant, that is, the set of invertible linear maps $A:\mathbb{R}^{2n} \rightarrow\mathbb{R}^{2n}$ such that $A^*\omega = \omega$. Here $A^*$ is the pullback of the differential map $A$.
(a) Show that a matrix $A$ is in $Sp(n, \mathbb{R})$ iff it takes that standard basis of $\mathbb{R}^{2n}$ into a symplectic basis.
(b) Show that $A \in Sp(n, \mathbb{R})$ iff $A^TJA = J$ where $J$ is the symplectic matrix
(c)Show that $Sp(n, \mathbb{R})$is an embedded Lie subgroup of $GL(2n, \mathbb{R})$ and determine its dimension.
(d) Determine the Lie algebra of $Sp(n, \mathbb{R})$
(e)Is $Sp(n, \mathbb{R})$ compact?
I think I've managed to answer both (a) and (b), but I'm having some difficulty calculating the dimension and its Lie algebra on (c) and (d). (e) is easy, tho: $Sp(n, \mathbb{R})$ contains a copy of $Sp(2, \mathbb{R})$ and this is equal to $Sl(2, \mathbb{R})$ which isn't compact, so $Sp(n, \mathbb{R})$ cannot be compact.
Can you help me with questions (c) and (d)? Thank you