How Do I describe the Lie Algebra of this Symplectic Group?

Question

Question

Write $(2n\times 2n)$-matrices in block form

$A=\begin{bmatrix}a&b\\c&d\end{bmatrix} \in Mat_{\mathbb{R}}(2n)$

where each of $a,b,c,d$ is an $n\times n$ block. Define

$J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$

where 1 is a shorthand for an $n\times n$ identity matrix. Let

$Sp(2n)= \{ A\in Mat_\mathbb{R}(2n)| AJA^T=J \}$,

so that $Sp(2n)=F^{-1}J$, where $F: Mat_{\mathbb{R}}(2n)\rightarrow Skew_{\mathbb{R}}(2n)$ is the map $F(A)=AJA^T$. Here, the codomain is the set of real $n\times n$ matrices satisfying $C=-C^T$.

(I have already shown that the differential $D_AF: Mat_{\mathbb{R}}(2n)\rightarrow Skew_{\mathbb{R}}(2n)$ is $AJH^T+HJA^T$)

Show that $Sp(2n)$ is a matrix Lie group and describe its Lie algebra $\mathfrak{sp} (2n)$.

Note I showed that $Sp(2n)$ is a matrix Lie group by showing group closure, identity, associativity, inverses and smoothness of group operations. However, I can’t seem to describe Lie algebra $\mathfrak{sp} (2n)$.

Answer 1

Let $A(t)$ be a curve in $Sp(2n)$ with $A(0)=Id$ and $A'(0)=X$. Differentiating the defining equation $$ \frac{d}{dt}\Big|_{t=0} (A(t)JA(t)^T) = \frac{d}{dt}\Big|_{t=0} J, $$ we get $$ X J + JX^T = 0. $$ That is the defining equation for the Lie algebra ${\mathfrak s}{\mathfrak p}(2n)$, in $Mat_{\mathbb R}(2n)$.

Answer 2

I haven't enough "reputation" to add a comment to Three Aggies, but I wanted to note additionally that $-JX^T= (XJ)^T$ so that the characterization $XJ + JX^T = 0$ can be rewritten as $XJ = (XJ)^T$.