sp(2n,R) — real symplectic Lie algebra

For an integer $n\geq1$ , the real symplectic Lie algebra $\mathfrak{sp}(2n,\mathbb R)$ consists of matrices preserving the standard alternating form infinitesimally:

\mathfrak{sp}(2n,\mathbb R)=\{X\in M_{2n}(\mathbb R):X^T J+JX=0\},\qquad J=\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}.

The bracket is $[X,Y]=XY-YX$ . Transpose is used without conjugation.

Block description and size convention

Writing matrices in four equal blocks, the defining equation gives

X=\begin{pmatrix}A&B\\C&-A^T\end{pmatrix},\qquad B=B^T,\quad C=C^T.

Thus there are $n^2+2\frac{n(n+1)}2=n(2n+1)$ independent real parameters. This is the split real symplectic algebra.

The number $2n$ is the full matrix size; the compact quaternionic algebra $\mathfrak{sp}(n)$ is a different real form.