sp(2n,C) — complex symplectic Lie algebra

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

\mathfrak{sp}(2n,\mathbb C)=\{X\in M_{2n}(\mathbb C):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 complex parameters. This is a complex Lie algebra; its underlying real dimension doubles.

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