sp(2,C) — complex symplectic Lie algebra

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

\mathfrak{sp}(2,\mathbb C)=\{X\in M_{2}(\mathbb C):X^T J+JX=0\},\qquad J=\begin{pmatrix}0&I_{1}\\-I_{1}&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 $1+2\cdot1=3$ independent complex parameters. This is a complex Lie algebra; its underlying real dimension doubles. In matrix size two the condition is exactly trace zero, giving $\mathfrak{sp}(2,\mathbb C)=\mathfrak{sl}(2,\mathbb C)$ .

The number $2$ is the full matrix size; the compact quaternionic algebra $\mathfrak{sp}(1)$ is a different real form.