Sp(2,C) - Knowlpedia

$\mathrm{Sp}(2,\mathbb C)$ is the symplectic group

\mathrm{Sp}(2,\mathbb C)=\{A\in\mathrm{GL}(2,\mathbb C):A^{\mathsf T}JA=J\},\qquad J=\begin{pmatrix}0&I_{1}\\-I_{1}&0\end{pmatrix}.

It preserves the displayed nondegenerate alternating bilinear form.

Dimensions and structure

This group has real dimension $6$ , complex dimension $3$ . The parameter $1$ is the number of symplectic pairs, while the matrix size is $2$ . Its tangent matrices have block form $\left(\begin{smallmatrix}P&Q\\R&-P^{\mathsf T}\end{smallmatrix}\right)$ with $Q=Q^{\mathsf T}$ and $R=R^{\mathsf T}$ . Counting these blocks gives the dimension. This is a connected simply connected complex Lie group. For two-by-two matrices $A^{\mathsf T}JA=(\det A)J$ , so this is precisely SL(2,C).

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

The groups coincide because a 2 by 2 matrix satisfies $A^{\mathsf T}JA=(\det A)J$ . See $\operatorname{SL}(2,\mathbb C)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.