Sp(2n,R) - Knowlpedia

For every integer $n\ge1$ , $\mathrm{Sp}(2n,\mathbb R)$ is the symplectic group

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

It preserves the displayed nondegenerate alternating bilinear form.

Dimensions and structure

This group has real dimension $n(2n+1)$ . The parameter $n$ is the number of symplectic pairs, while the matrix size is $2n$ . 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 the noncompact split real form, distinct from compact quaternionic Sp(n).

References

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