The real symplectic group $\text{Sp}(2n,\mathbb R)$ is the group of real $2n\times 2n$ matrices that preserve the skew-symmetric bilinear form \(\omega(x,y) = \langle x,\Omega y\rangle\) where $\langle\cdot,\cdot\rangle$ is the Euclidean inner product and \(\Omega = \begin{bmatrix}0 & I_n \\ -I_n & 0\end{bmatrix}\) for $I_n$ the $n\times n$ identity matrix. That is, $A\in \text{Sp}(2n, \mathbb R)$ if and only if $\omega(Ax,Ay) = \omega(x,y)$ for all $x,y \in \mathbb R^n$. Equivalently, $A\in \text{Sp}(2n, \mathbb R)$ if and only if \(-\Omega A^T \Omega = A^{-1}\) where , as usual, $A^T$ is the transpose of $A$ and $A^{-1}$ is its inverse.
This group is a matrix Lie group because it is a subgroup of the general linear group $\text{GL}(n,\mathbb C)$.
Wikidata ID: Q936434