The symplectic group

SymplecticGroup{T}

The manifold of real symplectic matrices, of size $2n×2n$ for some $n∈ℕ$ is given by

\[\mathrm{Sp}(2n, ℝ) = \bigl\{ p ∈ ℝ^{2n×2n}\ \big|\ p^\mathrm{T}J_{2n}p = J_{2n}\bigr \}\]

where $J_{2n} = \begin{pmatrix} 0_n & I_n\\ -I_n & 0_n\end{pmatrix}$ denotes the SymplecticElement.

This yields the SymplecticGroup together with the MatrixMultiplicationGroupOperation as the group operation.

The corresponding Lie algebra is given by the HamiltonianMatrices

\[\mathfrak so(2n, ℝ) = \bigl\{ X ∈ ℝ^{2n×2n}\ \big|\ X^+ = -X\bigr \},\]

where $⋅^+$ denotes the symplectic_inverse.

See [BZ21, Section 2] for more information.