MathGloss

The real symplectic group Sp(2n,R)\text{Sp}(2n,\mathbb R) is the group of real 2n×2n2n\times 2n matrices that preserve the skew-symmetric bilinear form ω(x,y)=x,Ωy\omega(x,y) = \langle x,\Omega y\rangle where ,\langle\cdot,\cdot\rangle is the Euclidean inner product and Ω=[0InIn0]\Omega = \begin{bmatrix}0 & I_n \\ -I_n & 0\end{bmatrix} for InI_n the n×nn\times n identity matrix. That is, ASp(2n,R)A\in \text{Sp}(2n, \mathbb R) if and only if ω(Ax,Ay)=ω(x,y)\omega(Ax,Ay) = \omega(x,y) for all x,yRnx,y \in \mathbb R^n. Equivalently, ASp(2n,R)A\in \text{Sp}(2n, \mathbb R) if and only if ΩATΩ=A1-\Omega A^T \Omega = A^{-1} where , as usual, ATA^T is the transpose of AA and A1A^{-1} is its inverse.

This group is a matrix Lie group because it is a subgroup of the general linear group GL(n,C)\text{GL}(n,\mathbb C).

Wikidata ID: Q936434