The real symplectic group Sp(2n,R) is the group of real 2n×2n matrices that preserve the skew-symmetric bilinear form ω(x,y)=⟨x,Ωy⟩ where ⟨⋅,⋅⟩ is the Euclidean inner product and Ω=[0−InIn0] for In the n×n identity matrix. That is, A∈Sp(2n,R) if and only if ω(Ax,Ay)=ω(x,y) for all x,y∈Rn. Equivalently, A∈Sp(2n,R) if and only if −ΩATΩ=A−1 where , as usual, AT 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 GL(n,C).
Wikidata ID: Q936434