Linear Symplectic Manifold
There is a standard linear model, namely a symplectic vector space R2n. Let R2n have the basis {v1, ... ,v2n}. Then we define our symplectic form ω so that for all 1 ≤ i ≤ n we have ω(vi,vn+i) = 1, ω(vn+i,vi) = −1, and ω is zero for all other pairs of basis vectors. In this case the symplectic form reduces to a simple quadratic form. If In denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the (2n × 2n) block matrix:
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