Symplectic Manifold - Linear Symplectic Manifold

Linear Symplectic Manifold

There is a standard linear model, namely a symplectic vector space R2n. Let R2n have the basis {v₁, ... ,v_2n}. Then we define our symplectic form ω so that for all 1 ≤ i ≤ n we have ω(v_i,v_n+i) = 1, ω(v_n+i,v_i) = −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 I_n denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the (2n × 2n) block matrix: