2.2. SYMPLECTIC STRUCTURE ON R2n 15 It is convenient to recast σ, employing a very useful matrix: DEFINITION. We introduce the 2n × 2n matrix (2.2.2) J := 0 I −I 0 . LEMMA 2.2 (Properties of J and σ). (i) We have (2.2.3) σ(z, w) = Jz, w for all z, w ∈ R2n, where ·, · now means the standard inner product on R2n. (ii) The bilinear form σ is antisymmetric: (2.2.4) σ(z, w) = −σ(w, z) and nondegenerate: (2.2.5) if σ(z, w) = 0 for all w, then z = 0. (iii) Also (2.2.6) J2 = −I, JT = −J = J−1. We leave the simple proofs to the reader. We now bring in the terminology of differential forms, reviewed in Ap- pendix B: NOTATION. We introduce for x = (x1,...,xn) and ξ = (ξ1,...,ξn) the 1-forms dxj and dξj for j = 1,...,n and then write (2.2.7) σ = dξ ∧ dx = n j=1 dξj ∧ dxj. Observe also that (2.2.8) σ = dω for ω := ξdx = n j=1 ξjdxj. Since d2 = 0, it follows that (2.2.9) dσ = 0.

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