Symplectic vector space

Determinant form

Over a field 𝕂 the vector space 𝑉 =𝕂2 may be endowed with symplectic structure under the so-called determinant form geoalg

𝜔(\vtwo𝑥𝑦,\vtwo𝑧𝑤)=det[𝑥𝑧𝑦𝑤]=𝑥𝑤−𝑦𝑧

named for the Matrix determinant. This turns out to be a special case of the Standard symplectic space.

Properties

1. Given 𝐴 ∈End⁡𝑉, 𝜔(𝐴𝑣,𝐴𝑤) =det(𝐴)𝜔(𝑣,𝑤), and thus Sp2⁡(𝕂) =SL2⁡(𝕂).

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