Determinant form

Over a field $𝕂$ the vector space $𝑉 = 𝕂 2$ may be endowed with symplectic structure under the so-called determinant form #m/def/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

Given $𝐴 \in E n d 𝑉$ , $𝜔 (𝐴 𝑣, 𝐴 𝑤) = d e t (𝐴) 𝜔 (𝑣, 𝑤)$ , and thus $S p 2 (𝕂) = S L 2 (𝕂)$ .

#state/develop | #lang/en | #SemBr