Symplectic Group
If V = W, then a symplectic map is called a linear symplectic transformation of V. In particular, in this case one has that ω(ƒ(u),ƒ(v)) = ω(u,v), and so the linear transformation ƒ preserves the symplectic form. The set of all symplectic transformations forms a group and in particular a Lie group, called the symplectic group and denoted by Sp(V) or sometimes Sp(V,ω). In matrix form symplectic transformations are given by symplectic matrices.
Read more about this topic: Symplectic Vector Space
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