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
Famous quotes containing the word group:
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)