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:
“Even in harmonious families there is this double life: the group life, which is the one we can observe in our neighbour’s household, and, underneath, another—secret and passionate and intense—which is the real life that stamps the faces and gives character to the voices of our friends. Always in his mind each member of these social units is escaping, running away, trying to break the net which circumstances and his own affections have woven about him.”
—Willa Cather (1873–1947)