Symplectic Groups

This lecture introduces the symplectic groups, which are linear transformations preserving a non-degenerate alternating bilinear form on a vector space. The instructor explains the properties of symplectic forms, the dimensionality constraints, and the standard symplectic group SP2m. The lecture covers the isomorphism between the symplectic group in dimension 2 and SL2, the equivalence of symplectic forms under change of basis, and the transitive action of the symplectic group on nonzero vectors. Additionally, the lecture touches upon the connectivity of the symplectic group and its subgroup relationship with the special linear group.