Symplectic Geometry, Hopf Algebraic Structures of Classical and Quantum q -Deformations of SU(2) Algebra

By deforming the symplectic structure on S 2 , we get the q -deformation of SU(2) algebra at classical level, SU q,ℏ →0 (2), in a Hamiltonian approach. Furthermore, we construct a set of operators on the line bundle over the deformed symplectic manifold such that they form SU q,ℏ →0 (2) in Lie brackets and set up a nontrivial Hopf algebra with a parameter q only in such a classical Hamiltonian system. We also show that the deformations from S 2 to are a set of quasiconformal transformations. The quantization via geometric approach of the system gives rise to the quantum q -deformed algebra SU q,ℏ (2), which has a Hopf algebraic structure with two independent parameters q and ℏ .