Current Algebraic Structures over Manifolds: Poisson Algebras, q-Deformations and Quantization

Poisson algebraic structures on current manifolds (of maps from a finite dimensional Riemannian manifold into a 2-dimensional manifold) are investigated in terms of symplectic geometry. It is shown that there is a one to one correspondence between such current manifolds and Poisson current algebras with three generators. A geometric meaning is given to q-deformations of current algebras. The geometric quantization of current algebras and quantum current algebraic maps is also studied.