  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.
