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

TL;DR

This paper explores the geometric and algebraic structures of current manifolds, establishing a correspondence with Poisson current algebras, and investigates q-deformations and quantization methods within this framework.

Contribution

It introduces a novel correspondence between current manifolds and Poisson current algebras with three generators, and provides geometric interpretations of q-deformations and quantization.

Findings

One-to-one correspondence between current manifolds and Poisson current algebras.

Geometric interpretation of q-deformations of current algebras.

Development of geometric quantization for current algebras.

Abstract

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.