Decomposition of noncommutative U(1) gauge potential

We investigate the decomposition of noncommutative gauge potential \(\hat{A_{i}}\), and find it has inner structure, namely, \(\hat{A_{i}}\) can be decomposed in two parts \(\hat{b_{i}}\) and \(\hat{a_{i}}\), here \(\hat{b_{i}}\) satisfies gauge transformations while \(\hat{a_{i}}\) satisfies adjoint transformations, so dose the Seiberg-Witten mapping of noncommutative U(1) gauge potential. By means of Seiberg-Witten mapping, we construct a mapping of unit vector field between noncommutative space and ordinary space, and find the noncommutative U(1) gauge potential and its gauge field tenser can be expressed in terms of the unit vector field. When the unit vector field has non singularity point, noncommutative gauge potential and gauge field tenser will equal to ordinary gauge potential and gauge field tenser.