Decomposition of Noncommutative U(1) Gauge Potential

We investigate the decomposition of noncommutative gauge potential A-circumflex i, and find that it has inner structure, namely, A-circumflex i can be decomposed in two parts, b-circumflex i and a-circumflex i, where b-circumflex i satisfies gauge transformations while a-circumflex i satisfies adjoint transformations, so dose the Seiherg-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 tensor can be expressed in terms of the unit vector field. When the unit vector field has no singularity point, noncommutative gauge potential and gauge field tensor will equal ordinary gauge potential and gauge field tensor