2-DIMENSIONAL NONCOMMUTATIVE ANALYSIS, MAURER-CARTAN EQUATION AND RIEMANN-HILBERT PROBLEM

In this paper a new analytic method is given, in which the quantum group theory and the classical analysis are fused into a noncommutative analysis. In this noncommutative analysis the Maurer-Cartan equation on the quantum plane can be solved. The classical Riemann-Hilbert problem can be extended to the noncommutative analysis, and by using this Riemann-Hilbert problem, some nontrivial new solutions of the quantum Maurer-Cartan equation can be generated.