Differential Calculus on the Quantum Sphere and Deformed Self-Duality Equation

We discuss the left-covariant 3-dimensional differential calculus on the quantum sphere $SU_q (2)/U(1)$. The $SU_q (2)$-spinor harmonics are treated as coordinates of the quantum sphere. We consider the gauge theory for the quantum group $SU_q (2)\times U(1)$ on the deformed Euclidean space $E_q (4)$. A $q$-generalization of the harmonic-gauge-field formalism is suggested . This formalism is applied for the harmonic (twistor) interpretation of the quantum-group self-duality equation (QGSDE). We consider the zero-curvature representation and the general construction of QGSDE-solutions in terms of the analytic prepotential.