Derivation-based Noncommutative Geometry And Its Applications In Integrable Systems

Derivation-based noncommutative diferential calculi are an importance part of non-commutative geometry, they have very important applications in noncommutative gaugetheory, integrable systems and so on. In this paper, we propose a class of derivation-baseddeformed noncommutative diferential calculi from a noncommutative modular space,thenwe get the connection and curvature theory in the sense of noncommutative. By means ofthis theory, we give out the zero-curvature representation of the diferential, diferential-diference and diference nonlinear evolution equations in an unifying manner.