The homotopy category and derived category of N-complexes

In this paper complexes with N-nilpotent differentials are considered. We describe the loop functor Ω and the suspension functor Σ   in the category CN(A)CN(A) of N  -complexes on an abelian category AA to provide an effective construction of left and right triangles in the homotopy category KN(A)KN(A) of N  -complexes, and prove that KN(A)KN(A) together with the functors Ω, Σ   and the classes of left and right triangles mentioned above is a pretriangulated category. We also investigate different homologies and get another example of pretriangulated categories, i.e., the derived category DN(A)DN(A) of N-complexes. Some basic properties are given.