$${\mathsf{N}}$$ -Complexes as Functors, Amplitude Cohomology and Fusion Rules

Abstract

We consider \({\mathsf{N}}\) -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated \({\mathsf{N}}\) -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive \({\mathsf{N}}\) -complexes is proved to be isomorphic to an Ext functor of an indecomposable \({\mathsf{N}}\) -complex inside the abelian functor category. Finally we show that for the monoidal structure of \({\mathsf{N}}\) -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for \({\mathsf{N}}\) -complexes can be determined.