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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. [ABSTRACT FROM AUTHOR]

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology. [ABSTRACT FROM AUTHOR]

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt(DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived. [ABSTRACT FROM AUTHOR]