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The homology theory of colored posets, defined by B. Everitt and P. Turner, is generalized. Two graph categories are defined and Khovanov type graph cohomology are interpreted as Ext* groups in functor categories associated to these categories. The connection, described by J. H. Przytycki, between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained from the point of view of homological algebra in functor categories.
This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra , where is the category of morphisms between projective objects in a finitary hereditary exact category . When is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra , which is spanned by isoclasses of indecomposable objects in . As applications, we demonstrate that contains a Lie subalgebra isomorphic to the central extension...
For any positive integer , let be a linearly oriented quiver of type with vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories and , where and are the two extriangulated categories corresponding to the representation category of and the morphism category of projective representations of , respectively. As a by-product,...
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