ABSTRACT:

We give a unified approach via incidence algebras to several types of representations present in literature: distributive, square-free, with finitely many orbits, or with finitely many invariant subspaces. We introduce deformations of incidence algebras of posets, classify them and their square-free representations in terms of cohomology of the simplicial realization of the poset. These deformations turn out to be precisely the algebras with two important properties: semidistributive and locally hereditary; as consequence of this and the topological methods, we obtain characterizations of incidence algebras: they are precisely algebras with a faithful square-free representation or equivalently, acyclic algebras with a faithful distributive module. As a consequence, we obtain that any distributive acyclic representation can be presented as the defining representation of an incidence algebra. Among other applications, we rederive several results in the literature of incidence algebras as well classify generic distributive or square-free representations in the acyclic case, and give consequences on representation and Grothendieck rings of incidence algebras, which turn out to categorify interesting semigroup algebras.