On Incidence Algebras and their Representations

Miodrag C. Iovanov, Gerard D. Koffi

Published 2017-02-10Version 1

We provide a unified approach, via incidence algebras, to the classification of several important types of representations: distributive, thin, with finitely many orbits, or with finitely many invariant subspaces, as well as to several types of algebras such as semidistributive, with finitely many ideals, or locally hereditary. The key tool is the introduction of a deformation theory of posets and incidence algebras. We show that these deformations of incidence algebras of posets are precisely the locally hereditary semidistributive algebras, and they are classified in terms of the cohomology of the simplicial realization of the poset. A series of applications is derived. First, we show that incidence algebras are exactly the acyclic algebras admitting a faithful distributive module, equivalently, algebras admitting a faithful thin representation, or a faithful distributive module. Second, we give a complete classification of all thin and all distributive representations of incidence algebras. Furthermore, we extend this to arbitrary algebras and provide a method to completely classify thin representations over an any algebra. As a consequence, we obtain the following "generic classification": we show that any thin representation of any algebra, and any distributive representation of an acyclic algebra can be presented, by choosing suitable bases and after canceling the annihilator, as the defining representation of an incidence algebra. Other applications include consequences on the structure of representation and Grothendieck rings of incidence algebras, and an answer to a conjecture of Bongartz and Ringel (the "no gap" or "accessibility" conjecture), in a particular case. Several results in literature are also re-derived, and several open questions are formulated at the end.