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On incidence algebras and their representations

### Miodrag C. Iovanov and Gerard D. Koffi

Vol. 316 (2022), No. 1, 131–167
##### Abstract

We provide a unified approach, via a deformation theory for incidence algebras that we introduce, to several types of representations with finiteness conditions, as well as to the combinatorial algebras which produce them. We show that for finite-dimensional algebras over infinite fields, modules with finitely many orbits, or with finitely many invariant subspaces, or that are distributive, coincide (and further coincide with thin modules in the acyclic case). Incidence algebras produce examples of such modules via their principal projective modules, and we show that algebras which are locally hereditary, and whose indecomposable projectives are distributive, or equivalently, which have finitely many ideals, are precisely the deformations of incidence algebras. New characterizations of incidence algebras are obtained, such as that they are exactly the algebras which have a faithful thin module. As a main consequence, we show that every thin module comes from an incidence algebra, i.e., if $V$ is thin (and, in particular, if $V$ is distributive and $A$ is acyclic), then $A∕\mathrm{ann}\left(V\right)$ is an incidence algebra and $V$ can be presented as its defining representation. As applications, other results in the literature are rederived and a positive answer to the accessibility question of Ringel and Bongartz, in the distributive case, is given.

 M. C. Iovanov dedicates this work to the memory of his late friend and former student, G. D. Koffi.
##### Keywords
incidence algebra, distributive, thin representation, finitely many orbits, deformations, cohomology, poset, simplicial realization
##### Mathematical Subject Classification
Primary: 06A11, 16G20, 16S80
Secondary: 05E45, 16T30, 18G99, 55U10
##### Milestones
Accepted: 15 December 2020
Published: 26 February 2022
##### Authors
 Miodrag C. Iovanov Department of Mathematics University of Iowa Iowa City, IA United States Gerard D. Koffi Department of Mathematics University of Texas, Arlington Arlington, TX United States