2024/11/23 by Ales M. Bouhada, Min Huang, Bouhada, Ales +4
Arts and Humanities · #FOS: Mathematics #Philosophy, Science, and History #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2411.15449
openalex publication_date 2024/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discover a new connection between Koszul theory and representation theory. Let \La be a quadratic algebra defined by a locally finite quiver with relations. Firstly, we give a combinatorial description of the local Koszul complexes and the quadratic dual \La^!, which enables us to describe the linear projective resolutions and the colinear injective coresolutions of graded simple \La-modules in terms of \La^!. As applications, we obtain a new class of Koszul algebras and a stronger version of the Extension Conjecture for finite dimensional Koszul algebras with a noetherian Koszul dual. Then we construct two Koszul functors, which induce a 2-real-parameter family of pairs of derived Koszul functors between categories derived from graded \La-modules and those derived from graded \La^!-modules. In case \La is Koszul, each pair of derived Koszul functors are mutually quasi-inverse, one of the pairs is Beilinson, Ginzburg and Soergel's Koszul duality. If \La and \La^! are locally bounded on opposite sides, then the Koszul functors induce two equivalences of bounded derived categories: one for finitely piece-supported graded modules, and one for finite dimensional graded modules. And if \La and \La^! are both locally bounded, then the bounded derived category of finite dimensional graded \La-modules has almost split triangles with the Auslander-Reiten translations and the Serre functors given by composites of derived Koszul functors.