A Representation theoretic perspective of Koszul theory

We discover a new connection between Koszul theory and representation theory. Let $\text{La}$ 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 ${\text{La}}^{!}$, which enables us to describe the linear projective resolutions and the colinear injective coresolutions of graded simple $\text{La}$-modules in terms of ${\text{La}}^{!}$. 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 $\text{La}$-modules and those derived from graded ${\text{La}}^{!}$-modules. In case $\text{La}$ 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 $\text{La}$ and ${\text{La}}^{!}$ 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 $\text{La}$ and ${\text{La}}^{!}$ are both locally bounded, then the bounded derived category of finite dimensional graded $\text{La}$-modules has almost split triangles with the Auslander-Reiten translations and the Serre functors given by composites of derived Koszul functors.