# Koszul Algebras and Flow Lattices

## TL;DR

This paper constructs Koszul algebras associated with graphs to realize integer cut and flow lattices homologically, introducing $q$-analogs and establishing new duality and invariance results.

## Contribution

It introduces a homological algebraic framework linking Koszul algebras to graph lattices, including $q$-analogs and a new matrix-tree theorem extension.

## Key findings

- Koszul algebras encode integer cut and flow lattices.
- $q$-analogs depend on spanning trees and relate to classical invariants.
- A $q$-analog of the matrix-tree theorem and duality results are established.

## Abstract

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $\Gamma$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{\Gamma,T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{\Gamma,T}$ modules is isomorphic to the Euclidean lattice $\mathbb Z^{E(\Gamma)}$, and we describe the sublattices of integer cuts and integer flows on $\Gamma$ in terms of the representation theory of $A_{\Gamma,T}$. The grading on $A_{\Gamma,T}$ gives rise to $q$-analogs of the lattices of integer cuts and flows; these $q$-lattices depend non-trivially on the choice of spanning tree. We give a $q$-analog of the matrix-tree theorem, and prove that the $q$-flow lattice of $(\Gamma_1,T_1)$ is isomorphic to the $q$-flow lattice of $(\Gamma_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of $\Gamma_1$ to the edges of $\Gamma_2$ taking the spanning tree $T_1$ to the spanning tree $T_2$. This gives a $q$-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.