Koszul algebras and flow lattices

We provide a homological algebraic realization of the lattices integer cuts and flows graphs. To finite 2-edge-connected graph Γ with spanning tree T, we associate dimensional Koszul algebra AΓ,T. Under construction, planar dual graphs trees are associated algebras. The Grothendieck group category finitely-generated AΓ,T modules is isomorphic to Euclidean lattice ZE(Γ), describe sublattices on in terms representation theory grading gives rise q-analogs flows; these q-lattices depend non-trivially choice tree. give q-analog matrix-tree theorem, prove that q-flow (Γ1,T1) (Γ2,T2) if only there cycle preserving bijection from edges Γ1 Γ2 taking T1 T2. This classical theorem Caporaso-Viviani Su-Wagner.