Koszul algebras and flow lattices

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra A_Γ,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_Γ,T modules is isomorphic to the Euclidean lattice Z^E(Γ), and we describe the sublattices of integer cuts and integer flows on Γ in terms of the representation theory of A_Γ,T. The grading on A_Γ,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 (Γ₁,T₁) is isomorphic to the q-flow lattice of (Γ₂,T₂) if and only if there is a cycle preserving bijection from the edges of Γ₁ to the edges of Γ₂ taking the spanning tree T₁ to the spanning tree T₂. This gives a q-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.