Koszul Algebras and Flow Lattices

Zsuzsanna Dancso, Anthony Licata

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 $\mathrm{\Gamma }$ with a spanning tree T, we associate a finite dimensional Koszul algebra $A_{\mathrm{\Gamma },T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{\mathrm{\Gamma },T}$-modules is isomorphic to the Euclidean lattice ${\mathbb{Z}}^{E(\mathrm{\Gamma })}$, and we describe the sublattices of integer cuts and integer flows on $\mathrm{\Gamma }$ in terms of the representation theory of $A_{\mathrm{\Gamma },T}$. The grading on $A_{\mathrm{\Gamma },T}$ 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 $({\mathrm{\Gamma }}_1,T_1)$ is isomorphic to the $q$-flow lattice of $({\mathrm{\Gamma }}_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of ${\mathrm{\Gamma }}_1$ to the edges of ${\mathrm{\Gamma }}_2$ 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.