## 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 $$\Gamma$$ 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.

This paper is available as a pdf (732kB) file. It is also on the arXiv: arxiv.org/abs/1905.03067.

 Thursday, March 12, 2020