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- W2947204151 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." @default.
- W2947204151 created "2019-06-07" @default.
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- W2947204151 date "2019-05-08" @default.
- W2947204151 modified "2023-10-06" @default.
- W2947204151 title "Koszul Algebras and Flow Lattices" @default.
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