Koszul代数与流格
Koszul Algebras and Flow Lattices
摘要：
我们提供了图的整数割和整数流的格的同调代数实现。对于具有生成树$T$的有限2-边连通图$Γ$，我们给出了一个有限维Koszul代数$A_{Γ,T}$ .在这种构造下，具有对偶生成树的平面对偶图是相伴的Koszul对偶代数。有限生成$A_{Γ,T}$模范畴的Grothendieck群同构于欧氏格$\mathbb Z^{E(Γ)}$，利用$A_{Γ,T}$的表示理论刻画了$Γ$上的整数割和整数流的子格.对$A_{Γ,T}$的分级产生了$q$ -整数割流格的类似物;这些$q$ -格依赖于生成树的选择。我们给出了矩阵树定理的$q$ -模拟，并证明了$(Γ_1,T_1)$的$q$ -流格同构于$(Γ_2,T_2)$的$q$ -流格当且仅当存在从$Γ_1$的边到$Γ_2$的边的保圈双射，它取生成树$T_1$到生成树$T_2$ .这给出了Caporaso-Viviani和Su-Wagner的经典定理的$q$ -模拟。
英文摘要：
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 $\mathbb Z^{E(Γ)}$, 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 $(Γ_1,T_1)$ is isomorphic to the $q$-flow lattice of $(Γ_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of $Γ_1$ to the edges of $Γ_2$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
