图的Hall代数与有根树
Hall algebras of graphs and rooted trees
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中文总结 AI 辅助
本文研究由有根树、无向图和有向图的2-Segal集产生的Hall代数，建立Green定理类比，描述本原元素并给出生成元与关系表示，同时引入与Tutte多项式相关的Hall多项式作为新的图不变量。
中文摘要 AI 辅助
通过2-Segal集理论，Hall代数可以与一大类组合结构相关联。本文研究了由有根树、无向图和有向图的2-Segal集产生的Hall代数。在每种情形下，我们建立了Green定理的类比，赋予Hall代数一个扭曲的双代数结构，描述了本原元素，并推导出由生成元和关系给出的表示。作为应用，我们将无向图的Hall代数实现为某个拓扑空间的上同调环。在有向图的情形下，我们引入了Hall多项式，定义为本原元素空间的Poincaré多项式。Hall多项式是底层无向图的一个不变量，我们证明它与Tutte多项式密切相关。此外，我们给出了具有相同Tutte多项式但不同Hall多项式的图的例子，从而表明Hall多项式编码了与Tutte多项式不同的信息。
英文摘要
Hall algebras can be associated with a broad class of combinatorial structures through the theory of 2-Segal sets. In this paper, we study the Hall algebras arising from the 2-Segal sets of rooted trees, undirected graphs, and directed graphs. In each case, we establish an analogue of Green's theorem giving a twisted bialgebra structure to the Hall algebra, describe the primitive elements, and derive a presentation by generators and relations. As an application, we realize the Hall algebra of an undirected graph as the cohomology ring of a topological space. In the case of directed graphs, we introduce the Hall polynomial, defined as the Poincar{é} polynomial of the space of primitive elements. The Hall polynomial is an invariant of the underlying undirected graph which we show to be closely related to the Tutte polynomial. We furthermore give examples of graphs with equal Tutte polynomials but distinct Hall polynomials showing thus that the Hall polynomial encodes different information from that of the Tutte polynomial.
