三价平面图的微分分次代数及其表示
Differential graded algebras for trivalent plane graphs and their representations
AI总结：
本文将Casals--Murphy dg-algebra推广至非交换系数并证明其函子性，证明其秩$r$表示对应于图面着色，并首次对无限族Legendrian曲面明确计算了表示与可构造层模空间间的双射。
摘要：
对于嵌入球面中的任何三价平面图，Casals和Murphy关联了一个微分分次代数（dg-algebra），其中底层的分次代数在交换环上是自由结合的。我们的第一个结果是构造了Casals--Murphy dg-algebra向非交换系数的推广，并证明了在交换环情形下先前未验证的各种函子性。我们的第二个结果是证明该dg-algebra在域$\mathbb{F}$上的秩$r$表示对应于用Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$的元素对图的面进行着色，使得相邻面横截，且至多相差$\operatorname{PGL}_{2r}(\mathbb{F})$的自然作用。在组合学底层，该dg-algebra是Legendrian 2-weaves的Legendrian卫星的完全非交换Legendrian接触dg-algebra的计算，尽管我们在本文中未作此证明。该图着色问题验证了对于Legendrian 2-weaves，Legendrian接触dg-algebra的秩$r$表示对应于微局部秩$r$的可构造层。这是首次对无限族Legendrian曲面明确计算表示与层的模空间之间的双射。
英文摘要：
To any trivalent plane graph embedded in the sphere, Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals--Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank $r$ representations of this dg-algebra, over a field $\mathbb{F}$, correspond to colorings of the faces of the graph by elements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so that bordering faces are transverse, up to the natural action of $\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that for Legendrian 2-weaves, rank $r$ representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank $r$. This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.
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