发表机构
California Institute of Technology（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过图解双模的导出张量积范畴化 Verma 模的 R-矩阵，构造辫子函子并证明辫群作用，为纽结补空间建立 Khovanov 同调，并与环形 Khovanov 同调比较。
AI 中文摘要
本文研究了 KLRW 代数到 $\mathfrak{sl}_2$ 的 Verma 模表示的张量积的推广。我们的动机是构建 $S^3$ 中纽结补空间的 Khovanov 同调理论（同时范畴化纽结补空间的 Gukov-Manolescu 双变量级数），这将在本工作的第二部分完成。我们将 Verma 模的 R-矩阵的范畴化构造为通过图解双模的导出张量积给出的函子，并显式计算它们的投射分解。我们还证明了这些辫子函子在相关范畴上诱导出辫群的作用。然后，我们描述了如何将 $\mathfrak{sl}_2$ 的有限维表示中的辫子股纳入，从而建立作为辫子补空间上 Khovanov 同调的函子。在平凡纽结的情形下，这给出了 $S^1\times D^2$ 中的纽结同调，我们通过几个例子将其与环形 Khovanov 同调进行比较，并表明它们密切相关，猜想它们具有相同的维数。最后，我们提出了 Verma 模着色辫子股的杯和帽的范畴化方案，这将在下一篇论文中进一步构建。
英文摘要
In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.
