论文标题
单位元素和链接同源性扭曲
Unipotent Elements and Twisting in Link Homology
论文作者
论文摘要
令$ \ Mathcal {u} $为复杂的还原组$ g $的一能力。修复对立的Borel子组$ b_ \ pm \ subseteq g $，带有Unipoltent $ u_ \ pm $。发送$ x_+x_- \ mapsto x_+x_-x _+^{ - 1} $的地图，以限制在u_ \ pm $ in u_ \ pm $限制从$ u_+u_+u_+u_- \ cap gb _+cap gb _+$ cap gb _+to $ \ nto $ \ mathcal {u} \ cap {ub cap gb _+$ $ g $ g $ g $ g，我们猜想限制地图形成了这些品种之间同质等效性的一半，因此诱导了其紧凑型共同支持的同构的同构。指出该地图对于$ b_+ \ cap gb_+ g^{ - 1} $的某些动作是等价的，我们证明对此同构的类似模拟存在。奇怪的是，这是基于调节理论的工具Khovanov-Rozansky同源性的一定双重性。
Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$ restricts to a map from $U_+U_- \cap gB_+$ into $\mathcal{U} \cap gB_+$, for any $g$. We conjecture that the restricted map forms half of a homotopy equivalence between these varieties, and thus, induces a weight-preserving isomorphism between their compactly-supported cohomologies. Noting that the map is equivariant with respect to certain actions of $B_+ \cap gB_+g^{-1}$, we prove for type $A$ that an equivariant analogue of this isomorphism exists. Curiously, this follows from a certain duality in Khovanov-Rozansky homology, a tool from knot theory.
