McKay对应、上同调Hall代数与范畴化
McKay correspondence, cohomological Hall algebras and categorification
AI总结：
本文利用导出McKay对应建立了二维ADE型Kleinian奇点消解上层范畴的Hall代数与仿射预投影代数表示Hall代数之间的同构，并给出其范畴化解释及A型情形下的Yangian刻画。
摘要：
设 $π\colon Y\to X$ 表示二维ADE型Kleinian奇点 $X$ 的典范消解。在本文中，我们在依赖于极化 $ω$ 和斜率 $μ$ 的关于 $ζ$ 的某些条件下，分别建立了 $Y$ 上具有固定斜率 $μ$ 的 $ω$-半稳定真支撑层（properly supported sheaves）的上同调Hall代数与K理论Hall代数，和零斜率的仿射ADE型预投影代数的 $ζ$-半稳定有限维表示之间的同构。这些同构由导出McKay对应诱导。此外，它们被解释为相应范畴化Hall代数之间单项等价（monoidal equivalence）的去范畴化版本。在A型情形下，我们给出了 $Y$ 上具有固定斜率 $μ$ 的 $ω$-半稳定真支撑层的上同调、K理论和范畴化Hall代数的更精细描述：例如，在上同调情形下，该代数可以用有限型ADE Dynkin图的Yangian来给出。
英文摘要：
Let $π\colon Y\to X$ denote the canonical resolution of the two dimensional Kleinian singularity $X$ of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of $ω$-semistable properly supported sheaves on $Y$ with fixed slope $μ$ and $ζ$-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on $ζ$ depending on the polarization $ω$ and $μ$. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide finer descriptions of the cohomological, K-theoretical and categorified Hall algebra of $ω$-semistable properly supported sheaves on $Y$ with fixed slope $μ$: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.
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