论文标题
共同厅代数及其通过扭转对的代表
Cohomological Hall algebras and their representations via torsion pairs
论文作者
论文摘要
在本文中，我们提供了一种在稳定的$ \ infty $ - 类别$ c $ c $的核心（K理论，分类）霍尔代数和相应的左和右表示的核心和左表示。更确切地说，代数与扭转部分相关，而表示形式与无扭转部分相关。 左右动作使我们能够构建Borel-Moore同源性内态环和无扭转物体堆栈的K理论的典型亚代构，其“正零件”恢复了共同体厅代数和K Theoricenty Hall algebra和与Torsion Part $ t $相关的K Theoricenty Hall algebra。这提供了一个新的方向，可能会导致克服共同霍尔代数理论的长期局限性，以产生整个代数的“积极部分”。我们还提供了一个几何标准，以确保两个不同操作员之间的换向器消失。 在Quiver案中，我们获得了颤抖的二维共同体霍尔代数对我们框架中Nakajima Quiver品种的共同点的作用。除了箭袋外，我们还将我们的框架应用于光滑的投影复杂表面上的两个扭转对，我们研究了相应的霍尔代数及其与之相关的表示。 Finally, we slightly modify our method to construct representations of the cohomological Hall algebra of zero-dimensional sheaves on $S$ on the Borel-Moore homology of the moduli spaces of Pandharipande-Thomas stable pairs on surfaces and on relative Hilbert schemes of points (and we obtain similar results at the level of K-theory and bounded derived category).
In this paper, we provide a way of attaching to a torsion pair $(T,F)$ on the heart of a stable $\infty$-category $C$ a cohomological (K-theoretical, categorified) Hall algebra and corresponding left and right representations. More precisely, the algebra is associated to the torsion part, while the representation is associated to the torsion-free part. The left and right actions enable us to construct canonical subalgebras of the endomorphism ring of the Borel-Moore homology and K-theory of the moduli stack of torsion-free objects, whose "positive parts" recover the cohomological Hall algebra and the K-theoretical Hall algebra associated to the torsion part $T$, respectively. This provides a new direction that might lead to overcome the long-standing limitation of the theory of cohomological Hall algebras to just produce "positive parts" of whole algebras. We also provide a geometric sufficient criterion ensuring the vanishing of the commutator between two different operators. In the quiver case, we obtain the action of the two-dimensional cohomological Hall algebra of a quiver on the cohomology of Nakajima quiver varieties within our framework. Besides the quiver case, we also apply our framework to two torsion pairs on a smooth projective complex surface, and we investigate the corresponding Hall algebras and their representations associated to them. Finally, we slightly modify our method to construct representations of the cohomological Hall algebra of zero-dimensional sheaves on $S$ on the Borel-Moore homology of the moduli spaces of Pandharipande-Thomas stable pairs on surfaces and on relative Hilbert schemes of points (and we obtain similar results at the level of K-theory and bounded derived category).
