论文标题
BPS躺在代数和前注射COHA上的不正当过滤
BPS Lie algebras and the less perverse filtration on the preprojective CoHA
论文作者
论文摘要
堆栈$ \ mathfrak {m}（π_q）的繁殖形态的前提代数$π_q$的表示形式是从一般2-Calabi-yau类别中堆栈的形态学的局部模型。我们表明，沿这种形态的双重化复合物的派生直接图像是纯粹的，并且在贝林森 - 伯恩斯坦 - 伯尼斯坦 - 埃尔德格伯 - 盖伯分解定理的意义上承认了分解。 我们使用此分解引入了$ \ mathfrak {m}（m}（π_q）$的Borel-Moore同源性的新变形过滤。我们表明，由$ \ mathfrak {m}（π_q）$构建的共同体霍尔代数的零件过滤量是同构的。该谎言代数是通过针对3-卡拉比YAU类别的临界同源霍尔代数的Kontsevich-Soibelman理论定义的。然后，我们将此谎言代数提升到$π_q$ - 模块的粗型束空间的类别中的lie代数对象，并使用此代数结构来证明上述分解定理中出现的汇总结果。特别是，我们证明了可分离的$π_q$ - 模块的奇异空间的相交共同体提供“ Cuspidal共同体” - $ \ Mathfrak {g} _ {π_Q} $的典范生成器的一个构想完整的子空间。
The affinization morphism for the stack $\mathfrak{M}(Π_Q)$ of representations of a preprojective algebra $Π_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem. We introduce a new perverse filtration on the Borel-Moore homology of $\mathfrak{M}(Π_Q)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of $\mathfrak{M}(Π_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $\mathfrak{g}_{Π_Q}$. This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of $Π_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable $Π_Q$-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for $\mathfrak{g}_{Π_Q}$.
