BPS李代数与预投射CoHA上的较少反常层滤过
BPS Lie algebras and the less perverse filtration on the preprojective CoHA
AI总结：
本文在预投射代数表示栈的Borel-Moore同调上引入新反常层滤过，证明其上同调霍尔代数滤过零片同构于BPS李代数的泛包络代数，并利用李代数结构证明半稳定模的交叉上同调提供尖点上同调。
摘要：
预投射代数$Π_Q$的表示栈$\mathfrak{M}(Π_Q)$的仿射化态射，是泛2-Calabi-Yau范畴中对象栈到良好模空间态射的一个局部模型。我们证明了沿此态射的对偶复形的导出正像是纯的，并且在Beilinson-Bernstein-Deligne-Gabber分解定理的意义下允许一个分解。利用该分解，我们在$\mathfrak{M}(Π_Q)$的Borel-Moore同调上引入了一种新的反常层滤过。我们证明了由$\mathfrak{M}(Π_Q)$的Borel-Moore同调构建的上同调霍尔代数上所得滤过的第零片，同构于相关BPS李代数$\mathfrak{g}_{Π_Q}$的泛包络代数。该李代数是通过3-Calabi-Yau范畴的临界上同调霍尔代数的Kontsevich-Soibelman理论定义的。随后，我们将该李代数提升为$Π_Q$-模粗模空间上反常层范畴中的一个李代数对象，并利用该代数结构证明关于上述分解定理中出现的直和项的结果。特别地，我们证明了半稳定$Π_Q$-模的奇异空间的交叉上同调提供了“尖点上同调”——这是$\mathfrak{g}_{Π_Q}$的典范生成子的一个猜想完备子空间。
英文摘要：
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 $\mathfrak{g}_{Π_Q}$.
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