AI 中文总结
研究$\mathbf{A}^2$的幂零上同调霍尔代数的交换性，通过结合对仿射BPS李代数上李括号的两个约束进行证明，还描述了特定作用下的等变幂零CoHAs，得到相关包络代数。
AI 中文摘要
本文证明了$\mathbf{A}^2$的半幂零和全幂零上同调霍尔代数（CoHAs）都是交换的。这一结果令人惊讶，与之前Davison研究的无幂零条件的$\mathbf{A}^2$的CoHA形成强烈对比，后者与$\mathbf{C}^*$上的微分算子李代数$W_{1+\infty}$相关且高度非交换。证明结合了对仿射BPS李代数上李括号的两个约束：关于反常滤过是滤过的，关于上同调次数是分次的。在约旦箭图和幂零CoHAs的情况下，这些约束迫使李括号消失。还描述了在一维环面以权重1缩放$\mathbf{A}^2$的第一个坐标、权重 -1缩放第二个坐标的作用下的等变幂零CoHAs。此时得到与李代数$W_{1+\infty}^+$上的幂零和半幂零滤过相关的里斯李代数的包络代数，这让人想起Davison对等变非幂零CoHA的描述。
英文摘要
In this paper, we prove that both the seminilpotent and the fully nilpotent CoHAs of $\mathbf{A}^2$ are commutative. This result is in strong contrast with the CoHA of $\mathbf{A}^2$ without nilpotency conditions, previously studied by Davison, which is related to the Lie algebra $W_{1+\infty}$ of differential operators on $\mathbf{C}^*$. The latter is highly noncommutative. Our proof combines two constraints on the Lie bracket on the affinized BPS Lie algebra: it is filtered with respect to the perverse filtration and it is graded with respect to the cohomological degree. In the case of the Jordan quiver and nilpotent CoHAs, these constraints force the Lie bracket to vanish. We also describe the equivariant nilpotent CoHAs in the presence of the action of a one-dimensional torus rescaling the first coordinate of $\mathbf{A}^2$ with weight $1$ and the second with weight $-1$. In this case, one obtains enveloping algebras of Rees Lie algebras associated with the nilpotent and the seminilpotent filtrations on the Lie algebra $W_{1+\infty}^+$, reminiscent of the description of the equivariant non-nilpotent CoHA given by Davison.
Commentsv2: a few minor corrections. v1:20 pages
