AI 中文总结
为具有典范三次势的三倍箭图的循环幂零上同调霍尔代数建立洗牌代数模型，借此关联其与量子库仑分支代数，证明特化后的超交换性，给出两者生成元，刻画BPS李代数，得到Kac多项式新公式并证明相关猜想。
AI 中文摘要
我们为具有典范三次势的三倍箭图的循环幂零上同调霍尔代数（CoHA）给出了一个明确的洗牌代数模型。结果，我们（1）将循环幂零CoHA与相应箭图规范理论的量子库仑分支代数联系起来；（2）表明循环幂零CoHA在\(\hbar = 0\)特化后是超交换的；（3）给出了循环幂零CoHA和完整预投射CoHA的生成元；（4）通过某些次数和整除条件得到了完整预投射CoHA的BPS李代数的明确刻画。这给出了关于箭图的Kac多项式的一个新公式，用某些多项式向量空间的维数表示。我们还证明了关于局部洗牌代数的球生成的一个猜想，并表明对于ADE箭图，循环幂零CoHA是杨式代数的Drinfeld - Gavarini对偶的正半部分。
英文摘要
We give an explicit shuffle algebra model for the loop-nilpotent cohomological Hall algebra (CoHA) of a tripled quiver with canonical cubic potential. As consequences, we (1) relate the loop-nilpotent CoHA to the quantized Coulomb branch algebra of the corresponding quiver gauge theory, (2) show that the loop-nilpotent CoHA is supercommutative after specialization at $\hbar=0$, (3) give generators for both the loop-nilpotent CoHA and the full preprojective CoHA, and (4) obtain an explicit characterization of the BPS Lie algebra of the full preprojective CoHA via certain degree and divisibility conditions. This gives a new formula for the Kac polynomials of the quiver in terms of the dimensions of certain vector spaces of polynomials. We also prove a conjecture on the spherical generation of the localized shuffle algebra and show that for ADE quivers, the loop-nilpotent CoHA is the positive half of Drinfeld-Gavarini dual of the Yangian.
