We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ andtheir infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

我们对中岛海森堡算子$P_{\pm 1}$的交换以及它们在量子环形代数$U_{q_1,q_2}(\ddot {gl_1})$中作用于希尔伯特方案的格罗腾迪克群的无穷多个对应物进行分类10、24、26、32]。通过将我们的结果与[26]相结合，我们可以得到对希尔伯特方案的派生类别的几何分类$U_{q_1,q_2}(\ddot {gl_1})$作用。我们的主要技术工具是通过最小模型程序的镜头，对某些三元组和四元组的嵌套希尔伯特方案进行详细的几何研究，通过表明这些嵌套希尔伯特方案是规范的或半除数的对数终端奇点。