标题： 格拉斯曼流形的量子K-理论来自杨-Baxter代数 显示英文标题

标题： Quantum K--theory of Grassmannians from a Yang-Baxter algebra

摘要： 在先前的一篇论文中，两位作者定义了一个$5$-顶点 Yang-Baxter 代数（一个 Hopf 代数），它作用于 Grassmannian 的等变量子 K-环的和$\mathrm{Gr}(k;n)$，其中$k$从$0$变化到$n$。我们几何地构造了作用于量子 K-环上的算子来描述这个作用。特别是，定义 Yang-Baxter 代数的$R$矩阵对应于左 Weyl 群的作用。最重要的是，我们利用 Grassmannian 的量子 K-理论的“量子=经典”陈述来证明 Yang-Baxter 代数生成元作用的显式几何解释。单值矩阵的对角线元素由明确定义的类的量子 K-乘法给出，非对角线元素由某些推拉卷积给出。我们利用这一点找到了量子 K-环中固定点类的量化，对应于 Yang-Baxter 代数的 Bethe 向量。在每个量子 K-环上，我们证明了两个 Frobenius 结构（一个来自几何，另一个来自可积系统构造）是一致的。我们讨论了几个应用，包括扩展仿射 Weyl 群在量子 K-理论环上的作用（扩展了 Seidel 作用），一个量子化的局部化映射（相对于量子 K-乘积是一个环同态），以及一个图形演算来乘以余标准商丛的 Hizrebruch$\lambda_y$类。 在附录中，我们说明当$n=2$时的结果。 显示英文摘要

摘要： In an earlier paper, two of the authors defined a $5$-vertex Yang-Baxter algebra (a Hopf algebra) which acts on the sum of the equivariant quantum K-rings of Grassmannians $\mathrm{Gr}(k;n)$, where $k$ varies from $0$ to $n$. We construct geometrically defined operators on quantum K-rings describing this action. In particular, the $R$-matrix defining the Yang-Baxter algebra corresponds to the left Weyl group action. Most importantly, we use the `quantum=classical' statement for the quantum K-theory of Grassmannians to prove an explicit geometric interpretation of the action of generators of the Yang-Baxter algebra. The diagonal entries of the monodromy matrix are given by quantum K-multiplications by explicitly defined classes, and the off-diagonal entries by certain push-pull convolutions. We use this to find a quantization of the classes of fixed points in the quantum K-rings, corresponding to the Bethe vectors of the Yang-Baxter algebra. On each of the quantum K-rings, we prove that the two Frobenius structures (one from geometry, and the other from the integrable system construction) coincide. We discuss several applications, including an action of the extended affine Weyl group on the quantum K-theory ring (extending the Seidel action), a quantum version of the localization map (which is a ring homomorphism with respect to the quantum K-product), and a graphical calculus to multiply by Hirzebruch $\lambda_y$ classes of the dual of the tautological quotient bundle. In an Appendix we illustrate our results in the case when $n=2$.