Commutative Subalgebras and Bethe Ansatz for Quantum Affine and Toroidal Algebras via the Shuffle Approach

这个研究项目是数学三个领域的交叉：代数表示理论、可积系统和几何表示理论。数学的前两个分支起源于物理学，而数学的后一个分支则涉及纯代数概念在几何中的应用。表征理论关注向量空间的对称性研究，例如我们的三维空间（更一般地说，无限维空间）与附加结构。这些对称性通常可以被认为是代数结构。以下两种情况特别令人感兴趣：(1)当存在足够多的对交换对称时，对交换对称在研究可积系统中是非常重要的；(2)底层向量空间产生于几何对象的情况在几何表示理论中是至关重要的。在这个项目中，首席研究员计划在被称为量子环面代数和仿射杨算子的代数的特殊情况下探索这些概念。这些结合代数可以看作是李代数的变形，并提供了近几十年来被广泛研究的经典量子仿射代数和杨代数的推广。本课题致力于量子环面代数和仿射杨算子的研究。PI的计划如下：(1)开发ADE型所有量子环面/仿射代数的洗牌实现。(2)统一所有已知的不同的表示结构，并提供更广泛的洗牌类型模块类。(3)通过shuffle实现研究量子环面代数的极大交换子代数，并对著名的Bethe ansatz问题提出了一种新的（shuffle）方法，该方法涉及这类极大交换子代数在有趣的表示类中的对角化。(4)将上述极大交换子代数与Nakajima颤变和仿射Laumon空间的量子上同调和量子k理论的研究联系起来。(5)将上述结论推广到仿射杨量的加性情况。(6)研究了“退化”菱形各顶点对应的参数代数位移的量化，并描述了这些代数与KZ方程和Casimir方程菱形右侧的关系。

This research project lies in the intersection of three fields of mathematics: algebraic representation theory, integrable systems, and geometric representation theory. The former two branches of mathematics originate from physics, while the last deals with applications of purely algebraic concepts to geometry. Representation theory concerns the study of symmetries of a vector space such as our three-dimensional space (more generally, an infinite dimensional space) with additional structures. These symmetries can be often thought of as algebraic structures. The following two cases are of particular interest: (1) the case of pair-wise commuting symmetries, when sufficiently many exist, is of central importance in the study of integrable systems; (2) the case when the underlying vector space arises from geometric objects is of central importance in geometric representation theory. In this project the principal investigator plans to explore these concepts in the particular cases of algebras known as quantum toroidal algebras and affine Yangians. These associative algebras can be viewed as deformations of Lie algebras and provide generalizations of the classical quantum affine algebras and Yangians that have been studied extensively in recent decades. This project is devoted to the study of quantum toroidal algebras and affine Yangians. The PI's plan is as follows: (1) Develop shuffle realizations of all quantum toroidal/affine algebras of ADE type. (2) Unify all known different constructions of their representations and provide a wider class of shuffle type modules. (3) Study the maximal commutative subalgebras of quantum toroidal algebras via the shuffle realization, and develop a new (shuffle) approach to the well-known Bethe ansatz problem, concerning diagonalization of such maximal commutative subalgebras in interesting classes of representations. (4) Relate the aforementioned maximal commutative subalgebras to the study of quantum cohomology and quantum K-theory of Nakajima quiver varieties and affine Laumon spaces. (5) Generalize all the above to the additive case of affine Yangians. (6) Study quantizations of the shift of the argument algebras corresponding to all vertices of the "degeneration" rhombus and describe the relation of these algebras to the right hand sides of the KZ equation and Casimir equation rhombi.