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

本研究项目涉及三个数学领域：代数表示论、可积系统和几何表示论。数学的前两个分支起源于物理学，而最后一个分支则研究纯代数概念在几何中的应用。表示理论涉及对具有附加结构的向量空间的对称性的研究，例如我们的三维空间(更广泛地说，无限维空间)。这些对称性通常可以被认为是代数结构。以下两种情况特别值得注意：(1)当存在足够多的成对交换对称时，在可积系统的研究中具有中心重要性；(2)当基础向量空间由几何对象产生时，在几何表示理论中具有中心重要性。在这个项目中，首席研究人员计划在称为量子环状代数和仿射延吉安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，是近几十年来被广泛研究的经典量子仿射代数和延安代数的推广。本项目致力于量子环状代数和仿射延安的研究。PI的计划如下：(1)发展所有ADE型量子环/仿射代数的混洗实现。(2)统一其表示的所有已知不同构造，并提供更广泛的洗牌型模类。(3)利用Shuffle实现研究量子环状代数的极大交换子代数，并发展了一种新的方法来解决著名的Bethe ansatz问题，涉及此类极大交换子代数在感兴趣的表示类中的对角化。(4)将上述极大交换子代数与Nakajima箭簇和仿射Laumon空间的量子上同调和量子K-理论相联系。(5)将上述结果推广到仿射延安的加性情形。(6)研究与“退化”菱形的所有顶点对应的自变量代数的移位的量子化，并刻画这些代数与KZ方程和Casimir方程菱形右端的关系。

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.