论文标题
量子仿射和环形代数的洗牌方法
Shuffle approach towards quantum affine and toroidal algebras
论文作者
论文摘要
这些是东京海洋科学与技术大学作者在2019年3月的第二周提供的关于崩溃代数的崩溃课程的详细讲义。这些注释包括三章，提供了一个单独的处理方法，可提供：$ \ mathfrak的量子循环代数（slfrak {sl}} _n $以及他们的超级和2- paremer earter eral earliz of tor earter of tor of tor of tor of tor of tor of tor of他们$ \ mathfrak {gl} _1 $，以及$ \ mathfrak {sl} _n $的量子环形代数。我们提供了相应的``阳性''亚词法以及通勤亚代词的缩影实现，并为环形代数的某些组合表示。涉及的关键技术之一是``专业化地图''。每章旨在强调理论的不同方面：在第一章中，我们使用Shuffle代数来构建一个新的PBWD基础家庭，用于$ a $ a $量子循环代数及其整体形式；在第二章中，我们提供了对Fock模块的几何解释，并使用交换性次数的缩影描述来构建Heisenberg代数的动作，对Equivariant $ k $ - 希尔伯特·希尔伯特（Hilbert of Hilbert）的观点构建；在上一章中，我们使用Miki的同构词与$ \ Mathfrak {Sl} _n $的量子环形代数的顶点和组合表示相关联，并使用Shuffle实现来显式计算Bethe bethe commuttative conmutighative subergebras及其限制。后一种结构的灵感来自恩里克斯（Enriquez）的工作，该作品将散装代数与量子贴合的代数的相关函数相关。
These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of $\mathfrak{sl}_n$ (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of $\mathfrak{gl}_1$, and the quantum toroidal algebras of $\mathfrak{sl}_n$. We provide the shuffle realization of the corresponding ``positive'' subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of ``specialization maps''. Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type $A$ quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant $K$-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of $\mathfrak{sl}_n$ using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras.
