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Exotic quantum groups, Lie superalgebras and integrable systems

奇异量子群、李超代数和可积系统

基本信息

批准号：
EP/K014412/1
负责人：
Alessandro Torrielli
金额：
$ 12.4万
依托单位：
University of Surrey
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK014412%2F1
关键词：
Exotic quantum groups Lie superalgebras

项目摘要

In recent years, motivated by the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered which had not been investigated before. These exotic algebraic constructions extend the standard framework of quantum groups to situations where new exciting effects manifest themselves. Integrable systems have the property that their evolution equations admit an exact solution via reduction to a linear problem. When these systems are combined with Lie superalgebras - namely, Lie algebras for which a grading exists identifying even and odd generators - unconventional features emerge. This has been established in part through the work of the PI. The Hopf algebra describing tensor products of these algebras, for instance, acquires non-trivial deformations, whose consequences have not yet been fully understood. Furthermore, the systems in question exhibit a symmetry enhancement which is not manifest from the Hamiltonian formulation. This "secret" symmetry results in novel higher-level quantities being conserved during the time evolution. A complete mathematical formulation of these phenomena has yet to be developed, and it is very much sought for in order to understand potential implications for branches of mathematics such as algebra, geometry, the theory of knot and link invariants and integrable systems. The aim of this research project is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between algebra and integrable systems. One of these problems is the so-called non-ultralocality of Poisson structures, governing the formulation of integrable systems in their semiclassical approximation. Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years. We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras. Any progress in this direction will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.We plan to attack the problem by combining a thorough study of a very diverse set of representations of these exotic algebras together with the development of new techniques to treat quantum superalgebras, and to derive from this combination a universal mathematical formulation which captures the common features of these representations and generalizes them. From this formulation, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems, following a top-down approach. The tools will primarily consist of the representation theory of Lie algebras and superalgebras and the technology of finite and infinite dimensional Hopf algebras, Lie bialgebras and their associated symplectic structures. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry (Hamiltonian structures), topology (knot invariants, Grassmannian manifolds) and mathematical physics.

近年来，由于对一类特殊的可积系统的研究，人们发现了一些以前没有研究过的新的数学结构。这些奇异的代数结构将量子群的标准框架扩展到新的激发效应显现的情况。可积系统具有这样的性质，即它们的发展方程可以通过化为线性问题而得到精确解。当这些系统与李超代数相结合时--也就是说，李代数存在一个等级来识别偶数和奇数生成元--非常规的特征就出现了。这在一定程度上是通过PI的工作建立的。例如，描述这些代数的张量积的霍普夫代数获得了非平凡的变形，其后果尚未完全理解。此外，所讨论的系统表现出对称性增强，这是不明显的哈密顿公式。这种“秘密”对称性导致了新的更高层次的量在时间演化过程中被守恒。一个完整的数学公式，这些现象尚未开发，它是非常寻求，以了解潜在的影响，如代数，几何，理论的数学分支，结和链接不变量和可积系统。这个研究项目的目的是了解这种奇异的结构，并利用这种新的理解来攻击代数和可积系统之间的接口具有挑战性的问题。这些问题之一是所谓的非超局部性的泊松结构，管理制定的可积系统在其半经典近似。非超局部性使得这些系统的解的代数解释变得更加模糊，这是一个多年来一直挑战数学家的难题。我们认为，在这个方向上取得重大进展的关键是严格理解的基础奇异代数。在这个方向上的任何进展将有一个重大的长期影响的数学界，并在英国和国际上的科学环境。我们计划攻击的问题相结合的一个非常多样化的表示这些奇异代数的一组彻底的研究与发展的新技术来处理量子超代数，并从这种组合中导出一个通用的数学公式，该公式抓住了这些表示的共同特征并将其推广。从这个公式中，我们计划得出量子群的新结果，并将其应用于可积系统中的非超局部性问题，遵循自上而下的方法。这些工具将主要包括李代数和超代数的表示理论以及有限和无限维Hopf代数，李双代数及其相关辛结构的技术。该项目的跨学科性质，结合了来自不同数学领域的思想和技术，将在广泛的主题中产生新的结果，从群论到几何（哈密顿结构），拓扑学（结不变量，格拉斯曼流形）和数学物理。