Exotic quantum groups, Lie superalgebras and integrable systems

近年来，在对一类特殊可积系统的研究的推动下，人们发现了一些以前从未被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，即它们的发展方程通过归结为一个线性问题而存在一个精确解。当这些系统与李超代数结合时--也就是，存在一个等级来识别偶数和奇数生成元的李代数--就会出现非常规的特征。这在一定程度上是通过国际和平研究所的工作建立起来的。例如，描述这些代数的张量积的Hopf代数获得了非平凡的变形，其结果还没有完全被理解。此外，所讨论的系统表现出从哈密顿公式中看不到的对称性增强。这种“秘密的”对称性导致了新的更高能级的量在时间演化过程中是守恒的。这些现象的一个完整的数学公式还没有发展出来，为了理解数学分支的潜在含义，如代数、几何、纽结和链接不变量理论以及可积系统，人们非常想要它。这个研究项目的目的是理解这种奇异的结构，并用这种新的理解来解决代数和可积系统之间的界面上具有挑战性的问题。其中一个问题是所谓的泊松结构的非超局域性，它支配着可积系统在半经典近似下的表述。非超局域性使得这些系统的解的代数解释变得非常模糊，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对基本的奇异代数有一个严格的理解。在这个方向上的任何进展都将对数学界、英国和国际的科学环境产生重大的长期影响。我们计划通过对这些奇异代数的一组非常不同的表示进行彻底研究，以及开发处理量子超代数的新技术来解决这个问题，并从这种组合中推导出一个通用的数学公式，该公式捕捉这些表示的共同特征并对其进行推广。从这个公式出发，我们计划得到关于量子群的新结果，并将它们应用于可积系统中的非超局域性问题，遵循自顶向下的方法。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，结合了不同数学领域的思想和技术，将在从群论到几何学(哈密尔顿结构)、拓扑学(纽结不变量、格拉斯曼流形)和数学物理的广泛主题上产生新的结果。

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.