Exotic quantum groups, Lie superalgebras and integrable systems

近年来，在对一类特殊的可积系统的研究的推动下，人们发现了以前没有被研究过的新的、显著的数学结构。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，即它们的演化方程可以通过简化成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，李代数存在一个分级来识别偶数和奇数生成器——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，其结果尚未被完全理解。此外，所讨论的系统表现出一种对称增强，这在哈密顿公式中是不明显的。这种“秘密的”对称性导致了在时间演化过程中新的更高层次的量被守恒。这些现象的一个完整的数学公式还没有被开发出来，为了理解数学分支的潜在含义，如代数、几何、结和连杆不变量理论和可积系统，人们非常寻求它。本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泊松结构的非超局域性，它控制着可积系统在半经典近似中的表述。非超局部性使得这些系统解的代数解释变得极其模糊，是一个多年来一直困扰数学家的难题。我们相信，在这个方向上取得重大进展的关键是对潜在的奇异代数的严格理解。在这个方向上的任何进展都将对数学界以及英国和国际上的科学环境产生重大的长期影响。我们计划通过结合对这些奇异代数的一组非常多样化的表示的深入研究以及处理量子超代数的新技术的发展来解决这个问题，并从这种组合中得出一个通用的数学公式，该公式捕捉了这些表示的共同特征并对其进行了推广。从这个公式中，我们计划推导出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维和无限维霍普夫代数、李双代数及其相关辛结构的技术。该项目的跨学科特征，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（结不变量，格拉斯曼流形）和数学物理。

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.