Exotic quantum groups, Lie superalgebras and integrable systems

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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.