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As a result, finite differential forms do not naturally commute with the functions on a discrete space, one needs the kind of generalisation explained above even though the functions themselves commute. Similarly, we can endow finite groups with braided Lie algebra structures.
This is one of many potential applications of the subject to other areas of pure mathematics. An introduction for mathematicians is in these LMS lecture notes.
If you know a bit of Fourier theory then what it corresponds to is gravity or curvature in momentum space Fourier conjugate to space time. The problem of quantum gravity then amounts to developing a theory with gravity and cogravity consistent with one another. If you have some exposure to quantum theory then you may know that the whole point of the correspondence principle in quantum mechanics is that certain macroscopic concepts like position and momentum coordinates have analogues as noncommuting operators.
But just how much of the macroscopic or geometrical world has its analogue in the quantum domain? The question was posed with the birth of quantum mechanics in the first decades of the 20th century.
Hopf Algebras and Quantum Groups - CRC Press Book
It is remarkable that it has taken the rest of the century to find a reasonable answer to this question, but I believe that it is now emerging. 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.
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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.