Quantum groups in action

Lead Research Organisation: University of Central Lancashire
Department Name: Jeremiah Horrocks Institute

Abstract

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Description This grant has currently produced two major outcomes, both still in progress.

1) A study of "quantum graphs" in complete generality, unifying and extending previous results in the literature, preprint at https://arxiv.org/abs/2203.08716
These mathematical objects arise from both quantum information / communication considerations in Communication Science; from category theory considerations in an approach to formalising quantum information theory ideas; and from purely mathematical constructions, as generalisations of graphs/networks to a noncommutative setting. It has been realised that these approaches are all the same, at least under certain conditions. Our work removes any such conditions, by generalising one of the definitions. We also present a purely "algebraic" construction to showing these equivalences. We study Quantum Isomorphisms of these objects, in the sense of compact quantum groups, again extending previous knowledge by showing how to make sense of this construction in the quantum communication sense of a quantum graph.

2) A detailed study of the "approximation property" for Locally Compact Quantum Groups. The latter are a generalisation of locally compact groups-- objects which encode the symmetries of "continuous" mathematical objects-- where the associated algebra of functions on the group may be non-commutative. This provides the correct framework for studying a general form of Fourier Theory (or Pontryagin Duality), and such objects, especially in the compact/discrete form, arise as "quantum" symmetries of other objects (such as graphs, discussed above). The underlying mathematical language is that of Operator Algebras, itself a large area of study. The approximation property can be seen as a very weak of providing a "finite" approximation to some structure, in a way that the overall structure can be reconstructed from these finite parts. There is classically a strong interplay between this property for groups (where it has nice stability results, leading to many examples) and a related property for operator algebras built out of these groups, which helps to understand these algebras. Our work extends both these facets-- examples and preservation under various constructions, and links with operator algebras-- to the quantum group setting.
Exploitation Route We believe that the work on quantum groups should be of much use to other researchers. We have taken pains to present our work as a self-contained construction, and we hope that this, along with our advances, will form a useful reference for students, and established researchers, who wish to learn about quantum graphs. In particular, the correspondence between the differing viewpoints on quantum graphs allows one to move between one perspective where it is very easy to generate examples, and another perspective, where it is very easy to perform calculations. Finding efficient ways to exploit this correspondence would seem to be key to more fully exploiting the correspondence.
Sectors Digital/Communication/Information Technologies (including Software)