Periodic Reporting for period 1 - HADG (Hopf algebroids in quantum differential geometry)
Reporting period: 2021-12-01 to 2023-11-30
Quantum groups or `Hopf algebras’ were enormously impactful in mathematics since the 1980s, with many connections to knot theory, integrable systems and quantum or `noncommutative’ geometry (where a continuum geometry is replaced by an algebra of coordinates which is then allowed to be noncommutative as in quantum theory). In subsequent years, people wondered about extending these structures to quantum groupoids or `Hopf algebroids’. The underlying classical concept of a `groupoid’ here is characterised by having only a partially defined product law, and such objects occur widely throughout mathematics and physics. Their quantum version should have a similarly important role, for example in Physics on quantum spacetime.
The project aimed uniquely to push forward the theory of Hopf algebroids and their interface with noncommutative differential geometry. Pure mathematical advances underly many scientific and technological discoveries down the line, and these then impact society. In the present case, these could ultimately be for example to quantum computing and to quantum gravity, where it is believed by many that the spacetime continuum has to be replaced by a quantum spacetime but where the physical consequences of this hypothesis cannot be explored without new mathematical tools.
The conclusion is that the overall scientific objectives were met establishing a sound foundation and key results on the theory of Hopf algebroids and its interface with quantum differential geometry. The project provides important new results in the field, with potential for further advances and applications in other areas. Training objectives for the Researcher were also met.
The project aimed uniquely to push forward the theory of Hopf algebroids and their interface with noncommutative differential geometry. Pure mathematical advances underly many scientific and technological discoveries down the line, and these then impact society. In the present case, these could ultimately be for example to quantum computing and to quantum gravity, where it is believed by many that the spacetime continuum has to be replaced by a quantum spacetime but where the physical consequences of this hypothesis cannot be explored without new mathematical tools.
The conclusion is that the overall scientific objectives were met establishing a sound foundation and key results on the theory of Hopf algebroids and its interface with quantum differential geometry. The project provides important new results in the field, with potential for further advances and applications in other areas. Training objectives for the Researcher were also met.
The project resulted in 5 research level mathematics publications. By the end of the period, one of these was already published in an algebra journal and another in a mathematical physics journal, with the other three remaining as preprints pending the journal publication process. A key main result was to correct the very definition of the antipode or `inversion’ for a Hopf algebroid, where pre-existing definitions turned out to be too restrictive. With our new weaker notion, we showed that there was a natural Hopf algebroid associated to a quantum principal bundle (previously this was only partially known, at a bialgebroid level), at least for the main classes of interest. These included the case where the fibre of the bundle is a Drinfeld q-deformation quantum group coordinate algebra. The work also showed how such Hopf algebroids can be constructed in the case of a trivial quantum principal bundle by a certain Hopf algebroid twisting procedure. Another main work studied `action Hopf algebroids’, which in nice cases can be seen as the quantum version of an algebra of differential operators on a quantum space. This paper also introduced for the first time a non-Abelian cohomology of Hopf algebroids and, using this, a theory of coquasi-Hopf algebroids. A third paper in this sequence extended the process of putting an `oid’ into key results in Hopf algebra theory, namely to the construction of a certain prebraided category of modules associated to a Hopf algebroid. Such modules for ordinary Hopf algebras are critical for the construction of their noncommutative differential structure, so the same result in the Hopf algebroid case is a step towards their own noncommutative differential geometry. Another paper constructs examples of Hopf algebroids for two classes of quantum principal bundles, namely quantum instanton bundles and a class of quantum homogeneous spaces. A final paper proposes a new formulation of a missing higher category notion of a 2-quantum group, as an application of Hopf algebroids.
All works were disseminated in their preprint form on the arXiv repository. Results were also disseminated in person, with the Researcher giving 14 conference and seminar talks about the results of the project. Also, in July 2023, the Researcher successfully organised a high-level conference `Hopf algebras and noncommutative geometry’ at Queen Mary University of London, with 18 speakers from the UK, USA, Belgium, France, Germany and Italy. This provided a snapshot of the state of the art in the field and of the work of the project in relation to it.
All works were disseminated in their preprint form on the arXiv repository. Results were also disseminated in person, with the Researcher giving 14 conference and seminar talks about the results of the project. Also, in July 2023, the Researcher successfully organised a high-level conference `Hopf algebras and noncommutative geometry’ at Queen Mary University of London, with 18 speakers from the UK, USA, Belgium, France, Germany and Italy. This provided a snapshot of the state of the art in the field and of the work of the project in relation to it.
The project has greatly progressed the state of the art, both in the mathematics of Hopf algebroids themselves and in their interface to quantum differential geometry. The project extended the very axioms of a Hopf algebroid, where existing notions were too restrictive for the intended role in quantum differential geometry. The project then advanced the mathematics behind this role. The project also broke a log-jam in how to put the `oid’ in previous results about Hopf algebras in relation to Drinfeld theory, notably introducing twisting of Hopf algebroids, a formulation of coquasi-Hopf algebroids and a prebraided category associated to a Hopf algebroid. The project results thereby laid the foundations for many future applications extending key lines of development successfully used for ordinary Hopf algebras, as well as applications in quantum differential geometry itself.