H2020Индивидуална стипендия2021–2023

HADG · Hopf algebroids in quantum differential geometry

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2021-12-01 → 2023-11-30
Финансиране от ЕС: 212 934 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Хопфовите алгеброиди развиват математическата теория за квантовите групи, като например заменят стандартната геометрия с алгебри от координати. Тези инструменти помагат за разбирането на квантовото пространство-време, което е важно за развитието на квантовите изчисления и квантовата гравитация.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Hopf algebroids in quantum differential geometry

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.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

While cohomology theories of various kinds are known on algebras, here we explore the much harder problem of what is the ‘homotopy’ of an algebra as a geometric object? For example, when is an algebra ‘simply connected’? The project will make sense of this notion using a constructive approach to noncommutative differential geometry in which the possibly noncommutative algebra A is extended to a graded algebra of ‘differential forms’. The Experienced Researcher will first develop and study a recent proposal of a Hopf algebroid D_A of ‘differential operators’ associated to this data, the existence of which is implied by the More-Eilenberg theorem applied to the category of bimodules on A equipped with flat bimodule connections. In the classical case of functions on a smooth manifold, this would be a version of the path groupoid and Morita equivalent to π_1. He will then relate it to a proposed new construction of a universal (co)measuring bialgebra adapted to the differential graded case as a generalised ‘diffeomorphism group’ and to a proposed new notion of differential ‘character variety’ defined by each Hopf algebra H as the moduli of flat connections up to equivalence on quantum principal bundles over A with fibre H. Classically, the holonomy associated to a flat connection identifies this as maps from π_1 to the fibre group modulo conjugation. Using these ingredients, the further aim will be to arrive at a quantum differential geometric picture of the Turaev-Viro invariant of 3-manifolds and generalise it to a suitable class of differential algebras A. The project will also study an analogue of D_A in Connes’ spectral triple approach to noncommutative geometry based on an axiomatic ‘Dirac operator’, explore generalisations at the level of 2-categories and Hopf monads and look for applications to algebraic models of quantum gravity, where both diffeomorphism invariance and ‘loops’ are expected to play a fundamental role.

Оригинален текст от CORDIS (на английски).

Участници

QUEEN MARY UNIVERSITY OF LONDON · LONDONКоординаторОбединеното кралство