Summary
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.
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| Web resources: | https://cordis.europa.eu/project/id/101027463 |
| Start date: | 01-12-2021 |
| End date: | 30-11-2023 |
| Total budget - Public funding: | 212 933,76 Euro - 212 933,00 Euro |
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Original description
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.Status
CLOSEDCall topic
MSCA-IF-2020Update Date
28-04-2024
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