Poisson Algebras, deformations and resolutions of singularities
Objective
Funding Scheme
MCERG  European Reintegration Grants (ERG)
Coordinator
UNIVERSITY OF KENT
Address
The Registry Canterbury
Ct2 7nz Canterbury, Kent
United Kingdom
Activity type
Higher or Secondary Education Establishments
EU Contribution
€ 45 000
Administrative Contact
Robert James Shank (Dr.)
Project information
POISSONALGEBRAS
Grant agreement ID: 207330
Status
Closed project

Start date
1 October 2007

End date
30 September 2010
Funded under:
FP7PEOPLE

Overall budget:
€ 45 000

EU contribution
€ 45 000
Final Report Summary  POISSONALGEBRAS (Poisson algebras, deformations and resolutions of singularities)
In the 20th century, Poisson brackets have led to the notion of Poisson algebras: a Poisson algebra is a commutative associative algebra together with a Poisson bracket; that is, a Lie bracket that satisfies Leibniz's rule (the bracket is a derivation in each argument). The modern formulation of Poisson algebras is due among others to Lichnerowicz, Kirillov and Weinstein. Poisson algebras have connections with many areas of mathematics and physics (differential geometry, Lie groups and representation theory, noncommutative geometry, integrable systems, singularities, quantum field theory, ...) and so, because of its wide range of applications, their study is of great interest for both the mathematicians and theorical physicists. Currently, this subject is one of the most active in both Mathematics and Mathematical Physics.
One way to approach Poisson algebras is via (deformation) quantisation. In Physics, quantisation is the transition from Classical to Quantum Mechanics. Mathematically, (deformation) quantisation is the transition from Poisson algebras (Poisson geometry) to noncommutative algebras (noncommutative geometry, that is, geometry of ``noncommutative spaces''). Roughly speaking, the idea is to use the Poisson bracket in order to deform the commutative product on the Poisson algebra under considerationthe elements of this algebra being the observables of classical mechanicsand obtain a noncommutative product suitable for quantum mechanics. The existence of such deformation quantisations is a longstanding problem. The proof of the Formality Conjecture by Kontsevich led to the existence and classification of deformation quantisations of arbitrary Poisson manifolds. Because of its importance, this result has an impact on many areas of mathematics.
The main aim of the project was to study Poisson algebras, their quantisations and their resolutions of singularities. Significant positive results have been obtained by the researcher during this project. The main achievements are the following.
1. Introduction of new techniques from automaton theory in order to study the representation theory of quantum algebras.
2. Observation and study of a strong link between the torusorbits of symplectic leaves of Poisson matrix varieties and torusinvariant prime ideals in quantum matrices. Moreover the researcher and his collaborators showed that the latter were also linked to the theory of total positivity in the sense of Lusztig. This new and unexpected connection between totally nonnegative matrices and quantum matrices has allowed the development of new algorithms to study totally nonnegative matrices.
3. Development of a ``Schubert cells" approach to quantum flag varieties. Moreover the researcher and his collaborators gave a geometrical description of quantum Schubert cells.
4. New connections between Poisson cohomology and Hochschild cohomology in the singular case.
5. Study of quantum cluster algebras.
To summarise, this project led to several new and unexpected results that have been published in a wide range of international leading journals. For all these reasons, this project has been truly successful.
Project information
POISSONALGEBRAS
Grant agreement ID: 207330
Status
Closed project

Start date
1 October 2007

End date
30 September 2010
Funded under:
FP7PEOPLE

Overall budget:
€ 45 000

EU contribution
€ 45 000
Deliverables
Deliverables not available
Publications
Publications not available
Project information
POISSONALGEBRAS
Grant agreement ID: 207330
Status
Closed project

Start date
1 October 2007

End date
30 September 2010
Funded under:
FP7PEOPLE

Overall budget:
€ 45 000

EU contribution
€ 45 000
Project information
POISSONALGEBRAS
Grant agreement ID: 207330
Status
Closed project

Start date
1 October 2007

End date
30 September 2010
Funded under:
FP7PEOPLE

Overall budget:
€ 45 000

EU contribution
€ 45 000