Flags, Quivers and Invariant Theory in Lie Representation Theory

Projektinformationen

LIEGRITS

ID Finanzhilfevereinbarung: 505078

Projekt abgeschlossen

Startdatum 1 Februar 2004

Enddatum 31 Januar 2008

Finanziert unter

• Human resources and Mobility in the specific programme for research, technological development and demonstration "Structuring the European Research Area" under the Sixth Framework Programme 2002-2006

Gesamtkosten € 2 770 654,00

EU-Beitrag € 2 770 654,00

Koordiniert durch

UNIVERSITEIT ANTWERPEN
Belgium

Final Activity Report Summary - LIEGRITS (Flags, Quivers and Invariant Theory in Lie Representation Theory)

The scientific objective of the project was to apply new emerging theories, like the geometry of path algebras, asymptotics of representations, crystal theory and non-commutative geometry, to Lie representation theory. A second goal was to allow young researchers to enter deeply in the field and acquire a polyvalent quality by the multidisciplinary aspect.

This four year project was very successful in reaching these objectives, as could be inferred by a quick look at some numbers. There were more than 1,000 publications by the team members, of which over 450 resulted directly from the project and 182 involved recruited researchers, 22 workshops and associated events were organised and about 500 person months of hired researchers were realised.

The scientific breakthroughs of the project covered almost completely the milestones set out in the planning and, in addition, several important new developments were derived by the results. A few, representative, scientific results were the following:

1. relating a certain class of finite dimensional representations of the loop algebra to certain Schubert varieties in the affine Grassmanian. The Kirillov-Reshetikkin modules and Weyl modules were in fact all Demazure modules, thus solving the Chari-Pressley conjecture.
2. in the simply-laced case a full theory of Demazure flags was established.
3. a complex classification of orbital varieties which were hyper surfaces in the nil radical of a parabolic for classical Lie algebras was obtained.
4. the Shapovalev-Kac determinant was calculated under some new conditions.
5. unanticipated significant new developments appeared in the study of invariants for parabolic actions. A slice theorem for parabolics of type A was proved and its relation to maximal Poisson commutative subalgebras similar to the mathematics of the Toda lattice was obtained.
6. new connections between representation theory and analytic number theory led to important information on L-functions.
7. non-commutative geometry allowed for gluing of infinite families of moduli spaces into a non-commutative variety. Moduli spaces of canonical linear control systems were glued and the resulting variety was identified as a specific open of an infinite Grasmannian.
8. development of the theory of cluster categories and algebras including results on cluster quivers of minimal type.
9. a solution at the Kac-Kazhdan conjecture in the general super case was obtained.
10. verification of the De Concini-Kac-Procesi conjecture on the dimension of irreducible representations of quantum groups at roots of unity.
11. the affine version of the Vogan conjecture was established in the theory of vertex algebras and conformal Lie algebras via a twisted affine generalisation of Kostant’s theory of multiplets.
12. iterated twisted algebras were introduced and an invariance under twisting theorem was proved via the relation of previously unconnected results from Hopf algebra theory and physics.
13. the training programme for young researchers was very intensive.

A broad network of regular seminars was set up, enriched with specific training activities, with some of them being organised by the young researchers themselves. Several workshops, out of 22 that were organised in total, focussed entirely on the aspect of training. Many specific courses were organised either in or outside the local doctoral programmes. Associated activities included a Socrates intensive programme, lasting three years, after the second project year and the creation of an international advanced Master’s degree with courses taught by international experts from the Liegrits network. Almost all experienced researchers (ER) in the project, with one exception, continued their research career obtaining grants or positions in Europe or the United States. All early stage researchers (ESR) either obtained their Ph.D. or were about to obtain it by the time of the project completion. The young researchers kept regular contact with their colleagues at other nodes and set up joint activities leading to fruitful cooperations and enhancing the internationalisation aspect of the project research.