Generic points in representation varieties with applications to Lie theory and 0-Schur algebras

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

Representation theory of quivers (RQT)and Lie theory (LT) are closelyrelated, for instance via root systems.This proposalexplores new connectionsbetween RTQ, LT,quantised enveloping algebras (QEA) and quantised Schur algebras ( QSA).The applicant will first study generic phenomena inrepresentation varieties, in particular, the existence of open orbits, and genericpoints in varieties of pairs of projective representations. Then she will work oninterpreting the generic phenomena in the setting of LT, QEA and QSA.The main applications will be to show the existence of Richardson elementsfor some classes of Lie algebras, in particular for seaweed Lie algebras. Otherapplications include a natural interpretation of combinatorial properties ofroot systems of simple Lie algebras and a new construction of crystal bases.

Planned Impact

The project combines different research areas in mathematics, it will establish new connections and open up new research areas for the future. The research results achieved will be reported in seminars and presented at conferences. Finally the results will be sent to leading journals in mathematics to be published. The applicant has a wide network of international research contacts, who will benefit from the results of this project. Via the cooperation and communication between the applicant and her contacts, this project will contribute to international academic cooperation between UK, and for instance China, Germany, France, Switzerland and Norway.

Publications

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Bernt Tore Jensen And Xiuping Su (Authors) (2016) Richardson elements of Seaweed Lie algebras of classic type in ArXiv: 1601.01755 (to be resubmitted to Advances in Mathematics, after being edited w.r.t. Referee's comments)

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Bernt Tore Jensen, Xiuping Su And Guiyu Yang (Authors) Degenerate 0-Schur algebras in to be submitted to ArXiv and a research journal to be considered for publication soon.

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Bernt Tore Jensen, Xiuping Su And Guiyu Yang(Authors) Idempotents and Gabriel quivers with relations for 0-Schur algebras in In preparation

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Jensen B (2015) A geometric realisation of 0-Schur and 0-Hecke algebras in Journal of Pure and Applied Algebra

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Jensen B (2016) Projective modules of 0-Schur algebras in Journal of Algebra

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Jie Du, Bernt Tore Jensen And Xiuping Su (authors) (2015) Presenting Hecke endomorphism algebras by Hasse quivers with relations in Journal of Pure and Applied Algebra)

 
Description The research project was successful and the main goals were achieved. In particular I have achieved the following.


1. The geometric construction of 0-Schur algebras and the structure of 0-Schur algebras.

Since the study of quantum groups in physics, the subject of quantised algebras has been an important research area in mathematics. The properties of a quantised algebra depends on an extra parameter q. The 0-version of the algebra (i.e. when q=0) has in particular a quite different feature from the generic version.

A geometric construction of the 0-Schur algebra leads to a monoid structure and thus makes the 0-algebra easier to study. For instance, we give a natural construction of an important class of modules (i.e. projective modules) and study the relations between the indecomposable projective modules via homomorphisms, which can lead to a good presentation of the algebras using oriented graphs (i.e. quivers).

2. On generic equivalence: generic equivalence studies relation between sub-varieties of generic points in different algebraic varieties. We apply generic equivalence to show that the existence of Richardson elements for quiver Lie algebras by reducing the problem to about existence of open orbits of simpler varieties with a simpler group action.

3. The existence of Richardson elements for seaweed Lie algebra/double parabolic algebras for classic Lie algebras. Our results on Richardson elements for seaweed Lie algebras of classical type demonstrate a striking phenomenon similar to the classification of simple Lie algebras. That is , Richardson elements exists for seaweed Lie algebras of classical type and they do not necessarily exist for seaweed Lie algebras of infinite type.
Exploitation Route The geometric methods used in the study of algebraic structures.

The new connections between representation theory of quivers/associative algebras and Lie theory.

The study of generic relations between different algebraic varieties, using generic equivalence.
Sectors Other

 
Description Affine cellular and affine Schur algebras (bulit on the research work in the project, the research assistant Dr Yang applied for and successfully obtained a grant from China).
Amount ¥400,000 (CNY)
Funding ID NSFC(natural science foundation of China) 11671234 
Organisation National Natural Science Foundation of China 
Sector Public
Country China
Start 01/2017 
End 01/2021
 
Description 0-Schur algebras 
Organisation Norwegian University of Science and Technology (NTNU)
Country Norway 
Sector Academic/University 
PI Contribution Each has equal contribution to the projects.
Collaborator Contribution Collaboration in research.
Impact Two papers published published in Journal of Pure and Applied algebra, and Journal of Algebra respectively; one is submitted and one is in preparation.
Start Year 2011
 
Description 0-Schur algebras 
Organisation Shandong University
Country China 
Sector Academic/University 
PI Contribution Each has equal contribution to the projects.
Collaborator Contribution Collaboration in research.
Impact Two papers published published in Journal of Pure and Applied algebra, and Journal of Algebra respectively; one is submitted and one is in preparation.
Start Year 2011
 
Description Hecek endomorphism algebras 
Organisation Norwegian University of Science and Technology (NTNU)
Department Department of Mathematical Sciences
Country Norway 
Sector Academic/University 
PI Contribution We contributed equally to the project.
Collaborator Contribution We contributed equally to the project.
Impact One paper is published on ArXiv and is submitted to a research journal to be considered for publication.
Start Year 2014
 
Description Hecek endomorphism algebras 
Organisation University of New South Wales
Country Australia 
Sector Academic/University 
PI Contribution We contributed equally to the project.
Collaborator Contribution We contributed equally to the project.
Impact One paper is published on ArXiv and is submitted to a research journal to be considered for publication.
Start Year 2014
 
Description Richardson elements 
Organisation Norwegian University of Science and Technology (NTNU)
Country Norway 
Sector Academic/University 
PI Contribution We contributed equally to the project.
Collaborator Contribution We contribute equally to the project.
Impact One paper (42 pages) is published in Algebra and Representation Theory and one is published on ArXiv and is to be submitted to a research journal soon.
Start Year 2012
 
Description BLOC (oxford, Sheffield) 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach National
Primary Audience Participants in your research and patient groups
Results and Impact The project supported both the PI and the post-doc to attend the colloqium. The PI gave a talk at the Artin-BLOC meeting at the satellite meeting
at the University of Sheffield and the post-doc attend a BLOC meeting at Oxford.

No direct impacts, but via communication with peers, the research results reached a broad audience.
Year(s) Of Engagement Activity 2012
 
Description ICRA (Bielefeld) 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact To learn from peers and to disseminate the research results.

Research results reach a broad audience.
Year(s) Of Engagement Activity 2012
 
Description Visit from Australia 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact Research cooperation.

A preliminary project was develop. I was invited then to visit UNSW and we have a paper in preparation.
Year(s) Of Engagement Activity 2012,2013
 
Description Visit from Norway 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Participants in your research and patient groups
Results and Impact Dr Dag Madsen visited me at Bath for research collaboration.

From this visit, we produced a ongoing project on homological algebra.
Year(s) Of Engagement Activity 2012
 
Description Visits from China 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Explore possibility of collaboration.

I was invited to give a talk at Xiamen University in China and I am invited to visit ShangDong University of Science and Technology. Research results reach a broad audience.
Year(s) Of Engagement Activity 2011,2013