Representation theory of Lie algebras and quantum groups

About the Project

These projects are open to students worldwide, but have no funding attached. Therefore, the successful applicant will be expected to fund tuition fees at the relevant level (home or international) and any applicable additional research costs. Please consider this before applying.

Lie algebras originally appeared in the work of the Norwegian mathematician Sophus Lie as infinitesimal symmetries of geometric objects called nowadays manifolds. In this project we explore algebraic properties of Lie algebras, of relevant Lie and algebraic groups, and of their generalisations called quantum groups which are neither commutative nor cocommutative Hopf algebras. We also study representation theory of these algebraic objects, i.e. their actions on linear spaces.

Representation theory of quantum groups at roots of unity.

Quantum groups are Hopf algebra deformations of universal enveloping algebras of complex finite-dimensional semi-simple Lie algebras depending on a deformation parameter q.

Representation theory of quantum groups at roots of unity, i.e. when q is a root of unity, has some striking similarities with representation theory of semi-simple Lie algebras over fields of prime characteristic. The main difficulty in the study of these representations is a fundamental difference between the field of real numbers and fields of prime characteristic: the latter ones are not ordered. As a consequence, classical results by H. Weyl on classification of finite-dimensional irreducible representations of finite-dimensional semi-simple Lie algebras in terms of highest weights have no analogues over fields of prime characteristic. Similar phenomenon occurs in the case of quantum groups at roots of unity.

In monograph [1] a long-standing De Concini-Kac-Procesi conjecture on dimensions of irreducible representations of quantum groups of unity is proved. The proof is based on a new Weyl group combinatorics developed in [1]. In this project we are going to further pursue the study of the structure of representations of quantum groups of unity basing on this combinatorics.

q-W-algebras.

q-W-algebras are examples of algebras obtained by the so-called quantum Poisson reduction using quantum groups. Originally, they appeared in physics as candidates for symmetry algebras in quantum field theory. In representation theory of quantum groups they play the same role as the symmetric group in the classical Schur-Weyl duality between irreducible finite-dimensional representations of the general linear group and of the symmetric group. They are also key ingredients of the q-version of the conjectural geometric Langlands correspondence. In this project algebraic properties of q-W-algebras and their applications in representation theory will be studied.

For this project distance mode is possible subject to the background of the applicant but supervision in person is strongly preferable.

Informal enquiries can be made by contacting Dr A Sevastyanov (a.sevastyanov@abdn.ac.uk).

Decisions will be based on academic merit. The successful applicant should have, or expect to obtain, a UK Honours Degree at 2.1 (or equivalent) in Pure Mathematics. An MSc in Pure Mathematics is preferable.

We encourage applications from all backgrounds and communities, and are committed to having a diverse, inclusive team.

Application Procedure:

Formal applications can be completed online: https://www.abdn.ac.uk/pgap/login.php.

You should apply for Degree of Doctor of Philosophy in Mathematics to ensure your application is passed to the correct team for processing.

Please clearly note the name of the lead supervisor and project titleon the application form. If you do not include these details, it may not be considered for the project.

Your application must include: A personal statement, an up-to-date copy of your academic CV, and clear copies of your educational certificates and transcripts and 3 academic references.

Please note: you do not need to provide a research proposal with this application.

If you require any additional assistance in submitting your application or have any queries about the application process, please don't hesitate to contact us at researchadmissions@abdn.ac.uk

Funding Notes

This is a self-funding project open to students worldwide. Our typical start dates for this programme are February or October.

Fees for this programme can be found here Finance and Funding | Study Here | The University of Aberdeen.