Research interests:

Representation theory, Lie theory, Lie algebras, quantum algebra, deformation theory, Langlands duality, geometric and quantum Langlands correspondences, geometric representation theory.

Publications

We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloured Kac-Moody algebras.

We formulate the GQE conjecture which predicts that every representation in the category O integebrable of a K-M algebra can be deformed into a representation of an associated GQE algebra. We give various evidences for this conjecture and make a first step towards its resolution by proving that Kac-Moody algebras without Serre relations can be deformed into GQE algebras without Serre relations.

In case the conjecture holds, we establish an analog result for coloured K-M algebras, we prove that the deformed representation theories are parallel to the classical one, we explicit a deformed Serre presentation for GQE algebras, we prove that the latter are the representatives of a natural class of formal deformations of K-M algebras and are trivial in finite type. As an application, we explain in terms of interpolation both classical and quantum Langlands dualities between representations of Lie algebras, and we propose a new approach which aims at proving a conjecture of Frenkel-Hernandez.

In general, we prove that representations of two isogenic coloured K-M algebras can be interpolated by representations of a third one. Observing that standard quantum algebras satisfy the GQE conjecture, we give a new proof of the previously mentioned classical Langlands duality (the first proofs are due to Littelmann and McGerty).

[2] Algèbres enveloppantes quantiques généralisées, algèbres de Kac-Moody colorées et interpolation de Langlands

PhD thesis, Université Paris Diderot - Paris 7 (2013)

PDF

We propose in this thesis a new deformation process of Kac-Moody (K-M) algebras and their representations. The direction of deformation is given by a collection of numbers, called a colouring. The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras.PhD thesis, Université Paris Diderot - Paris 7 (2013)

We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloured Kac-Moody algebras.

We formulate the GQE conjecture which predicts that every representation in the category O integebrable of a K-M algebra can be deformed into a representation of an associated GQE algebra. We give various evidences for this conjecture and make a first step towards its resolution by proving that Kac-Moody algebras without Serre relations can be deformed into GQE algebras without Serre relations.

In case the conjecture holds, we establish an analog result for coloured K-M algebras, we prove that the deformed representation theories are parallel to the classical one, we explicit a deformed Serre presentation for GQE algebras, we prove that the latter are the representatives of a natural class of formal deformations of K-M algebras and are trivial in finite type. As an application, we explain in terms of interpolation both classical and quantum Langlands dualities between representations of Lie algebras, and we propose a new approach which aims at proving a conjecture of Frenkel-Hernandez.

In general, we prove that representations of two isogenic coloured K-M algebras can be interpolated by representations of a third one. Observing that standard quantum algebras satisfy the GQE conjecture, we give a new proof of the previously mentioned classical Langlands duality (the first proofs are due to Littelmann and McGerty).

[1] Groupes Quantiques d’Interpolation de Langlands de Rang 1

International Mathematics Research Notices 2013, no. 6, 1268–1323

PDF Preprint arXiv: 1107.1992

We study a certain family, parameterized by an positive integer g, of double deformations of the envelopping algebra U(sl2), in the spirit of E. Frenkel and D. Hernandez (arXiv:0809.4453). We prove that each of these double deformations simultaneously deforms two rank 1 quantum groups. We show this interpolating property explains the Langlands duality for the representations of the quantum groups in rank 1. Hence we prove a conjecture of E. Frenkel and D. Hernandez (arXiv:0809.4453) in this case : we prove for all g the existence of representations which simultaneously deform two Langlands dual representations. We also study more generaly the finite rank representation theory of this family of double deformations.
International Mathematics Research Notices 2013, no. 6, 1268–1323

PDF Preprint arXiv: 1107.1992

Contact details

Email: ab2177 AT cam.ac.uk

Department of Pure Mathematics and Mathematical Statistics,

Centre for Mathematical Sciences,

University of Cambridge,

Wilberforce Road,

Cambridge,

CB3 0WB,

United Kingdom

Centre for Mathematical Sciences,

University of Cambridge,

Wilberforce Road,

Cambridge,

CB3 0WB,

United Kingdom

Office: E2.07

Telephone: +44 1223 3 37982

Telephone: +44 1223 3 37982

St John's College,

Cambridge,

CB2 1TP,

United Kingdom.

Cambridge,

CB2 1TP,

United Kingdom.

Office: H1, Second Court

Telephone: +44 1223 3 37721

Telephone: +44 1223 3 37721