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Representations of quantum affine algebras and applications

Total Cost €


EC-Contrib. €






Project "QAffine" data sheet

The following table provides information about the project.


Organization address
address: RUE MICHEL ANGE 3
city: PARIS
postcode: 75794

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country France [FR]
 Project website
 Total cost 1˙182˙000 €
 EC max contribution 1˙182˙000 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-CoG
 Funding Scheme ERC-COG
 Starting year 2015
 Duration (year-month-day) from 2015-09-01   to  2020-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. They can be defined as quantizations of affine Kac-Moody algebras or as affinizations of finite type quantum groups (Drinfeld Theorem). The representation theory of quantum affine algebras is very rich. It has been studied intensively during the past twenty five years from different point of views, in particular in connections with various fields in mathematics and in physics, such as geometry (geometric representation theory, geometric Langlands program), topology (invariants in small dimension), combinatorics (crystals, positivity problems) and theoretical physics (Bethe Ansatz, integrable systems). In particular, the category C of finite-dimensional representations of a quantum affine algebra is one of the most studied object in quantum groups theory. However, many important and fundamental questions are still unsolved in this field. The aim of the research project is to make significant advances in the understanding of the category C as well as of its applications in the following five directions. They seem to us to be the most promising directions for this field in the next years : 1. Asymptotical representations and applications to quantum integrable systems, 2. G-bundles on elliptic curves and quantum groups at roots of 1, 3. Categorications (of cluster algebras and of quantum groups), 4. Langlands duality for quantum groups, 5. Proof of (geometric) character formulas and applications. The resources would be used for the following : (1) Hiring of 2 PhD students (in 2015 and 2017). (2) Hiring of 2 Postdocs (in 2015 and 2017). (3) Invitations and travel for ongoing and future scientific collaborations. (4) Organization of a summer school in Paris on quantum affine algebras.


year authors and title journal last update
List of publications.
2019 Bittmann, Léa
Quantum Grothendieck rings as quantum cluster algebras
published pages: , ISSN: , DOI:
2019 Hernandez, David
Quantum periodicity and Kirillov-Reshetikhin modules
published pages: , ISSN: , DOI:
Progress in Mathematics (to appear) 2020-03-11
2019 Elie Casbi
Newton-Okounkov bodies for categories of modules over quiver Hecke algebras
published pages: , ISSN: , DOI:
2019 Hernandez, David; Leclerc, Bernard
Quantum affine algebras and cluster algebras
published pages: , ISSN: , DOI:
Progress in Mathematics (to appear) 2020-03-11
2019 Hernandez, David
Stable maps, Q-operators and category O
published pages: , ISSN: , DOI: 2020-03-11
2019 Casbi, Elie
Dominance order and monoidal categorification of cluster algebras
published pages: , ISSN: , DOI:
Pacific Journal of Mathematics (to appear) 2020-03-11
2019 David Hernandez
Grothendieck ring isomorphims, cluster algebras and Kazhdan-Lusztig polynomials
published pages: , ISSN: , DOI:
Oberwolfach Reports 2020-03-11
2019 d\'Andecy, L. Poulain; Walker, R.
Affine Hecke algebras of type D and generalisations of quiver Hecke algebras
published pages: , ISSN: , DOI:
2019 Léa Bittmann
Anneaux de Grothendieck quantiques, algèbres amassées et catégorie O affine quantique
published pages: , ISSN: , DOI:
2019 Inoue, Rei; Ishibashi, Tsukasa; Oya, Hironori
\"Cluster realizations of Weyl groups and higher Teichm\"\"uller theory\"\"\"
published pages: , ISSN: , DOI:
2017 J-H. Jung and M. Kim
Supersymmetric polynomials and the center of the walled Brauer algebra
published pages: , ISSN: , DOI:
arXiv 2020-01-17
2016 David Hernandez, Bernard Leclerc
Cluster algebras and category forrepresentations of Borel subalgebras of quantum affine algebras
published pages: 2015-2052, ISSN: 1937-0652, DOI: 10.2140/ant.2016.10.2015
Algebra & Number Theory 10/9 2020-01-17
2017 Ruari Walker
Convolution Products and R-Matrices for KLR Algebras of Type B
published pages: , ISSN: , DOI:
arXiv 2020-01-17
2017 Loic Poulain d\'Andecy, Ruari Walker
Affine Hecke algebras and generalisations of quiver Hecke algebras for type B
published pages: , ISSN: , DOI:
arXiv 2020-01-17
2018 David Hernandez
Advances in R-matrices and their applications
published pages: , ISSN: 0303-1179, DOI:
Astérisque (to appear) 2020-01-17
2017 Léa Bittmann
Asymptotics of Standard Modules of Quantum Affine Algebras
published pages: , ISSN: 1386-923X, DOI: 10.1007/s10468-018-9818-0
Algebras and Representation Theory 2020-01-17
2016 David Hernandez, Bernard Leclerc
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules
published pages: 1113-1159, ISSN: 1435-9855, DOI: 10.4171/JEMS/609
Journal of the European Mathematical Society 18/5 2020-01-17
2015 Edward Frenkel, David Hernandez
Baxter’s relations and spectra of quantum integrable models
published pages: 2407-2460, ISSN: 0012-7094, DOI: 10.1215/00127094-3146282
Duke Mathematical Journal 164/12 2020-01-17
2018 Edward Frenkel, David Hernandez
Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers
published pages: 361-414, ISSN: 0010-3616, DOI: 10.1007/s00220-018-3194-9
Communications in Mathematical Physics 362/2 2020-01-17
2018 David Hernandez
Spectra of quantum integrable systems and cluster algebras, Langlands duality and category O
published pages: , ISSN: 1660-8933, DOI:
Oberwolfach Reports (to appear) 2020-01-17
2017 David Hernandez
Cyclicity and R-matrices
published pages: , ISSN: , DOI:
arXiv 2020-01-17
2017 Yoshiyuki Kimura and Hironori Oya
Twist automorphisms on quantum unipotent cells and dual canonical bases
published pages: , ISSN: , DOI:
2018 Hernandez, David; Oya, Hironori
Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
published pages: , ISSN: , DOI:
arXiv 1 2020-01-17
2017 Hironori Oya
The Chamber Ansatz for quantum unipotent cells
published pages: , ISSN: , DOI:
Transformation Groups (to appear) 2020-01-17

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