Applications of Hecke Algebras: Representations, Knots and Physics
ANR project JCJC ANR-18-CE40-0001
Summary
Between representation theory, knot theory and integrable systems, interactions are plentiful.
This is one of the several fruitful (and two-way) bridges between mathematics and theoretical physics.
If one had to identify a common ground for these interactions, for example in the sense of a common algebraic structure,
quantum groups would certainly be a good answer. Hecke algebras, considered in a large sense, could be another.
Hecke algebras originally appeared in the theory of modular forms in the 30's.
Since then the name "Hecke algebras" has been progressively used to refer to a wide variety of objects,
appearing and extensively studied in several areas of mathematics. Classes of examples of Hecke algebras of special interest
for this project are:
- Centralisers (endomorphisms algebras) of induced representations;
- Deformations of Coxeter groups and (complex) reflection groups;
- Quotients of group algebras of (generalised) braid groups;
- Centralisers of tensor representations of quantum groups.
Quite remarkably, all the classes of examples above have a lot in common and this is the main reason why Hecke algebras
are so important in modern mathematics: they can be studied from many points of view and they have applications in many
different fields.
The project is centered on the study of Hecke algebras and concerns their applications to/interplay between different areas of
mathematics and physics. We will focus on three main areas, where Hecke algebras and related algebras play an important role:
- Representation theory of different kind of Hecke algebras (and generalisations);
- Low-dimensional topology, mainly the study of braid groups and invariants of links;
- Theoretical physics (integrable systems, statistical models, quantum field theory).
One major objective of this project is to study these three thematics simultaneously, especially focusing on interactions between them,
and considering Hecke algebras as bridges between these areas.
Participants
Events
- February 2019: Réunion de lancement, Reims, 14-15 février 2019. Orateurs: A. Gainutdinov, L. Poulain d'Andecy, E. Wagner
- March 2019: Winter Braids IX , March, 4th - 7th 2019
- March 2021: Winter school of the ANR project,
"Applications of Hecke and related algebras: Representations, Integrability and Physics", Les Houches, February 27th - 3rd March 2023
- July 2023: Final conference of the ANR project, 16th - 21st July 2023, Spetses.
Publications
- N. Jacon,
Two maps on affine type A crystals and Hecke algebras,
preprint ArXiv:2102.05335
- A. Lafay, A. M. Gainutdinov, J. L. Jacobsen,
Uq(sln) web models and Zn spin interfaces,
preprint ArXiv:2101.00282
- O. Brunat, J.-B. Gramain, N. Jacon,
On unitriangular basic sets for symmetric and alternating groups,
preprint ArXiv:2011.00815
- M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel,
Mapping Class Group Representations From Non-Semisimple TQFTs,
preprint ArXiv:2010.14852
- N. Crampé, L. Frappat, J. Gaboriaud, L. Poulain d'Andecy, E. Ragoucy, L. Vinet,
The Askey-Wilson algebra and its avatars,
J. Phys. A 54 (2021) 063001. ArXiv:2009.14815.
- N. Crampé, L. Vinet, M. Zaimi,
Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of Uq(sl2),
preprint ArXiv:2008.04905.
- E. Chavli, M. Chlouveraki,
The freeness and trace conjectures for parabolic Hecke subalgebras,
preprint ArXiv:2007.11535.
- N. Crampé, L. Poulain d'Andecy, L. Vinet,
A Calabi-Yau algebra with E6 symmetry and the Clebsch-Gordan series of sl(3),
preprint ArXiv:2005.13444.
- N. Crampé, L. Poulain d'Andecy,
Baxterisation of the fused Hecke algebra and R-matrices with gl(N)-symmetry,
preprint ArXiv:2004.05035.
- J. Belletête, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, T. S. Tavares,
Topological defects in periodic RSOS models and anyonic chains,
preprint ArXiv:2003.11293.
- J. Berger, A. M. Gainutdinov, I. Runkel,
Monadic cointegrals and applications to quasi-Hopf algebras,
J. Pure Appl. Algebra 225 (2021) 106678. ArXiv:2003.13307.
- N. Crampé, L. Poulain d'Andecy,
Fused braids and centralisers of tensor representations of Uq(glN),
preprint ArXiv:2001.11372.
- N. Jacon, C. Lecouvey,
Cores of Ariki-Koike algebras,
Documenta Mathematica (in press) ArXiv:1912.08461
- M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel,
3-dimensional TQFTs from non-semisimple modular categories,
preprint ArXiv:1912.02063.
- A. M. Gainutdinov, J. Haferkamp, C. Schweigert,
Davydov-Yetter cohomology, comonads and Ocneanu rigidity,
preprint ArXiv:1910.06094.
- N. Jacon, C. Lecouvey,
Keys and Demazure crystals for Kac-Moody algebras,
Journal of Combinatorial Algebra 4 (2020) 325-358. ArXiv:1909.09520
- L.-H. Robert, E. Wagner,
State sums for some super quantum link invariants,
Topology and Geometry: A Collection of Papers Dedicated to
Vladimir G. Turaev (in press). ArXiv:1909.02305
- N. Crampé, W. van de Vijver, L. Vinet,
Racah problems for the oscillator algebra, the Lie algebra sln, and multivariate Krawtchouk polynomials,
preprint ArXiv:1909.1264.
- N. Crampé, L. Frappat, L. Vinet,
Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra ,
J. Phys. A 52 (2019), no. 42, 424001, 11 pp. ArXiv:1906.03936.
- N. Crampé, L. Poulain d'Andecy, L. Vinet,
Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2), Trans. Amer. Math. Soc. 373 (2020),
no. 7, 4907-4932. ArXiv:1905.06346
- M. Chlouveraki, D. Goundaroulis, A. Kontogeorgis, S. Lambropoulou, A generalized skein relation for Khovanov homology and a categorification of the Θ-invariant,
Proc. Roy. Soc. Edinburgh Sect. A (in press). ArXiv:1904.07794
- L. Poulain d'Andecy, S. Rostam,
Morita equivalences for cyclotomic Hecke algebras of type B and D,
Bulletin de la SMF (in press). ArXiv:1903.01580.
- L.-H. Robert, E. Wagner,
A quantum categorification of the Alexander polynomial,
Geometry and Topology (in press). ArXiv:1902.05648
- D. Moussard, E. Wagner,
A Fox-Milnor theorem for the Alexander polynomial of knotted 2-spheres in S4,
Journal of the Mathematical Society of Japan 72 (3) (2020) 891–907. ArXiv:1901.00474
- L. Poulain d'Andecy, R. Walker,
Affine Hecke algebras of type D and generalisations of quiver Hecke algebras,
J. Algebra 552 (2020), 1–37. ArXiv:1901.09978
- J. Berger, A. M. Gainutdinov, I. Runkel,
Modified traces for quasi-Hopf algebras,
J. of Algebra 548 (2020) 96-119. ArXiv:1812.10445
- J. Guilhot, J. Parkinson,
Balanced representations, the asymptotic Plancherel formula, and
Lusztig’s conjectures for C2,
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 969-1031. ArXiv:1803.11067
- J. Belletête, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, T. S. Tavares,
Topological defects in lattice models and affine Temperley-Lieb algebra,
preprint ArXiv:1811.02551.
- A. Beliakova, C. Blanchet, A. M. Gainutdinov,
Modified trace is a symmetrised integral,
Selecta Mathematica (2021) DOI: 10.1007/s00029-021-00626-5. ArXiv:1801.00321
Contact
Loïc Poulain d'Andecy
loic.poulain-dandecy(at)univ-reims.fr
Laboratoire de Mathématiques de Reims (LMR) - UMR 9008
U.F.R. Sciences Exactes et Naturelles
Moulin de la Housse - BP 1039
51687 REIMS cedex 2 FRANCE