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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - QAffine (Representations of quantum affine algebras and applications)

Teaser

Summary

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. Categorifications (of cluster algebras and of quantum groups),
4. Langlands duality for quantum groups,
5. Proof of (geometric) character formulas and applications.

Work performed

We have focused in this period on the following directions which are the most relevant
for the current developments in the field.
For the direction 1 (Asymptotical representations and applications to quantum
integrable systems), we proved (with E. Frenkel) a system of relations in the Grothendieck
ring of the category O of representations of the Borel subalgebra of an untwisted quantum
affine algebra. Consequently we obtained a direct and uniform proof of the Bethe Ansatz
equations for a large class of quantum integrable models, under a mild genericity
condition.
For the direction 4 (Langlands duality for quantum groups) : some authors have
shown that solutions of the system above can be attached to certain affine opers for the
Langlands dual affine Kac-Moody algebra. Hence our result (with E. Frenkel) provides
strong evidences for a conjecture of Feigin-Frenkel linking the spectra of quantum KdV
Hamiltonians and affine opers for the Langlands dual affine algebra.
For the direction 3 (Categorifications of cluster algebras and quantum groups), with B. Leclerc we showed
that the Grothendieck ring of a certain monoidal subcategory of O has the structure of a
cluster algebra of infinite rank, with an initial seed consisting of prefundamental
representations. In particular, the celebrated Baxter relations for the 6-vertex model get
interpreted as Fomin-Zelevinsky mutation relations.

I co-organized a conference at the CIRM (France) in September 2016 which focused
directly on various parts on the project. I am an organizer of a weekly seminar at the
Institut Henri PoincarÃ© with regular speakers directly related to the project.

Final results

The two main results are (see above) :

1) Frenkel-Hernandez : the proof of a system of relations in the Grothendieck
ring of the category O implying the Bethe Ansatz equations for a large class of
quantum integrable models, under a mild genericity condition.

We expect to get informations on the partition function of the quantum integrable
model as it can expressed in terms of eigenvalues of transfer-matrices satisfied
by the Bethe Ansatz equations. We also expect new insights on the
Langlands duality conjectured by Feigin-Frenkel for affine opers (see above).

2) Hernandez-Leclerc : we showed that the Grothendieck ring of a certain monoidal
subcategory of O has the structure of a cluster algebra. In particular, the
celebrated Baxter relations for the 6-vertex model get interpreted as
Fomin-Zelevinsky mutation relations.

We expect to get new informations on the structure of the Grothendieck ring
as the cluster algebra structure implies conjectural relations of
tensor product decompositions. In particular we established the relations
corresponding to the first step relations (these relations were used to
drop the generecity condition for Bethe Ansatz equations).