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

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

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.

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 from the beginning of the project to the end of the period covered by the report and main results achieved so far

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 Poincaré with regular speakers directly related to the project.

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 Poincaré with regular speakers directly related to the project.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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).

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).