## 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.

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 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 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.

## 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).