## Periodic Reporting for period 4 - QAffine (Representations of quantum affine algebras and applications)

Reporting period: 2020-03-01 to 2021-02-28

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 these directions, significant advances have been mage.

The main results of the project are :

(A) Frenkel-Hernandez : the proof of a remarkable system of relations implying the conjectural Bethe Ansatz equations for quantum integrable models.

(B) Hernandez-Leclerc and Hernandez : the Grothendieck ring of a certain monoidal subcategory O has the

structure of a cluster algebra. Certain cluster relations were then categorified in terms of new R-matrices obtained from algebraic stable maps.

(C) Hernandez-Oya and Fujita-Hernandez-Oh-Oya : we established isomorphisms of quantum Grothendieck rings

preserving canonical basis. Combining with recent results in terms of generalized Schur-Weyl dualities, we established the validity of a

Kazdhan-Lusztig algorithm for characters of simple modules in a monoidal category of representations in type B (this was conjectured in 2002) and

a related positivity conjecture for general types.

Two phD-thesis funded by the project were defended.

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 these directions, significant advances have been mage.

The main results of the project are :

(A) Frenkel-Hernandez : the proof of a remarkable system of relations implying the conjectural Bethe Ansatz equations for quantum integrable models.

(B) Hernandez-Leclerc and Hernandez : the Grothendieck ring of a certain monoidal subcategory O has the

structure of a cluster algebra. Certain cluster relations were then categorified in terms of new R-matrices obtained from algebraic stable maps.

(C) Hernandez-Oya and Fujita-Hernandez-Oh-Oya : we established isomorphisms of quantum Grothendieck rings

preserving canonical basis. Combining with recent results in terms of generalized Schur-Weyl dualities, we established the validity of a

Kazdhan-Lusztig algorithm for characters of simple modules in a monoidal category of representations in type B (this was conjectured in 2002) and

a related positivity conjecture for general types.

Two phD-thesis funded by the project were defended.

The main results obtained are the following.

1) Frenkel-Hernandez [1] : 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 0 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.

2) Hernandez-Leclerc [3] and Hernandez [16] : 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 genericity condition for Bethe Ansatz equations).

These relations were then categorified in terms of new R-matrices obtained from algebraic stable maps in [16].

3) Hernandez-Oya [12] and Fujita-Hernandez-Oh-Oya [28] : we established in [12] isomorphisms of quantum Grothendieck rings

in types A and B preserving canonical basis. This was extended in [28] to more general types and larger categories.

Combining with recent results of Kashiwara and its collaborators, we established the validity of a Kazdhan-Lusztig algorithm for the

(q)-characters of simple modules in a monoidal category of representations in type B (this was conjectured in 2002) and

a related positivity conjecture for general types.

I co-organized a conference Algebraic Combinatorics in Representation Theory at the CIRM (Luminy, 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 in Paris with

regular speakers directly related to the project.

I co-organized a thematic trimester on Representation Theory at the Institut Henri Poincaré in Paris in 2020 at the end of the project.

There was a winter shool at the CIRM in Luminy in January 2020.

I gave several talks in international conferences, seminars and colloquia on the work in this ERC project

(Corea, USA, Germany, Japan, Taiwan, Italy, Brazil...) as well as a course at ESI in Vienna and a Bourbaki seminar in Paris.

1) Frenkel-Hernandez [1] : 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 0 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.

2) Hernandez-Leclerc [3] and Hernandez [16] : 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 genericity condition for Bethe Ansatz equations).

These relations were then categorified in terms of new R-matrices obtained from algebraic stable maps in [16].

3) Hernandez-Oya [12] and Fujita-Hernandez-Oh-Oya [28] : we established in [12] isomorphisms of quantum Grothendieck rings

in types A and B preserving canonical basis. This was extended in [28] to more general types and larger categories.

Combining with recent results of Kashiwara and its collaborators, we established the validity of a Kazdhan-Lusztig algorithm for the

(q)-characters of simple modules in a monoidal category of representations in type B (this was conjectured in 2002) and

a related positivity conjecture for general types.

I co-organized a conference Algebraic Combinatorics in Representation Theory at the CIRM (Luminy, 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 in Paris with

regular speakers directly related to the project.

I co-organized a thematic trimester on Representation Theory at the Institut Henri Poincaré in Paris in 2020 at the end of the project.

There was a winter shool at the CIRM in Luminy in January 2020.

I gave several talks in international conferences, seminars and colloquia on the work in this ERC project

(Corea, USA, Germany, Japan, Taiwan, Italy, Brazil...) as well as a course at ESI in Vienna and a Bourbaki seminar in Paris.

The results explained above are advances and clearly beyond the state of the art. For instance the

following points are completely new : the QQ-relations in the Grothendieck of

representations, the monoidal categorification of cluster algebras in the category O,

the categorifications of exchange relation from algebraic stable maps, the proof of long-standing problems

on quantum Grothendieck rings in non simply-laced types (positivity and Kazhdan-Lusztig conjectures).

following points are completely new : the QQ-relations in the Grothendieck of

representations, the monoidal categorification of cluster algebras in the category O,

the categorifications of exchange relation from algebraic stable maps, the proof of long-standing problems

on quantum Grothendieck rings in non simply-laced types (positivity and Kazhdan-Lusztig conjectures).