David Hernandez

The Kirillov-Reshetikhin conjecture and solutions of T-systems

Abstract We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima in [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.], [H. Nakajima, t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Th. 7 (2003), 259–274.] with geometric arguments (main result of [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

Cluster algebras and quantum affine algebras

Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.

Quantum Grothendieck rings and derived Hall algebras

We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall algebra of the derived category of a quiver Q of the same Dynkin type. Along the way, we study for each choice of orientation Q a tensor subcategory whose t-deformed Grothendieck ring is isomorphic to the positive part of a quantum enveloping algebra of the same Dynkin type, where the classes of simple objects correspond to Lusztigs dual canonical basis.

A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules

We describe a cluster algebra algorithm for calculating q-characters of Kirillov-Reshetikhin modules for any untwisted quantum affine algebra. This yields a geometric q-character formula for tensor products of Kirillov-Reshetikhin modules. In simply laced type this formula extends Nakajimas formula for q-characters of standard modules in terms of homology of graded quiver varieties.

Simple tensor products

Let ℱ be the category of finite-dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S1⊗⋅⋅⋅⊗SN of simple objects of ℱ is simple if and only Si⊗Sj is simple for any i

Langlands Duality for Finite-Dimensional Representations of Quantum Affine Algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandez (Math Ann, to appear) and Frenkel and Reshetikhin (Commun Math Phys 197(1):1–32, 1998). We prove this duality for the Kirillov–Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct “interpolating (q, t)-characters” depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.

Monoidal Categorifications of Cluster Algebras of Type A and D

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and D, we show that these categories provide monoidal categorifications of cluster algebras of the same type. The proof is purely representation-theoretical, in the spirit of Hernandez and Leclerc (Duke Math. J. 154, 265–341, 2010).

t-analogues des opérateurs d'écrantage associés aux q-caractères

Frenkel and Reshetikhin introduced screening operators related to q-characters of finite dimensional representations of quantum affine algebras. We propose t-analogs of screening operators related to Nakajimas q,t-characters with the same properties of symmetry. Nakajima introduced non-commutative rings, so we propose bimodules. Our construction uses only combinatorial definition of q,t-characters, and therefore can be extended to the non-simply laced case.