Classification and structure of certain representations of quantum affine algebras
Estudamos representacoes de dimensao finita para uma algebra afim quantizada a partir de dois pon... more Estudamos representacoes de dimensao finita para uma algebra afim quantizada a partir de dois pontos de vista distintos. Na primeira parte deste trabalho estudamos o limite graduado de uma certa subclasse de representacoes irredutiveis. Seja V uma representacao de dimensao finita para uma algebra do tipo A e suponha que V e isomorfa ao produto tensorial de uma afinizacao minimal por partes cujo peso maximo e a soma de distintos pesos fundamentais por modulos de Kirillov--Reshetikhin cujos pesos maximos sao o dobro de um peso fundamental. Provamos que V admite limite graduado L e que L e isomorfo a um modulo de Demazure de nivel dois bem como ao produto de fusao dos limites graduados de cada um dos supramencionados fatores tensoriais de V. Provamos ainda que, se a algebra for do tipo classica (resp. G), o limite graduado das afinizacoes minimais (regulares) (resp. modulos de Kirillov--Reshetikhin) sao isomorfos ao modulos CV para alguma R^+ particao descrita explicitamente. Na segunda parte provamos que um modulo para a algebra afim quantizada do tipo B e posto n e manso se, e somente se, ele e fino. Em outras palavras, os geradores da subalgebra de Cartan afim sao diagonalizaveis se, e somente se, os autoespacos generalizados associados tem dimensao um. Classificamos tais modulos e descrevemos seus respectivos q-caracteres. Em alguns casos, o q-caracter e descrito por super standard Young tableaux do tipo (2n|1) Abstract
We study the graded limits of simple U_q(s̃l̃_n+1)-modules which are isomorphic to tensor product... more We study the graded limits of simple U_q(s̃l̃_n+1)-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fix fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of fi... more We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type A. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients.
We study finite-dimensional representations of quantum affine algebras of type B_N. We show that ... more We study finite-dimensional representations of quantum affine algebras of type B_N. We show that a module is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their q-characters. In some cases, the q-characters are described by super standard Young tableaux of type (2N|1).
We study a class modules, called Chari-Venkatesh modules, for the current superalgebra 𝔰𝔩(1|2)[t]... more We study a class modules, called Chari-Venkatesh modules, for the current superalgebra 𝔰𝔩(1|2)[t]. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules.
We study a class modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak... more We study a class modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak{sl}(1|2)[t]$. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules.
In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl... more In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules. In the language of cluster algebras our Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots, x_n'$ of an appropriate cluster algebra. Our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of ...
Journal de l’École polytechnique — Mathématiques, 2019
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of fi... more We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite-dimensional representations of a quantum affine algebra of type A. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients. Résumé (Produits tensoriels et q-caractères de HL-modules et catégorifications monoïdales) Dans ce travail, nous étudions certaines sous-catégories monoïdales (introduites par David Hernandez et Bernard Leclerc) de représentations de dimension finie d'une algèbre affine de type A. Nous classifions l'ensemble des représentations premières de ces sous-catégories, et donnons des conditions nécessaires et suffisantes pour que le produit tensoriel des deux représentations premières soit irréductible. Dans le cas où le produit tensoriel est réductible, nous décrivons une factorisation en modules premiers des facteurs simples. En conséquence, nous prouvons que ces sous-catégories monoïdales sont des catégorifications monoïdales d'algèbres amassées de type A avec coefficients.
We study graded limits of simple Uq( sln+1)-modules which are isomorphic to tensor products of Ki... more We study graded limits of simple Uq( sln+1)-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fixed fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.
Journal of the Institute of Mathematics of Jussieu, 2015
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to$\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for$\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154(2010), 265–341;Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to$\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chariet al.(arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result ...
Transactions of the American Mathematical Society, 2016
We study finite-dimensional representations of quantum affine algebras of type BN . We show that ... more We study finite-dimensional representations of quantum affine algebras of type BN . We show that a module is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their q-characters. In some cases, the q-characters are described by super standard Young tableaux of type (2N |1).
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to sl_n+1. After a suitable twist, the limit is a module for sl_n+1[t], i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to sl_n+1[t]--stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. More generally, we prove that any level two Demazure module is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine sl_n+1.
Journal of the Institute of Mathematics of Jussieu, 2015
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to sln+1. After a suitable twist, the limit is a module for sln+1[t], i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in [28], , in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to sln+1[t]-stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. It was proved in [18] that a stable Demazure module is isomorphic to the fusion product of stable, prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine sln+1.
Classification and structure of certain representations of quantum affine algebras
Estudamos representacoes de dimensao finita para uma algebra afim quantizada a partir de dois pon... more Estudamos representacoes de dimensao finita para uma algebra afim quantizada a partir de dois pontos de vista distintos. Na primeira parte deste trabalho estudamos o limite graduado de uma certa subclasse de representacoes irredutiveis. Seja V uma representacao de dimensao finita para uma algebra do tipo A e suponha que V e isomorfa ao produto tensorial de uma afinizacao minimal por partes cujo peso maximo e a soma de distintos pesos fundamentais por modulos de Kirillov--Reshetikhin cujos pesos maximos sao o dobro de um peso fundamental. Provamos que V admite limite graduado L e que L e isomorfo a um modulo de Demazure de nivel dois bem como ao produto de fusao dos limites graduados de cada um dos supramencionados fatores tensoriais de V. Provamos ainda que, se a algebra for do tipo classica (resp. G), o limite graduado das afinizacoes minimais (regulares) (resp. modulos de Kirillov--Reshetikhin) sao isomorfos ao modulos CV para alguma R^+ particao descrita explicitamente. Na segunda parte provamos que um modulo para a algebra afim quantizada do tipo B e posto n e manso se, e somente se, ele e fino. Em outras palavras, os geradores da subalgebra de Cartan afim sao diagonalizaveis se, e somente se, os autoespacos generalizados associados tem dimensao um. Classificamos tais modulos e descrevemos seus respectivos q-caracteres. Em alguns casos, o q-caracter e descrito por super standard Young tableaux do tipo (2n|1) Abstract
We study the graded limits of simple U_q(s̃l̃_n+1)-modules which are isomorphic to tensor product... more We study the graded limits of simple U_q(s̃l̃_n+1)-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fix fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of fi... more We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type A. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients.
We study finite-dimensional representations of quantum affine algebras of type B_N. We show that ... more We study finite-dimensional representations of quantum affine algebras of type B_N. We show that a module is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their q-characters. In some cases, the q-characters are described by super standard Young tableaux of type (2N|1).
We study a class modules, called Chari-Venkatesh modules, for the current superalgebra 𝔰𝔩(1|2)[t]... more We study a class modules, called Chari-Venkatesh modules, for the current superalgebra 𝔰𝔩(1|2)[t]. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules.
We study a class modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak... more We study a class modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak{sl}(1|2)[t]$. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules.
In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl... more In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules. In the language of cluster algebras our Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots, x_n'$ of an appropriate cluster algebra. Our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of ...
Journal de l’École polytechnique — Mathématiques, 2019
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of fi... more We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite-dimensional representations of a quantum affine algebra of type A. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients. Résumé (Produits tensoriels et q-caractères de HL-modules et catégorifications monoïdales) Dans ce travail, nous étudions certaines sous-catégories monoïdales (introduites par David Hernandez et Bernard Leclerc) de représentations de dimension finie d'une algèbre affine de type A. Nous classifions l'ensemble des représentations premières de ces sous-catégories, et donnons des conditions nécessaires et suffisantes pour que le produit tensoriel des deux représentations premières soit irréductible. Dans le cas où le produit tensoriel est réductible, nous décrivons une factorisation en modules premiers des facteurs simples. En conséquence, nous prouvons que ces sous-catégories monoïdales sont des catégorifications monoïdales d'algèbres amassées de type A avec coefficients.
We study graded limits of simple Uq( sln+1)-modules which are isomorphic to tensor products of Ki... more We study graded limits of simple Uq( sln+1)-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fixed fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.
Journal of the Institute of Mathematics of Jussieu, 2015
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to$\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for$\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154(2010), 265–341;Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to$\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chariet al.(arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result ...
Transactions of the American Mathematical Society, 2016
We study finite-dimensional representations of quantum affine algebras of type BN . We show that ... more We study finite-dimensional representations of quantum affine algebras of type BN . We show that a module is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their q-characters. In some cases, the q-characters are described by super standard Young tableaux of type (2N |1).
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to sl_n+1. After a suitable twist, the limit is a module for sl_n+1[t], i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to sl_n+1[t]--stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. More generally, we prove that any level two Demazure module is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine sl_n+1.
Journal of the Institute of Mathematics of Jussieu, 2015
We study the classical limit of a family of irreducible representations of the quantum affine alg... more We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to sln+1. After a suitable twist, the limit is a module for sln+1[t], i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in [28], , in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to sln+1[t]-stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. It was proved in [18] that a stable Demazure module is isomorphic to the fusion product of stable, prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine sln+1.
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