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

David Hernandez; Bernard Leclerc

Displaying similar documents to “A cluster algebra approach to $q$ -characters of Kirillov–Reshetikhin modules”

Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

Some Results About Holonomic $ε$ -Modules

Jan-Erik Björk (1983)

Recherche Coopérative sur Programme n°25

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Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let $U$ be a dg- $A$ -module, $B$ the endomorphism dg-algebra of $U$ . We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃 (C, d)$ of $C$ , $𝐃 (B, d)$ of $B$ and $𝐃 (A, d)$ of $A$ . We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

The Grothendieck ring of quantum double of quaternion group

Hua Sun, Jia Pang, Yanxi Shen (2024)

Czechoslovak Mathematical Journal

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Let $𝕜$ be an algebraically closed field of characteristic $p \neq 2$ , and let $Q_{8}$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D (𝕜 Q_{8})$ of group algebra $𝕜 Q_{8}$ . Moreover, we investigate the tensor product decomposition rules of all simple $D (𝕜 Q_{8})$ -modules. Finally, we describe the Grothendieck ring $G_{0} (D (𝕜 Q_{8}))$ by generators with relations.

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$ -module. The paper investigates conditions which imply that the module ${Ext}_{A}^{*} (X)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$ -module. The notion of $C$ -tilting $R$ -modules is introduced as the relative setting of the notion of tilting $R$ -modules with respect to $C$ . Some properties of tilting and $C$ -tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$ -tilting $R$ -module is $C$ -projective. Finally, we investigate some kernel subcategories related to $C$ -tilting modules.

Separable $ℵ_{k}$ -free modules with almost trivial dual

Daniel Herden, Héctor Gabriel Salazar Pedroza (2016)

Commentationes Mathematicae Universitatis Carolinae

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An $R$ -module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$ -module of countable infinite rank $R^{(ω)}$ . For every natural number $k > 1$ , we construct arbitrarily large separable $ℵ_{k}$ -free $R$ -modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Annihilator ideals of finite dimensional simple modules of two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$

Yu Wang, Xiaoming Li (2023)

Czechoslovak Mathematical Journal

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Let $U$ be the two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$ and $F (U)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F (U)$ in the case when $r s^{- 1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

On $τ$ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of $τ$ -extending modules by $τ$ -rational submodules and study some properties of such modules. It is shown that the set of all $τ$ -rational left ideals of $_{R} R$ is a Gabriel filter. An $R$ -module $M$ is called $τ$ -extending if every submodule of $M$ is $τ$ -rational in a direct summand of $M$ . It is proved that $M$ is $τ$ -extending if and only if $M = R e j_{M} E (R / τ (R)) \oplus N$ , such that $N$ is a $τ$ -extending submodule of $M$ . An example is given to show that the direct sum of $τ$ -extending modules need not...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If $X$ is a smooth scheme over a perfect field of characteristic $p$ , and if $𝒟_{X}^{(\infty)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $𝒟_{X}^{(\infty)}$ on an $𝒪_{X}$ -module $ℰ$ is equivalent to giving an infinite sequence of $𝒪_{X}$ -modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$ , endowed with Frobenius liftings. We also show that it...

Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal of a commutative Noetherian ring $R$ . It is shown that the $R$ -modules $H_{I}^{j} (M)$ are $I$ -cofinite for all finitely generated $R$ -modules $M$ and all $j \in ℕ_{0}$ if and only if the $R$ -modules ${Ext}_{R}^{i} (N, H_{I}^{j} (M))$ and ${Tor}_{i}^{R} (N, H_{I}^{j} (M))$ are $I$ -cofinite for all finitely generated $R$ -modules $M$ , $N$ and all integers $i, j \in ℕ_{0}$ .

The correspondence between Barsotti-Tate groups and Kisin modules when $p = 2$

Tong Liu (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a finite extension over $ℚ_{2}$ and $𝒪_{K}$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over $𝒪_{K}$ .

Displaying similar documents to “A cluster algebra approach to q -characters of Kirillov–Reshetikhin modules”

Displaying similar documents to “A cluster algebra approach to $q$ -characters of Kirillov–Reshetikhin modules”