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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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.

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...

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 $𝒪_{K}$ .

On the composition structure of the twisted Verma modules for $𝔰𝔩 (3, ℂ)$

Libor Křižka, Petr Somberg (2015)

Archivum Mathematicum

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We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $𝔰𝔩 (3, ℂ)$ , including the explicit structure of singular vectors for both $𝔰𝔩 (3, ℂ)$ and one of its Lie subalgebras $𝔰𝔩 (2, ℂ)$ , and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$ -modules on the Schubert cells in the full flag manifold for $SL (3, ℂ)$ .

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

Lifting $D$ -modules from positive to zero characteristic

João Pedro P. dos Santos (2011)

Bulletin de la Société Mathématique de France

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We study liftings or deformations of $D$ -modules ( $D$ is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic $D$ -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given $D$ -module in positive characteristic. At the end we compare...

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors...

Melkersson condition on Serre subcategories

Reza Sazeedeh, Rasul Rasuli (2016)

Colloquium Mathematicae

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Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition $C_{}$ , defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class $_{}$ consisting of all modules satisfying $C_{}$ . If and are ideals of R, we get a necessary and sufficient condition for to satisfy $C_{}$ and $C_{}$ simultaneously. We also...

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

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Given a tuple $(𝒳_{1}, ..., 𝒳_{k})$ of irreducible characters of $G L_{n} (F_{q})$ we define a star-shaped quiver $Γ$ together with a dimension vector $v$ . Assume that $(𝒳_{1}, ..., 𝒳_{k})$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $𝒳_{1} \otimes \dots \otimes 𝒳_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(Γ, v)$ . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied,...

Some results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...

Right coideal subalgebras of $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

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We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t > 4$ ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ . In particular, the total number of right coideal subalgebras that contain the coradical equals $(2 n)!!$ ; the order of the Weyl group defined by the root system of type $B_{n}$ .

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

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Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤 = Lie G$ . Let $(e, h, f)$ be an $𝔰 𝔩_{2}$ -triple in $𝔤$ with $e$ being a long root vector in $𝔤$ . Let $(\cdot, \cdot)$ be the $G$ -invariant bilinear form on $𝔤$ with $(e, f) = 1$ and let $χ \in 𝔤^{*}$ be such that $χ (x) = (e, x)$ for all $x \in 𝔤$ . Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$ ; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Otmar Venjakob (2002)

Journal of the European Mathematical Society

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This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $Λ$ of a $p$ -adic analytic group $G$ . For $G$ without any $p$ -torsion element we prove that $Λ$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $Λ$ -module. This is classical when $G = ℤ_{p}^{k}$ for some integer $k \geq 1$ , but was previously unknown in the non-commutative case. Then the category...

Irregularity of an analogue of the Gauss-Manin systems

Céline Roucairol (2006)

Bulletin de la Société Mathématique de France

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In $𝒟$ -modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+} (𝒪 e^{g})$ of a $𝒟$ -module twisted by the exponential of a polynomial $g$ by another polynomial $f$ , where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$ -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$ -minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$ -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${(H_{I}^{i} (M))}_{𝔪}$ is a minimax $R_{𝔪}$ -module for all $𝔪 \in Max (R)$ and for all $i < n$ , then the set ${Ass}_{R} (H_{I}^{n} (M))$ is finite. Also, if $H_{I}^{i} (M)$ is...

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”