Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Peng Shan

Displaying similar documents to “Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras”

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].

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $X_{i}$ , i∈ I, and $Y_{j}$ , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $\prod_{i \in I} X_{i}$ onto $\prod_{j \in J} Y_{j}$ . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_{i}$ onto $Y_{b (i)}$ , i∈ I.

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...

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- $n$ commutative algebras equipped with an ordered basis is an affine scheme $𝔅_{n}$ of finite type over $ℤ$ , with geometrically connected fibers. It is smooth if and only if $n \leq 3$ . It is reducible if $n \geq 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$ ). The relative dimension of $𝔅_{n}$ is $\frac{2}{27} n^{3} + O (n^{8 / 3})$ . The subscheme parameterizing étale algebras is isomorphic to ${GL}_{n} / S_{n}$ , which is of dimension only $n^{2}$ . For $n \geq 8$ , there exist algebras that are not limits of étale algebras. The dimension...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

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

David Hernandez, Bernard Leclerc (2016)

Journal of the European Mathematical Society

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We describe a cluster algebra algorithm for calculating $q$ -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra $U_{q} (\hat{𝔤})$ . This yields a geometric $q$ -character formula for tensor products of Kirillov–Reshetikhin modules. When $𝔤$ is of type $A, D, E$ , this formula extends Nakajima’s formula for $q$ -characters of standard modules in terms of homology of graded quiver varieties.

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

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 $𝒪_{K}$ .

A generalization of a formalized theory of fields of sets on non-classical logics

Helena Rasiowa

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Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮 *$ ..................................................................................................................... 5 § 3. $𝒮 *$ -algebras.....................................................................................................................

Trivialization of $𝒞 (X)$ -algebras with strongly self-absorbing fibres

Marius Dadarlat, Wilhelm Winter (2008)

Bulletin de la Société Mathématique de France

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Suppose $A$ is a separable unital $𝒞 (X)$ -algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$ -injective $C^{*}$ -algebra $𝒟$ . We show that $A$ and $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack (2013)

Annales de l’institut Fourier

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Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G .$ We prove that the Bruhat-Tits building $𝔅^{1} (H)$ of $H$ can be affinely and $G$ -equivariantly embedded in the Bruhat-Tits building $𝔅^{1} (G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{'}$ be maps from $𝔅^{1} (H)$ to $𝔅^{1} (G)$ which preserve the Moy–Prasad filtrations....

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

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We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

Geometric theta-lifting for the dual pair ${𝕊𝕆}_{2 m}, 𝕊 p_{2 n}$

Sergey Lysenko (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $> 2$ . Consider the dual pair $H = {SO}_{2 m}, G = {Sp}_{2 n}$ over $X$ with $H$ split. Write ${Bun}_{G}$ and ${Bun}_{H}$ for the stacks of $G$ -torsors and $H$ -torsors on $X$ . The theta-kernel ${Aut}_{G, H}$ on ${Bun}_{G} \times {Bun}_{H}$ yields theta-lifting functors $F_{G} : D ({Bun}_{H}) \to D ({Bun}_{G})$ and $F_{H} : D ({Bun}_{G}) \to D ({Bun}_{H})$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...

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...