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 of non-negative integers there are only a finite number of isomorphism classes of indecomposables X 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 < if and only if g l d i m K Q / I < if and only if g l d i m K Q g / I g < .

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 ( n ) = ( μ ) n 𝔖 n : a highest weight equivalence between 𝒪 p and 𝒪 σ ( p ) for σ 𝔖 and an action of 𝔖 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...

On strongly affine extensions of commutative rings

Nabil Zeidi (2020)

Czechoslovak Mathematical Journal

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A ring extension R S is said to be strongly affine if each R -subalgebra of S is a finite-type R -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if R is a quasi-local ring of finite dimension, then R S is integrally closed and strongly affine if and only if R S is a Prüfer extension (i.e. ( R , S ) is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

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 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 * 𝐑 m k 𝐑 m * k 𝐑 m for k = 1 , 3 .

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

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We investigate derived equivalences between subalgebras of some Φ -Auslander-Yoneda algebras from a class of n -angles in weakly n -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by n -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to n -angle cases. Finally, we give an explicit example to illustrate our result.

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 n + 1 and n , where n + 1 and 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 i I X i onto j 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 ) ) 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 , with geometrically connected fibers. It is smooth if and only if n 3 . It is reducible if n 8 (and the converse holds, at least if we remove the fibers above 2 and 3 ). The relative dimension of 𝔅 n is 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 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 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 (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 ( 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.

Drinfeld doubles via derived Hall algebras and Bridgeland's Hall algebras

Fan Xu, Haicheng Zhang (2021)

Czechoslovak Mathematical Journal

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Let 𝒜 be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev’s theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of 𝒜 via its derived Hall algebra and Bridgeland’s Hall algebra of m -cyclic complexes.

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 ) 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 universal tropicalization and the Berkovich analytification

Jeffrey Giansiracusa, Noah Giansiracusa (2022)

Kybernetika

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Given an integral scheme X over a non-archimedean valued field k , we construct a universal closed embedding of X into a k -scheme equipped with a model over the field with one element 𝔽 1 (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by previous work of the authors, and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification X an . Moreover, using the scheme-theoretic...

Matrix representation of finite effect algebras

Grzegorz Bińczak, Joanna Kaleta, Andrzej Zembrzuski (2023)

Kybernetika

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In this paper we present representation of finite effect algebras by matrices. For each non-trivial finite effect algebra E we construct set of matrices M ( E ) in such a way that effect algebras E 1 and E 2 are isomorphic if and only if M ( E 1 ) = M ( E 2 ) . The paper also contains the full list of matrices representing all nontrivial finite effect algebras of cardinality at most 8 .

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.....................................................................................................................