Singular localization of $\mathfrak {g}$-modules and applications to representation theory

Erik Backelin; Kobi Kremnitzer

Displaying similar documents to “Singular localization of $𝔤$ -modules and applications to representation theory”

Restriction to Levi subalgebras and generalization of the category $𝒪$

Guillaume Tomasini (2013)

Annales de l’institut Fourier

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The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple. ...

On the Bernstein-Walsh-Siciak theorem

Rafał Pierzchała (2012)

Studia Mathematica

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By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set $K \subset ℂ^{N}$ can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the...

Quasianalytic functions in the sense of Bernstein

W. Pleśniak

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CONTENTS1. Introduction................................................................................................................ 32. The extremal function.............................................................................................. 83. Some lemmas on polynomials............................................................................. 124. Category theorems in topological groups........................................................... 165. Best approximation...

Mixed norm condition numbers for the univariate Bernstein basis

Tom Lyche, Karl Scherer (2006)

Banach Center Publications

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We study mixed norm condition numbers for the univariate Bernstein basis for polynomials of degree n, that is, we measure the stability of the coefficients of the basis in the $l_{q}$ -sequence norm whereas the polynomials to be represented are measured in the $L_{p}$ -function norm. The resulting condition numbers differ from earlier results obtained for p = q.

On the $K$ -theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Patrick Polo (1995)

Annales de l'institut Fourier

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Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤 = Lie (G)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$ . For $μ \in 𝔥^{*}$ , let $J_{μ}$ be the corresponding minimal primitive ideal, let $U_{μ} = U / J_{μ}$ , and let $𝒯_{U_{μ}} : K_{0} (U_{m} u) \to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$ -algebras $U_{μ}$ . When $μ$ is regular, Hodges has shown that $K_{0} (U_{μ}) ≅ K_{0} (X)$ . In this case $K_{0} (U_{μ})$ is generated by the classes corresponding...

Singular Blocks of the Category O.

Ronald S. Irving (1990)

Mathematische Zeitschrift

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Bad properties of the Bernstein numbers

Albrecht Pietsch (2008)

Studia Mathematica

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We show that the classes $_{p}^{b e r n} : = T : (b ₙ (T)) \in l_{p}$ associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, $_{p}^{b e r n} \circ_{q}^{b e r n} ⊈_{r}^{b e r n}$ for 1/r = 1/p + 1/q.

Branching problems and $𝔰𝔩 (2, ℂ)$ -actions

Pavle Pandžić, Petr Somberg (2015)

Archivum Mathematicum

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We study certain $𝔰𝔩 (2, ℂ)$ -actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(𝔤, 𝔭)$ , $(𝔤^{'}, 𝔭^{'})$ of Lie algebras and their parabolic subalgebras.

The Enright Functor on the Bernstein-Gelfand-Gelfand Category ... .

A. Joseph (1982)

Inventiones mathematicae

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Schroeder-Bernstein Quintuples for Banach Spaces

Elói Medina Galego (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p,q,r,s,t) in ℕ for X to be isomorphic to Y whenever ⎧ $X X^{p} \oplus Y^{q}$ , ⎨ ⎩ $Y^{t} X^{r} \oplus Y^{s}$ . Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition...

Basic relations valid for the Bernstein spaces $B ²_{σ}$ and their extensions to larger function spaces via a unified distance concept

P. L. Butzer, R. L. Stens, G. Schmeisser (2014)

Banach Center Publications

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Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces $B_{σ}^{p}$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from $B_{σ}^{p}$ . The difficult situation of derivative-free error estimates is also covered. ...

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 $SL (3, ℂ)$ .

Separable functors for the category of Doi Hom-Hopf modules

Shuangjian Guo, Xiaohui Zhang (2016)

Colloquium Mathematicae

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Let $̃ (_{k}) {(H)}_{A}^{C}$ be the category of Doi Hom-Hopf modules, $̃ {(_{k})}_{A}$ be the category of A-Hom-modules, and F be the forgetful functor from $̃ (_{k}) {(H)}_{A}^{C}$ to $̃ {(_{k})}_{A}$ . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

Base change for Picard-Vessiot closures

Andy R. Magid (2011)

Banach Center Publications

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The differential automorphism group, over F, Π₁(F₁) of the Picard-Vessiot closure F₁ of a differential field F is a proalgebraic group over the field $C_{F}$ of constants of F, which is assumed to be algebraically closed of characteristic zero, and its category of $C_{F}$ modules is equivalent to the category of differential modules over F. We show how this group and the category equivalence behave under a differential extension E ⊃ F, where $C_{E}$ is also algebraically closed.

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...

Sets with the Bernstein and generalized Markov properties

Mirosław Baran, Agnieszka Kowalska (2014)

Annales Polonici Mathematici

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It is known that for $C^{\infty}$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty}$ determining. In this paper we give examples of sets which are not $C^{\infty}$ determining, but have the Bernstein and generalized Markov properties.

Displaying similar documents to “Singular localization of 𝔤 -modules and applications to representation theory”

Displaying similar documents to “Singular localization of $𝔤$ -modules and applications to representation theory”