On generalized Weyl operators

The ``generalized Weyl' operators between two Hilbert spaces are taken to be those with closed range for which the null space and that of the adjoint are of equal Hilbert space dimension. We show that products of two of these which happen to have closed range, and finite rank perturbation of these, are also generalized Weyl.

     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

Fixed rings of quantum generalized Weyl algebras

Abstract

Generalized Weyl algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite order automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings including global dimension, the Calabi–Yau property, rigidity, and simplicity.     

GENERALIZED CIRCLES AND THEIR CONFORMAL MAPPING IN A SUBSPACE OF A WEYL SPACE

The authors give the necessary and sufficient conditions for a generalized circle in a Weyl hypersurface to be generalized circle in the enveloping Weyl space. They then obtain the neccessary and sufficient conditions under which a generalized concircular transformation of one Weyl space onto another induces a generalized transformation oil its subspaces. Finally, it is shown that any totally geodesic or totally umbilical hypersurface of a generalized concircularly flat Weyl space is also generalized concircularly flat.     

广义Weyl型李代数Wgl_n(F)上的2-上同调群

设F是一特征为零的域,W是F上的广义Weyl代数,gl_n(F)为F上的一般线性李代教,则结合代数Wgl_n(F)上具有一个诱导的李代数结构,本文讨论了李代数Wgl_n(F)的2-上同调群的结构.     

Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.     

Good Reduction of Good Filtrations at Places

We consider filtered or graded algebras A over a field K. Assume that there is a discrete valuation O_v of K with m_v its maximal ideal and k_v:=O_v/m_v its residue field. Let Λ be O_v-order such that ΛK=A and the Λ-reduction of A at the place . As in many examples of quantized algebras A comes with a specific filtration that reduces well with respect to the valuation filtration defined by Λ on A and the reduction relates to the part of degree zero in the associated graded algebra. Hence several lifting properties fellow from valuation like theory, also for modules with good filtrations. Presented by A. Verschoren Mathematics Subject Classifications (2000) 16W35, 16W70, 16W60, 06B23, 06B25. Toukaiddine Petit: Author supported by the Scientific Programme NOG of the European Science Foundation. Freddy Oystaeyen: Acknowledging the EC project Liegrits MCRTN 505078.     

Isomorphism problems and groups of automorphisms for generalized Weyl algebras

We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of by finding sets of generators for the group of automorphisms.

     

一类特殊广义Kac-Moody代数虚根决定的反射与Weyl群间的关系(英)

对有限型李代数g（A）,相应于每个根α的反射r_α均在g（A）的Weyl群W中．当g（A）为可对称化的不定型Kac－Moody代数时,若α为一虚根且（α,α）＜0,则亦可定义反射r_α并有r_α∈－W或r_α是－W中元与一个图自同构之积（见[3]）．本文给出了一类秩为3的广义Kac－Moody代数的虚根系,然后讨论了一类特殊的广义Kac－Moody代数的虚根决定的反射与Weyl群之间的关系．     

Ixr(a)的有限型Ixor(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型IC or (a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数IXr(a)的Weyl群W同构于有限型IXr(a)的未定Weyl群.     

IX_r(a)的有限型IX_r~°(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型I(?)r(a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数,IXr(a)的Weyl群W同构于有限型I(?)r(a)的未定Weyl群.     

Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.     

Weyl Spectrum of Upper Triangular Operator Matrices

This paper is concerned with general n × n upper-triangular operator matrices with given diagonal entries. The characterizations of perturbations of their left(resp. right) Weyl spectrum and Weyl spectrum are given, based on the space decomposition technique. Moreover, some sufficient and necessary conditions are given under which the left(resp. right) Weyl spectrum and the Weyl spectrum of such operator matrix, respectively, coincide with the union of the left(resp. right) Weyl spectrum and the Weyl spectrum of its diagonal entries.     

Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions

We obtain estimates for the counting function of the Neumann Laplacian on a planar domain bounded by the graph of a lower semicontinuous L₁-function. These estimates imply necessary and sufficient conditions for the validity of the classical one-term Weyl formula for the counting function and, under certain restrictions, give an order sharp remainder estimate in this formula.     

Weighted inequalities for some one-sided operators

We give a characterization of the pairs of weights such that the Weyl fractional integral operator maps into weak , or . For the case we give necessary and sufficient conditions for the weak type of a maximal operator that includes as particular cases the Weyl fractional integral, the dual of the Hardy operator and the fractional one-sided maximal operator. As a consequence we give a new characterization of the pairs of weights for which the fractional one-sided maximal operator is bounded.

     

Some Weyl Modules for Simple Algebraic Groups of Type B4 and D4

Let G be a simply connected, semisimple algebraic group of type B₄ or D₄ over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.