Generalized Weyl Algebras Research Articles

On Morita Equivalence for Simple Generalized Weyl Algebras

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges’ result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for n ≥ 3.

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n).

Non-finitely generated projective modules over generalized Weyl algebras

We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules.

Locally finite simple weight modules over twisted generalized Weyl algebras

We present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. When a certain commutative subalgebra is finitely generated over an algebraically closed field we obtain a classification of a class of locally finite simple weight modules as those induced from simple modules over a subalgebra isomorphic to a tensor product of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra.

ISOMORPHISMS BETWEEN QUANTUM GENERALIZED WEYL ALGEBRAS

We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R = k[h].

Generalized Laurent polynomial rings as quantum projective 3-spaces

Given a ring R, we introduce the notion of a generalized Laurent polynomial ring over R. This class includes the generalized Weyl algebras. We show that these rings inherit many properties from the ground ring R. This construction is then used to create two new families of quadratic global dimension four Artin–Schelter regular algebras. We show that in most cases the second family has a finite point scheme and a defining automorphism of finite order. Nonetheless, a generic algebra in this family is not finite over its center.

Simple modules of the Witten–Woronowicz algebra

It was proved in [V.V. Bavula, D.A. Jordan, Trans. Amer. Math. Soc. 353 (2) (2001) 769–794] that Witten's second deformation and Woronowicz's deformation are isomorphic algebras. For a class of generalized Weyl algebras including the two deformations above, simple modules are being classified (up to irreducible elements of certain noncommutative Euclidean rings).

Derivations of generalized Weyl algebras

A class of the associative and Lie algebras A[D] = A⊗F[D] of Weyl type are studied, where A is a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] is the polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A such that A is D-simple. The derivations of these associative and Lie algebras are precisely determined.

Some associative algebras related to $U(\mathfrak g)$ and twisted generalized Weyl algebras

We prove that both Mickelsson step algebras and Orthogonal Gelfand-Zetlin algebras are twisted generalized Weyl algebras. Using an analogue of the Shapovalov form we construct all weight simple graded modules and some classes of simple weight modules over a twisted generalized Weyl algebra, improving the results from [6], where a particular class of algebras was considered and only special modules were classified.

Hochschild homology and cohomology of generalized Weyl algebras

Nous calculons l'homologie et la cohomologie de Hochschild des algebres de Weyl generalisees introduites par V. Bavula. Nous repondons a une question de Bavula-Jordan concernant les generateurs du groupe d'automorphismes d'une telle algebre. Ce calcul explique aussi des resultats connus sur les invariants des algebres de Weyl et des quotients primitifs.

FINITE-DIMENSIONAL REPRESENTATIONS OF QUANTUM GROUP

ABSTRACT Let not be a root of unity and be any Laurent polynomial in . In this paper, we define a class of generalized Weyl algebra which is denoted by . A necessary and sufficient condition for being a Hopf algebra is obtained. All finite-dimensional simple -modules are classified, and the center of is given.

Pure Injective Envelopes of Finite Length Modules over a Generalized Weyl Algebra

We investigate certain pure injective modules over generalised Weyl algebras. We consider pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finite-dimensional algebras and hereditary orders.

Representations of Twisted Generalized Weyl Constructions

We study bounded and unbounded *-representations of Twisted Generalized Weyl Algebras and algebras similar to them for different choices of involutions.

Involutions on generalized Weyl algebras preserving the principal grading

All *-representations of generalized Weyl algebras by bounded operators are classified with respect to the * preserving principal grading.

Generalized Weyl Algebras Are Tensor Krull Minimal

Formulae for calculating the Krull dimension of noetherian rings obtained by the authors and their collaborators are used to calculate Krull dimension for certain classes of algebras. An F-algebra T is said to be tensor Krull minimal (TKM) with respect to a class of F-algebras Ω if K(T⊗B)=K(T)+K(B), for each B∈Ω. We show that generalized Weyl algebras over affine commutative F-algebras, where F is an uncountable algebraically closed field, are TKM with respect to the class of countably generated left noetherian F-algebras. This simplifies the task of calculating many Krull dimensions. In addition, we develop an improved formula for the Krull dimension of a skew Laurent extension D[x,x−1;σ], where D is a polynomial algebra over an algebraically closed field, and σ is an affine automorphism. Finally, we calculate the Krull dimension of the noetherian down–up algebras introduced by Benkart.

THE FINITE LENGTH MODULES FOR THIN -GRADED RINGS

A -graded ring R = is called thin, provided R 0 = K is a skew field and each K–K-bimodule Ri has left and right dimension at most 1. Such rings occur, for example, as fiber products of skew polynomial rings, as numerical skew monoid rings or as factor rings or localizations of generalized Weyl algebras. We show that in case R is (graded) affine with no homogeneous units in nonzero degree, the representation type (finite, “twisted tame infinite,” “twisted wild”) can be read off from the combinatorial structure, that is, from the vanishing of products of homogeneous elements of certain degrees. Those rings R which are of twisted wild representation type admit by our definition a representation embedding from the category of finite dimensional modules over some twisted wild algebra T(K, σ K ⊕τ K) into the category mod R of finite length R-modules. The rings which we call here “twisted tame” come close to twisted versions of string algebras, for which we obtain the classification of the indecomposable finite length modules as a version of the Butler–Ringel theorem. Finally, for rings which are either not affine or which have homogeneous units in nonzero degree, the representation type does not depend on the combinatorial structure alone, as examples show.

Krull Dimension of Generalized Weyl Algebras with Noncommutative Coefficients

We obtain formulae for the Krull dimension of the generalized Weyl algebra T=R(σ,a), where R is a left noetherian ring.

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 U ( s l 2 ) U(\mathfrak {sl}_2) introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of U ( s l 2 ) U(\mathfrak {sl}_2) by finding sets of generators for the group of automorphisms.

The Extension Group of the Simple Modules Over the First Weyl Algebra

The aim of this paper is to prove that for certain generalized Weyl algebras A (including the first Weyl algebra A1 over a field of characteristic zero) and for every simple left (right) A-module M, there are infinitely many non-isomorphic simple left (right) A-modules {Ni} such that Ext1A (M, Ni) ≠ 0 (respectively Ext1A(Ni, M) ≠ 0.

Simple Holonomic Modules over the Second Weyl Algebra A2

For simple generalized Weyl algebras Λ of Gelfand–Kirillov dimension 4, a class including the second Weyl algebra A2, some simple factor algebras of the universal enveloping algebra of the Lie algebra sl(2)×sl(2) and of Uqsl(2)×Uqsl(2), etc., the simple holonomic Λ-modules are classified up to pairs of irreducible elements of certain noncommutative Euclidean ring.