Generalized Weyl Algebras Research Articles

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work.

New simple [formula omitted]-modules from Weyl algebra modules

We construct a class of modules over the affine Lie algebra slˆ2 from the modules over the degree-2 Weyl algebra K2 and modules over the 2-dimensional solvable Lie algebra b. We determine the irreducibility and isomorphisms of these modules with examples given by Bavula's construction of modules for generalized Weyl algebras in [5]. Finally, we show these irreducible slˆ2-modules are generally new.

Inner generalized Weyl algebras and their simplicity criteria

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA [Formula: see text] a derivative series of IGWAs, [Formula: see text], is attached where [Formula: see text] is an arbitrary ordinal. In general, all rings [Formula: see text] are distinct. A new construction of rings, the inner [Formula: see text]-extension of a ring, is introduced (where [Formula: see text] and [Formula: see text] are endomorphisms of a ring [Formula: see text] and [Formula: see text]).

Weight modules over Bell–Rogalski algebras

We study a class of Z-graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order.

Derivations of quantum and involution generalized Weyl algebras

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of [Formula: see text], and a family of algebras which localize to the group algebra of the infinite group with generators [Formula: see text] and [Formula: see text], subject to the relation [Formula: see text].

Fixed rings of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, such as Weyl algebras, primitive quotients of U(sl2), and multiparameter quantized Weyl algebras. In this work, we study invariants of TGWAs under diagonal automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply our results particularly to k-finitistic TGWAs of type (A1)n and A2, and study properties of their fixed rings, including the noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring. As an auxiliary result, in order to study invariants of tensor products of TGWAs, we prove that the class of regular, μ-consistent TGWAs is closed under tensor products.

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs.

K-teoría bivariante de álgebras localmente convexas Z-graduadas

In the present work, we describe some results about the K-theory of Z-graded algebras. First, in the context of C* algebras, we begin with the Pimsner-Voiculescu sequence for crossed products and its generalizations. We will see that there are results analog to these in the context of locally convex algebras and we conclude with results for generalized Weyl algebras.

Pointed Hopf actions on quantum generalized Weyl algebras

We study actions of pointed Hopf algebras in the Z-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the Z-grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weyl algebras are commutative rings.

Reflexive hull discriminants and applications

We introduce the reflexive hull discriminant as a tool to study noncommutative algebras that are finitely generated, but not necessarily free, over their centers. As an example, we compute the reflexive hull discriminants for quantum generalized Weyl algebras and use them to determine automorphism groups and other properties, recovering results of Su{\'a}rez-Alvarez, Vivas, and others.

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.

Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })\). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when \(R=\mathbb {C}[z]\). As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over \(\mathfrak {sl}_{2}\) is a tensor product of two Weyl algebra modules.

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

LD-stability for Goldie rings

The lower transcendence degree , introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras , the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD -stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras , finite W -algebras of type A , generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups . We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.

An extension of [formula omitted] related to the alternating group and Galois orders

In 2010, V. Futorny and S. Ovsienko gave a realization of U(gln) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S1×S2×⋯×Sn, where Sj is the symmetric group on j variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted A(gln), and show that it is a Galois ring. For n=2, we show that it is a generalized Weyl algebra, and for n=3 provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over A(gln). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A(gln), and the positive solution to Noether's problem for the alternating group.

Bivariant K-theory of generalized Weyl algebras

We compute the isomorphism class in \mathfrak{KK}^{\mathrm {alg}} of all noncommutative generalized Weyl algebras A=\mathbb C[h](\sigma, P) ,where \sigma(h)=qh+h_0 is an automorphism of \mathcal C[h] , except when q\neq 1 is a root of unity. In particular, we compute the isomorphism class in \mathfrak{KK}^{\mathrm {alg}} of the quantum Weyl algebra, the primitive factors B_{\lambda} of U(\mathfrak{sl}_2) and the quantum weighted projective lines \mathcal{O}(\mathbb{WP}_q(k, l)) .

Simple [formula omitted]-graded domains of Gelfand–Kirillov dimension two

Let k be an algebraically closed field and A a Z-graded finitely generated simple k-algebra which is a domain of Gelfand–Kirillov dimension 2. We show that the category of Z-graded right A-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of U(sl2).

Fixed rings of quantum generalized Weyl algebras

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.

Classification of twisted generalized Weyl algebras over polynomial rings

Let R be a polynomial ring in m variables over a field of characteristic zero. We classify all rank n twisted generalized Weyl algebras over R, up to Zn-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of U(g) where g=gln, sln, or sp2n, algebras related to U(slˆ2) and a finite W-algebra associated to sl4. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case n=2, which can be solved using methods from previous work by the authors [17], [15]. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation.

Generalized Weyl algebras and diskew polynomial rings

The aim of the paper is to extend the class of generalized Weyl algebras (GWAs) to a larger class of rings (they are also called GWAs) that are determined by two ring endomorphisms rather than one as in the case of ‘old’ GWAs. A new class of rings, the diskew polynomial rings, is introduced that is closely related to GWAs (they are GWAs under a mild condition). Simplicity criteria are given for GWAs and diskew polynomial rings.