Non-finitely generated projective modules over generalized Weyl algebras

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.

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.

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)) .

Galois orders, introduced in 2010 by V. Futorny and S. Ovsienko, form a class of associative algebras that contain many important examples, such as the enveloping algebra of gl_n (as well as its quantum deformation), generalized Weyl algebras, and shifted Yangians. The main motivation for introducing Galois orders is they provide a setting for studying certain infinite dimensional irreducible representations, called Gelfand-Tsetlin modules. Principal Galois orders, defined by J. Hartwig in 2017, are Galois orders with an extra property, which makes them easier to study. All of the mentioned examples are principal Galois orders. In 2019, B. Webster defined principal flag orders which are Morita equivalent to principal Galois orders and further simplifies their study. The purpose of this dissertation is twofold: (1) To introduce a new example of a Galois order, A(gl_n), which is an extension of the enveloping algebra of gl_n such that the ``Weyl group'' of A(gl_n) is the alternating group; (2) To describe some techniques to study such objects including tensor products and morphisms between standard flag orders with conjectured application to the orthogonal Lie algebra.

Involutions on generalized Weyl algebras preserving the principal grading

Let \( K \) be an algebraically closed field of characteristic zero. The generalized Weyl algebra \( A_{n,f} \) is defined by generators \( x_1, x_2, \dots, x_n, y_1, \dots, y_n, z_1, \dots, z_n \) subject to certain commutation relations and additional structure determined by a collection of functions \( f = (f_1, \dots, f_n) \). We focus on the structure of left and right ideals in \( A_{n,f}(K) \), particularly proving that every left or right ideal can be generated by two elements. The proof is based on showing that if a left ideal can be generated by three elements, it can be reduced to two elements by applying the Noetherian property of the ring and an iterative reduction process. This result complements the simplicity of \( A_{n,f}(K) \), as established in prior work.

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.

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A_1; (b) simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and (c) simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A-infinity modules over A_1 to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $C^2/\Gamma$, where $\Gamma$ is a finite cyclic subgroup of $SL(2,C)$.

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.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1 → M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

In this research, the concepts injective and projective semi modules have been generalized to generalized-injective(projective) and essentially-injective semimodules, also the relationship of these concepts with almost-injective(projective) semimodules have been discussed. The semiring having the properties that every quasi- injective semimodule is injective and every almost self-injective semimodule is injective have been defined. Some results between these concepts have been obtained like, every almost-injective is generalized-injective semimodule, the definitions, where a semi module is indecomposable have been discussed. Dually for almost-projective semimodule. Also obtained that, the direct sum of two quasi-injective Ȑ-semimodule is quasi-injective when Ȑ is QI-semiring, this fact has been expanded to AQI-semiring.
 Conclusion: In this paper, several generalization of injective and projective semi modules have been defined and some concepts with related to almost injective(projective)semimodules have been presented, also the concepts QI-semiring, AQI-semiring defined, some relationship between these concepts and almost injective(projective) semimodules discussed.

Let G be a finite group with a Sylow 2-subgroup P which is either quaternion or semi-dihedral. Let k be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial kG -modules, whose restrictions to P are isomorphic to the direct sum of the known exotic endotrivial kP -modules and some projective modules. This provides a description of the group T (G) of endotrivial kG -modules.

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.