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 $U(\mathfrak{sl}_2)$, 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. In particular, this is true for $\Bbbk$-finitistic TGWAs of type $(A_1)^n$ and $A_2$. We apply this theorem to study properties of the fixed ring, such as 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.
We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter …
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus,…
We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure module for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on it…
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 $A(R,σ,tt')\to A(R,σ,t)\otimes_R A(R,σ,t')$. 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]…
In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings.
The aim of this work is to investigate the structure of some skew twisted algebras, when the coefficient ring is a localization of the polynomial ring over the field of characteristic zero, and an involution is provided. A parallel construction of the rational twisted generalized Weyl algebras is given. We propose a method and explicit formulas for a constructive description of these algebras and their involution-symmetric invariant subalgebras based on the Gelfand-Zeitlin realization of the universal enveloping algebra of some complex Lie algebras. As concrete examples we discuss special unit…
Global and local Weyl modules for the untwisted multiloop Lie algebras were defined by Chari, the first and the second author via homological properties. In this paper we extended the ideas to give a categorical definition of the Weyl modules for twisted multiloop algebras. Our methods led us to describe an identification of the finite--dimensional highest weight modules for twisted multiloop algebras with suitably chosen finite--dimensional highest weight modules for untwisted multiloop algebras.
To each multiquiver $Γ$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras $\mathcal{A}(Γ)$ carry a canonical representation by differential operators and that $\mathcal{A}(Γ)$ is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of $Γ$ for when this representation is faithful or locally surjective. By forgetting some of the structure of $Γ$ one obtains a Dynkin diagram, $D(Γ)…
Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional Hopf algebras in positive characteristic, and include bosonizations of Nichols algebras of Jordan type in a general setting as well as their liftings when $p=3$. Our techniques are applications of twisted tensor product resolutions and Anick resolutions in combination with May spectral sequences.
In this article, we work with unital twisted partial action of $\mathbb{Z}$ on an unital ring $R$ and we introduce the twisted partial skew power series rings and twisted partial skew Laurent series rings. We study primality, semi-primality and prime ideals in these rings. We completely describe the prime radical in partial skew Laurent series rings. Moreover, we study the Goldie property in partial skew power series rings and partial skew Laurent series rings and we describe conditions of the semiprimality of twisted partial skew power series 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 $\mathbb{Z}^n$-graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of $U(\mathfrak{g})$ where $\mathfrak{g}=\mathfrak{gl}_n$, $\mathfrak{sl}_n$, or $\mathfrak{sp}_{2n}$, algebras related to $U(\widehat{\mathfrak{sl}}_2)$ and a finite W-algebra associated to $\mathfrak{sl}_4$. To accomplish this classification we first show that the problem is equi…
The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old' GWAs. A new class of rings, the {\em diskew polynomial rings}, is introduced that is closely related to GWAs (they are GWAs under a mild condition). The, so-called, ambiskew polynomial rings are a small subclass of the class of diskew polynomial rings. Semisimplicity criteria are given for generalized Weyl algebras and diskew polynomial rings.
In this paper we show that each non-zero ideal of a twisted generalized Weyl algebra (TGWA) $A$ intersects the centralizer of the distinguished subalgebra $R$ in $A$ non-trivially. We also provide a necessary and sufficient condition for the centralizer of $R$ in $A$ to be commutative, and give examples of TGWAs associated to symmetric Cartan matrices satisfying this condition. By imposing a certain finiteness condition on $R$ (weaker than Noetherianity) we are able to make an Ore localization which turns out to be useful when investigating simplicity of the TGWA. Under this mild assumption we…
Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information about $G$ is encoded in all the twisted group rings of $G$ over $R$. We investigate this problem over fields. We start with abelian groups and show how the results depend on the roots of unity in $R$. In order to deal with non-abelian groups we construct a generalization of a Schur cover which exists also when $R$ is not an algebraically closed field, but st…
A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by V.Mazorchuk and L.Turowska (1999) it is claimed that certain consistency conditions for s_i and t_i are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R…
Twisted generalized Weyl algebras (TGWAs) are defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac-Moody algebras can be defined. In this paper we introduce the class of locally finite TGWAs, and show that one can associate to such an algebra a polynomial Cartan matrix (a notion extending the usual generalized Cartan matrices appearing in Kac-Moody algebra theory) and that the corresponding generalized Serre relations hold in the TGWA. We also give an explicit construction of a family of locally finite TGWAs dependi…
We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to study the problem in particular over any field of characteristic $0$. We prove that there are finite groups $G$ and $H$ which can not be distinguished by their rational twisted group algebras, while $G$ and $H$ can be identified by their semi-simple twisted group algebras over other fields. This is in contrast with the fact that the structural information on $G$…
We give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple Lie algebra over a ring. In particular, the universal central extensions of twisted multiloop Lie tori are obtained by the descent construction.
We study actions by filtered automorphisms on classical generalized Weyl algebras (GWAs). In the case of a defining polynomial of degree two, we prove that the fixed ring under the action of a finite cyclic group of filtered automorphisms is again a classical GWA, extending a result of Jordan and Wells. Partial results are provided for the case of higher degree polynomials. In addition, we establish a version of Auslander's theorem for finite cyclic groups of filtered automorphisms acting on classical GWAs.
If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest.
We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories.
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 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, rigidity, and simplicity.
We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.
In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin-Tate-van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.
Homological smoothness and twisted Calabi-Yau property of generalized Weyl algebras over polynomial algebras in two variables is studied. A necessary and sufficient condition to be homologically smooth is given. The Nakayama automorphisms of such algebras are also computed in terms of the Jacobian determinants of defining automorphisms.
Let U be the group of units of an infinite twisted group algebra K_λG over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus' theorem.
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras wh…
In this short note, we construct a right adjoint to the functor which associates to a ring $R$ equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an $R$-module to the group of semilinear $R$-module automorphisms of the module. As an immediate corollary, we provide a novel proof of the classical observation that modules over a twisted group ring are modules over the base ring together with a semilinear action.
In this paper we prove that if R is a left Noetherian and left regular ring such that all finitely generated projective left R-modules are stably free, then the same is true for the completion R[[x;σ,δ]] of any Ore extension R[x;σ,δ] of R. Additionally assumes certain natural conditions for σand δfor ensure that this completion are good defined. The proof compares the Grothendieck groups K_{0} of R and S using some results find in [3], [6], [7] and [9].
We classify reflexive graded right ideals, up to isomorphism and shift, of generic cubic three dimensional Artin-Schelter regular algebras. We also determine the possible Hilbert functions of these ideals. These results are obtained by using similar methods as for quadratic Artin-Schelter algebras. In particular our results apply to the enveloping algebra of the Heisenberg-Lie algebra from which we deduce a classification of right ideals of an invariant ring of the first Weyl algebra.
We study $(σ,τ)$-derivations of a group ring $RG$ where $G$ is a group with center having finite index in $G$ and $R$ is a semiprime ring with $1$ such that either $R$ has no torsion elements or that if $R$ has $p$-torsion elements, then $p$ does not divide the order of $G$ and let $σ,τ$ be $R$-linear endomorphisms of $RG$ fixing the center of $RG$ pointwise. We generalize Main Theorem $1.1$ of \cite{Chau-19} and prove that there is a ring $T\supset R$ such that $\mathcal{Z}(T)\supset\mathcal{Z}(R)$ and that for the natural extensions of $σ, τ$ to $TG$ we get $H^1(TG,{}_σTG_τ)=0$, where ${}_σT…
The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational clas…
We prove that the Weyl algebra over $\mathbb{C}$ cannot be a fixed ring of any domain under a nontrivial action of a finite group by algebra automorphisms, thus settling a 30-year old problem. In fact, we prove the following much more general result. Let $X$ be a smooth affine variety over $\mathbb{C}$, let $D(X)$ denote the ring of algebraic differential operators on $X,$ and let $Γ$ be a finite group. If $D(X)$ is isomorphic to the ring of $Γ$-invariants of a $\mathbb{C}$-domain $R$ on which $Γ$ acts faithfully by $\mathbb{C}$-algebra automorphisms, then $R$ is isomorphic to the ring of diff…
In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra $\A_h$ generated by elements $x,y,$ which satisfy the relation $yx-xy=h$ for some $0\neq h\in \FF[x]$. We generalize this construction to $\A_h(\B)$ by working over the fixed $\FF$-algebra $\B$ instead of $\FF$. We describe the polynomial identities for $\A_h(\B)$ over the infinite field $\FF$ in case $h\in\B[x]$ satisfies certain restrictions.
Given a projective scheme $X$ over a field $k$, an automorphism $σ$ of $X$, and a $σ$-ample invertible sheaf $L$, one may form the twisted homogeneous coordinate ring $B = B(X, L, σ)$, one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of $B$, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field $k$ of characteristic zero, we prove that that the primitive ideals of $B$ are characterized by the usual Dixmier-Moegli…
Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.
There is an error in the computation of the truncated point schemes of the degenerate Sklyanin algebra S(1, 1, 1); they are larger than was claimed in Proposition 3.13 of the above paper. We provide a description of the correct truncated point schemes. Results about the corresponding point parameter ring associated to these schemes are given afterward.
In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are often algebras over a finite field, allows us to detect many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes.
