In this paper we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew $PBW$ extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand-Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre's conjecture will be discussed for a particular class of skew $PBW$ extensions. The questions about the noetherianity and the Zariski cancellation property of Artin-Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.
Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.
In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = σ(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = σ(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.
In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew quantum polynomials. In general, if we have an endomorphism on a generic skew $PBW$ extension and there are some $x_i,x_j,x_u$ such that the endomorphism is not zero on this elements and the principal coefficients are invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the ring of coefficients. Of course, this is valid for quantum polyno…
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this short paper we study the d-Hermite condition about stably free modules for skew PBW extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and close related with these questions, …
In this paper we describe the prime ideals of some important classes of skew PBW extensions, using the classical technique of extending and contracting ideals. Skew PBW extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others.
We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-gr…
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand-Kirillov conjecture for the skew q…
In [1] the author gives a description of Poisson brackets on some algebras of quantum polynomials $\mathcal{O}_q$, which is called\textit{ general algebra of quantum polynomials}. The main of this paper is to present a generalization of [1] through a description of Poisson brackets on some skew PBW extensions of a ring $A$ by the extensions $\mathcal{O}^{r,n}_{q,δ}$, which are a generalization of $\mathcal{O}_q$ and show some examples of skew PBW extension where we can apply this description.
In this paper we compute the center of many noncommutative algebras that can be interpreted as skew $PBW$ extensions. We show that, under some natural assumptions on the parameters that define the extension, either the center is trivial, or, it is of polynomial type. As an application, we provided new examples of noncommutative algebras that are cancellative.
Many rings and algebras arising in quantum mechanics, algebraic analysis, and non-commutative algebraic geometry can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. In the present paper we study two aspects of these non-commutative rings: its finitely generated projective modules from a matrix-constructive approach, and the construction of the Gröbner theory for its left ideals and modules. These two topics could be interesting in future eventual applications of skew $PBW$ extensions in functional linear systems and in non-commutative algebraic geometry.
In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others. As a corollary we will give a new short proof of the Poincaré-Birkhoff-Witt theorem about the bas…
We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
For any differential graded (DG for short) Poisson algebra $A$ given by generators and relations, we give a "formula" for computing the universal enveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has a Poincaré-Birkhoff-Witt basis provided that $A$ is a graded commutative polynomial algebra. As an application of the PBW-basis, we show that a DG symplectic ideal of a DG Poisson algebra $A$ is the annihilator of a simple DG Poisson $A$-module, where $A$ is the DG Poisson homomorphic image of a DG Poisson algebra $R$ whose underlying algebra structure is a graded commutative polynomi…
In this paper we compute the $Tor$ and $Ext$ modules over skew $PBW$ extensions. If $A$ is a bijective skew $PBW$ extension of a ring $R$, we give presentations of $Tor_r^{A}(M,N)$, where $M$ is a finitely generated centralizing subbimodule of $A^m$, $m\geq 1$, and $N$ is a left $A$-submodule of $A^l$, $l\geq 1$. In the case of $Ext_A^{r}(M,N)$, $M$ is a left $A$-submodule of $A^m$ and $N$ is a finitely generated centralizing subbimodule of $A^l$. As application of these computations, we test stably-freeness, reflexiveness, and we will compute also the torsion, the dual and the grade of a give…
Koszul and homogeneous Koszul algebras were defined by Priddy in \cite{Priddy1970}. There exist some relations between these algebras and the skew PBW extensions introduced in \cite{LezamaGallego}. In this paper we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.
The aim of this paper is to investigate a general notion of $σ$-PBW extensions over Armendariz rings. As an application, the properties of being Baer, quasi-Baer, p.p. and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions.
As the case of free $\Bbbk$-algebras and $PBW$ algebras, given a bijective skew $PBW$ extension $A$, we will show that it is possible transfer Gröbner bases between $A$ and its associated graded ring.
In this article we give a description for the centralizer of the coefficient ring $R$ in the skew PBW extension $σ(R)<x_1,x_2,\cdots,x_n>.$ We give an explicit description in the quasi-commutative case and state a necessary condition in the general case. We also consider the PBW extension $σ(\mathcal{A})<x_1,x_2,\cdots,x_n>$ of the algebra of functions with finite support on a countable set, describing the centralizer of $\mathcal{A}$ and the center of the skew PBW extension.
Let R be a commutative algebra. In this paper we show that constant skew PBW extensions of a generalized Koszul algebra R are also generalized Koszul. Let A be a semi-commutative skew PBW extension of R such that A is R-augmented. We show also that if R is augmented Koszul then A is R-augmented Koszul.
In this paper we show that the tensor product of skew PBW extensions is a skew PBW ex- tension. We also characterize the enveloping algebra of a skew PBW extension. Finally, we establish sufficient conditions to guarantee the property of being skew Calabi-Yau algebra over skew PBW extensions.
A deformation $U$, of a graded $K$-algebra $A$ is said to be of PBW type if $grU$ is $A$. It has been shown for Koszul and $N$-Koszul algebras that the deformation is PBW if and only if the relations of $U$ satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a {\it finite} and explicitly determined linear algebra problem. We extend these results to an arbitrary graded $K$-algebra, using the notion of central extensions of algebras and a homological constant attached to $A$ which we call the {\it complexity} of $A$.
We establish necessary or sufficient conditions to guarantee that skew Poincaré-Birkhoff-Witt extensions are NI or NJ rings. Our results extend those corresponding for skew polynomial rings and establish similar properties for other families of noncommutative rings such as universal enveloping algebras and examples of differential operators.
In this paper, we introduce the concept of $Σ$-semicommutative ring, for $Σ$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $Σ$-rigid, nil-reversible and rings satisfying the $Σ$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $Σ$-semicommutative rings. We prove that if a ring $R$ is $Σ$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-…
In this paper we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended, in particular, a Zariski topology will be constructed. Assuming additionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field, we will prove an adapted version of the Hilbert's Nullstellensatz theorem that covers the classical one. The Gröbner bases of skew PBW extensions will be use…
We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.
We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincar\' e-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the $\mathbb{Z}_2$-graded PBW deformations of quantum exterior algebras. We also determine the pos…
We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier \cite{OuyangBirkenmeier2012} in the setting of noncommutative rings having PBW bases. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, differential operators on noncommutative spaces, noncommutative algebraic geometry, and theore…
We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter-Drinfeld structure has universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum D…
In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of $Σ$-skew CN rings and $Σ$-skew reflexive (RNP) rings, for $Σ$ a finite family of ring endomorphisms of a ring $R$. Under certain conditions of compatibility, we study the transfer of the $Σ$-skew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corre…
In this article we prove finite generation of the cohomology of quotients of a PBW algebra A by relating it to the cohomology of quotients of a quantum symmetric algebra S which is isomorphic to the associated graded algebra of A. The proof uses a spectral sequence argument and a finite generation lemma adapted from Friedlander and Suslin.
We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.
In this paper we extend some results obtained by Artamonov and Sabitov for quantum polynomials to skew quantum polynomials and quasi-commutative bijective skew PBW extensions. Moreover, we find a counterexample to the conjecture proposed in [6]
We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to …
In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.
We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.
We compute point schemes of some regular algebras using (Wolfram) Mathematica. These algebras are Ore extensions of regular graded skew Clifford algebras of global dimension 3.
For ungraded quotients of an arbitrary $\mathbb{Z}$-graded ring, we define the general PBW property, that covers the classical PBW property and the $N$-type PBW property studied via the $N$-Koszulity by several authors ([BG1], BG2], [FV]). In view of the noncommutative Gröbner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [Gol] concerning Gröbner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of ap…
We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.
In this paper, we study the Ext-algebras of graded skew extensions. For a connected graded algebra $A$ and a graded automorphism $σ$, we analyze the Yoneda product of the Ext-algebra of graded skew extension $A[z;σ]$, and prove this Ext-algebra is an $R$-smash product of the Ext-algebra of $A$ and the one of polynomial algebra $k[z]$ as an associative algebra.
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.
In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are included at the end of the paper.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is de…
Results of Vancliff, Van Rompay and Willaert in 1998 prove that point modules over a regular graded Clifford algebra (GCA) are determined by (commutative) quadrics of rank at most two that belong to the quadric system associated to the GCA. In 2010, Cassidy and Vancliff generalized the notion of a GCA to that of a graded skew Clifford algebra (GSCA). The results in this article show that prior results may be extended, with suitable modification, to GSCAs. In particular, using the notion of μ-rank introduced recently by the authors, the point modules over a regular GSCA are determined by (nonco…
Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abeli…
For a skew version of a graded $(A_\infty)$ hypersurface singularity $A$, we study the stable category of graded maximal Cohen-Macaulay modules over $A$. As a consequence, we see that $A$ has countably infinite Cohen-Macaulay representation type and is not a noncommutative graded isolated singularity.
