Some open problems in the context of skew PBW extensions and semi-graded rings

TL;DR

This paper explores open problems in non-commutative algebra and algebraic geometry using skew PBW extensions and semi-graded rings, focusing on conjectures, isomorphisms, and properties like noetherianity.

Contribution

It analyzes key open problems and reformulates conjectures within the framework of skew PBW extensions and semi-graded rings, providing new insights and partial solutions.

Findings

Analysis of isomorphisms related to the Gelfand-Kirillov conjecture

Discussion of Serre's conjecture for specific skew PBW extensions

Reformulation of noetherianity and Zariski cancellation questions for semi-graded rings

Abstract

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.

Equations

$$\mathrm{GKdim}(A):=\underset{V}{\sup }\overline{\underset{n\to \infty }{\lim }}{\log }_n{\dim }_K{V}^n,$$

$$\mathrm{qgr}-A\simeq \mathrm{qgr}-\Gamma (\pi (A){)}_{\ge 0},$$

$$x_{i}r-c_{i,r}{x}_i\in R.$$

$$x_j{x}_i-c_{i,j}{x}_i{x}_j\in R+R{x}_1+\cdots +R{x}_n.$$

$$x_j{x}_i=c_{ij}{x}_i{x}_j+a_{ij}^{(1)}{x}_1+\cdots +a_{ij}^{(n)}{x}_n+d_{ij},\text{for every}1\le i,j\le n.$$

$$x_{i}r=c_{i,r}{x}_i.$$

$$x_j{x}_i=c_{i,j}{x}_i{x}_j.$$

$$F_n(B):=B_0\oplus \cdots \oplus B_n.$$

$$Ho{m}_{\mathrm{qsgr}-\mathrm{B}}(M,N):=\underrightarrow{\lim }Ho{m}_{\mathrm{sgr}-B}(M^{\prime },N/N^{\prime }),$$

$$\mathrm{qgr}-\underline{B}\simeq \mathrm{qgr}-\Gamma (\pi (B){)}_{\ge 0}.$$

$$\underline{B}$$

$$f_0+\cdots +f_d$$

$$\mathrm{qgr}-Gr(B)\simeq \mathrm{qgr}-\underline{B}\simeq \mathrm{qgr}-\Gamma (\pi (B){)}_{\ge 0}.$$

$$\underset{d=0}{\overset{\infty }{⨁}}Ho{m}_{\mathrm{sgr}-B}(B,B(d))=\underline{B}$$

$$f_0+\cdots +f_d$$

$$\underset{d=0}{\overset{\infty }{⨁}}Ho{m}_{\mathrm{sgr}-B}(B,B(d))=\underline{B}$$

$$f_0+\cdots +f_d$$

$$Q(\mathcal{U}(\mathcal{G}))\cong Q(A_n(L[s_1,\dots ,s_k])),$$

$$Q(Q_{\text{q},\sigma }^{k,n}(R))\cong Q({\text{Q}}_{\text{q},\sigma }[x_1,\dots ,x_n]),\text{with}\text{Q}:=Q(R).$$

$$UF{U}^{-1}=\left[\begin{array}{cc}0& 0\\ 0& I_r\end{array}\right],$$

$$UF{U}^{-1}=\left[\begin{array}{cc}0& 0\\ 0& I_r\end{array}\right]\text{and}r=dim(⟨F⟩).$$

$$U_k^{\prime }{F}_k{U}_k^{\prime -1}=\left[\begin{array}{cc}{\alpha }_k& 0\\ 0& F_{k+1}\end{array}\right],\text{ where }{\alpha }_k\in \{0,1\}.$$

$$F^{(1)}=\left[\begin{array}{c}-t^{2}q{x}^2\\ \left(-ta+2t\right)x-2a+2\\ tx+2\\ -1\end{array}\right],$$

$$F^{(2)}=\left[\begin{array}{c}-2tx+2\\ -t^{2}q{x}^2+\left(ta-4t\right)x+2a-1\\ -tx-2\\ tx+2\end{array}\right],$$

$$F^{(3)}=\left[\begin{array}{c}-tx-2\\ \left(-2t^{2}qa+3t^{2}q\right)x^2+\left(a^{2}t-8ta+8t\right)x+2a^2-3a+1\\ t^{2}q{x}^2+\left(-ta+4t\right)x-2a+2\\ \left(ta-2t\right)x+2a-2\end{array}\right],$$

$$F^{(4)}=\left[\begin{array}{c}-t^3{q}^3{x}^3+\left(-q^2{t}^2-5t^{2}q\right)x^2-5tx+2\\ -t^3{q}^3{x}^3+\left(-q^2{t}^2-3t^{2}q\right)x^2+\left(-ta+t\right)x-2a+2\\ tx+2\\ t^{2}q{x}^2+2tx-1\end{array}\right].$$

$$U=\left[\begin{array}{cccc}tx+1& 0& t^{2}q{x}^2+2tx-1& t^{2}q{x}^2+3tx\\ 1& -tx-2& \left(-ta+2t\right)x-2a+2& -t^{2}q{x}^2-2tx+2\\ tx-1& 1& t^{2}q{x}^2+a-1& t^{2}q{x}^2+2tx-1\\ 1& 0& tx& tx+1\end{array}\right],$$

$$U^{-1}=\left[\begin{array}{cccc}tx& -1& -tx-2& 0\\ a-1& -tx+a-1& -t^{2}q{x}^2+\left(ta-4t\right)x+2a-1& t^3{q}^3{x}^3-\left(-q+a-4\right)t^{2}q{x}^2+\left(-3ta+3t\right)x+1\\ -1& -1& -tx-2& t^{2}q{x}^2+3tx\\ 0& 1& tx+2& -t^{2}q{x}^2-2tx+1\end{array}\right],$$

$$UF{U}^{-1}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$$

$$0\to B\stackrel{{\varphi }_2=\left[\begin{array}{ccc}-x_3& x_2& x_1\end{array}\right]}{\to }{B}^3\stackrel{{\varphi }_1=\left[\begin{array}{ccc}1+x_2& -x_1& 0\\ x_3& -1& x_1\\ 0& x_3& 1-x_2\end{array}\right]}{\to }{B}^3\stackrel{{\varphi }_0=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]}{\to }B\stackrel{ϵ}{\to }K\to 0.$$

$$0\to Ho{m}_B(K,B)\stackrel{{ϵ}^{\ast }}{\to }Ho{m}_B(B,B)\stackrel{{\varphi }_0^{\ast }}{\to }Ho{m}_B(B^3,B)\stackrel{{\varphi }_1^{\ast }}{\to }Ho{m}_B(B^3,B)\stackrel{{\varphi }_2^{\ast }}{\to }Ho{m}_B(B,B)\to 0.$$

$$0\to B\stackrel{{\varphi }_0^{\ast }=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]}{\to }{B}^3\stackrel{{\varphi }_1^{\ast }=\left[\begin{array}{ccc}1+x_2& -x_1& 0\\ x_3& -1& x_1\\ 0& x_3& 1-x_2\end{array}\right]}{\to }{B}^3\stackrel{{\varphi }_2^{\ast }=\left[\begin{array}{ccc}-x_3& x_2& x_1\end{array}\right]}{\to }B\to 0.$$

Full text

Key words and phrases. Gelfand-Kirillov conjecture, Serre’s conjecture, Artin-Schelter regular algebras, Zariski cancellation problem, skew $PBW$ extensions, semi-graded rings.

2010 Mathematics Subject Classification. Primary: 16S36. Secondary: 16U20, 16D40, 16E05, 16E65, 16S38, 16S80, 16W70, 16Z05.

1 Introduction

Commutative algebra and algebraic geometry are source of many interesting problems of non-commutative algebra and geometry. For example, the famous theorem of Serre of commutative projective algebraic geometry states that the category of coherent sheaves over the projective $n$-space $\mathbb{P}^{n}$ is equivalent to a category of noetherian graded modules over a graded commutative polynomial ring. The study of this equivalence for non-commutative algebras gave origin to an intensive study of the so-called non-commutative projective schemes associated to non-commutative finitely graded noetherian algebras, and produced a beautiful result, due to Artin and Zhang, and also proved independently by Verevkin, known as the non-commutative version of Serre’s theorem. Another famous example that has to be mentioned is the Zariski cancellation problem on the algebra of commutative polynomials $K[x_{1},\dots,x_{n}]$ over a field $K$, this problem asks if given a commutative $K$-algebra $B$ and an isomorphism $K[x_{1},\dots,x_{n}][t]\cong B[t]$, follows that $K[x_{1},\dots,x_{n}]\cong B$. This problem has been reformulated for non-commutative algebras and occupied the attention of many researchers in the last years. In this paper we will present some famous open problems of non-commutative algebra and non-commutative algebraic geometry, but interpreting them from the approach of skew $PBW$ extensions and semi-graded rings. For some of these problems we will include some advances. Skew $PBW$ extension were define first in [36], many important algebras coming from mathematical physics are particular examples of skew $PBW$ extensions: The enveloping algebra $\cal{U}(\cal{G})$, where $\cal{G}$ is a finite dimensional Lie algebra, the algebra of $q$-differential operators, the algebra of shift operators, the additive analogue of the Weyl algebra, the multiplicative analogue of the Weyl algebra, the quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, $3$-dimensional skew polynomial algebras, the dispin algebra, the Woronowicz algebra, the $q$-Heisenberg algebra, among many others (see [38] for the definition of these algebras). On the other hand, the semi-graded rings generalize the finitely graded algebras and the skew PBW extensions, this general class of non-commutative rings was introduced in [37].

The problems that we will analyze and formulate from the general approach of skew $PBW$ extensions and semi-graded rings are: 1) Investigate the Serre-Artin-Zhang-Verevkin theorem (see [10]) for semi-graded rings. 2) Study the Gelfand-Kirillov conjecture ([28]) for bijective skew $PBW$ extensions over Ore domains. Moreover, investigate for them the commutation between the center and the total ring of fractions. 3) Give a matrix-constructive proof of the Quillen-Suslin theorem (see [53], [46] and [55]) for a particular class of skew polynomials rings. Include an algorithm that computes a basis of a given finitely generated projective module. 4) Investigate if the semi-graded Artin-Schelter regular algebras ([9]) are noetherian domains. 5) Investigate the cancellation property (see [13]) for a particular class of semi-graded Artin-Schelter regular algebras of global dimension $3$.

Despite of the paper is a revision of previous works, written with the purpose of reformulating some famous open problems in a more general context, some new results are included. The first problem was solved in [37] assuming that the semi-graded rings are domains, in the present paper we will prove an improved result avoiding the restriction about the absence zero divisors. With respect to the problem 4), we give an advance showing that a particular class of semi-graded Artin-Schelter regular algebras are noetherian domains. Moreover, all of examples of semi-graded Artin-Schelter regular algebras exhibited in the present paper are noetherian domains. The new results of the paper are concentrated in Theorem 1.24, Corollary 1.25 and Theorem 4.14.

The precise description of the classical famous open problems, and the new reformulation of them in the context of the present paper, will be given below in each section. We have included basic definitions and facts about finitely graded algebras, skew $PBW$ extensions and semi-graded rings, needed for a better understanding of the paper. If not otherwise noted, all modules are left modules; $B$ will denote a non-commutative ring; $Z(B)$ is the center of $B$; $M_{s}(B)$ be the ring of $s\times s$ matrices over $B$, and $GL_{s}(B):=\{F\in M_{s}(B)\mid F\,\text{is invertible in}\,M_{s}(B)\}$ be the general linear group over $S$; $K$ will be a field.

1.1 Finitely graded algebras

Our first problem concerns with the generalization of a famous theorem of non-commutative projective algebraic geometry over finitely graded algebras. In this subsection we review the theorem and some well-known basic facts of finitely graded algebras (see [51]).

Definition 1.1**.**

Let $K$ be a field. It is said that a $K$-algebra $A$ is finitely graded if the following conditions hold:

• •

$A$* is $\mathbb{N}$-graded: $A=\bigoplus_{n\geq 0}A_{n}$.*

• •

$A$* is connected, i.e., $A_{0}=K$.*

• •

$A$* is finitely generated as $K$-algebra.*

The most remarkable examples of finitely graded algebras for which non-commutative projective algebraic geometry has been developed are the quantum plane, the Jordan plane, the Sklyanin algebra and the multi-parametric quantum affine $n$-space (see [51]).

Let $A$ be a finitely graded $K$-algebra and $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ be a $\mathbb{Z}$-graded $A$-module which is finitely generated. Then,

1. (i)

For every $n\in\mathbb{Z}$, $\dim_{K}M_{n}<\infty$. 2. (ii)

The Hilbert series of $A$ is defined by

$h_{A}(t):=\sum_{n=0}^{\infty}(\dim_{K}A_{n})t^{n}$. 3. (iii)

The Gelfand-Kirillov dimension of $A$ is defined by

[TABLE]

where $V$ ranges over all frames of $A$ and $V^{n}:=\,_{K}\langle v_{1}\cdots v_{n}|v_{i}\in V\rangle$ (a frame of $A$ is a finite dimensional $K$-subspace of $A$ such that $1\in V$; since $A$ is a $K$-algebra, then $K\hookrightarrow A$, and hence, $K$ is a frame of $A$ of dimension $1$). 4. (iv)

A famous theorem of Serre on commutative projective algebraic geometry states that the category of coherent sheaves over the projective $n$-space $\mathbb{P}^{n}$ is equivalent to a category of noetherian graded modules over a graded commutative polynomial ring. The study of this equivalence for non-commutative finitely graded noetherian algebras is known as the non-commutative version of Serre’s theorem and is due to Artin, Zhang and Verevkin. In the next numerals we give the ingredients needed for the formulation of this theorem (see [10], [57]). 5. (v)

Suppose that $A$ is left noetherian. Let ${\rm gr}-A$ be the abelian category of finitely generated $\mathbb{Z}$-graded left $A$-modules. The abelian category ${\rm qgr}-A$ is defined in the following way: The objects are the same as the objects in ${\rm gr}-A$, and we let $\pi:{\rm gr}-A\to{\rm qgr}-A$ be the identity map on the objects. The morphisms in ${\rm qgr}-A$ are defined in the following way:

$Hom_{\rm{qgr}-A}(\pi(M),\pi(N)):=\underrightarrow{\lim}Hom_{{\rm gr}-A}(M_{\geq n},N/T(N))$,

where the direct limit is taken over maps of abelian groups

$Hom_{{\rm gr}-A}(M_{\geq n},N/T(N))\to Hom_{{\rm gr}-A}(M_{\geq n+1},N/T(N))$

induced by the inclusion homomorphism $M_{\geq n+1}\to M_{\geq n}$; $T(N)$ is the torsion submodule of $N$ and an element $x\in N$ is torsion if $A_{\geq n}x=0$ for some $n\geq 0$. The pair $(\rm{qgr}-A,\pi(A))$ is called the non-commutative projective scheme associated to $A$, and denoted by $\rm{qgr}-A$. Thus, ${\rm qgr}-A$ is a quotient category, ${\rm qgr}-A={\rm gr}-A/{\rm tor}-A$. 6. (vi)

$($Serre’s theorem$)$ Let $A$ be a commutative finitely graded $K$-algebra generated in degree $1$. Then, there exists an equivalence of categories

${\rm qgr}-A\simeq{\rm coh}({\rm proj}(A))$.

In particular,

${\rm qgr}-K[x_{0},\dots,x_{n}]\simeq{\rm coh}(\mathbb{P}^{n})$. 7. (vii)

Suppose that $A$ is left noetherian and let $i\geq 0$, it is said that $A$ satisfies the $\chi_{i}$ condition if for every finitely generated $\mathbb{Z}$-graded $A$-module $M$, $\dim_{K}(\underline{Ext}_{A}^{j}(K,M))<\infty$ for any $j\leq i$; the algebra $A$ satisfies the $\chi$ condition if it satisfies the $\chi_{i}$ condition for all $i\geq 0$ (since $K$ is finitely generated as $A$-module, then $\dim_{K}(\underline{Ext}_{A}^{j}(K,M))=\dim_{K}(Ext_{A}^{j}(K,M))$). 8. (viii)

$($Serre-Artin-Zhang-Verevkin theorem$)$ If $A$ is left noetherian and satisfies $\chi_{1}$, then $\Gamma(\pi(A))_{\geq 0}$ is left noetherian and there exists an equivalence of categories

[TABLE]

where $\Gamma(\pi(A))_{\geq 0}:=\bigoplus_{d=0}^{\infty}Hom_{{\rm qgr}-A}(\pi(A),s^{d}(\pi(A)))$ and $s$ is the autoequivalence of ${\rm qg}r-A$ defined by the shifts of degrees.

1.2 Skew $PBW$ extensions

The second ingredient needed for setting some of the problems that we will discuss in the present paper are the skew $PBW$ extensions introduced first in [36]. Skew $PBW$ extensions are non-commutative rings of polynomial type and cover many examples of quantum algebras and rings coming from mathematical physics: Habitual ring of polynomials in several variables, Weyl algebras, enveloping algebras of finite dimensional Lie algebras, algebra of $q$-differential operators, many important types of Ore algebras, algebras of diffusion type, additive and multiplicative analogues of the Weyl algebra, dispin algebra $\mathcal{U}(osp(1,2))$, quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, Woronowicz algebra $\mathcal{W}_{\nu}(\mathfrak{sl}(2,K))$, Manin algebra $\mathcal{O}_{q}(M_{2}(K))$, coordinate algebra of the quantum group $SL_{q}(2)$, $q$-Heisenberg algebra H${}_{n}(q)$, Hayashi algebra $W_{q}(J)$, differential operators on a quantum space $D_{\textbf{q}}(S_{\textbf{q}})$, Witten’s deformation of $\mathcal{U}(\mathfrak{sl}(2,K))$, multiparameter Weyl algebra $A_{n}^{Q,\Gamma}(K)$, quantum symplectic space $\mathcal{O}_{q}(\mathfrak{sp}(K^{2n}))$, some quadratic algebras in 3 variables, some 3-dimensional skew polynomial algebras, particular types of Sklyanin algebras, among many others. For a precise definition of any of these rings and algebras see [38]

Definition 1.2** ([36]).**

Let $R$ and $A$ be rings. We say that $A$ is a skew $PBW$ extension of $R$ $($also called a $\sigma-PBW$ extension of R$$) if the following conditions hold:

1. (i)

$R\subseteq A$. 2. (ii)

There exist finitely many elements $x_{1},\dots,x_{n}\in A$ such $A$ is a left $R$-free module with basis

${\rm Mon}(A):=\{x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\mid\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}\}$, with $\mathbb{N}:=\{0,1,2,\dots\}$.

The set ${\rm Mon}(A)$ is called the set of standard monomials of $A$. 3. (iii)

For every $1\leq i\leq n$ and $r\in R-\{0\}$ there exists $c_{i,r}\in R-\{0\}$ such that

[TABLE] 4. (iv)

For every $1\leq i,j\leq n$ there exists $c_{i,j}\in R-\{0\}$ such that

[TABLE]

Under these conditions we will write $A:=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$.

Associated to a skew $PBW$ extension $A=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ there are $n$ injective endomorphisms $\sigma_{1},\dots,\sigma_{n}$ of $R$ and $\sigma_{i}$-derivations, as the following proposition shows.

Proposition 1.3** ([36], Proposition 3).**

Let $A$ be a skew $PBW$ extension of $R$. Then, for every $1\leq i\leq n$, there exist an injective ring endomorphism $\sigma_{i}:R\rightarrow R$ and a $\sigma_{i}$-derivation $\delta_{i}:R\rightarrow R$ such that

$x_{i}r=\sigma_{i}(r)x_{i}+\delta_{i}(r)$,

for each $r\in R$.

From Definition 1.2 (iv), there exists a unique finite set of constants $c_{ij},d_{ij},a_{ij}^{(k)}\in R$ such that

[TABLE]

A particular case of skew $PBW$ extension is when all derivations $\delta_{i}$ are zero. Another interesting case is when all $\sigma_{i}$ are bijective and the constants $c_{ij}$ are invertible.

Definition 1.4** ([36]).**

Let $A$ be a skew $PBW$ extension.

1. (a)

$A$* is quasi-commutative if the conditions (iii) and (iv) in Definition 1.2 are replaced by*

1. (iii’)

For every $1\leq i\leq n$ and $r\in R-\{0\}$ there exists $c_{i,r}\in R-\{0\}$ such that

[TABLE] 2. (iv’)

For every $1\leq i,j\leq n$ there exists $c_{i,j}\in R-\{0\}$ such that

[TABLE] 2. (b)

$A$* is bijective if $\sigma_{i}$ is bijective for every $1\leq i\leq n$ and $c_{i,j}$ is invertible for any $1\leq i<j\leq n$.*

Observe that quasi-commutative skew $PBW$ extensions are $\mathbb{N}$-graded rings, but arbitrary skew $PBW$ extensions are semi-graded rings as we will see in the next subsection. Actually, the main motivation for constructing the non-commutative algebraic geometry of semi-graded rings is due to arbitrary skew $PBW$ extensions.

Many properties of skew $PBW$ extensions have been studied in previous works (see [2], [3], [38], [47], [48], [49], [50]). The next theorem establishes two ring theoretic results for skew $PBW$ extensions.

Theorem 1.5** ([38], [2]).**

Let $A$ be a bijective skew $PBW$ extension of a ring $R$.

1. (i)

$($Hilbert Basis Theorem$)$* If $R$ is a left $($right$)$ Noetherian ring then $A$ is also left $($right$)$ Noetherian.* 2. (ii)

$($Ore’s theorem$)$* If $R$ is a left Ore domain $R$. Then $A$ is also a left Ore domain.*

1.3 Semi-graded rings

The third notion needed for setting the problems analyzed in the paper are the semi-graded rings that we will recall in this subsection. In [37] were proved some properties of them, in particular, it was showed that graded rings, finitely graded algebras and skew $PBW$ extensions are particular cases of this type of non-commutative rings.

Definition 1.6**.**

Let $B$ be a ring. We say that $B$ is semi-graded $(SG)$ if there exists a collection $\{B_{n}\}_{n\geq 0}$ of subgroups $B_{n}$ of the additive group $B^{+}$ such that the following conditions hold:

1. (i)

$B=\bigoplus_{n\geq 0}B_{n}$. 2. (ii)

For every $m,n\geq 0$, $B_{m}B_{n}\subseteq B_{0}\oplus\cdots\oplus B_{m+n}$. 3. (iii)

$1\in B_{0}$.

The collection $\{B_{n}\}_{n\geq 0}$ is called a semi-graduation of $B$ and we say that the elements of $B_{n}$ are homogeneous of degree $n$. Let $B$ and $C$ be semi-graded rings and let $f:B\to C$ be a ring homomorphism, we say that $f$ is homogeneous if $f(B_{n})\subseteq C_{n}$ for every $n\geq 0$.

Definition 1.7**.**

Let $B$ be a $SG$ ring and let $M$ be a $B$-module. We say that $M$ is a $\mathbb{Z}$-semi-graded, or simply semi-graded, if there exists a collection $\{M_{n}\}_{n\in\mathbb{Z}}$ of subgroups $M_{n}$ of the additive group $M^{+}$ such that the following conditions hold:

1. (i)

$M=\bigoplus_{n\in\mathbb{Z}}M_{n}$. 2. (ii)

For every $m\geq 0$ and $n\in\mathbb{Z}$, $B_{m}M_{n}\subseteq\bigoplus_{k\leq m+n}M_{k}$.

We say that $M$ is positively semi-graded, also called $\mathbb{N}$-semi-graded, if $M_{n}=0$ for every $n<0$. Let $f:M\to N$ be an homomorphism of $B$-modules, where $M$ and $N$ are semi-graded $B$-modules, we say that $f$ is homogeneous if $f(M_{n})\subseteq N_{n}$ for every $n\in\mathbb{Z}$.

Definition 1.8**.**

Let $B$ be a $SG$ ring and $M$ be a semi-graded module over $B$. Let $N$ be a submodule of $M$, we say that $N$ is a semi-graded submodule of $M$ if $N=\bigoplus_{n\in\mathbb{Z}}N_{n}$.

Definition 1.9**.**

Let $B$ be a ring. We say that $B$ is finitely semi-graded $(FSG)$ if $B$ satisfies the following conditions:

1. (i)

$B$* is $SG$.* 2. (ii)

There exists finitely many elements $x_{1},\dots,x_{n}\in B$ such that the subring generated by $B_{0}$ and $x_{1},\dots,x_{n}$ coincides with $B$. 3. (iii)

For every $n\geq 0$, $B_{n}$ is a free $B_{0}$-module of finite dimension.

Moreover, if $M$ is a $B$-module, we say that $M$ is finitely semi-graded if $M$ is semi-graded, finitely generated, and for every $n\in\mathbb{Z}$, $M_{n}$ is a free $B_{0}$-module of finite dimension.

Remark 1.10**.**

Observe if $B$ is $FSG$, then $B_{0}B_{p}=B_{p}$ for every $p\geq 0$, and if $M$ is finitely semi-graded, then $B_{0}M_{n}=M_{n}$ for all $n\in\mathbb{Z}$.

From the definitions above we get the following conclusions.

Proposition 1.11** ([37]).**

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $SG$ ring and $I$ be a proper two-sided ideal of $B$ semi-graded as left ideal. Then,

1. (i)

$B_{0}$* is a subring of $B$. Moreover, for any $n\geq 0$, $B_{0}\oplus\cdots\oplus B_{n}$ is a $B_{0}-B_{0}$-bimodule, as well as $B$.* 2. (ii)

$B$* has a standard $\mathbb{N}$-filtration given by*

[TABLE] 3. (iii)

The associated graded ring $Gr(B)$ satisfies

$Gr(B)_{n}\cong B_{n}$, for every $n\geq 0$ $($isomorphism of abelian groups$)$. 4. (iv)

Let $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$ be a semi-graded $B$-module and $N$ a submodule of $M$. The following conditions are equivalent:

1. (a)

$N$* is semi-graded.* 2. (b)

For every $z\in N$, the homogeneous components of $z$ are in $N$. 3. (c)

$M/N$* is semi-graded with semi-graduation given by*

$(M/N)_{n}:=(M_{n}+N)/N$, $n\in\mathbb{Z}$. 5. (v)

$B/I$* is $SG$.*

Proposition 1.12** ([37]).**

(i)* Any $\mathbb{N}$-graded ring is $SG$.*

(ii)* Let $K$ be a field. Any finitely graded $K$-algebra is a $FSG$ ring.*

(iii)* Any skew $PBW$ extension is a $FSG$ ring.*

Remark 1.13**.**

(i) In [37] was proved that the previous inclusions are proper.

(ii) The class of $FSG$ rings also includes properly the multiple Ore extensions introduced in [58].

1.4 Problem 1

• •

Investigate the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings.

This problem was partially solved in [37] where it was assumed that the semi-graded left noetherian ring $B$ is a domain, however, the Serre-Artin-Zhang-Verevkin theorem for finitely graded algebras does not include this restriction. Next we will present the main ingredients of an improved proof where the domain restriction has been removed.

Let $B=\bigoplus_{n\geq 0}B_{n}$ be a $SG$ ring that satisfies the following conditions:

(C1) $B$ is left noetherian.

(C2) $B_{0}$ is left noetherian.

(C3) For every $n$, $B_{n}$ is a finitely generated left $B_{0}$-module.

(C4) $B_{0}\subset Z(B)$.

Proposition 1.14** ([37], Proposition 5.2).**

Let ${\rm sgr}-B$ be the collection of all finitely generated semi-graded $B$-modules, then ${\rm sgr}-B$ is an abelian category where the morphisms are the homogeneous $B$-homomorphisms.

Definition 5.3 in [37] can be improved in the following way.

Definition 1.15**.**

Let $M$ be an object of ${\rm sgr}-B$.

1. (i)

For $s\geq 0$, $B_{\geq s}$ is the least two-sided ideal of $B$ that satisfies the following conditions:

1. (a)

$B_{\geq s}$* contains $\bigoplus_{p\geq s}B_{p}$.* 2. (b)

$B_{\geq s}$* is semi-graded as left ideal of $B$.* 2. (ii)

An element $x\in M$ is torsion if there exist $s,n\geq 0$ such that $B_{\geq s}^{\ n}x=0$. The set of torsion elements of $M$ is denoted by $T(M)$. $M$ is torsion if $T(M)=M$ and torsion-free if $T(M)=0$.

Theorem 1.16** ([37], Theorem 5.5).**

The collection ${\rm stor}-B$ of torsion modules forms a Serre subcategory of ${\rm sgr}-B$, and the quotient category

${\rm qsgr}-B:={\rm sgr}-B/{\rm stor}-B$**

is abelian.

Remark 1.17**.**

Recall from the general theory of abelian categories (see [54]) that the quotient functor

$\pi:{\rm sgr}-B\to{\rm qsgr}-B$**

is exact and defined by

$\pi(M):=M$* and $\pi(f):=\overline{f}$,*

with $M$ and $M\xrightarrow{f}N$ in ${\rm sgr}-B$ and the morphisms in the category ${\rm qsgr}-B$ are defined by

[TABLE]

where the direct limit is taken over all $M^{\prime}\subseteq M$, $N^{\prime}\subseteq N$ in ${\rm sgr}-B$ with $M/M^{\prime}\in{\rm stor}-B$ and $N^{\prime}\in{\rm stor}-B$ (see [29], [24] and also [54] Proposition 2.13.4).

Definition 1.18** ([37], Definition 6.1).**

Let $M$ be a semi-graded $B$-module, $M=\bigoplus_{n\in\mathbb{Z}}M_{n}$. Let $i\in\mathbb{Z}$, the semi-graded module $M(i)$ defined by $M(i)_{n}:=M_{i+n}$ is called a shift of $M$, i.e.,

$M(i)=\bigoplus_{n\in\mathbb{Z}}M(i)_{n}=\bigoplus_{n\in\mathbb{Z}}M_{i+n}$.

The next proposition shows that the shift of degrees is an autoequivalence.

Proposition 1.19** ([37], Proposition 6.3).**

Let $s:{\rm sgr}-B\to{\rm sgr}-B$ defined by

$M\mapsto M(1)$,

$M\xrightarrow{f}N\mapsto M(1)\xrightarrow{f(1)}N(1)$,

$f(1)(m):=f(m)$, $m\in M(1)$.

Then,

1. (i)

$s$* is an autoequivalence.* 2. (ii)

For every $d\in\mathbb{Z}$, $s^{d}(M)=M(d)$. 3. (iii)

$s$* induces an autoequivalence of ${\rm qsgr}-B$ also denoted by $s$.*

Proposition 1.20**.**

$s\pi=\pi s$.

Proof.

We have

${\rm sgr}-B\xrightarrow{\pi}{\rm qsgr}-B\xrightarrow{s}{\rm qsgr}-B$ and ${\rm sgr}-B\xrightarrow{s}{\rm sgr}-B\xrightarrow{\pi}{\rm qsgr}-B$,

so, for $M$ in ${\rm sgr}-B$, $s\pi(M)=s(\pi(M))=\pi(M)(1)=M(1)$ and $\pi s(M)=\pi(M(1))=M(1)$; for $M\xrightarrow{f}N$ in ${\rm sgr}-B$, $s\pi(f)=s(\overline{f})=\overline{f(1)}$ and $\pi s(f)=\pi(f(1))=\overline{f(1)}$. ∎

Definition 1.21**.**

Let $s$ be the autoequivalence of ${\rm qsgr}-B$ defined by the shifts of degrees. We define

$\Gamma(\pi(B))_{\geq 0}:=\bigoplus_{d=0}^{\infty}Hom_{{\rm qsgr}-B}(\pi(B),s^{d}(\pi(B)))$.

The domain condition on $B$ in Lemma 6.10 of [37] has been removed in the following simplified version.

Lemma 1.22**.**

Let $B$ be a ring that satisfies (C1)-(C4).

1. (i)

$\Gamma(\pi(B))_{\geq 0}$* is a $\mathbb{N}$-graded ring.* 2. (ii)

Let $\underline{B}:=\bigoplus_{d=0}^{\infty}Hom_{{\rm sgr}-B}(B,s^{d}(B))$. Then, $\underline{B}$ is a $\mathbb{N}$-graded ring and there exists a ring homomorphism $\underline{B}\to\Gamma(\pi(B))_{\geq 0}$. 3. (iii)

For any object $M$ of ${\rm sgr}-B$

$\Gamma(M)_{\geq 0}:=\bigoplus_{d=0}^{\infty}Hom_{{\rm sgr}-B}(B,s^{d}(M))$**

is a graded $\underline{B}$-module, and

$\Gamma(\pi(M))_{\geq 0}:=\bigoplus_{d=0}^{\infty}Hom_{{\rm qsgr}-B}(\pi(B),s^{d}(\pi(M)))$**

is a graded $\Gamma(\pi(B))_{\geq 0}$-module. 4. (iv)

$\underline{B}$* has the following properties:*

1. (a)

$(\underline{B})_{0}\cong B_{0}$* and $\underline{B}$ satisfies (C2).* 2. (b)

$\underline{B}$* satisfies (C3). More generally, let $N$ be a finitely generated graded $\underline{B}$-module, then every homogeneous component of $N$ is finitely generated over $(\underline{B})_{0}$.* 3. (c)

$\underline{B}$* satisfies (C1).* 5. (v)

If $\underline{B}$ satisfies $\mathcal{X}_{1}$, then

1. (a)

$\Gamma(\pi(B))_{\geq 0}$* satisfies (C2).* 2. (b)

$\Gamma(\pi(B))_{\geq 0}$* satisfies (C3). More generally, let $N$ be a finitely generated graded $\Gamma(\pi(B))_{\geq 0}$-module, then every homogeneous component of $N$ is finitely generated over $(\Gamma(\pi(B))_{\geq 0})_{0}$.* 3. (c)

$\Gamma(\pi(B))_{\geq 0}$* satisfies (C1).*

Proof.

We include only the proof of the part (v) since the proof of the others are exactly as in [37].

(v) We set $\Gamma:=\Gamma(\pi(B))_{\geq 0}$. Then,

(a) $\Gamma$ satisfies (C2): From (1.9) we have $\Gamma_{0}=Hom_{{\rm qsgr}-B}(\pi(B),\pi(B))=Hom_{{\rm qsgr}-B}(B,B)=\underrightarrow{\lim}Hom_{{\rm sgr}-B}(I^{\prime},B/N^{\prime})$, where the direct limit is taken over all pairs $(I^{\prime},N^{\prime})$ in ${\rm sgr}-B$, with $I^{\prime},N^{\prime}\subseteq B$, $B/I^{\prime}\in{\rm stor}-B$ and $N^{\prime}\in{\rm stor}-B$. Since $\pi$ is a covariant functor, we obtain a ring homomorphism (taking in particular $I^{\prime}=B$ and $N^{\prime}=0$)

$(\underline{B})_{0}=Hom_{{\rm sgr}-B}(B,B)\xrightarrow{\gamma}Hom_{{\rm qsgr}-B}(B,B)=\Gamma_{0}$

$\gamma(f):=\pi(f)=\overline{f}$.

Since $B_{0}\cong(\underline{B})_{0}$ (isomorphism defined by $\alpha(x)=\alpha_{x},\alpha_{x}(b):=bx,x\in B_{0},b\in B$), then $\Gamma_{0}$, $\Gamma$ and $\underline{B}$ are $B_{0}$-modules. Actually, they are $B_{0}$-algebras: We check this for $\underline{B}$, the proof for $\Gamma(\pi(B))_{\geq 0}$ is similar, and from this we get also that $\Gamma_{0}$ is a $B_{0}$-algebra. If $f\in Hom_{{\rm sgr}-B}(B,B(n)),g\in Hom_{{\rm sgr}-B}(B,B(m))$, $x\in B_{0}$ and $b\in B$, then

$[x\cdot(f\star g)](b)=x\cdot(s^{n}(g)\circ f)(b)=xg(n)(f(b))$;

$[f\star(x\cdot g)](b)=[s^{n}(x\cdot g)\circ f](b)=(x\cdot g)(n)(f(b))=xg(n)(f(b))$.

Since $B_{0}$ is noetherian, in order to prove that $\Gamma_{0}$ is a noetherian ring, the idea is to show that $\Gamma_{0}$ is finitely generated as $B_{0}$-module, but since $\underline{B}$ satisfies $\mathcal{X}_{1}$, this follows from Proposition 3.1.3 (3) in [10]. Thus, $\Gamma_{0}$ is a commutative noetherian ring, and hence, $\Gamma$ satisfies (C2).

(b) $\Gamma$ satisfies (C3): Since $\Gamma$ is graded, $\Gamma_{d}$ is a $\Gamma_{0}$-module for every $d$, but by $\gamma$ in (a), the idea is to prove that $\Gamma_{d}$ is finitely generated over $B_{0}$, but again we apply Proposition 3.1.3 (3) in [10].

For the second part of (b), let $N$ be a $\Gamma$-module generated by a finite set of homogeneous elements $x_{1},\dots,x_{r}$, with $x_{i}\in N_{d_{i}}$, $1\leq i\leq r$. Let $x\in N_{d}$, then there exist $f_{1},\dots,f_{r}\in\Gamma$ such that $x=f_{1}\cdot x_{1}+\cdots+f_{r}\cdot x_{r}$, from this we can assume that $f_{i}\in\Gamma_{d-d_{i}}$, but as was observed before, every $\Gamma_{d-d_{i}}$ is finitely generated as $\Gamma_{0}$-module, so $N_{d}$ is finitely generated over $\Gamma_{0}$ for every $d$.

(c) $\Gamma$ satisfies (C1): By (iii), $\Gamma$ is not only $SG$ but $\mathbb{N}$-graded. The proof of (C1) is exactly as in the part (iv) of Lemma 6.10 in [37]. ∎

Proposition 1.23** ([10], Proposition 2.5).**

Let $S$ be a commutative noetherian ring and $\rho:C\to D$ be a homomorphism of $\mathbb{N}$-graded left noetherian $S$-algebras. If the kernel and cokernel of $\rho$ are right bounded, then $D\underline{\otimes}\,_{C}\,-$ defines an equivalence of categories ${\rm qgr}-C\simeq{\rm qgr}-D$, where $\underline{\otimes}$ denotes the graded tensor product.

The solution of Problem 1 is as follows.

Theorem 1.24**.**

Let $B$ be a $SG$ ring that satisfies (C1)-(C4) and assume that $\underline{B}$ satisfies the condition $\mathcal{X}_{1}$, then there exists an equivalence of categories

[TABLE]

Proof.

The ring homomorphism

[TABLE]

satisfies the conditions of Proposition 1.23, with $S=B_{0}$, $C=\underline{B}$ and $D=\Gamma(\pi(B))_{\geq 0}$. In fact, from the proof of Lemma 1.22 we know that $\underline{B}$ and $\Gamma(\pi(B))_{\geq 0}$ are $\mathbb{N}$-graded left noetherian $B_{0}$-algebras. Since $\underline{B}$ satisfies the condition $\mathcal{X}_{1}$, we can apply the proof of part S10 in Theorem 4.5 in [10] to conclude that the kernel and cokernel of $\rho$ are right bounded. ∎

We will see next that our Theorem 1.24 extends the Serre-Artin-Zhang-Verevkin equivalence (1.2).

Corollary 1.25**.**

Let $B$ be a $SG$ ring that satisfies (C1)-(C4). Then,

1. (i)

There is an injective homomorphism of $\mathbb{N}$-graded $B_{0}$-algebras $\eta:\underline{B}\to Gr(B)$. 2. (ii)

If $B_{0}=K$ is a field and $Gr(B)$ is left noetherian and satisfies $\mathcal{X}_{1}$, then $\underline{B}$ satisfies $\mathcal{X}_{1}$ and the following equivalences of categories hold:

[TABLE] 3. (iii)

If $B$ is finitely graded and satisfies $\mathcal{X}_{1}$, then $\underline{B}\cong B$ and the Serre-Artin-Zhang-Verevkin equivalence ${\rm qgr}-B\simeq{\rm qgr}-\Gamma(\pi(B))_{\geq 0}$ holds.

Proof.

(i) $\eta$ is defined by (see Proposition 1.11)

[TABLE]

with $f_{i}\in Hom_{{\rm sgr}-B}(B,B(i))$, $0\leq i\leq d$. It is clear thet $\eta$ is additive and $\eta(1)=1$; $\eta$ is multiplicative:

$\eta(f_{n}\star g_{m})=\eta(s^{n}(g_{m})\circ f_{n})=\overline{(s^{n}(g_{m})\circ f_{n})(1)}=\overline{s^{n}(g_{m})(f_{n}(1))}=\overline{g_{m}(f_{n}(1))}=\overline{f_{n}(1)g_{m}(1)}=\overline{f_{n}(1)}\ \overline{g_{m}(1)}=\eta(f_{n})\eta(g_{m})$.

$\eta$ is a $B_{0}$-homomorphism: Let $x\in B_{0}$ and $f_{d}\in Hom_{{\rm sgr}-B}(B,B(d))$, since $B_{0}\cong(\underline{B})_{0}$, then

$\eta(x\cdot f_{d})=\eta(f_{d}\circ\alpha_{x})=\overline{(f_{d}\circ\alpha_{x})(1)}=\overline{f_{d}(\alpha_{x}(1))}=\overline{f_{d}(x)}=\overline{x\cdot f_{d}(1)}=\overline{x}\cdot\overline{f_{d}(1)}=x\cdot\overline{f_{d}(1)}=x\cdot\eta(f_{d})$.

$\eta$ is injective: If $\overline{f_{0}(1)}+\cdots+\overline{f_{d}(1)}=\overline{0}$, then $\overline{f_{k}(1)}=0$ for every $0\leq k\leq d$, therefore $f_{k}(1)\in(B_{0}\oplus\cdots\oplus B_{k-1})\cap B_{k}$ since $f_{k}(1)\in B(k)_{0}=B_{k}$.

(ii) Since $\underline{B}$ and $Gr(B)$ are $\mathbb{N}$-graded left noetherian $K$-algebras ($K$ a field) and the kernel and cokernel of $\eta$ are right bounded, we apply Lemma 8.2 in [10] to conclude that $\underline{B}$ satisfies $\mathcal{X}_{1}$. Thus, from Theorem 1.24 we get the second equivalence of (1.12). Applying Proposition 1.23 to $\eta$ we obtain the first equivalence.

(iii) We define $\theta$ by

[TABLE]

with $f_{i}\in Hom_{{\rm sgr}-B}(B,B(i))$, $0\leq i\leq d$. As (i), we can prove that $\theta$ is an isomorphism of $B_{0}$-algebras. Thus, we get the Serre-Artin-Zhang-Verevkin equivalence ${\rm qgr}-B\simeq{\rm qgr}-\Gamma(\pi(B))_{\geq 0}$. ∎

Example 1.26**.**

In [37] was proved that the following examples of skew $PBW$ extensions are semi-graded rings (most of them non $\mathbb{N}$-graded) and satisfy the conditions (C1)-(C4), moreover, in each case, $\underline{B}$ satisfies the condition $\mathcal{X}_{1}$, therefore, for these algebras Theorem 1.24 is true. In every example $B_{0}=K$ is a field: Enveloping algebra of a Lie $K$-algebra $\mathcal{G}$ of dimension $n$, \mbox{{\cal U}}(\mathcal{G}); the quantum algebra $\mathcal{U}^{\prime}(so(3,K))$, with $q\in K-\{0\}$; the dispin algebra \mbox{{\cal U}}(osp(1,2)); the Woronowicz algebra \mbox{{\cal W}}_{\nu}(\mathfrak{sl}(2,K)), where $\nu\in K-\{0\}$ is not a root of unity; eight types of $3$-dimensional skew polynomial algebras.

2 Gelfand-Kirillov conjecture

In this section we will review some aspects of the quantum version of the Gelfand-Kirillov conjecture and we will formulate a related problem in the context of the skew $PBW$ extensions. We start recalling the classical conjecture and some well known cases where the conjecture has positive answer. In this section $Q(B)$ denotes the total ring of fraction of an Ore (left and right) domain $B$.

Conjecture 2.1** (Gelfand-Kirillov, [28]).**

Let $\mathcal{G}$ be an algebraic Lie algebra of finite dimension over a field $L$, with $char(L)=0$. Then, there exist integers $n,k\geq 1$ such that

[TABLE]

where $A_{n}(L[s_{1},\dots,s_{k}])$ is the general Weyl algebra over $L$.

Recall that $\mathcal{G}$ is algebraic if $\mathcal{G}$ is the Lie algebra of a linear affine algebraic group. A group $G$ is linear affine algebraic if $G$ is an affine algebraic variety such that the multiplication and the inversion in $G$ are morphisms of affine algebraic varieties.

Next we recall some examples of algebraic Lie algebras for which the classical conjecture (2.1) holds.

Example 2.2**.**

(i) The algebra of all $n\times n$ matrices over a field $L$ with ${\rm char}(L)=0$. The same is true for the algebra of matrices of null trace ([28], Lemma 7).

(ii) A finite dimensional nilpotent Lie algebra over a field $L$, with ${\rm char}(L)=0$. Moreover, in this case, $Q(Z(\mathcal{U}(\mathcal{G})))\cong Z(Q(\mathcal{U}(\mathcal{G})))$ ([28], Lemma 8).

(iii) A finite dimensional solvable algebraic Lie algebra over the field $\mathbb{C}$ of complex numbers (see [32], Theorem 3.2)

(iv) Any algebraic Lie algebra $\mathcal{G}$ over an algebraically closed field $L$ of characteristic zero, with $\dim(\mathcal{G})\leq 8$ (see [4]).

Remark 2.3**.**

(i) More examples can be found in [14], [23], [33] and [45].

(ii) There are examples of non-algebraic Lie algebras for which the conjecture is false. However, other examples show that the conjecture holds for some non-algebraic Lie algebras (see [28], Section 8).

We are interested in the quantum version of the Gelfand-Kirillov conjecture, i.e., in this case $\mathcal{U}(\mathcal{G})$ is replaced for a quantum algebra and the Weyl algebra $A_{n}(L[s_{1},\dots,s_{k}])$ in (2.1) is replaced by a suitable $n$-multiparametric quantum affine space $K_{\rm\textbf{q}}[x_{1},\dots,x_{n}]$ , as it is shown the following examples.

Example 2.4** ([15]).**

Let $K$ be a field and $B:=K[x_{1}][x_{2};\sigma_{2},\delta_{2}]\cdots[x_{n};\sigma_{n},\delta_{n}]$ be an iterated skew polynomial ring with some extra adequate conditions on $\sigma$’s and $\delta$’s. Then there exits $\textbf{\rm{q}}:=[q_{i,j}]\in M_{n}(K)$ with $q_{ii}=1=q_{ij}q_{ji}$, for every $1\leq i,j\leq n$, such that $Q(B)\cong Q(K_{\rm\textbf{q}}[x_{1},\dots,x_{n}])$.

Example 2.5** ([5], Theorem 3.5).**

Let $A_{n}^{Q,\Gamma}(K)$ be the multiparameter quantized Weyl algebra (see [38]); in particular, consider the case when there exists a parameter $q\in K^{*}$ that is non root of unity, such that every parameter in $Q=[q_{1},\dots,q_{n}]$ and $\Gamma=[\gamma_{ij}]$ is a power of $q$, and in addition, $q_{i}\neq 1$, $1\leq i\leq n$. Under these conditions, there exits $\textbf{\rm{q}}:=[q_{ij}]\in M_{2n}(K)$ with $q_{ii}=1=q_{ij}q_{ji}$, and $q_{ij}$ is a power of $q$, $1\leq i,j\leq n$, such that

$Q(A_{n}^{Q,\Gamma}(K))\cong Q(K_{\textbf{q}}[x_{1},\dots,x_{2n}])$,

$Z(Q(K_{\textbf{q}}[x_{1},\dots,x_{2n}]))=K$.

Example 2.6** ([5], Theorem 2.15).**

Let $U_{q}^{+}(sl_{m})$ be the quantum enveloping algebra of the Lie algebra of strictly superior triangular matrices of size $m\times m$ over a field $K$.

1. (i)

If $m=2n+1$, then

$Q(U_{q}^{+}(sl_{m}))\cong Q({\rm K}_{{\rm q}}[x_{1},\dots,x_{2n^{2}}])$,

where ${\rm K}:=Q(Z(U_{q}^{+}(sl_{m})))$ and ${\rm q}:=[q_{ij}]\in M_{2n^{2}}(K)$, with $q_{ii}=1=q_{ij}q_{ji}$, and $q_{ij}$ is a power of $q$ for every $1\leq i,j\leq 2n^{2}$. 2. (ii)

If $m=2n$, then

$Q(U_{q}^{+}(sl_{m}))\cong Q({\rm K}_{{\rm q}}[x_{1},\dots,x_{2n(n-1)}])$,

where ${\rm K}:=Q(Z(U_{q}^{+}(sl_{m})))$ and ${\rm q}:=[q_{ij}]\in M_{2n(n-1)}(K)$, with $q_{ii}=1=q_{ij}q_{ji}$, and $q_{ij}$ is a power of $q$ for every $1\leq i,j\leq 2n(n-1)$.

Moreover, in both cases

$Q(Z(U_{q}^{+}(sl_{m})))\cong Z(Q(U_{q}^{+}(sl_{m})))$.

2.1 Problem 2

• •

Study the Gelfand-Kirillov conjecture for bijective skew $PBW$ extensions over Ore domains. Moreover, investigate for them the isomorphism $Q(Z(A))\cong Z(Q(A))$.

The ring $Q^{k,n}_{\textbf{q},\sigma}(R)$ of skew quantum polynomials over $R$, also denoted by $R_{\textbf{q},\sigma}[x_{1}^{\pm 1},\dots,x_{k}^{\pm 1},x_{k+1},\dots,x_{n}]$, conforms a particular type of quasi-commutative bijective skew $PBW$ extension. In [2] was proved the following partial solution of Problem 2.

Theorem 2.7** ([2], Corollary 5.1).**

If $R$ is an Ore domain $($left and right$)$, then

[TABLE]

The precise definition of $R_{\textbf{q},\sigma}[x_{1}^{\pm 1},\dots,x_{k}^{\pm 1},x_{k+1},\dots,x_{n}]$ and $\textbf{Q}_{\textbf{q},\sigma}[x_{1},\dots,x_{n}]$ can be found in [2]. Observe that the $n$-multiparametric quantum affine space $K_{\rm\textbf{q}}[x_{1},\dots,x_{n}]$ was replaced in (2.2) by the $n$-multiparametric skew quantum space $\textbf{Q}_{\textbf{q},\sigma}[x_{1},\dots,x_{n}]$.

3 Serre’s problem

Given a ring, it is an interesting problem to investigate if the finitely generated projective modules over it are free. This problem becomes very important after the formulation in 1955 of the famous Serre’s problem about the freeness of finitely generated projective modules over the polynomial ring $K[x_{1},\dots,x_{n}]$, $K$ a field (see [6], [7], [11], [34]). The Serre’s problem was solved positively, and independently, by Quillen in USA, and by Suslin in Leningrad, USSR (St. Petersburg, Russia) in 1976 ([46], [55]).

For arbitrary skew $PBW$ extensions the problem has negative answer, for example, let $A:=R[x;\sigma]$ be the skew polynomial ring over $R:=K[y]$, where $K$ is a field and $\sigma(y):=y+1$, from [43], 12.2.11 we can conclude that there exist finitely generated projective modules over $A$ that are not free. However, in [8] was proved that if $K$ is a field, $A:=K[x_{1},\dots,x_{n};\sigma]$, $\sigma$ is an automorphism of $K$ of finite order, $x_{i}x_{j}=x_{j}x_{i}$ and $x_{i}r=\sigma(r)x_{i}$, for every $r\in K$ and $1\leq i,j\leq n$, then every finitely generated projective module over $A$ is free. The proof in [8] of this theorem (Quillen-Suslin theorem) is not construcitve, i.e., the proof shows the existence of a basis for every finitely generated projective module $M$ over $A$, but an explicit basis of $M$ was not constructed.

3.1 Problem 3

• •

Give a matrix-constructive proof of the Quillen-Suslin theorem for $A:=K[x_{1},\dots,x_{n};\sigma]$. Include an algorithm that computes a basis of a given finitely generated projective $A$-module.

In [21] was solved the previous problem for the case of only one variable, in this particular situation weaker hypotheses can be set, namely, $K$ is a division ring, $\sigma$ is not necessarily of finite order and a non trivial $\sigma$-derivation $\delta$ of $K$ can be added.

Theorem 3.1** ([21], Theorem 2.1).**

Let $K$ be a division ring and $A:=K[x;\sigma,\delta]$, with $\sigma$ bijective. Then $A$ is a $\mathcal{PF}$-ring, i.e., every finitely generated projective module over $A$ is free.

The proof of the previous theorem given in [21] is not only matrix-constructive, but also some algorithms that compute the basis of a given finitely generated projective $A$-module are exhibited and implemented in Maple. Next we review the main ingredients of the proof and include an illustrative example of algorithms.

Proposition 3.2** ([27]).**

Let $B$ be a ring. $B$ is a $\mathcal{PF}$-ring if and only if for every $s\geq 1$, given an idempotent matrix $F\in M_{s}(B)$, there exists an invertible matrix $U\in GL_{s}(B)$ such that

[TABLE]

where $r=dim(\langle F\rangle)$, $0\leq r\leq s$, and $\langle F\rangle$ represents the free left $B$-module $M$ generated by the rows of $F$. Moreover, the final $r$ rows of $U$ form a basis of $M$.

The matrix-constructive proof of Theorem 3.1 consists in constructing explicitly the matrix $U$ in Proposition 3.2 from the entries of the given idempotent matrix $F$ (recall that a left module $M$ over a ring $B$ is finitely generated projective if and only if $M$ is the left $B$-module generated by the rows of a idempotent matrix over $B$). In [21] were designed two algorithms for calculating the basis of a given finitely generated projective module: a constructive simplified version and a more complete computational version over a field. The computational version was implemented using ${\rm Maple}^{\circledR}$ 2016. For completeness, we include next the constructive simplified version (see [21], Algorithm 1):

**Algorithm for the Quillen-Suslin theorem:

Constructive version**

INPUT: $A:=K[x,\sigma,\delta]$; $F\in M_{s}(A)$ an idempotent matrix.

OUTPUT: Matrices $U$, $U^{-1}$ and a basis $X$ of $\langle F\rangle$, where

$UFU^{-1}=\begin{bmatrix}0&0\\ 0&I_{r}\end{bmatrix}\ \text{and}\ r=dim(\langle F\rangle).$

(3.2)

INITIALIZATION: $F_{1}:=F$.

FOR $k$ from $1$ to $n-1$ DO

1.

Follow the reduction procedures (B1) and (B2) in the proof of Theorem 3.1 in order to compute matrices $U_{k}^{\prime}$, $U_{k}^{\prime-1}$ and $F_{k+1}$ such that

$U_{k}^{\prime}F_{k}U_{k}^{\prime-1}=\begin{bmatrix}\alpha_{k}&0\\ 0&F_{k+1}\end{bmatrix},\text{ where }\alpha_{k}\in\{0,1\}.$

2.

$U_{k}:=\begin{bmatrix}I_{k-1}&0\\ 0&U_{k}^{\prime}\end{bmatrix}U_{k-1}$; compute $U_{k}^{-1}$.

3.

By permutation matrices modify $U_{n-1}$.

RETURN $U:=U_{n-1}$, $U^{-1}$ satisfying (3.2), and a basis $X$ of $\langle F\rangle$.

Example 3.3**.**

([21], Example 3.4) Let $A:=K[x,\sigma,\delta]$, $K:=\mathbb{Q}(t)$, $\sigma(f(t)):=f(qt)$ and $\delta(f(t)):=\tfrac{f(qt)-f(t)}{t(q-1)}$, where $q\in K-\{0,1\}$. We consider the idempotent matrix $F:=[F^{(1)}\ F^{(2)}\ F^{(3)}\ F^{(4)}]\in M_{4}(A)$, with $F^{(i)}$ the $ith$ column of $F$ and $a\in\mathbb{Q}$, where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Applying the algorithms we obtain

[TABLE]

[TABLE]

[TABLE]

Therefore, $r=2$ and the last two rows of $U$ conform a basis $X=\{\emph{{x}}_{1},\textbf{\emph{x}}_{2}\}$, of $\langle F\rangle$,

$\emph{{x}}_{1}=(tx-1,1,{t}^{2}q{x}^{2}+a-1,{t}^{2}q{x}^{2}+2\,tx-1)$, $\emph{{x}}_{2}=(1,0,tx,tx+1)$.

Remark 3.4**.**

For $n\geq 2$, the commutativity of $K$ in Problem 3 is necessary, in [34] (p. 36) is proved that if $B$ is a division ring, then $B[x,y]$ is not $\mathcal{PF}$.

4 Artin-Schelter open problems

Artin-Schelter regular algebras (shortly denoted as $AS$) were introduced by Michael Artin and William Schelter in [9]. In non-commutative algebraic geometry these algebras play the role of $K[x_{1},\dots,x_{n}]$ in commutative algebraic geometry, thus, in particular, the noetherian $AS$ algebras satisfy the condition $\mathcal{X}$ of Subsection 1.1 (see Theorem 12.6 in [54]), and hence, for them the Serre-Artin-Zhang-Verevkin theorem holds. Nowadays $AS$ algebras are intensively investigated, in the next subsection we present the definition and some properties and open problems on these algebras.

4.1 Artin-Schelter regular algebras

Definition 4.1** ([9]).**

Let $K$ be a field and $A$ be a finitely graded $K$-algebra. It says that $A$ is an Artin-Schelter regular algebra $(AS)$ if

1. (i)

${\rm gld}(A)=d<\infty$. 2. (ii)

${\rm GKdim}(A)<\infty$. 3. (iii)

$\underline{Ext}_{A}^{i}(_{A}K,_{A}A)\cong\begin{cases}0\ \ if\ i\neq d\\ K(l)_{A}\ \ if\ i=d\end{cases}$**

for some shift $l\in\mathbb{Z}$.

Remark 4.2**.**

(i) In [51] is showed that the third condition is equivalent to

$\underline{Ext}_{A}^{i}(K_{A},A_{A})\cong\begin{cases}0\ \ if\ i\neq d\\ {}_{A}K(l)\ \ if\ i=d.\end{cases}$**

In addition, since $K\cong A/A_{\geq 1}$ is finitely generated as left $A$-module, then we can replace $\underline{Ext}_{A}^{i}(_{A}K,_{A}A)$ by $Ext_{A}^{i}(_{A}K,_{A}A)$. The same is true for $\underline{Ext}_{A}^{i}(K_{A},A_{A})$.

(ii) Some key examples of Artin-Schelter regular algebras are: Let $A=K[x_{1},\dots,x_{n}]$ be the usual commutative polynomial ring in $n$ variables over the field $K$, with any weights $\deg(x_{i})=d_{i}\geq 1$, $1\leq i\leq n$, then $A$ is $AS$; if $A$ is a commutative $AS$ algebra, then $A$ is isomorphic to a weighted commutative polynomial ring; the Jordan and quantum planes are $AS$ of dimension $2$; more generally, the $n$-multiparametric quantum affine space $K_{\rm\textbf{q}}[x_{1},\dots,x_{n}]$ is $AS$. $AS$ algebras of dimension $\leq 3$ are completely classified. It is an interesting and intensively investigated open problem to classify the $AS$ algebras of dimension $4$ or $5$.

(iii) Artin-Schelter open problems: In [9] were formulated the following problems:

Any $AS$ algebra is noetherian? 2. 2.

Any $AS$ algebra is a domain?

There are some advances with respect to the previous open problems: Both questions above have positive answer for any $AS$ algebra of dimension $\leq 3$. In addition, the same is true for all known concrete examples of $AS$ algebras of higher dimension. If $A$ is $AS$ noetherian and its dimension is $\leq 4$, then $A$ is a domain.

4.2 Semi-graded Artin-Schelter regular algebras

$AS$ algebras are $\mathbb{N}$-graded and connected, recently Gaddis in [25] and [26] introduces the technique of homogenization and the notion of essentially regular algebras in order the study of the Artin-Schelter condition for non $\mathbb{N}$-graded algebras. On the other hand, with the purpose of giving new examples of $AS$ algebras, in [41] and [42] are defined the $\mathbb{Z}^{s}$-graded Artin-Schelter regular algebras and there in have been proved some results for the classification of $AS$ algebras of dimension $5$ with two generators. The semi-graded Artin-Schelter regular algebras that we will introduce in the present section is a new different approach to this problem, and extend the classical notion of Artin-Schelter regular algebra defined originally in [9]. In a forthcoming paper we will generalize some classical well-known results on Artin-Schelter regular algebras to the semi-graded case.

Definition 4.3**.**

Let $K$ be a field and $B$ be a $K$-algebra. We say that $B$ is a left semi-graded Artin-Schelter regular algebra $(\mathcal{SAS})$ if the following conditions hold:

1. (i)

$B$* is a $FSG$ ring with semi-graduation $B=\bigoplus_{p\geq 0}B_{p}$.* 2. (ii)

$B$* is connected, i.e., $B_{0}=K$.* 3. (iii)

${\rm lgld}(B):=d<\infty$. 4. (iv)

$Ext_{B}^{i}(_{B}(B/B_{\geq 1}),_{B}B)\cong\begin{cases}0\ \ if\ i\neq d\\ (B/B_{\geq 1})_{B}\ \ if\ i=d\end{cases}$**

Remark 4.4**.**

(i) If $K\cap B_{\geq 1}\neq 0$, then $B/B_{\geq 1}=0$; if $K\cap B_{\geq 1}=0$ then $B/B_{\geq 1}\cong K$, where this is an isomorphism of $K$-algebras induced by the canonical projection $\epsilon:B\to K$ (recall that $B_{\geq 1}$ is semi-graded). Moreover, ${}_{B}(B/B_{\geq 1})\cong_{B}K$ and $(B/B_{\geq 1})_{B}\cong K_{B}$.

(ii) Our definition extends (except for the ${\rm GKdim}$) the classical notion of Artin-Schelter algebra since in such case $B/B_{\geq 1}\cong K$, ${\rm lgld}(B)={\rm pd}(_{B}K)={\rm pd}(K_{B})={\rm rgld}(B)={\rm gld}(B)$ and $Ext_{B}^{i}(K,B)=\underline{Ext}_{B}^{i}(K,B)$ (see [51]). Thus, every Artin-Schelter regular algebra is $\mathcal{SAS}$.

4.3 Problem 4

• •

Is any $\mathcal{SAS}$ algebra with finite ${\rm GKdim}$ a left noetherian domain?

We will present next some examples of $\mathcal{SAS}$ algebras that are not Artin-Schelter, in every example, the algebra is a left-right noetherian domain.

For the matrix representation of homomorphisms of left modules in this section we will use the left row notation and for right modules the right column notation. In order to compute free resolutions we will apply Theorem 19 of [35] that has been implemented in the library SPBWE.lib developed in Maple by W. Fajardo in [19] (see also [20]). This library contains the packages SPBWETools, SPBWERings and SPBWEGrobner with utilities to define and perform calculations with skew $PBW$ extensions.

Example 4.5**.**

The Weyl algebra $A_{1}(K)$ is not Artin-Schelter, but it is a $\mathcal{SAS}$ algebra. In fact, the canonical semi-graduation of $A_{1}(K)$ is

$A_{1}(K)=K\oplus\,_{K}\langle x,y\rangle\oplus\,_{K}\langle x^{2},xy,y^{2}\rangle\oplus\cdots$**

Recall that ${\rm gld}(A_{1}(K))\leq 2$ ([43]); moreover, $A_{1}(K)_{\geq 1}=A_{1}(K)$, so $A_{1}(K)/A_{1}(K)_{\geq 1}=0$. Hence, the condition (iv) in Definition 4.3 trivially holds. This argument can be applied also to the general Weyl algebra $A_{n}(K)$ and the generalized Weyl algebra $B_{n}(K)$.

Example 4.6**.**

The quantum deformation $A_{1}^{q}(K)$ of the Weyl algebra is not Artin-Schelter, but it is a $\mathcal{SAS}$ algebra. Indeed, recall that $A_{1}^{q}(K)$ is the $K$-algebra defined by the relation $yx=qxy+1$, with $q\in K-\{0\}$ ($A_{1}^{q}(K)$ coincides with the additive analog of the Weyl algebra in two variables, as well as, with the linear algebra of $q$-differential operators in two variables). The semi-graduation of $A_{1}^{q}(K)$ is as in the previous example, moreover ${\rm gld}(A_{1}^{q}(K))\leq 2$ and $A_{1}^{q}(K)_{\geq 1}=A_{1}^{q}(K)$, so $A_{1}^{q}(K)/A_{1}^{q}(K)_{\geq 1}=0$ and the condition (iv) in Definition 4.3 holds.

Example 4.7**.**

Consider the algebra $\mathcal{J}_{1}:=K\langle x,y\rangle/\langle yx-xy+y^{2}+1\rangle$, according to Corollary 2.3.14 in [25], ${\rm gld}(\mathcal{J}_{1})=2$. The semi-graduation of $\mathcal{J}_{1}$ is as in Example 4.5, so $(\mathcal{J}_{1})_{\geq 1}=\mathcal{J}_{1}$. Thus, $\mathcal{J}_{1}$ is $\mathcal{SAS}$ but is not Artin-Schelter.

Example 4.8**.**

Now we consider the dispin algebra \mbox{{\cal U}}(osp(1,2)), which in this example we will be denoted simply as $B$ (see Example 1.26). Recall that $B$ is defined by the relations

$x_{1}x_{2}-x_{2}x_{1}=x_{1},\ \ \ x_{3}x_{1}+x_{1}x_{3}=x_{2},\ \ \ x_{2}x_{3}-x_{3}x_{2}=x_{3}$.

Since $B$ is not finitely graded, then $B$ is not Artin-Schelter. We will show that $B$ is $\mathcal{SAS}$. We know that ${\rm gld}(B)=3$ (see [38]); the semi-graduation of $B$ is given by

$B=K\oplus\,_{K}\langle x_{1},x_{2},x_{3}\rangle\oplus\,_{K}\langle x_{1}^{2},x_{1}x_{2},x_{1}x_{3},x_{2}^{2},x_{2}x_{3},x_{3}^{2}\rangle\oplus\cdots$**

Note that $B_{\geq 1}=\oplus_{p\geq 1}B_{p}$ and this ideal coincides with the two-sided ideal of $B$ generated by $x_{1},x_{2},x_{3}$; moreover, ${}_{B}(B/B_{\geq 1})\cong_{B}K$, where the structure of left $B$-module for $K$ is given by the canonical projection $\epsilon:B\to K$ (the same is true for the right structure, $(B/B_{\geq 1})_{B}\cong K_{B}$). With SPBWE we get the following free resolution of ${}_{B}K$:

[TABLE]

Now we apply $Hom_{B}(-,\,_{B}B)$ and we get the complex of right $B$-modules

[TABLE]

Note that $Hom_{B}(K,B)=0$: In fact, let $\alpha\in Hom_{B}(K,B)$ and $\alpha(1):=b\in B$, then $\alpha(x_{1}1)=\alpha(0)=0=x_{1}\alpha(1)=x_{1}b$, so $b=0$ (recall that $B$ is a domain) and from this $\alpha(k)=0$ for every $k\in K$, i.e., $\alpha=0$. Moreover, from the isomorphisms of right $B$-modules $Hom_{B}(B,B)\cong B$ and $Hom_{B}(B^{3},B)\cong B^{3}$ we obtain the complex

[TABLE]

So, $Ext_{B}^{0}(K,B)=Hom_{B}(K,B)=0$, $Ext_{B}^{1}(K,B)=0=Ext_{B}^{2}(K,B)$ and $Ext_{B}^{3}(K,B)=B/Im(\phi_{2}^{*})=B/B_{\geq 1}\cong K_{B}$. This shows that $B$ is $\mathcal{SAS}$.

Example 4.9**.**

The next examples are similar to the previous, in every case ${\rm gld}(B)=3$ and $B$ is $\mathcal{SAS}$. The free resolutions have been computed with SPBWE.

(a) Consider the universal enveloping algebra of the Lie algebra $\mathfrak{sl}(2,K)$, $B:=\mathcal{U}(\mathfrak{sl}(2,K))$. $B$ is the $K$-algebra generated by the variables $x,y,z$ subject to the relations

[TABLE]

The free resolutions are:

[TABLE]

[TABLE]

(b) Now let $B:=\mathcal{U}(\mathfrak{so}(3,K))$ be the $K$-algebra generated by the variables $x,y,z$ subject to the relations

[TABLE]

In this case we have

[TABLE]

[TABLE]

(c) Quantum algebra $B:=\mathcal{U}^{\prime}(so(3,K))$, with $q\in K-\{0\}$:

$x_{2}x_{1}-qx_{1}x_{2}=-q^{1/2}x_{3},\ \ \ x_{3}x_{1}-q^{-1}x_{1}x_{3}=q^{-1/2}x_{2},\ \ \ x_{3}x_{2}-qx_{2}x_{3}=-q^{1/2}x_{1}$.

In this case the free resolutions are:

[TABLE]

[TABLE]

(d) Woronowicz algebra B:=\mbox{{\cal W}}_{\nu}(\mathfrak{sl}(2,K)), where $\nu\in K-\{0\}$ is not a root of unity:

$x_{1}x_{3}-\nu^{4}x_{3}x_{1}=(1+\nu^{2})x_{1},\ \ \ x_{1}x_{2}-\nu^{2}x_{2}x_{1}=\nu x_{3},\ \ \ x_{3}x_{2}-\nu^{4}x_{2}x_{3}=(1+\nu^{2})x_{2}$.

The free resolutions are:

[TABLE]

[TABLE]

In the following example we study the semi-graded Artin-Schelter condition for the eight types of $3$-dimensional skew polynomial algebras considered in Example 1.26, five of them are $\mathcal{SAS}$ and the other three are not.

Example 4.10**.**

The first type coincides with the dispin algebra taking $\beta=-1$, and the fifth type corresponds to $\mathcal{U}(\mathfrak{so}(3,K))$, thus they are $\mathcal{SAS}$. For the other six types we compute next the free resolutions with SPBWE:

$x_{2}x_{3}-x_{3}x_{2}=0,\ \ x_{3}x_{1}-\beta x_{1}x_{3}=x_{2},\ \ x_{1}x_{2}-x_{2}x_{1}=0$* ($\mathcal{SAS}$):*

[TABLE]

[TABLE]

$x_{2}x_{3}-x_{3}x_{2}=x_{3},\ \ x_{3}x_{1}-\beta x_{1}x_{3}=0,\ \ x_{1}x_{2}-x_{2}x_{1}=x_{1}$* ($\mathcal{SAS}$):*

[TABLE]

[TABLE]

$x_{2}x_{3}-x_{3}x_{2}=x_{3},\ \ x_{3}x_{1}-\beta x_{1}x_{3}=0,\ \ x_{1}x_{2}-x_{2}x_{1}=0$* (not $\mathcal{SAS}$) :*

[TABLE]

[TABLE]

$x_{2}x_{3}-x_{3}x_{2}=0,\ \ x_{3}x_{1}-x_{1}x_{3}=0,\ \ x_{1}x_{2}-x_{2}x_{1}=x_{3}$* ($\mathcal{SAS}$):*

[TABLE]

[TABLE]

$x_{2}x_{3}-x_{3}x_{2}=-x_{2},\ \ x_{3}x_{1}-x_{1}x_{3}=x_{1}+x_{2},\ \ x_{1}x_{2}-x_{2}x_{1}=0$* (not $\mathcal{SAS}$) :*

[TABLE]

[TABLE]

$x_{2}x_{3}-x_{3}x_{2}=x_{3},\ \ x_{3}x_{1}-x_{1}x_{3}=x_{3},\ \ x_{1}x_{2}-x_{2}x_{1}=0$* (not $\mathcal{SAS}$):*

[TABLE]

[TABLE]

Next we present an algebra essentially regular in the sense of Gaddis (see [25] and [26]) but not $\mathcal{SAS}$. A $\mathbb{N}$-filtered algebra $A$ is essentially regular if and only if $Gr(A)$ is $AS$ (see Proposition 2.3.7 in [25]).

Example 4.11**.**

Consider the algebra $\mathcal{U}:=K\{x,y\}/\langle yx-xy+y\rangle$, by Corollary 2.3.14 in [25], $\mathcal{U}$ is essentially regular of global dimension $2$. Clearly $\mathcal{U}$ is not Artin-Schelter, actually we will show that $\mathcal{U}$ is not $\mathcal{SAS}$. The semi-graduation of $\mathcal{U}$ is as in Example 4.5, so $\mathcal{U}_{\geq 1}=\oplus_{p\geq 1}\mathcal{U}_{p}$ and this ideal coincides with the two-sided ideal of $\mathcal{U}$ generated by $x$ and $y$. Observe that ${}_{\mathcal{U}}(\mathcal{U}/\mathcal{U}_{\geq 1})\cong_{\mathcal{U}}K$, where the structure of left $\mathcal{U}$-module for $K$ is given by the canonical projection $\epsilon:\mathcal{U}\to K$ (the same is true for the right structure, $(\mathcal{U}/\mathcal{U}_{\geq 1})_{\mathcal{U}}\cong K_{\mathcal{U}}$). The following sequence is a free resolution of ${}_{\mathcal{U}}K$:

[TABLE]

This statement can be proved using SPBWE or simply by hand. In fact, $\epsilon$ is clearly surjective; $\phi_{1}$ is injective since if $\phi_{1}(u)=0$ for $u\in\mathcal{U}$, then $u\begin{bmatrix}y&1-x\end{bmatrix}=0$, so $u=0$ since $\mathcal{U}$ is a domain. $Im(\phi_{0})=\ker(\epsilon)=\mathcal{U}_{\geq 1}$ since

$\begin{bmatrix}u&v\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=ux+vy$, with $u,v\in\mathcal{U}$.

$Im(\phi_{1})\subseteq\ker(\phi_{0})$* since $\phi_{1}\phi_{0}=0$:*

$\begin{bmatrix}y&1-x\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=yx+(1-x)y=0$.

Now, $\ker(\phi_{0})\subseteq Im(\phi_{1})$: In fact, let $\begin{bmatrix}u&v\end{bmatrix}\in\ker(\phi_{0})$, then $ux+vy=0$; let

$u=u_{0}+u_{1}x+u_{2}y+u_{3}x^{2}+u_{4}xy+u_{5}y^{2}+\cdots$,

$v=v_{0}+v_{1}x+v_{2}y+v_{3}x^{2}+v_{4}xy+v_{5}y^{2}+\cdots$,

from $ux+vy=0$ we conclude that all terms of $u$ involving only $x^{\prime}s$ have coefficient equals zero, i.e., $u=py$ for some $p\in\mathcal{U}$, hence $vy=-pyx=-p(xy-y)=p(1-x)y$, but since $A$ is a domain, $v=p(1-x)$, whence,

$\begin{bmatrix}u&v\end{bmatrix}=p\begin{bmatrix}y&1-x\end{bmatrix}\in Im(\phi_{1})$.

Now we apply $Hom_{\mathcal{U}}(-,\,_{\mathcal{U}}\mathcal{U})$ and we get the complex of right $\mathcal{U}$-modules

[TABLE]

As in Example 4.8, $Hom_{\mathcal{U}}(K,\mathcal{U})=0$, $Hom_{\mathcal{U}}(\mathcal{U},\mathcal{U})\cong\mathcal{U}$ and $Hom_{\mathcal{U}}(\mathcal{U}^{2},\mathcal{U})\cong\mathcal{U}^{2}$, so we obtain the complex

[TABLE]

with $\phi_{0}^{*}$ is injective. Moreover, $Im(\phi_{0}^{*})=\ker(\phi_{1}^{*})$: Indeed, it is clear that $Im(\phi_{0}^{*})\subseteq\ker(\phi_{1}^{*})$; let $\begin{bmatrix}u\\ v\end{bmatrix}\in\ker(\phi_{1}^{*})$, then $yu+(1-x)v=0$, from this by a direct computation we get that $v=q^{\prime}y$ for some $q^{\prime}\in\mathcal{U}$, but again by a direct computation it is easy to show that given a polynomial $q^{\prime}$ there exists $q\in\mathcal{U}$ such that $q^{\prime}y=yq$, whence $v=yq$ for some $q\in\mathcal{U}$. Hence, $yu+(1-x)yq=0$ implies $y(u-xq)=0$, so $u=xq$. Therefore, $\ker(\phi_{1}^{*})\subseteq Im(\phi_{0}^{*})$ and we have proved the claimed equality.

Thus, $Ext_{\mathcal{U}}^{0}(K,\mathcal{U})=Hom_{\mathcal{U}}(K,\mathcal{U})=0$, $Ext_{\mathcal{U}}^{1}(K,\mathcal{U})=0$ and $Ext_{\mathcal{U}}^{2}(K,\mathcal{U})=\mathcal{U}/Im(\phi_{1}^{*})\ncong K_{\mathcal{U}}$. In fact, suppose there exists a right $\mathcal{U}$-module isomorphism $\mathcal{U}/Im(\phi_{1}^{*})\xrightarrow{\alpha}K_{\mathcal{U}}$, let $\alpha(\overline{1}):=\lambda$, then $\alpha(\overline{1})\cdot(1-x)=\lambda\cdot(1-x)$, so $\alpha(\overline{1}\cdot(1-x))=\lambda$, i.e., $\alpha(\overline{0})=0=\lambda$, whence $\alpha=0$, a contradiction. We conclude that $\mathcal{U}$ is not $\mathcal{SAS}$.

Now we will show an algebra that is $\mathcal{SAS}$ but is not essentially regular.

Example 4.12**.**

Let $\mathcal{S}$ be the algebra of Theorem 4.0.7 in [25] defined by

$B:=K\{x,y\}/\langle yx-1\rangle$.

$B$* has a semi-graduation as in Example 4.5. It is known that ${\rm gld}(B)=1$ (see [25], Proposition 4.1.1 and also [12]) and it is clear that $B/B_{\geq 1}=0$. Thus, the condition (iv) in Definition 4.3 trivially holds and $B$ is $\mathcal{SAS}$. Now observe that $Gr(B)\cong R_{yx}$, where $R_{yx}$ is defined in [25] by*

$R_{yx}:=K\{x,y\}/\langle yx\rangle$.

According to Corollary 2.3.14 in [25], $B$ is not essentially regular.

We conclude the list of examples with an algebra that is not essentially regular neither $\mathcal{SAS}$.

Example 4.13**.**

Let $B:=R_{yx}$. Observe that $B$ is a finitely graded algebra with graduation as in in Example 4.5; by a direct computation we get the following exact sequences

[TABLE]

[TABLE]

According to [25], ${\rm gld}(B)=2$, but note that $B/yB\ncong_{B}K$. In fact, suppose there exists a right $B$-module isomorphism $B/yB\xrightarrow{\alpha}K_{B}$, let $\alpha(\overline{1}):=\lambda$, then $\alpha(\overline{1})\cdot x=\lambda\cdot x$, so $\alpha(\overline{x})=0$, i.e., $\overline{x}=\overline{0}$ but clearly $x\notin yB$. This says that $B$ is not $\mathcal{SAS}$ neither essentially regular.

The previous examples induce the following general result.

Theorem 4.14**.**

Let $K$ be a field and $A:=\sigma(R)\langle x_{1},\dots,x_{n}\rangle$ be a bijective skew $PBW$ extension that satisfies the following conditions:

1. (i)

$R$* and $A$ are $K$-algebras.* 2. (ii)

${\rm lgld}(R)<\infty$. 3. (iii)

$R$* is a $FSG$ ring with semi-graduation $R=\bigoplus_{p\geq 0}R_{p}$.* 4. (iv)

$R$* is connected, i.e., $R_{0}=K$.* 5. (v)

For $1\leq i\leq n$, $\sigma_{i},\delta_{i}$ in Proposition 1.3 are homogeneous, and there exist $i,j$ such that the parameter $d_{ij}$ in $(\ref{equation1.2.1})$ satifies $d_{ij}\in K-\{0\}$.

Then, $B$ is a $\mathcal{SAS}$ algebra.

Proof.

First note that $A$ is $FSG$ and connected with semi-graduation

$A_{0}:=K$, $A_{p}:=_{K}\langle R_{q}x^{\alpha}\mid q+|\alpha|=p\rangle$ for $p\geq 1$,

with $x^{\alpha}$ as in Definition 1.2 and $|\alpha|:=\alpha_{1}+\cdots+\alpha_{n}$. Observe that if $r_{1},\dots,r_{m}$ generate $R$ as $K$-algebra, then $r_{1},\dots,r_{m},x_{1},\dots,x_{n}$ generate $A$ as $K$-algebra. Moreover, $\dim_{K}A_{p}<\infty$ for every $p\geq 0$.

We know that ${\rm lgld}(A)<\infty$ (see [38]). The condition (v) in the statement of the theorem says that $A/A_{\geq 1}=0$, so the condition (iv) in Definition 4.3 trivially holds. ∎

Remark 4.15**.**

From the general homological properties of skew $PBW$ extensions we know that if $R$ is a noetherian domain, then $A$ is a noetherian domain ([38]). This agrees with the question of Problem 4.

5 Zariski cancellation problem

The Zariski cancellation problem (ZCP) arises in commutative algebra and can be formulated in the following way: Let $K$ be a field, $A:=K[t_{1},\dots,t_{n}]$ be the commutative algebra of usual polynomials and $B$ be a commutative $K$-algebra,

if $A[t]\cong B[t]$, then $A\cong B$?

The ZCP has very interesting connections with some famous classical problems: The Automorphism problem, the Dixmier conjecture, the Jacobian conjecture, among some others, see a discussion in [18]. Recently, the ZCP has been considered for non-commutative algebras, in [13] Bell and Zhang studied the Zariski cancellation problem for some non-commutative Artin-Schelter regular algebras; other works on this problem are [16], [17], [40], [56]. In this section we will formulate a problem related to the Zariski cancellation problem in the context of $\mathcal{SAS}$ algebras.

Despite of the most important results about the cancellation problem are for algebras over fields (see for example Theorem 5.4), the general recent formulation of this problem is for $R$-algebras, where $R$ is an arbitrary commutative domain (Definition 5.1). Thus, the general definition below includes the algebras over fields as well as the particular case of $\mathbb{Z}$-algebras, i.e., arbitrary rings.

Definition 5.1** ([13]).**

Let $R$ be a conmutative domain and let $A$ be a $R$-algebra.

1. (i)

$A$* is cancellative if for every $R$-algebra $B$,*

$A[t]\cong B[t]\Rightarrow A\cong B$. 2. (ii)

$A$* is strongly cancellative if for any $d\geq 1$ and every $R$-algebra $B$,*

$A[t_{1},\dots,t_{d}]\cong B[t_{1},\dots,t_{d}]\Rightarrow A\cong B$. 3. (iii)

$A$* is universally cancellative if for any $R$-flat finitely generated commutative domain $S$ such that $S/I\cong R$ for some ideal $I$ of $S$, and any $R$-algebra $B$*

$A\otimes S\cong B\otimes S\Rightarrow A\cong B$,

where the tensor product is over $R$.

Remark 5.2**.**

(i) All isomorphisms in the previous definition are isomorphisms of $R$-algebras, and hence, these notions depend on the ring $R$.

(ii) Observe that the commutative Zariski cancellation problem asks if the $K$-algebra of polynomials $K[t_{1},\dots,t_{n}]$ is cancellative. Abhyankar-Eakin-Heinzer in [1] proved that $K[t_{1}]$ is cancellative (actually, they proved that every commutative finitely generated domain of Gelfand-Kirillov dimension one is cancellative, see Corollary 3.4 in [1]); Fujita in [22] and Miyanishi-Sugie in [44] proved that if $charK=0$, then $K[t_{1},t_{2}]$ is cancellative; if $charK>0$, Russell in [52] proved that $K[t_{1},t_{2}]$ is cancellative. Recently, in 2014, Gupta proved that if $n\geq 3$ and $charK>0$ then $K[t_{1},\dots,t_{n}]$ is not cancellative (see [30], [31]). The ZCP problem for $K[t_{1},\dots,t_{n}]$ remains open for $n\geq 3$ and $charK=0$.

Proposition 5.3** ([13], Remark 1.2).**

For any $R$-algebra $A$,

universally cancellative $\Rightarrow$ strongly cancellative $\Rightarrow$ cancellative.

For the investigation of the ZCP problem for a given non-commutative algebra $A$ have been used some subalgebras of $A$. The first one is the center.

Theorem 5.4** ([13], Proposition 1.3).**

Let $K$ be a field and $A$ be a $K$-algebra. If $Z(A)=K$, then $A$ is universally cancellative, and hence, cancellative.

In [39] was computed the center of many $K$-algebras interpreted as skew $PBW$ extensions, some of them with trivial center, and hence, such algebras are cancellative.

Another subalgebra involved in the study of the ZCP problem is the Makar-Limanov invariant.

Definition 5.5**.**

Let $A$ be a $R$-algebra.

1. (i)

${\rm Der}(A)$* denotes the collection of $R$-derivations of $A$ and ${\rm LND}(A)$ the collection of locally nilpotent $R$-derivations of $A$; $\delta\in{\rm Der}(A)$ is locally nilpotent if given $a\in A$ there exists $n\geq 1$ such that $\delta^{n}(a)=0$.* 2. (ii)

The Makar-Limanov invariant of $A$ is defined to be

[TABLE]

Theorem 5.6** ([13], Theorems 3.3 and 3.6).**

Let $A$ be a $R$-algebra that is a finitely generated domain of finite Gelfand-Kirillov dimension.

1. (i)

If ${\rm ML}(A[t])=A$, then $A$ is cancellative. 2. (ii)

If $charR=0$ and ${\rm ML}(A)=A$, then $A$ is cancellative.

The effective computation of ${\rm ML}(A)$ is in general a difficult task, therefore other strategies and techniques have been introduced in order to investigate the cancellation property for non-commutative algebras. For example, setting algebraic condition on $A$, it is interesting to know if $A$ becomes cancellative. In this direction recently have been proved the following results.

Theorem 5.7** ([40], Theorem 4.1).**

Let $A$ be a $R$-algebra. If $A$ is left $($or right$)$ artinian, then $A$ is strongly cancellative, and hence, cancellative.

Theorem 5.8** ([56], Theorem 0.1).**

Let $A$ be a $K$-algebra, $K$ a field with $charK=0$. If $A$ is a noetherian $($left and right$)$ $AS$ algebra generated in degree $1$ with ${\rm gld}(A)=3$ and $A$ is not $PI$, then $A$ is cancellative.

Observe that if in the previous theorem we could remove the condition not $PI$, then the classical commutative ZCP problem would be solved for $n=3$. With this, we can formulate our last problem in the next subsection.

5.1 Problem 5

• •

Let $B$ be a $\mathcal{SAS}$ algebra over a field $K$ with $char(K)=0$. Assume that $B$ is noetherian $($left and right$)$, generated in degree $1$ and ${\rm gld}(B)=3$. Is $B$ cancellative?