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Nakayama Automorphism of Some Skew PBW Extensions.

1 Introduction

There are two notions of Nakayama automorphisms: one for skew Calabi-Yau algebras and another for Frobenius algebras. In this paper, we focus on skew Calabi-Yau algebras, or equivalently, Artin-Schelter regular algebras in the connected graded case (Proposition 2.7). The Nakayama automorphism is one of important homological invariants for Artin-Schelter regular algebras algebras. Nakayama automorphisms have been studied by several authors. Reyes, Rogalski and Zhang in [1] proved three homological identities about the Nakayama automorphism and gave several applications. Liu, Wang and Wu in [2] proved that if R is skew Calabi-Yau with Nakayama automorphism v then the Ore extension A = R[x; [sigma], [delta]] has Nakayama automorphism v' such that v'\R = [[sigma].sup.-1]v and v'(x) = ux + b with u,b [member of] R and u invertible, the parameters u and b are still unknown. Zhu, Van Oystaeyen and Zhang computed the Nakayama automorphisms of trimmed double Ore extensions and skew polynomial extension R[x; [sigma]], where [sigma] is a graded algebra automorphism of R in terms of the homological determinant. Lu, Mao and Zhang in [3] and [4] used the Nakayama automorphism to study group actions and Hopf algebra actions on Artin-Schelter regular algebras of global dimension three and calculated explicitly the Nakayama automorphism of a class of connected graded Artin-Schelter regular algebras. Shen, Zhou and Lu in [5] described the Nakayama automorphisms of twisted tensor products of noetherian Artin-Schelter regular algebras. Liu and Ma in [6] gave an explicit formula to calculate the Nakayama automorphism of any Ore extension R[x; [sigma], [delta]] over a polynomial algebra R = K[[t.sub.1],..., [t.sub.m]] for an arbitrary m. In general, Nakayama automorphisms are known to be tough to compute.

Skew PBW extensions and quasi-commutative skew PBW extensions were defined in [7] and are non-commutative rings of polynomial type defined by a ring and a set of variables with relations between them. Skew PBW extensions include rings and algebras coming from mathematical physics such PBW extensions, group rings of polycyclic-by-finite groups, Ore algebras, operator algebras, diffusion algebras, some quantum algebras, quadratic algebras in three variables, some 3-dimensional skew polynomial algebras, some quantum groups, some types of Auslander-Gorenstein rings, some Koszul algebras, some Calabi-Yau algebras, some Artin-Schelter regular algebras, some quantum universal enveloping algebras, and others. There are some special subclasses of skew PBW extensions such as bijective, quasi-commutative, of derivation type and of endomorphism type (see Definition 2.2). Some noncommutative rings are skew PBW extensions but not Ore extensions (see [8],[9] and [10]).

Several properties of skew PBW extensions have been studied in the literature (see for example [11],[12],[13],[14],[15],[16]). It is known that quasi-commutative skew PBW extensions are isomorphic to iterated Ore extensions of endomorphism type (see [8, Theorem 2.3]). In [9] it was defined graded skew PBW extensions with the aim of studying Koszul property in these extensions.

The Nakayama automorphism has not been studied explicitly for skew PBW extensions. In this paper we focus on the Nakayama automorphism of quasi-commutative skew PBW extensions (Section 3) and non quasi-commutative skew PBW extensions of the form [sigma](K[[t.sub.1], ..., [t.sub.m]])<x> (Section 4). Since every graded quasi-commutative skew PBW extension is isomorphic to a graded iterated Ore extension of endomorphism type (see [17, Proposition 2.7]), we have that if A is a graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra R, then A is Artin-Schelter regular (see Proposition 3.1). Now, for B a connected graded algebra, B is Artin-Schelter regular if and only if B is graded skew Calabi-Yau (see [1]), and hence the Nakayama automorphism of Artin-Schelter regular algebras exists. Therefore, if R is an Artin-Schelter regular algebra with Nakayama automorphism v, then the Nakayama automorphism [micro] of a graded quasi-commutative skew PBW extension A exists, and we compute it using the Nakayama automorphism v together some especial automorphisms of R and A (see Theorem 3.1). The main results in Section 3 are Theorem 3.1, Corollary 3.2, Proposition 3.3, Corollary 3.3 and Examples 3.1, 3.2. Since Ore extensions of bijective type are skew PBW extensions. In Section 4 we use the results of [6] to calculate explicitly the Nakayama automorphism of some skew PBW extensions (Examples 4.1, 4.2 and Proposition 4.1).

We establish the following notation: the symbol N is used to denote the set of natural numbers including zero. If [alpha] := ([[alpha].sub.1],..., [[alpha].sub.n]) [member of] [N.sup.n] then |[alpha]| := [[alpha].sub.1] + *** + [[alpha].sub.n]. The letter K denotes a field. Every algebra is a K-algebra.

2 Preliminaries

In this section, we fix basic notations and recall definitions and properties for this paper.

Definition 2.1. Let R and A be rings. We say that A is a skew PBW extension over R, if the following conditions hold:

(i) R [[subset].bar] A;

(ii) there exist elements [x.sub.1],...,[x.sub.n] G A such that A is a left free R-module, with basis the basic elements

[mathematical expression not reproducible].

In this case, it is said also that A is a left polynomial ring over R with respect to {[x.sub.1], *** , [x.sub.n]} and Mon(A) is the set of standard monomials of A. Moreover, [x.sup.0.sub.1] *** [x.sup.0.sub.n]:= 1 [member of] Mon(A).

(iii) For each 1 [less than or equal to] i [less than or equal to] n and any r [member of] R \ {0}, there exists an element [c.sub.i,r] [member of] R \ {0} such that

[x.sub.i]r - [c.sub.i,r][x.sub.i] [member of] R. (1)

(iv) For any elements 1 [less than or equal to] i, j [less than or equal to] n, there exists [c.sub.i,j] [member of] R \ {0} such that

[x.sub.j][x.sub.i] - [c.sub.i,j][x.sub.i][x.sub.j] [member of] R + [Rx.sub.1] + *** + [Rx.sub.n]. (2)

Under these conditions, we will write A := [sigma](R)<[x.sub.1],...,[x.sub.n>.

The notation [sigma](R)<[x.sub.1],..., [x.sub.n> and the name of the skew PBW extensions are due to the next proposition.

Proposition 2.1 ([7], Proposition 3). Let A be a skew PBW extension of R. For each 1 [less than or equal to] i [less than or equal to] n, there exist an injective endomorphism [[sigma].sub.i] : R [right arrow] R and a [sigma]i-derivation [[delta].sub.i] : R [right arrow] R such that

[x.sub.i]r = [[sigma].sub.i](r)[x.sub.i] + [[delta].sub.i](r), r [member of] R. (3)

In the following definition we recall some sub-classes of skew PBW extensions. Examples of these sub-classes of algebras can be found in [15].

Definition 2.2. Let A be a skew PBW extension of R, [SIGMA] := {[[sigma].sub.1],..., [[sigma].sub.n]} and [DELTA] := {[[delta].sub.1],..., [[delta].sub.n]}, where [[sigma].sub.i] and [[delta].sub.i] (1 [less than or equal to] i [less than or equal to] n) are as in Proposition 2.1

(a) A is called quasi-commutative, if the conditions (iii) and (iv) in Definition 2.1 are replaced by

(iii') for each 1 [less than or equal to] i [less than or equal to] n and all r [member of] R \ {0}, there exists [c.sub.i.r] [member of] R \ {0} such that

[x.sub.i]r = [c.sub.i,r][x.sub.i]; (4)

(iv') for any 1 [less than or equal to] i, j [less than or equal to] n, there exists [c.sub.i,j] [member of] R \ {0} such that

[x.sub.j][x.sub.i] = [c.sub.i,j][x.sub.i][x.sub.j]. (5)

(b) A is called bijective, if [[sigma].sub.i] is bijective for each [[sigma].sub.i] [member of] [SIGMA], and [c.sub.i,j] is invertible for any 1 [less than or equal to] i [less than or equal to] j [less than or equal to] n.

(c) If [[sigma].sub.i] = [id.sub.R] for every [[sigma].sub.i] [member of] [SIGMA], we say that A is a skew PBW extension of derivation type.

(d) If [[delta].sub.i] = 0 for every [[delta].sub.i] [member of] [DELTA], we say that A is a skew PBW extension of endomorphism type.

The next proposition was proved in [9].

Proposition 2.2. Let R = [[direct sum].sub.m[greater than or equal to]0] [R.sub.m] be a N-graded algebra and let A = [sigma](R)<x.sub.1],... ,[x.sub.n]> be a bijective skew PBW extension of R satisfying the following two conditions:

(i) [[sigma].sub.i] is a graded ring homomorphism and [[delta].sub.i] : R(-1) [right arrow] R is a graded [[sigma].sub.i]-derivation for all 1 [less than or equal to] i [less than or equal to] n, where [[sigma].sub.i] and [[delta].sub.i] are as in Proposition 2.1.

(ii) [x.sub.j][x.sub.i] - [c.sub.i,j][x.sub.i][x.sub.j] [member of] [R.sub.2] + [R.sub.1][x.sub.1] + *** + [R.sub.1][x.sub.n, as in (2) and [c.sub.i,j] [member of] [R.sub.0].

For p [greater than or equal to] 0, let [A.sub.p] the K-space generated by the set

[r.sub.t][x.sup.[alpha]] |t+|[alpha]| = p, [r.sub.t] [member of] [R.sub.t] and [x.sup.[alpha]] [member of] Mon(A) .

Then A is a N-graded algebra with graduation

[mathematical expression not reproducible] (6)

Definition 2.3 ([9], Definition 2.6). Let A = [sigma](R)<[x.sub.1],... ,[x.sub.n]> be a bijective skew PBW extension of a N-graded algebra R = [[direct sum].sub.m[greater than or equal to]0] [R.sub.m]. We say that A is a graded skew PBW extension if A satisfies the conditions (i) and (ii) in Proposition 2.2.

Note that the family of graded iterated Ore extensions is strictly contained in the family of graded skew PBW extensions (see [9, Remark 2.11]). Examples of graded skew PBW extensions can be found in [9] and [10].

Proposition 2.3. Quasi-commutative skew PBW extensions with the trivial graduation of R are graded skew PBW extensions. If we assume that R has a different graduation to the trivial graduation, then A is graded skew PBW extension if and only if [[sigma].sub.i] is graded and [c.sub.i,j] [member of] R0, for 1 [greater than or equal to] i, j [greater than or equal to] n.

Proof. Let R = [R.sub.0] and r [member of] R = [R.sub.0]. From (4) we have that [x.sub.ir] = [c.sub.i,r][x.sub.i] = [[sigma].sub.i](r)[x.sub.i]. So, [[sigma].sub.i](r) = [c.sub.i,r] [member of] R = [R.sub.0] and [[delta].sub.i] = 0, for 1 [less than or equal to] i [less than or equal to] n. Therefore [[sigma].sub.i] is a graded ring homomorphism and [[delta].sub.i] : R(-1) [right arrow] R is a graded [[sigma].sub.i]-derivation for all 1 [less than or equal to] i [less than or equal to] n. From (5) we have that xjxi - [c.sub.i,j][x.sub.i][x.sub.j] = 0 [member of] [R.sub.2] + [R.sub.1][x.sub.1] + *** +[R.sub.1][x.sub.n= and [c.sub.i,j] [member of] R = [R.sub.0]. If R has a nontrivial graduation, then the result is obtained from the relations (4), (5) and Definition 2.3.

Let B = [[direct sum].sub.p[greater than or equal to]0] [B.sub.p] be an N-graded algebra. B is connected if [B.sub.0] = K. In [18, Definition 1.4] finitely graded algebra was presented. B is finitely graded if the following conditions hold:

(i) B is N-graded (positively graded): B = [[direct sum].sub.j[greater than or equal to]0] [B.sub.j].

(ii) B is connected.

(iii) B is finitely generated as algebra, i.e., there is a finite set of elements [x.sub.1],... ,[x.sub.n] [member of] B such that the set [mathematical expression not reproducible] spans Basa vector space.

Note that a finitely graded algebra B is finitely generated as K-algebra if and only if B = K<[x.sub.1],... ,[x.sub.m]>/I, where I is a proper homogeneous two-sided ideal of K<[x.sub.1],... ,[x.sub.m]>. A finitely graded algebra B is finitely presented if the two-sided ideal I is finitely generated.

The following properties of graded skew PBW extensions were proved in [9].

Remark 2.1 ([9], Remark 2.10). Let A = [sigma](R)<[x.sub.1],... ,[x.sub.n]> be a graded skew PBW extension. Then we have the following properties:

(i) A is a N-graded algebra and [A.sub.0] = [R.sub.0].

(ii) R is connected if and only if A is connected.

(iii) If R is finitely generated then A is finitely generated.

(iv) For (i), (ii) and (iii) above, we have that if R is a finitely graded algebra then A is a finitely graded algebra.

(v) If R is locally finite, then A as K-algebra is a locally finite.

(vi) A as R-module is locally finite.

(vii) If R is finitely presented then A is finitely presented.

The following proposition establishes the relation between graded skew PBW extensions and graded iterated Ore extensions.

Proposition 2.4 ([17], Proposition 2.7). Let A = [sigma](R)<[x.sub.1],..., [x.sub.n]> be a graded skew PBW extension. If A is quasi-commutative, then A is isomorphic to a graded iterated Ore extension of endomorphism type R[[z.sub.1]; [[theta].sub.1]] *** [[z.sub.n]; [[theta].sub.n]], where [[theta].sub.i] is bijective, for each i; [[theta].sub.1] = [[sigma].sub.1];

[[theta].sub.j] : R[[z.sub.1]; [[theta].sub.1]] *** [[z.sub.j-1]; [[theta].sub.j-1] [right arrow] R[[z.sub.1]; [[theta].sub.1]] *** [[z.sub.j-1]; [[theta].sub.j-1]

is such that [[theta].sub.j]([z.sub.i]) = [c.sub.i,j][z.sub.i] ([c.sub.i,j] [member of] [R.sub.0] as in (2)), 1 [less than or equal to] i < j [less than or equal to] n and [[theta].sub.i](r) = [[sigma].sub.i](r), for r [member of] R.

If M and N are graded B-modules, we use [mathematical expression not reproducible] to denote the set of all B-module homomorphisms h : M [right arrow] N such that h([M.sub.i]) [[subset].bar] [N.sub.i+d]. We set [Hom.sub.B](M, N) = [[direct sum].sub. d[member of]Z] [mathematical expression not reproducible], and we denote the corresponding derived functors by [mathematical expression not reproducible].

Definition 2.4. Let B = K[direct sum][B.sub.1] [direct sum] [B.sub.2] [direct sum] *** be a finitely presented graded algebra over K. The algebra B will be called Artin-Schelter regular, if B has the following properties:

(i) B has finite global dimension d: every graded B--module has projective dimension [less than or equal to] d.

(ii) B has finite Gelfand-Kirillov dimension.

(iii) B is Gorenstein, meaning that [mathematical expression not reproducible] if i [not equal to] d, and [mathematical expression not reproducible] K(l) for some l [member of] Z.

The enveloping algebra of a ring B is defined as [B.sup.e] := B [cross product] [B.sup.op]. We characterize the enveloping algebra of a skew PBW extension in [12]. If M is an B-bimodule, then M is an [B.sup.e] module with the action given by (a [cross product] b) * m = amb, for all m [member of] M, a,b [member of] B. Given automorphisms v, [tau] [member of] Aut(B), we can define the twisted [B.sup.e]-module [.sup.v][M.sup.[tau]] with the rule (a[cross product]b) * m = v(a)m[tau](b), for all m [member of] M, a,b [member of] B. When one or the other of v, [tau] is the identity map, we shall simply omit it, writing for example [M.sup.v] for [.sup.1][M.sup.v].

Proposition 2.5 ([19], Lemma 2.1). Let v, [sigma] and [empty set] be automorphisms of B. Then

(i) The map [.sup.v] [B.sup.[sigma]] [right arrow] [.sup.[empty set][sigma]] [B.sup.[empty set][sigma]], a [right arrow] [empty set](a) is an isomorphism of [B.sup.e]-modules. In particular,

[mathematical expression not reproducible]

(ii) B [congruent to] [B.sup.[sigma] as [B.sup.e]-modules if and only if [sigma] is an inner automorphism.

An algebra B is said to be homologically smooth, if as a [B.sup.e]-module, B has a finitely generated projective resolution of finite length.

Definition 2.5. An algebra B is called skew Calabi-Yau of dimension d if

(i) B is homologically smooth.

(ii) There exists an algebra automorphism v of B such that

[mathematical expression not reproducible]

as [B.sup.e] -modules. If v is the identity, then B is said to be Calabi-Yau.

A graded algebra B is called graded skew Calabi-Yau of dimension d, if

(i) B is homologically smooth.

(ii) There exists a graded automorphism v of B such that

[mathematical expression not reproducible]

as [B.sup.e]-graded modules, for some integer l. If v is the identity, then B is said to be graded Calabi-Yau.

The automorphism v is called the Nakayama automorphism of B. As a consequence of Proposition 2.5, we have that a skew Calabi-Yau algebra is Calabi-Yau if and only if its Nakayama automorphism is inner. If B is a Calabi-Yau algebra of dimension d, then the Hochschild dimension of B (that is, the projective dimension of A as an A-bimodule) is d (see [20, Proposition 2.2]).

Proposition 2.6. Let B be a skew Calabi-Yau algebra with Nakayama automorphism v. Then v is unique up to an inner automorphism, i.e, the Nakayama automorphism is determined up to multiplication by an inner automorphism of B.

Proof. Let B be a skew Calabi-Yau algebra with Nakayama automorphism v and let [micro] be another Nakayama automorphism, i.e., [mathematical expression not reproducible], then [mathematical expression not reproducible] as [B.sup.e] -modules. By Proposition 2.5-(i), [mathematical expression not reproducible]; by Proposition 2.5-(ii), [v.sup.-1][micro] is an inner automorphism of B. Let [v.sup.-1][micro] = [sigma] where [sigma] is an inner automorphism of B, so [micro] = v[sigma] for some inner automorphism [sigma] of B.

Reyes, Rogalski and Zhang in [1], proved that Artin-Schelter regular algebras and graded skew Calabi-Yau algebras coincide for the connected case.

Proposition 2.7 ([1], Lemma 1.2). Let B be a connected graded algebra. Then B is graded skew Calabi-Yau if and only if it is Artin-Schelter regular.

3 Nakayama automorphism of quasi-commutative skew PBW extensions

Since graded quasi-commutative skew PBW extensions are isomorphic to graded iterated Ore extensions of endomorphism type, it follows that graded quasi-commutative skew PBW extensions with coefficients in Artin-Schelter regular algebras are skew Calabi-Yau, and the Nakayama automorphism exists for these extensions. With this in mind, in this section we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.

The following properties of graded quasi-commutative skew PBW extensions were proved in [17, Theorem 3.6 and Theorem 4.5].

Proposition 3.1. Let A = [sigma](R)<[x.sub.1],... ,[x.sub.n]> be a graded quasic-ommutative skew PBW extension of R.

(i) If R is Artin-Schelter regular, then A is Artin-Schelter regular.

(ii) If R is a finitely presented graded skew Calabi-Yau algebra of global dimension d, then A is graded skew Calabi-Yau of global dimension d + n.

As a consequence of Proposition 2.7 and Proposition 3.1 we have the following property.

Corollary 3.1. Let R be an Artin-Schelter regular algebra of global dimension d. Then every graded quasi-commutative skew PBW extension A = [sigma](R)<[x.sub.1],..., [x.sub.n]> is graded skew Calabi-Yau of global dimension d + n.

Liu, Wang and Wu in [2, Theorem 3.3] proved that if R is a skew Calabi-Yau algebra with Nakayama automorphism v then the Ore extension A = R[x; [sigma], [delta]] has Nakayama automorphism v' such that v'|R = [[sigma].sup.-1]v and v'(x) = ux + b with u, b [member of] R and u invertible. If [delta] = 0 then they deduced the following result.

Proposition 3.2 ([2], Remark 3.4). With the above notation v'(x) = ux if [delta] = 0.

In the next theorem we show a way to calculate the Nakayama automorphism for a graded quasi-commutative skew PBW extension A of an Artin-Schelter regular algebra R.

Theorem 3.1. Let R be an Artin-Schelter regular algebra with Nakayama automorphism v. Then the Nakayama automorphism [micro] of a graded quasi-commutative skew PBW extension A = [sigma](R)<[x.sub.1],...,[x.sub.n]> is given by

[micro](r) = [([[sigma].sub.1] *** [[sigma].sub.n]).sup.-1] v(r), for r [member of] R, and

[mathematical expression not reproducible], for each 1 [less than or equal to] i [less than or equal to] n,

where [[sigma].sub.i] is as in Proposition 2.1, [u.sub.i],[c.sub.i,j] [member of] K\ {0}, and the elements [c.sub.i,j] are as in Definition 2.1.

Proof. Note that A is graded skew Calabi-Yau (see Corollary 3.1) and therefore the Nakayama automorphism of A exists. By Proposition 2.4 we have that A is isomorphic to a graded iterated Ore extension R[[x.sub.1]; [[theta].sub.1]] *** [[x.sub.n]; [[theta].sub.n]], where [[theta].sub.i] is bijective; [[theta].sub.1] = [[sigma].sub.1];

[[theta].sub.j] : R[[x.sub.1]; [[theta].sub.1]] *** [[x.sub.j-1]; [[theta].sub.j-1]] [right arrow] R[[x.sub.1]; [[theta].sub.1]] *** [[x.sub.j-1]; [[theta].sub.j-1]

is such that [[theta].sub.j]([x.sub.i]) = [c.sub.i,j][x.sub.i] ([c.sub.i,j] [member of] K as in Definition 2.1), 1 [less than or equal to] i < j [less than or equal to] n and [[theta].sub.i](r) = [[sigma].sub.i](r), for r [member of] R. Note that

[mathematical expression not reproducible] (7)

Now, since R is connected then by Remark 2.1, A is connected. So, the multiplicative group of R and also the multiplicative group of A is K \ {0}, therefore the identity map is the only inner automorphism of A. Let [micro]i the Nakayama automorphism of R[[x.sub.1]; [[theta].sub.1]] *** [[x.sub.i];[[theta].sub.i]] .

By Proposition 3.2 we have that the Nakayama automorphism [[micro].sub.1] of R[[x.sub.1]; [[theta].sub.1]] is given by [mathematical expression not reproducible] for r [member of] R, and [[micro].sub.1]([x.sub.1]) = [u.sub.1][x.sub.1] with [u.sub.1] [member of] K \ {0}; the Nakayama automorphism [[micro].sub.2] of R[[x.sub.1]; [[theta].sub.1]][[x.sub.2]; [[theta].sub.2]] is given by [mathematical expression not reproducible], for r [member of] R; [mathematical expression not reproducible] and [[micro].sub.2]([x.sub.2]) = [u.sub.2][x.sub.2], for [u.sub.2] [member of] K \ {0}; the Nakayama automorphism [[micro].sub.3] of R[[x.sub.1];[[theta].sub.1]][[x.sub.2];[[theta].sub.2]][[x.sub.3];[theta]3] is given by [mathematical expression not reproducible], for r [member of] R; [mathematical expression not reproducible] ; [mathematical expression not reproducible] and [[micro].sub.3]([x.sub.3]) = [u.sub.3][x.sub.3], for [u.sub.3] [member of] K \ {0}.

Continuing with the procedure we have that the Nakayama automorphism of A is given by

[mathematical expression not reproducible],

for r [member of] R, and

[mathematical expression not reproducible]

In general, for 1 [less than or equal to] i [less than or equal to] n, we have that

[mathematical expression not reproducible].

Note that [c.sub.i,i] = 1.

From Theorem 3.1 the following corollary is immediately obtained.

Corollary 3.2. Let A = [sigma](R)<[x.sub.1],... ,[x.sub.n]> be a graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra R with Nakayama automorphism v. Then A is graded Calabi-Yau if and only if [[sigma].sub.1] *** [[sigma].sub.n] = v and [mathematical expression not reproducible], for all 1 [less than or equal to] i [less than or equal to] n, where [u.sub.i] is as in Theorem 3.1.

Let K[[t.sub.1],... ,[t.sub.m]] be the polynomial algebra and [sigma] : K[[t.sub.1],... ,[t.sub.m]] [right arrow] K[[t.sub.1],..., [t.sub.m]] an algebra automorphism. We define

[mathematical expression not reproducible] (8)

Remark 3.1. Liu and Ma proved in [6] that if A = K[[t.sub.1],... ,[t.sub.m]][x;[sigma]] is an Ore extension then the Nakayama automorphism v of A is given by v([t.sub.i]) = [t.sub.i] and v(x) = det(M)x, where det(M) is the usual determinant of the matrix M in (8). Note that A = K[[t.sub.1],... ,[t.sub.m]][x;[sigma]] is a quasi-commutative skew PBW extension and if [sigma] is a graded automorphism then A is a graded quasi-commutative skew PBW extension.

Let h [member of] K \ {0}. The algebra of shift operators is defined by [S.sub.h]:= K[t][[x.sub.h]; [[sigma].sub.h]], where [[sigma].sub.h](p(t)) := p(t - h). Notice that [x.sub.h]t = (t - h)[x.sub.h] and for p(t) [member of] K[t] we have [mathematical expression not reproducible]. Thus, [S.sub.h] [congruent to] [sigma] (K[t])<[[x.sub.h]> is a quasi-commutative skew PBW extension of K[t].

Proposition 3.3. The algebra of shift operators is a Calabi-Yau algebra.

Proof. Since [x.sub.h]t = (t - h)[x.sub.h], then [[sigma].sub.h](t) := t - h, then [mathematical expression not reproducible]. By Remark 3.1 we have that the Nakayama automorphism v of [S.sub.h] is given by v(t) = t and v([x.sub.h]) = det(M)x = 1, i.e., the Nakayama automorphism of Sh is the identity map. Therefore [S.sub.h] is Calabi-Yau.

Corollary 3.3. Let A = K[[t.sub.1],... ,[t.sub.m]][x;[sigma]] with [sigma] be a graded algebra automorphism. Then u = det(M), where u is as in Theorem 3.1 and M is as in (8).

Proof. Note that A = K[[t.sub.1],..., [t.sub.m]][x; [sigma]] is a quasi-commutative skew PBW extension of the graded Calabi-Yau algebra K[[t.sub.1],..., [t.sub.m]]. Since [sigma] is graded then A is a graded quasi-commutative skew PBW extension. By Theorem 3.1 we have that the Nakayama automorphism [micro] of A is given by

[micro]([t.sub.i]) = [t.sub.i], for 1 [less than or equal to] i [less than or equal to] m, and

[micro](x) = ux, for some u [member of] K\ {0}.

By Remark 3.1 we have that the Nakayama automorphism of A is given by

v([t.sub.i]) = [t.sub.] and

v(x) = det(M)x.

Since K[[t.sub.1],... , [t.sub.m]] is connected, then by Remark 2.1 A is connected, so the multiplicative group of A is K \ {0}. Therefore the identity map is the only inner automorphism of A. As by Proposition 2.6 we have that the Nakayama automorphism is unique up to inner automorphisms, then v must be equal to [micro]. Sou = det(M).

Example 3.1. Let R be an Artin-Schelter regular algebra of global dimension d with Nakayama automorphism v. Let

A = R[[x.sub.1],... ,[x.sub.n]; [[sigma].sub.1],..., [[sigma].sub.n]]

be an iterated skew polynomial ring (see [21, Page 23]), with [[sigma].sub.i] graded. A is a skew PBW extension of R with relations [x.sub.i]r = [[sigma].sub.i](r)[x.sub.i] and [x.sub.j][x.sub.i] = [x.sub.i][x.sub.j], for r [member of] R and 1 [less than or equal to] i, j [less than or equal to] n. As R is graded and [c.sub.i],j = 1 [member of] [R.sub.o], then by Proposition 2.3 we have that A is a graded quasi-commutative skew PBW extension of R. Therefore, A is a graded skew Calabi-Yau algebra (Corollary 3.1). By Proposition 2.4, R[[x.sub.1],... ,[x.sub.n]; [[sigma].sub.1],..., [[sigma].sub.n]] [congruent to] R[[x.sub.1]; [[theta].sub.1]] * * * [[x.sub.n; [[theta].sub.n]], where [[theta].sub.j](r) = [[sigma].sub.j](r) and [[theta].sub.j]([x.sub.i]) = [x.sub.i] for i < j. Applying Theorem 3.1 we have that the Nakayama automorphism [micro] of A is given by [micro](r) = [([[sigma].sub.1] *** [[sigma].sub.n]).sup.-1] v(r), if r [member of] R and [mathematical expression not reproducible], [u.sub.i] [member of] K\ {0}, 1 [less than or equal to] i [less than or equal to] n.

Example 3.2. For a fixed q [member of] K - {0}, the algebra of linear partial q-dilation operators H, with polynomial coefficients, is the free algebra [mathematical expression not reproducible], n [less than or equal to] m, subject to the relations:

[t.sub.j][t.sub.i] = [t.sub.i][t.sub.j], 1 [less than or equal to] i < j [less than or equal to] n;

[mathematical expression not reproducible]

The algebra H is a graded quasi-commutative skew PBW extension of K[[t.sub.1],...,[t.sub.n]], where K[[t.sub.1],... ,[t.sub.n]] is endowed with usual graduation. According to Proposition 2.1, we have that

[mathematical expression not reproducible]

By Proposition 2.4, H is isomorphic to a graded iterated Ore extension of endomorphism type

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

Since K[[t.sub.1],... ,[t.sub.n]] is a graded Calabi-Yau algebra, then its Nakayama automorphism v is the identity map. Applying Theorem 3.1, we have that the Nakayama automorphism [micro] of H is given by

[mathematical expression not reproducible]

[mathematical expression not reproducible]

So, det(M) = q. By Corollary 3.3 we have that the Nakayama automorphism [micro] of H is given by

[mathematical expression not reproducible]

Therefore, the Nakayama automorphism of H is completely determined by the parameter q.

4 Nakayama automorphism of some specific skew PBW extensions

Liu and Ma in [6] computed the Nakayama automorphism for Ore extensions over a polynomial algebra in n variables. Since Ore extensions of bijective type are skew PBW extensions, in this section we use these results to calculate explicitly the Nakayama automorphism of some skew PBW extensions of the form A = [sigma](K[[t.sub.1],..., [t.sub.m]])<x>, which are not quasicommutative.

Next we present [6, Theorem 4.2], but using the notation of skew PBW extensions.

Theorem 4.1. Let A = [sigma](K[[t.sub.1],..., [t.sub.m]])<x> be a skew PBW extension of derivation type. The Nakayama automorphism v of A is then given by v(r) = r or all r [member of] R and [mathematical expression not reproducible], where [delta] is as in Proposition 2.1.

Next we calculate the Nakayama automorphism of a skew PBW extension of derivation type using Theorem 4.1. This automorphism had already been calculated by other authors using other techniques (see for example [2] or [22]).

Example 4.1. Let A = K<x, y>/<yx - xy - [x.sup.2]> be the Jordan plane. Note that A = K[x][y;[delta]] is a graded skew PBW extension of K[x] of derivation type, with [delta](x) = [x.sup.2]. By Theorem 4.1 we have that the Nakayama automorphism of A is given by

[mathematical expression not reproducible]

If the Nakayama automorphism of a skew Calabi-Yau algebra A is the identity then A is Calabi-Yau. Thus, from Theorem 4.1 the following are obtained immediately.

Corollary 4.1. Let A = [sigma](K[[t.sub.1], ..., [t.sub.m]])<x> be a skew PBW extension. Then A is Calabi-Yau if and only if [sigma] = id and [mathematical expression not reproducible].

Example 4.2. The first Weyl algebra [A.sub.1](K) = K[t][x; [partial derivative]/[partial derivative]t] is a skew PBW extension of derivation type which is not graded. Note that xt = tx + 1, then [delta](t) = 1 and [mathematical expression not reproducible]. By Corollary 4.1 we have that [A.sub.1](K) is a Calabi-Yau algebra, which was already known but using other techniques (see for example [23]).

Liu and Ma in [6, Lemma 4.4] proved that for the Ore extension K[[t.sub.1],..., [t.sub.m]][x; [sigma], [delta] with [sigma] [not equal to] id, there is a unique k in the quotient field K[[t.sub.1],..., [t.sub.m]][[x; [sigma], [delta]].sub.q] of K[[t.sub.1],..., [t.sub.m]][x; [sigma], [delta]] such that

[delta](h) = k([sigma](h) - h) for all h [member of] R. (9)

Next we present [6, Theorem 4.5], but using the notation of skew PBW extensions.

Theorem 4.2. Let A = [sigma](K[[t.sub.1],... ,[t.sub.m])<x> be a graded skew PBW extension with [sigma] [not equal to] id. Then the Nakayama automorphism v of A is given by

[mathematical expression not reproducible]

where M is as in (8), k is as in (9) and [[sigma].sub.q] is the extension of [sigma] to K[[[t.sub.1],..., [t.sub.m]].sub.q].

Lu, Mao and Zhang in [3] calculated the Nakayama automorphism for the following classes of algebras:

A(1) = K<[t.sub.1], [t.sub.2], [t.sub.3]>/<[t.sub.2][t.sub.1] - [p.sub.12][t.sub.1][t.sub.2], [t.sub.3][t.sub.1] - [p.sub.13][t.sub.1][t.sub.3],[t.sub.3][t.sub.2] - [p.sub.23][t.sub.2][t.sub.3]>,

A(2) = K<[t.sub.1], [t.sub.2],[t.sub.3]>/<[t.sub.1][t.sub.2] - [t.sub.2][t.sub.1], [t.sub.1][t.sub.3] - [t.sub.3][t.sub.1], [t.sub.3][t.sub.2] - p[t.sub.2][t.sub.3] - [t.sub.1.sup.2]>,

A(3) = K<[t.sub.1],[t.sub.2],[t.sub.3]>/<([t.sub.2] + [t.sub.1])[t.sub.1] - [t.sub.1][t.sub.2], [t.sub.3][t.sub.1] - q[t.sub.1][t.sub.3], [t.sub.3][t.sub.2] - q([t.sub.2] + [t.sub.1])[t.sub.3]>,

A(4) = K<[t.sub.1], [t.sub.2],[t.sub.3]>/<([t.sub.2] + [t.sub.1])[t.sub.1] - [t.sub.1][t.sub.2], [t.sub.3][t.sub.1] - p[t.sub.1][t.sub.3], [t.sub.3][t.sub.2] - p[t.sub.2][t.sub.3]>,

A(5) = K<[t.sub.1],[t.sub.2],[t.sub.3]>/<([t.sub.2] + [t.sub.1])[t.sub.1] - [t.sub.1][t.sub.2], ([t.sub.3] + [t.sub.2] + [t.sub.1])[t.sub.1] - [t.sub.1][t.sub.3], ([t.sub.3] + [t.sub.2] + [t.sub.1])[t.sub.2] - ([t.sub.2] + [t.sub.1])[t.sub.3]>,

where K is an algebraically closed field of characteristic zero and [p.sub.12],[p.sub.13], [p.sub.23],p,q [member of] K \ {0}. Notice that these algebras are graded skew PBW extensions. In the following proposition we calculate the Nakayama automorphism of A(2) using Theorem 4.2.

Proposition 4.1. The Nakayama automorphism v of A(2) is given by v([t.sub.1]) = [t.sub.1], v([t.sub.2]) = [p.sup.-1][t.sub.2], v([t.sub.3]) = p[t.sub.3].

Proof. Since [t.sub.2][t.sub.1] = [t.sub.1][t.sub.2]; [t.sub.3][t.sub.1] = [t.sub.1][t.sub.3] and [t.sub.3][t.sub.2] = p[t.sub.2][t.sub.3] - [t.sup.2.sub.1], then A(2) is a graded skew PBW extension of K[[t.sub.1],[t.sub.2]], i.e., A(2) = [sigma](K[[t.sub.1],[t.sub.2]])<[t.sub.3]>. Note that [sigma]([t.sub.1]) = [t.sub.1], [sigma]([t.sub.2]) = p[t.sub.2], [delta]([t.sub.1]) = 0 and [delta]([t.sub.2]) = -[t.sup.2.sub.1], where [sigma] and [delta] are as in Proposition 2.1. Thus,

[mathematical expression not reproducible]

Note that [mathematical expression not reproducible], [mathematical expression not reproducible] and [mathematical expression not reproducible]. Then by Theorem 4.2 v([t.sub.1]) = [[sigma].sup.-1]([t.sub.1]) = [t.sub.1], [mathematical expression not reproducible]

The Nakayama automorphism of Proposition 4.1 coincides with the Nakayama automorphism in [3, Equation (E1.5.2)].

5 Conclusions

There are some special subclasses of skew PBW extensions such as bijective, quasi-commutative, of derivation type and of endomorphism type (see Definition 2.2). Some noncommutative rings are skew PBW extensions but not Ore extensions (see [8],[9] and [10]). We illustrate each one of the properties studied here with some of these examples of skew PBW extensions.

From Proposition 3.1, if A is a graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra R, then A is Artin-Schelter regular. Now, for connected graded algebras, an algebra is Artin-Schelter regular if and only if this is graded skew Calabi-Yau (see 2.7). Therefore, if R is an Artin-Schelter regular algebra with Nakayama automorphism v, then the Nakayama automorphism [micro] of a graded quasi-commutative skew PBW extension A exists, and we compute it using the Nakayama automorphism v together some especial automorphisms of R and A (see Theorem 3.1). As a consequence of Theorem 3.1, we have that if A = [sigma](R)<[x.sub.1],... ,[x.sub.n]> is a graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra R with Nakayama automorphism v, then A is graded Calabi-Yau if and only if [[sigma].sub.1] * * * [[sigma].sub.n] = v and [mathematical expression not reproducible], for all 1 [less than or equal to] i [less than or equal to] n (see Corollary 3.2). Another consequence of Theorem 3.1 is the fact that for A = K[[t.sub.1],..., [t.sub.m]][x; [sigma]] with [sigma] be a graded algebra automorphism, u = det(M), i.e., the element u is calculable (see Corollary 3.3).

Most authors give a description of the Nakayama automorphism of some specific Artin-Schelter regular algebras in terms of some parameters that can not be calculated explicitly (see for example [2],[5] and [24]). Since Ore extensions of bijective type are skew PBW extensions, in Section 4 is calculated explicitly the Nakayama automorphism of some skew PBW extensions (Examples 4.1, 4.2 and Proposition 4.1).

Acknowledgements

The authors express their gratitude to Professor Oswaldo Lezama for valuable suggestions and helpful comments for the improvement of the paper. Armando Reyes was supported by the research fund of Facultad de Ciencias, Universidad Nacional de Colombia, Bogota, Colombia, HERMES CODE 41535.

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Hector Suarez (1) and Armando Reyes (2)

Received: 11-11-2018 | Accepted: 27-05-2019 | Online: 31-05-2019

MSC: 16W50, 16S37, 16W70, 16S36, 13N10.

doi:10.17230/ingciencia.15.29.6

(1) Universidad Pedagogica y Tecnologica de Colombia, hector.suarez@uptc.edu.co, https://orcid.org/0000-0003-4618-0599, Escuela de Matematicas y Estadistica, Boyaca, Colombia.

(2) Universidad Nacional de Colombia, mareyesv@unal.edu.co, https://orcid.org/0000-0002-5774-0822, Departamento de Matematicas, Bogota, Colombia.
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