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Ciencia en Desarrollo
Print version ISSN 01217488
Ciencia en Desarrollo vol.6 no.2 Tunja July/Dec. 2015
Some Relations between NKoszul, ArtinSchelter Regular and CalabiYau Algebras with Skew PBW Extensions
Algunas relaciones entre álgebras NKoszul, ArtinSchelter regular y CalabiYau con extensiones PBW torcidas
H. Suárez^{a,*}
O. Lezama^{b}
A. Reyes^{b}
^{a} Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia.
^{b} Seminario de Álgebra Constructiva SAC^{2}, Departamento de Matemáticas, Universidad Nacional de Colombia, sede Bogotá, Colombia.
^{*} Autor de correspondencia: hector.suarez@uptc.edu.co
Recepción: 05ene15 Aceptación: 15jun15
Abstract
Some authors have studied relations between ArtinSchelter regular algebras, NKoszul algebras and CalabiYau algebras (resp. skew CalabiYau) of dimension d. In this paper we want to show through examples and counterexamples some relations between these classes of algebras with skew PBW extensions. In addition, we also exhibit some examples of the preservation of these properties by Ore extensions.
Key words: Skew PBW extensions, CalabiYau algebras, NKoszul algebras, AS regular algebras, Ore extensions.
Resumen
Algunos autores han estudiado las relaciones entre las álgebras ArtinSchelter regular, las álgebras N Koszul y las álgebras CalabiYau (resp. skew CalabiYau) de dimensión d. En este artículo queremos mostrar a través de ejemplos y contraejemplos algunos relaciones entre estas clases de álgebras y las extensiones PBW torcidas. Además, mostraremos algunos ejemplos de preservación de estas propiedades en las extensiones de Ore.
Palabras clave: Skew PBW extensions, CalabiYau algebras, NKoszul algebras, AS regular algebras, Ore extensions.
1. Introduction
Recently there have been defined some special classes of algebras such as NKoszul algebras, CalabiYau algebras and skew PBW extensions. Koszul algebras, which in this article are called 2Koszul algebras were introduced by Stewart B. Priddy in [34]. Later in 2001, Roland Berger in [3] introduces a generalization of Kozsul algebras, which are then called generalized Koszul algebras or NKoszul algebras. In [17] Victor Ginzburg defined dCalabiYau algebras or CalabiYau algebras of dimension d (or simply CalabiYau algebras). Then in [6], Roland Berger and Rachel Taillefer introduced the definition of graded CalabiYau algebra. As a generalization of CalabiYau algebras, were also defined the skew CalabiYau algebras. On the other hand, the skew PBW extensions were introduced in 2011 by Oswaldo Lezama and Claudia Gallego in [16].
In the current literature, it has been studied certain relations between Artin Schelter regular algebras, NKoszul algebras, CalabiYau algebras and skew CalabiYau algebras. Our aim is to show through a serie of examples some relationships between the above algebras and skew PBW extensions. Unless otherwise specified, throughout this article, K will represent a fixed but arbitrary field.
2. Definitions and Elementary Properties
2.1. AS Regular Algebras
Regular algebras were defined by Michael Artin and William Schelter in [2]. They studied the regular algebras of global dimension three which are generated by elements of degree one and classified into thirteen types.
Definition 1 ([2] ). Let A = ⊕A_{1} ⊕A_{2} ⊕··· be a finitely presented graded algebra over K. The algebra A will be called regular if it has the following properties:
(i) A has finite global dimension d: every graded Amodule has projective dimension ≤ d.
(ii) A has finite GelfandKirillov dimension (GKdim), i.e., A has polynomial growth.
(iii) A is Gorenstein ,i.e., Ext^{q}_{A}(, A) = 0 if q # d, and Ext^{d}_{A}(, A) ≅ .
In the current literature these algebras are called ArtinSchlter regular algebras (AS regular algebras).
Most of the authors do not consider the condition (ii) in the definition of AS regular algebras. We say that A has polynomial growth if there exist c ∈ ℝ^{+ }and r ∈ℕ such that for all n ∈ℕ, dim_{K} A_{n} ≤cn^{r}.
2.2. NKoszul Algebras
Koszul algebras were defined by Stewart B. Priddy in [34], later in 2001, Roland Berger in [3] introduces a generalization of Koszul algebras which are called generalized Koszul algebras or NKoszul algebras. Koszul algebras defined by Stewart B. Priddy correspond to 2Koszul algebras in this paper.
Definition 2 ([3] ). The generalized Koszul algebras are graded algebras A = ⊕A_{1} ⊕A_{2} ⊕··· which are generated in degrees 0 and 1 such that there is a graded projective resolution of
such that for any i ≥ 0, P^{i }is generated in degree δ(i), where
for some N ≥2.
If N = 2, NKoszul algebras is usually called Koszul. In this situation, Definition 2 coincides with that given by Stewart Priddy in [34].
2.3. CalabiYau Algebras of Dimension d
CalabiYau algebras of dimension d or dCalabiYau algebras were defined by Victor Ginzburg in [17].
Definition 3 ([17], Definition 3.2.4). A Kalgebra A is called a CalabiYau algebra of dimension d if
(i) A is homologically smooth; that is, A has a finite resolution of finitely generated projective Abimodules;
(ii) Ext^{i}_{ABim}(A, A⊗A) ≅ , as Abimodules.
The space A ⊗ A is endowed with two Abimodule structures: the outer structure defined by a ·(x ⊗ y) ·b = ax ⊗yb, and the inner structure defined by a ·(x⊗y) ·b = xb⊗ay. Consequently, the Hom spaces Hom_{AA}(M,A ⊗A) of Abimodule morphisms from M to A ⊗ A endowed with the outer structure are again Abimodules using the inner structure of A ⊗A, and the same is true for the Hochschild cohomology spaces H^{k}(A,A⊗A). For A^{e }= A⊗A^{op}, the enveloping algebra of A, each Abimodule M is a left A^{e}module for the action (a⊗b).m = amb and right A^{e}module for the action m.(a⊗b) = bma.
Let A = ⊕_{n∈Z}An be a Zgraded algebra, and M = _{i∈Z }M_{i} be a graded Abimodule. For any integer l, M(l) is a graded Abimodule whose degree i component is M(l)_{i} = M_{i+l}.
Definition 4. A graded algebra A is called a graded CalabiYau algebra of dimension d if
(i) A has a finite resolution of finitely generated graded projective Abimodules, and
(ii) Ext^{i}_{Ae }(A,A ⊗A)≅, as graded Abimodules; for some integer l.
It follows from Definition 4 that every graded CalabiYau algebra of dimension d is CalabiYau of dimension d (see [6], Proposition 4.3).
Let M be an Abimodule, ν , μ: A →A two automorphism, the skew Abimodule ^{ν}M^{μ }is equal to M as a vector space whit a ·m ·b = ν (a)mμ(b).
Definition 5. Let A be a algebra. A is called skew CalabiYau of dimension d if there exists an automorphism ν of A such that
(i) A is homologically smooth; and
(ii) Ext^{i}_{Ae }(A,A^{e}) ≅ 0 when i # d and Ext^{d}_{Ae }(A,A^{e}) ≅ ^{1}A^{ν} as A^{e}modules.
In this case, ν is called the Nakayama Automorphism of A. The Nakayama automorphism is unique up to an inner automorphism. A νskew CalabiYau algebra A is CalabiYau in the sense of Ginzburg if and only if ν is an inner automorphism of A (see [30], Definition 1.1). So every CalabiYau algebra is skew CalabiYau.
2.4. Skew PBW Extensions
Skew PBW extensions or σPBW extensions were defined in 2011 by Oswaldo Lezama and Claudia Gallego in [16].
Definition 6. Let R and A be rings. We say that A is a skew PBW extension of R if the following conditions hold:
(i) R ⊆ A.
(ii) There exist elements x_{1},...,x_{n} in A such that A is a left free Rmodule, with basis,
(iii) For each 1 ≤ i ≤ n and any r ∈ R {0} there exists an element c_{i,r }∈R {0} such that
(iv) For any elements 1 ≤i, j ≤n, there exists c_{i,j }∈ R {0} such that
Proposition 1 ([16], Proposition 3). Let A be a skew PBW extension of R. Then, for every 1 ≤ i ≤n, there exist an injective ring endomorphism σ_{i} : R →R and a σ_{i}derivation δ_{i} : R →R such that
for each r ∈R.
In this case we write A := σ(R)⟨x_{1},...,x_{n}⟩.
We say that A is a bijective if σ_{i} is bijective for each 1 ≤i ≤ n and c_{i,j }is invertible for any 1 ≤i < j ≤ n (see [16], Definition 4).
3. Relations, Examples and Counterexamples
Some authors have found some interesting relations between AS regular algebras, NKoszul algebras and CalabiYau algebras. Some examples of these relations are the following:
(i) Roland Berger and Nicolas Marconnet in Proposition 5.2 of [8] show that if A = T (V)/⟨R⟩ is a connected graded algebra such that the space V of generators is concentrated in degree 1, the space R of relations lives in degrees ≥2, the global dimension d of A is 2 or 3, and that A is AS regular (the polynomial growth imposed by Artin and Schelter is often removed and in fact, it is not necessary), then A is NKoszul if d = 3, and 2Kozul if d = 2.
(ii) Roland Berger y Rachel Taillefer in Proposition 4.3 of [6] show than if A is a connected Ngraded CalabiYau algebra then A is AS regular algebra, and in Proposition 5.4 they prove that if A is AS regular Calgebra of global dimension 3 (with polynomial growth), then A is CalabiYau if and only if A is of type A in the classification of Artin and Schelter given in [2].
(iii) Let be of characteristic zero, V be an ndimensional space with n ≥ 1, w be a nonzero homogeneous potential of V of degree N + 1 with N ≥ 2, and A = A(w) be the potential algebra defined by w (so that the space of generators of A is V); Roland Berger and Andrea Solotar in Theorem 2.6 of [4] prove that if the space of relations R (i.e. the subspace of V^{⊗N }generated by the relations ∂x(w), x ∈ X)of A is ndimensional, then A is 3CalabiYau if and only if A is NKoszul of global dimension 3 and dimR_{N+1 }= 1, where R_{N+1} = (R ⊗ V) ∩ (V ⊗R) ⊆ V⊗^{(N+1)}.
(iv) Manuel Reyes, Daniel Rogalski and James Zhang in Lemma 1.2 of [37] show that if A is a connected graded algebra, then A is graded skew CalabiYau if and only if A is AS regular.
3.1. Examples
In the current literature there are not explicit relations between skew PBW extensions with AS regular algebras, NKoszul algebras or CalabiYau algebras. Next we will show some examples of algebras that are AS regular, or NKoszul, or CalabiYau, or a combination of these types, that are skew PBW extensions.
3.1.1. AS regular + NKoszul + CalabiYau
Below are some examples of algebras that are AS regular, NKoszul and CalabiYau, and in addition, they are also skew PBW extensions.

The polynomial algebra A = [x,y] is a connected graded Noetherian algebra of global dimension 2. It follows that A is ASregular with GKdim(A) = 2 (see [40], Theorem 3.5), A is 2Koszul algebra (see [8], Proposition 5.2). Moreover, A is CalabiYau of dimension 2 (see [28] ), and A isaskew PBW extension (see [16], Example 5).

Let A = [x1,...,x_{n}] be the polynomial algebra in n variables. Then A is a 2Kozsul algebra (see [31], Example 1.6), A is a skew PBW extension (see [16], Example 5), A is CalabiYau of dimension n (see [9], page 18) and therefore, AS regular (see [6], Proposition 4.3).

Let A = ⟨x,y,z⟩/⟨yz  zy,zx  xz,xy  yx + z^{2}⟩ which is of type S' in the classification of threedimensional AS regular algebras given in [2]. According to [8], A is 3CalabiYau (see [45], Example 3.6), and by Proposition 5.2 of [8] A is 2Koszul. We note that A ℝ σ([z] )⟨x,y⟩ and therefore A isaskew PBW extension.

For any n ≥ 2, let A be a nondegenerate noncommutative quadric graded algebra in n variables x_{1},...,x_{n} of degree 1. Let z be an extra variable of degree 1. Let B be an algebra defined by a nonzero cubic potential w in the variables x_{1},...,x_{n}, z. Assume that the graded algebra B is isomorphic to a skew polynomial algebra A[z;σ;δ]over A in the variable z, defined by a 0degree homogeneous automorphism σ of A and a 1degree homogeneous σderivation δ of A. Then B is 2Koszul and 3CalabiYau (see [4], Proposition 4.1). B isaskew PBW extension.
3.1.2. AS Regular + NKoszul
The following are some examples of AS regular NKoszul algebras which are skew PBW extensions. It is not clear if these algebras are CalabiYau or not, since we have no clear criteria for making claims in this regard.

The algebra A = ⟨x,y,z⟩/⟨αβxy + aαβyx, α zx + axz,yz + aβzy⟩ is AS regular of global dimension 3 of type S_{1} (see [2], Theorem 3.10). Moreover, A is 2Koszul (see [8], Proposition 5.2), and A isaskew PBW extension.
A may be or not CalabiYau, depends on the coeffcients a, α and β (see [6], Proposition 5.4). 
The quantum plane A = ⟨x,y⟩/⟨yx  cxy⟩ (c # 0) is an AS regular algebra of global dimension 2 (see [2], page 172), Moreover A is a skew PBW extension as well as 2Koszul (see [8], Proposition 5.2). For example, if c = 1 then the quantum plane A is a 2CalabiYau algebra.

The Jordan plane A = ⟨x,y⟩/⟨yx  xy  x^{2}⟩ is an AS regular algebra of global dimension 2 (see [2], page 172). Since A is a quadratic algebra and ⟨yx  xy  x^{2}⟩ is a principal ideal, it follows that A is 2Koszul (see [15], page 7), A ℝ σ([x] )⟨y⟩ and therefore A is a skew PBW extension. The Jordan plane A is not CalabiYau (see [30]).
3.1.3. Skew CalabiYau algebras
The following is an example of skew CalabiYau algebra that is skew PBW extension. Multiparameter quantum affne nspaces O_{q}(^{n}) can be obtained by iterated Ore extensions. Let n ≥1 and q be a matrix (q_{ij})_{nxn} whit entries in a field where q_{ii }= 1y q_{ij}q_{ji} = 1 for all 1 ≤i, j ≤n. Then quantum affne nspace O_{q}(^{n}) is defined to be algebra generated by x_{1}, ···, x_{n} with the relations x_{j}x_{i} = q_{ij}x_{i}x_{j} for all 1 ≤i, j ≤n. The algebra O_{q}(^{n})isskew CalabiYau whit the Nakayama automorphism ν such that ν (x_{i}) = (Π_{j=1 }q_{ji})xi (see [30], Proposition 4.1). This algebra is a skew PBW extension (see [29] ).
The Jordan plane A = ⟨x, y⟩/⟨yx xy x^{2}⟩ is skew CalabiYau, but not CalabiYau (see [30]).
3.1.4. The universal enveloping algebra and the Sridharan enveloping algebra of Lie algebra
Let be a finite dimensional Lie algebra over with basis {x_{1}, ···, x_{n}}. The universal enveloping algebra of , denoted (), is a PBW extension of since x_{i}r rx_{i} = 0, x_{i}x_{j} x_{j}x_{i} = [x_{i}, x_{j}] ∈ = + x_{1} + ···+ x_{n}, r_{i} ∈, for 1 ≤i, j ≤n. JiWei He, Fred Van Oystaeyen and Yinhuo Zhang showed that for the 3dimensional Lie algebra with basis {x, y, z}, () is a CalabiYau algebra if and only if the Lie bracket is given by [x, y] = ax + by + wz, [x, z] = cx + vy bz,[y, z] = ux cy + az, where a, b, c, u, v, w ∈; and if is a finite dimensional Lie algebra, () is CalabiYau of dimension 3 if and only if is isomorphic to one of the following Lie algebras (see [22], Proposition 4.5 and Proposition 4.6 ):
(i) The 3dimensional simple Lie algebra sl(2, );
(ii) has a basis {x, y, z}such that [x, y] = y, [x, z] = z and [y, z] = 0;
(iii) The Heisenberg algebra, that is; has a basis {x, y, z} such that [x, y] = z and [x, z] = [y, z] = 0;
(vi) The 3dimensional abelian Lie algebra.
We note that if is a finite dimensional Lie algebra over a field and () is the universal enveloping algebra of , then () is a skew PBW extension (see [16] ); in particular, universal enveloping CalabiYau algebra () of dimension 3 is a skew PBW extension.
Let be a finite dimensional Lie algebra, and let f ∈Z^{2}(, ) be an arbitrary 2cocycle, that is, f : ×→ such that f (x, x) = 0 and
for all x, y, z ∈.
The Sridharan enveloping algebra of is defined to be the associative algebra _{f} () = T ()/I, where I is the twoside ideal of T () generated by the elements
For x ∈, we still denote by x its image in _{f}(). _{f}() is a filtered algebra with the associated graded algebra gr(_{f}()) being a polynomial algebra.
Let be a field and algebraically closed whit characteristic zero. If is a Lie Kalgebra of dimension three then, the Sridharan enveloping algebra _{f}(), for f ∈Z^{2}(, ), is isomorphic to one of ten following associative algebras, defined by three generator x, y, z and the following commutation relations (see [32], Theorem 1.3):
where α ∈ {0}. Therefore the Sridharan enveloping algebra _{f}()isaskew PBW extension.
Let be a finite dimensional Lie algebra. Then for any 2cocycle f ∈Z^{2}(, ), the following statements are equivalent (see [22], Theorem 5.3).
(i) The Sridharan enveloping algebra _{f}() is CalabiYau of dimension d.
(ii) The universal enveloping algebra () is CalabiYau of dimension d.
Let _{f}() be a Sridharan enveloping algebra of a finite dimensional Lie algebra . Then _{f}() is CalabiYau of dimension 3 if and only if _{f}() is isomorphic to ⟨x,y,z⟩/⟨R⟩ with the commuting relations R listed in the following table (see [22], Theorem 5.5):
where {x,y}= xy yx.
From the above discussion we have the following result.
Proposition 2. Let _{f}() be a Sridharan enveloping algebra of a finite dimensional Lie algebra . If _{f}() is CalabiYau of dimension 3 then () is a skew PBW extension.
The Sridharan enveloping algebra of an ndimensional abelian Lie algebra is nCalabiYau; in particular the Weyl algebra A_{n} is 2nCalabiYau (see [9], Theorem 6.5) as well as a skew PBW extension (see [16], Example 5).
3.2. Counterexamples
Next we will show some examples of algebras that are AS regular, or NKoszul, or CalabiYau, but are not skew PBW extensions.

A = ⟨x,y,z⟩/⟨xyyx z^{2 },yz z x^{2 },zx xz  y^{2}⟩ is AS regular of global dimension 3 of type A (see [2], page 173). A is 2Koszul (see [8], Proposition 5.2) and CalabiYau of dimension 3 (see [6], Proposition 5.4).

A = ⟨x,y⟩/⟨x+ xy+ yx + yxy,xy + yx+ xyx + y^{3}⟩ is AS regular of global dimension 3 of type A (see [2], Theorem 3.10), A is 3Koszul (see [8], Proposition 5.2) and CalabiYau of dimension 3 (see [6], Proposition 5.4).

A = ⟨x,y⟩/⟨yx⟩ is not AS regular algebra. A is the only graded algebra of global dimension 2 and GKdimension 2 which is not Noetherian (see [2], page 172). A is 2Koszul (see [15], page 7), A is not 2CalabiYau (see [6], Proposition 4.3)

The exterior algebra A = K⟨x_{1},···,x_{n}⟩/⟨x_{k}^{2},x_{i}x_{j} + x_{j}x_{i}⟩k,i<j in n variables is an 2Koszul algebra (see [31], Example 1.6).

If A = ⟨x_{1},···,x_{n}⟩/I is an quadratic algebra and I is principal, then A is 2Koszul (see [15], page 7). It depends on the ideal I whether A is CalabiYau or not.

Consider V of dimension 1, V = x and w = x^{N+1}. Then, dimR = dimR_{N+1} = 1, A(w)is NKoszul (since the global dimension of A(w)is infinite, and A(w) is not 3CalabiYau (see [4], Example 2.12).
4. Some Properties Preserved by Ore Extensions
Suppose σ : A → A is a graded algebra automorphism and δ : A(1) → A is a graded σderivation. If B := A[z;σ,δ] is the associated Ore extension, then B isaskew PBW extension. In this case we have B = A[z,σ; δ] = σ(A)⟨x⟩ (see [16], Example 5).
Below we list some properties that are preserved by Ore extensions:

If A is a connected graded algebra then B is a connected graded algebra.

If A is homologically smooth, then so is B (see [30], Proposition 3.1).

B is 2Koszul if and only if A is 2Koszul (see [33], Corollary 1.3).

Let A = ⟨x_{1},...,x_{n}⟩/⟨f⟩ where f = (x_{1},..., x_{n})M(x_{1},···,x_{n})^{t }and M is an n × n matrix. Then A is CalabiYau of dimension 2 if and only if M is invertible and antisymmetric (see [24], Corollary 1).
Let δ be a graded derivation of the free algebra ⟨x_{1},...,x_{n}⟩ of degree 1. If δ( f ) = 0, then δ induces a graded derivation on A. Let B = A[z; ] be the Ore extension of A defined by the graded derivation . Then B is a graded CalabiYau algebra of dimension 3 (see [21], Proposition 1.3). 
If A is ν skew CalabiYau projective algebra of dimension d, then B is skew CalabiYau of dimension d + 1 and the Nakayama automorphism ν' of B satisfies that ν' = σ^{1}ν and A ν' (z) = uz+ b, with u,b ∈ A and u invertible (see [30], Theorem 3.3).

Let A be a 2Koszul AS regular algebra of global dimension d with the Nakayama automorphism ξ. Then B = A[z,ξ] is a CalabiYau algebra of dimension d + 1 (see [25], Theorem 3.3).

Let A be a νskew CalabiYau algebra of dimension d and σ ∈ Aut(A), then A[x;σ] and A[x^{±1};σ] are CalabiYau algebras of dimension d + 1 (see [18], Theorema 1.1). Furthermore, if A[x; σ] is CalabiYau, then A[x^{±1}; σ] is CalabiYau.

Now we present an example of skew CalabiYau algebra that is not CalabiYau (see [30] ), and then, we consider the corresponding Ore extension. Let A = ⟨x,y⟩/⟨yx  xy x^{2}⟩ be the Jordan plane, A is AS regular algebra of dimension 2 and therefore A is 2Koszul, A = [x] [y,δ1] with δ_{1}(x) = x^{2}. It follows that A is skew CalabiYau but not CalabiYau. A has Nakayama automorphism given by ν (x) = x and ν (y) = 2x + y, B = A[z;ν] is an Ore extension of Jordan plane. Then B is skew CalabiYau with the Nakayama automorphism ν' such that ν' (x) = x and ν' (y) = y. B = [x,z] [y; δ] where δ is given by δ(x) = x^{2 }and δ(z) = 2xz. So, ν' (z) = z. It follows that B is CalabiYau, which was already proved by Berger and Pichereau in [5].

In [44], AS regular algebras of dimension 5 generated by two generators of degree 1 with three generating relations of degree 4 are classified under some generic condition. There are nine types such AS regular algebras in this classification list. Among them, the algebras D and G are given by iterated Ore extensions (see [44], Section 5.2).
The algebra D is skew CalabiYau with the Nakayama automorphism ν given by ν(x) = p^{3}q^{4}x; ν (y) = p^{3}q^{4}y. D is CalabiYau if and only if that p,q satisfy the system of equations (see [30], Theorem 4.3)
The algebra G is skew CalabiYau with the Nakayama automorphism ν given by ν (x) = gx; ν (y) = g^{1}y. D is CalabiYau if and only if g = 1.
They study and classification of AS regular algebras of dimension five with two generators under an additional Z^{2}grading uses Gröbner basis computations (see [48]).

Let be a field, let n be an even natural number ≥ 2, and let A be the associative algebra defined by generators x_{1},...,x_{n} subject to the single relation
where the bracket stands for the commutator, ν is a linear combination of the x_{i}'s, and λ ∈ . Then the filtered algebra A is 2Koszul. Furthermore A is 2CalabiYau if and only if ν = 0 (see [9], Theorem 6.4). So, if σ_{2} = i[x_{1}] and δ_{2}([x_{1}] ) ⊆ , then the skew PBW extension σ()⟨x_{1},x_{2}⟩ ℝ [x_{1}] [x_{2}; σ_{2},δ_{2}]is 2CalabiYau.
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