# Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

^{2}, Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Bogotá 111321, Colombia

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Semi-Graded Rings and Modules

**Definition**

**1.**

- (i)
- $B={\u2a01}_{n\ge 0}{B}_{n}$.
- (ii)
- For every $m,n\ge 0$, ${B}_{m}{B}_{n}\subseteq {B}_{0}\oplus \cdots \oplus {B}_{m+n}$.
- (iii)
- $1\in {B}_{0}$.

**Definition**

**2.**

- (i)
- $M={\u2a01}_{n\in \mathbb{Z}}{M}_{n}$.
- (ii)
- For every $m\ge 0$ and $n\in \mathbb{Z}$, ${B}_{m}{M}_{n}\subseteq {\u2a01}_{k\le m+n}{M}_{k}$.

**Definition**

**3.**

**Definition**

**4.**

- (i)
- B is $SG$.
- (ii)
- There exist finitely many elements ${x}_{1},\cdots ,{x}_{n}\in B$ such that the subring generated by ${B}_{0}$ and ${x}_{1},\cdots ,{x}_{n}$ coincides with B.
- (iii)
- For every $n\ge 0$, ${B}_{n}$ is a free ${B}_{0}$-module of finite dimension.

**Proposition**

**1.**

- (i)
- ${B}_{0}$ is a subring of B. Moreover, for any $n\ge 0$, ${B}_{0}\oplus \cdots \oplus {B}_{n}$ is a ${B}_{0}-{B}_{0}$-bimodule, as well as B.
- (ii)
- B has a standard $\mathbb{N}$-filtration given by:$${F}_{n}\left(B\right):={B}_{0}\oplus \cdots \oplus {B}_{n}.$$
- (iii)
- The associated graded ring $Gr\left(B\right)$ satisfies:$$Gr{\left(B\right)}_{n}\cong {B}_{n},foreveryn\ge 0(isomorphismofabeliangroups).$$
- (iv)
- Let $M={\u2a01}_{n\in \mathbb{Z}}{M}_{n}$ be a semi-graded B-module and N a submodule of M. The following conditions are equivalent:
- (a)
- N is semi-graded.
- (b)
- For every $z\in N$, the homogeneous components of z are in N.
- (c)
- $M/N$ is semi-graded with semi-graduation given by:$${(M/N)}_{n}:=({M}_{n}+N)/N,n\in \mathbb{Z}.$$

**Remark**

**1.**

**Definition**

**5.**

**Remark**

**2.**

#### 1.2. Skew $PBW$ Extensions

**Definition**

**6**

**(**[11]

**).**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:

- (i)
- $R\subseteq A$.
- (ii)
- There exist finitely many elements ${x}_{1},\cdots ,{x}_{n}\in A$ such A is a left R-free module with basis:$$\mathrm{Mon}\left(A\right):=\{{x}^{\alpha}={x}_{1}^{{\alpha}_{1}}\cdots {x}_{n}^{{\alpha}_{n}}\mid \alpha =({\alpha}_{1},\cdots ,{\alpha}_{n})\in {\mathbb{N}}^{n}\},with\mathbb{N}:=\{0,1,2,\cdots \}.$$The set $\mathrm{Mon}\left(A\right)$ is called the set of standard monomials of A.
- (iii)
- For every $1\le i\le n$ and $r\in R-\left\{0\right\}$, there exists ${c}_{i,r}\in R-\left\{0\right\}$ such that:$${x}_{i}r-{c}_{i,r}{x}_{i}\in R.$$
- (iv)
- For every $1\le i,j\le n$, there exists ${c}_{i,j}\in R-\left\{0\right\}$ such that:$${x}_{j}{x}_{i}-{c}_{i,j}{x}_{i}{x}_{j}\in R+R{x}_{1}+\cdots +R{x}_{n}.$$

**Example**

**1.**

**H**${}_{n}\left(q\right)$, Hayashi algebra ${W}_{q}\left(J\right)$, differential operators on a quantum space ${D}_{\mathit{q}}\left({S}_{\mathit{q}}\right)$, Witten’s deformation of $\mathcal{U}\left(\mathfrak{sl}\right(2,K\left)\right)$, multi-parameter Weyl algebra ${A}_{n}^{Q,\Gamma}\left(K\right)$, quantum symplectic space ${\mathcal{O}}_{q}\left(\mathfrak{sp}\left({K}^{2n}\right)\right)$, some quadratic algebras in 3 variables, some 3-dimensional skew polynomial algebras, particular types of Sklyanin algebras, homogenized enveloping algebra $\mathcal{A}\left(\mathcal{G}\right)$, and Sridharan enveloping algebra of 3-dimensional Lie algebra $\mathcal{G}$, among many others. For a precise definition of any of these rings and algebras, see [6,12,13,14,15].

**Proposition**

**2**

**(**[11]

**).**Let A be a skew $PBW$ extension of R. Then, for every $1\le i\le n$, there exists an injective ring endomorphism ${\sigma}_{i}:R\to R$ and a ${\sigma}_{i}$-derivation ${\delta}_{i}:R\to R$ such that:

**Definition**

**7**

- (a)
- A is quasi-commutative if the conditions (iii) and (iv) in Definition 6 are replaced by:
- (iii’)
- For every $1\le i\le n$ and $r\in R-\left\{0\right\}$, there exists ${c}_{i,r}\in R-\left\{0\right\}$ such that:$${x}_{i}r={c}_{i,r}{x}_{i}.$$
- (iv’)
- For every $1\le i,j\le n$, there exists ${c}_{i,j}\in R-\left\{0\right\}$ such that:$${x}_{j}{x}_{i}={c}_{i,j}{x}_{i}{x}_{j}.$$

- (b)
- A is bijective if ${\sigma}_{i}$ is bijective for every $1\le i\le n$, and ${c}_{i,j}$ is invertible for any $1\le i,j\le n$.
- (c)
- A is constant if the condition (ii) in Definition 6 is replaced by: For every $1\le i\le n$ and $r\in R$,$${x}_{i}r=r{x}_{i}.$$
- (d)
- A is pre-commutative if the condition (iv) in Definition 6 is replaced by: For any $1\le i,j\le n$ there exists ${c}_{i,j}\in R\phantom{\rule{4pt}{0ex}}\backslash \phantom{\rule{4pt}{0ex}}\left\{0\right\}$ such that:$${x}_{j}{x}_{i}-{c}_{i,j}{x}_{i}{x}_{j}\in R{x}_{1}+\cdots +R{x}_{n}.$$
- (e)
- A is called semi-commutative if A is quasi-commutative and constant.

**Remark**

**3.**

**Definition**

**8.**

**Definition**

**9.**

- (i)
- For $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{n})\in {\mathbb{N}}^{n}$, $\left|\alpha \right|:={\alpha}_{1}+\cdots +{\alpha}_{n}$.
- (ii)
- For $X={x}^{\alpha}\in Mon\left(A\right)$, $exp\left(X\right):=\alpha $ and $deg\left(X\right):=\left|\alpha \right|$.
- (iii)
- Let $0\ne f\in A$, and $t\left(f\right)$ is the finite set of terms that conform f, i.e., if $f={c}_{1}{X}_{1}+\cdots +{c}_{t}{X}_{t}$, with ${X}_{i}\in Mon\left(A\right)$ and ${c}_{i}\in R-\left\{0\right\}$, then $t\left(f\right):=\{{c}_{1}{X}_{1},\cdots ,{c}_{t}{X}_{t}\}$.
- (iv)
- Let f be as in(iii), then $deg\left(f\right):=max{\{deg\left({X}_{i}\right)\}}_{i=1}^{t}.$

**Theorem**

**1**

**(**[12]

**).**Let A be an arbitrary skew $PBW$ extension of the ring R. Then, A is a $\mathbb{N}$-filtered ring with filtration given by:

**Proposition**

**3**

**Theorem**

**2**

**(**[8]

**).**Let $A=\sigma \left(R\right)\langle {x}_{1},\cdots ,{x}_{n}\rangle $ be an arbitrary skew $PBW$ extension. Then:

**Remark**

**4.**

## 2. Finitely Semi-Graded Algebras

#### 2.1. Definition

**Definition**

**10.**

- (i)
- B is an $FSG$ ring with semi-graduation $B={\u2a01}_{p\ge 0}{B}_{p}$.
- (ii)
- For every $p,q\ge 1$, ${B}_{p}{B}_{q}\subseteq {B}_{1}\oplus \cdots \oplus {B}_{p+q}$.
- (iii)
- B is connected, i.e., ${B}_{0}=K$.
- (iv)
- B is generated in degree 1.

**Remark**

**5.**

**Proposition**

**4.**

**Proof.**

#### 2.2. Examples of $FSG$ Algebras

**Example**

**2**

**.**Note that a skew $PBW$ extension of the field K is an $FSG$ algebra if and only if it is constant and pre-commutative. Thus, we have:

**Example**

**3**

**.**The following algebras are $FSG$ but not skew $PBW$ extensions of the base field K (however, in every example below, the algebra is a skew $PBW$ extension of some other subring):

**Example**

**4**

**.**The following $FSG$ algebras are not skew $PBW$ extensions:

## 3. Some Invariants Associated to $\mathit{FSG}$ Algebras

#### 3.1. The Hilbert Series

**Theorem**

**3**

**(**[10]

**).**Let A and B be connected graded algebras finitely generated in degree 1. Then, $A\cong B$ as K-algebras if and only if $A\cong B$ as graded algebras.

**Corollary**

**1**

**(**[10]

**).**Let A be a connected graded algebra finitely generated in degree 1. If A has two graduations $A={\u2a01}_{n\ge 0}{A}_{n}={\u2a01}_{n\ge 0}{B}_{n}$, then there exists an algebra automorphism $\varphi :A\to A$ such that $\varphi \left({A}_{n}\right)={B}_{n}$ for every $n\ge 0$. In particular, ${dim}_{K}{A}_{n}={dim}_{K}{B}_{n}$ for every $n\ge 0$, and the Hilbert series of A is well-defined. Moreover, if $Aut\left(A\right)=Au{t}_{\mathbf{gr}}\left(A\right)$, then ${A}_{n}={B}_{n}$ for every $n\ge 0$.

**Proposition**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**5.**

#### 3.2. The Yoneda Algebra

#### 3.3. The Poincaré Series

**Theorem**

**5.**

**Proof.**

**Corollary**

**4.**

**Proof.**

## 4. Koszulity

#### 4.1. Semi-Graded Koszul Algebras

**Proposition**

**6**

**(**[30]

**).**If L is a semidirect product of the family ${\left\{{L}_{\omega}\right\}}_{\omega \in \Omega}$, then L is distributive if and only if for all $\omega \in \Omega $, ${L}_{\omega}$ is distributive.

**Proposition**

**7**

**(**[26]

**).**Let V be a vector space and ${X}_{1},\cdots ,{X}_{n}\subseteq V$ be a finite collection of subspaces of V. The following conditions are equivalent:

- (i)
- The collection ${X}_{1},\cdots ,{X}_{n}$ is distributive.
- (ii)
- There exists a basis $\mathcal{B}:={\left\{{\omega}_{i}\right\}}_{i\in \mathcal{C}}$ of V such that each of the subspaces ${X}_{i}$ is the linear span of a set of vectors ${\omega}_{i}$.
- (iii)
- There exists a basis $\mathcal{B}$ of V such that $\mathcal{B}\cap {X}_{i}$ is a basis of ${X}_{i}$ for every $1\le i\le n$.

**Definition**

**11.**

**Theorem**

**6.**

**Proof.**

**Definition**

**12.**

- (i)
- B is finitely presented with $I=\langle {b}_{1},\cdots ,{b}_{m}\rangle $ and ${b}_{i}\in {F}_{\ge 1}$ for $1\le i\le m$.
- (ii)
- $L\left(B\right)$ is distributive.

**Remark**

**6.**

**Theorem**

**7.**

**Proof.**

**Example**

**6.**

**Example**

**7.**

- ${A}_{1}{I}_{2}$ is K-generated by $D=\{{x}^{3}-xyx,{x}^{2}y-x{y}^{2},y{x}^{2},yxy\}$, and D is K-linearly independent; therefore, ${dim}_{K}\left({A}_{1}{I}_{2}\right)=4$.
- ${I}_{2}{A}_{1}$ is K-generated by $C=\{{x}^{3}-{x}^{2}y,y{x}^{2}-yxy,xyx,{y}^{2}x\}$, which is K-linearly independent; therefore, ${dim}_{K}\left({I}_{2}{A}_{1}\right)=4$.

#### 4.2. Poincaré Series of Skew $PBW$ Extensions

**Theorem**

**8.**

**Proof.**

**Example**

**8.**

## 5. Point Modules and the Point Functor for $\mathbf{FSG}$ Rings

**Definition**

**13.**

- (i)
- A point module for B is a finitely $\mathbb{N}$-semi-graded B-module $M={\u2a01}_{n\in \mathbb{N}}{M}_{n}$ such that M is cyclic, generated in degree 0, i.e., there exists an element ${m}_{0}\in {M}_{0}$ such that $M=B{m}_{0}$, and ${dim}_{{B}_{0}}\left({M}_{n}\right)=1$ for all $n\ge 0$.
- (ii)
- Two point modules M and ${M}^{\prime}$ for B are isomorphic if there exists a homogeneous B-isomorphism between them.
- (iii)
- $P\left(B\right)$ is the collection of isomorphism classes of point modules for B.

**Theorem**

**9.**

**Proof.**

**Definition**

**14.**

**Remark**

**7.**

**Theorem**

**10.**

**Proof.**

**Definition**

**15.**

**Theorem**

**11.**

**Proof.**

**Remark**

**8.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Skew PBW Extension | C | B | P | QC | SC |
---|---|---|---|---|---|

Classical polynomial ring | ✓ | ✓ | ✓ | ✓ | ✓ |

Ore extensions of bijective type | ★ | ✓ | ✓ | ★ | ★ |

Weyl algebra | ★ | ✓ | ✓ | ★ | ★ |

Particular Sklyanin algebra | ✓ | ✓ | ✓ | ✓ | ✓ |

Universal enveloping algebra of a Lie algebra | ✓ | ✓ | ✓ | ★ | ★ |

Homogenized enveloping algebra $\mathcal{A}\left(\mathcal{G}\right)$ | ✓ | ✓ | ✓ | ★ | ★ |

Tensor product | ✓ | ✓ | ✓ | ★ | ★ |

Crossed product | ★ | ✓ | ★ | ★ | ★ |

Algebra of q-differential operators | ★ | ✓ | ✓ | ★ | ★ |

Algebra of shift operators | ★ | ✓ | ✓ | ✓ | ★ |

Mixed algebra | ★ | ✓ | ★ | ★ | ★ |

Algebra of discrete linear systems | ★ | ✓ | ✓ | ✓ | ★ |

Linear partial differential operators | ★ | ✓ | ✓ | ★ | ★ |

Linear partial shift operators | ★ | ✓ | ✓ | ✓ | ★ |

Algebra of linear partial difference operators | ★ | ✓ | ✓ | ★ | ★ |

Algebra of linear partial q-dilation operators | ★ | ✓ | ✓ | ✓ | ★ |

Algebra of linear partial q-differential operators | ★ | ✓ | ✓ | ★ | ★ |

Algebras of diffusion type | ✓ | ✓ | ✓ | ★ | ★ |

Additive analogue of the Weyl algebra | ✓ | ✓ | ★ | ★ | ★ |

Multiplicative analogue of the Weyl algebra | ✓ | ✓ | ✓ | ✓ | ✓ |

Quantum algebra ${\mathcal{U}}^{\prime}\left(\mathfrak{so}(3,K)\right)$ | ✓ | ✓ | ✓ | ★ | ★ |

Dispin algebra | ✓ | ✓ | ✓ | ★ | ★ |

Woronowicz algebra | ✓ | ✓ | ✓ | ★ | ★ |

Complex algebra | ★ | ✓ | ★ | ★ | ★ |

Algebra $\mathbf{U}$ | ★ | ✓ | ★ | ★ | ★ |

Manin algebra | ★ | ✓ | ✓ | ★ | ★ |

q-Heisenberg algebra | ✓ | ✓ | ✓ | ★ | ★ |

Quantum enveloping algebra of $\mathfrak{sl}(2,\mathbb{K})$ | ★ | ✓ | ★ | ★ | ★ |

Hayashi’s algebra | ★ | ✓ | ★ | ★ | ★ |

The algebra of differential operators on a quantum space ${S}_{q}$ | ★ | ✓ | ★ | ★ | ★ |

Witten’s deformation of $\mathcal{U}\left(\mathfrak{sl}\right(2,\mathbb{K}\left)\right)$ | ★ | ✓ | ★ | ★ | ★ |

Quantum Weyl algebra of Maltsiniotis | ★ | ✓ | ★ | ★ | ★ |

Quantum Weyl algebra | ★ | ✓ | ★ | ★ | ★ |

Multi-parameter quantized Weyl algebra | ★ | ✓ | ★ | ★ | ★ |

Quantum symplectic space | ★ | ✓ | ★ | ★ | ★ |

Quadratic algebras in 3 variables | ★ | ✓ | ★ | ★ | ★ |

Cardinal | 3-Dimensional Skew Polynomial Algebras | C | B | P | QC | SC |
---|---|---|---|---|---|---|

$\left|\right\{\alpha ,\beta ,\gamma \left\}\right|=3$ | $yz-\alpha zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-\gamma yx=0$ | ✓ | ✓ | ✓ | ✓ | ✓ |

$yz-zy=z$, $zx-\beta xz=y$, $xy-yx=x$ | ✓ | ✓ | ✓ | ★ | ★ | |

$yz-zy=z,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=b,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=x$ | ✓ | ✓ | ★ | ★ | ★ | |

$\left|\right\{\alpha ,\beta ,\gamma \left\}\right|=2$, | $yz-zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=y,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=0$ | ✓ | ✓ | ✓ | ★ | ★ |

$\beta \ne \alpha =\gamma =1$ | $yz-zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=b,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=0$ | ✓ | ✓ | ★ | ★ | ★ |

$yz-zy=az,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=x$ | ✓ | ✓ | ✓ | ★ | ★ | |

$yz-zy=z,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=0$ | ✓ | ✓ | ✓ | ★ | ★ | |

$\left|\right\{\alpha ,\beta ,\gamma \left\}\right|=2,$ | $yz-\alpha zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=y+b,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-\alpha yx=0$ | ✓ | ✓ | ★ | ★ | ★ |

$\beta \ne \alpha =\gamma \ne 1$ | $yz-\alpha zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\beta xz=b,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-\alpha yx=0$ | ✓ | ✓ | ★ | ★ | ★ |

$\alpha =\beta =\gamma \ne 1$ | $yz-\alpha zy={a}_{1}x+{b}_{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-\alpha xz={a}_{2}y+{b}_{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-\alpha yx={a}_{3}z+{b}_{3}$ | ✓ | ✓ | ★ | ★ | ★ |

$yz-zy=x,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-xz=y,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=z$ | ✓ | ✓ | ✓ | ★ | ★ | |

$yz-zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-xz=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=z$ | ✓ | ✓ | ✓ | ★ | ★ | |

$\alpha =\beta =\gamma =1$ | $yz-zy=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-xz=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=b$ | ✓ | ✓ | ★ | ★ | ★ |

$yz-zy=-y,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-xz=x+y,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=0$ | ✓ | ✓ | ✓ | ★ | ★ | |

$yz-zy=az,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}zx-xz=x,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}xy-yx=0$ | ✓ | ✓ | ✓ | ★ | ★ |

Sridharan Enveloping Algebra of 3-Dimensional Lie Algebra $\mathcal{G}$ | ||||||||
---|---|---|---|---|---|---|---|---|

Type | $[\mathit{x},\mathit{y}]$ | $[\mathit{y},\mathit{z}]$ | $[\mathit{z},\mathit{x}]$ | C | B | P | QC | SC |

1 | 0 | 0 | 0 | ✓ | ✓ | ✓ | ✓ | ✓ |

2 | 0 | x | 0 | ✓ | ✓ | ✓ | ★ | ★ |

3 | x | 0 | 0 | ✓ | ✓ | ✓ | ★ | ★ |

4 | 0 | $\alpha y$ | $-x$ | ✓ | ✓ | ✓ | ★ | ★ |

5 | 0 | y | $-(x+y)$ | ✓ | ✓ | ✓ | ★ | ★ |

6 | z | $-2y$ | $-2x$ | ✓ | ✓ | ✓ | ★ | ★ |

7 | 1 | 0 | 0 | ✓ | ✓ | ★ | ★ | ★ |

8 | 1 | x | 0 | ✓ | ✓ | ★ | ★ | ★ |

9 | x | 1 | 0 | ✓ | ✓ | ★ | ★ | ★ |

10 | 1 | y | x | ✓ | ✓ | ★ | ★ | ★ |

$\mathcal{SK}$ Algebra | ${\mathit{P}}_{\mathit{A}}\left(\mathit{t}\right)$ |
---|---|

Classical polynomial algebra $K[{x}_{1},\cdots ,{x}_{n}]$ | ${(1+t)}^{n}$ |

Some Sridharan enveloping algebras of 3-dimensional Lie algebras | ${(1+t)}^{3}$ |

Particular Sklyanin algebra | ${(1+t)}^{3}$ |

L. Partial q-dilation operators $K[{t}_{1},\cdots ,{t}_{n}][{H}_{1}^{\left(q\right)},\cdots ,{H}_{m}^{\left(q\right)}]$ | ${(1+t)}^{n+m}$ |

Multiplicative analogue of the Weyl algebra ${\mathcal{O}}_{n}\left({\lambda}_{ji}\right)$ | ${(1+t)}^{n}$ |

Some 3-dimensional skew polynomial algebras | ${(1+t)}^{3}$ |

Multi-parameter quantum affine n-space | ${(1+t)}^{n}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lezama, O.; Gomez, J.
Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras. *Symmetry* **2019**, *11*, 881.
https://doi.org/10.3390/sym11070881

**AMA Style**

Lezama O, Gomez J.
Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras. *Symmetry*. 2019; 11(7):881.
https://doi.org/10.3390/sym11070881

**Chicago/Turabian Style**

Lezama, Oswaldo, and Jaime Gomez.
2019. "Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras" *Symmetry* 11, no. 7: 881.
https://doi.org/10.3390/sym11070881