Graded Monomial Ordering for $\mathbb{N}$-graded and $\mathbb{N}$-filtered Solvable Polynomial Algebras of $({\cal B},d(~))$-type

TL;DR

This paper establishes the existence of graded monomial orderings for certain $ $-graded and $ $-filtered solvable polynomial algebras of $({ mf B},d(~))$-type, linking algebraic structure with ordering properties.

Contribution

It proves that $ $-graded algebras of $({ mf B},d(~))$-type have graded monomial orderings and characterizes $ $-filtered algebras of this type via graded monomial orderings.

Findings

Existence of graded monomial orderings for $ $-graded algebras.

Characterization of $ $-filtered algebras via monomial orderings.

Connection between algebraic structure and ordering in solvable polynomial algebras.

Abstract

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.

Equations

$$A_p=K-span\{a^{\alpha }\in \mathcal{B}|d(a^{\alpha })=p\}$$

$$A_p{A}_q\subseteq A_{p+q},p,q\in \mathbb{N},$$

$$\begin{array}[]{l}a_{j}a_{i}\neq 0,~{}a_{j}a_{i}=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}~{}\hbox{implies}~{}d(a^{\alpha(s)})=m_{j}+m_{i},\\ \hbox{where}~{}\lambda_{s}\in K^{*},~{}a^{\alpha(s)}\in\mathcal{B},~{}1\leq i<j\leq n,\end{array}$$

$$F_{p}A=K-span\{a^{\alpha }\in \mathcal{B}|d(a^{\alpha })\le p\}$$

$$F_{p}A{F}_{q}A\subseteq F_{p+q}A,p,q\in \mathbb{N},$$

$$\begin{array}[]{l}a_{j}a_{i}\neq 0~{}\hbox{and}~{}a_{j}a_{i}=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}~{}\hbox{implies}~{}d(a^{\alpha(s)})\leq m_{j}+m_{i},\\ \hbox{where}~{}\lambda_{s}\in K^{*},~{}a^{\alpha(s)}\in\mathcal{B},~{}1\leq i<j\leq n.\end{array}$$

$$\begin{array}[]{l}a_{1}a_{2}=a_{2}a_{1},\\ a_{3}a_{1}=\lambda a_{1}a_{3}-\mu a_{2}^{2}a_{3}-f(a_{2}),\\ a_{3}a_{2}=a_{2}a_{3},\end{array}$$

$$d(f)=\left\{\begin{array}[]{ll}0,&\hbox{if}~{}f\in F_{0}A=K,\\ p,&\hbox{if}~{}f\in F_{p}A-F_{p-1}A~{}\hbox{for some}~{}p>0.\end{array}\right.$$

$$\begin{array}[]{cccc}\varphi:&\widetilde{A}&\smash{\mathop{\longrightarrow}\limits}&G(A)=\displaystyle\bigoplus_{p\in\mathbb{N}}F_{p}A/F_{p-1}A\\ &H&\mapsto&f+F_{p-1}A\end{array}$$

$$\begin{array}[]{cccc}\psi:&\widetilde{A}&\smash{\mathop{\longrightarrow}\limits}&A\\ &H&\mapsto&f\end{array}$$

$$\begin{array}[]{l}d(a^{\alpha})=d_{\rm gr}(\sigma(a^{\alpha}))=d_{\rm gr}(\widetilde{a^{\alpha}}),\\ \sigma(a^{\alpha})=\sigma(a_{1})^{\alpha_{1}}\cdots\sigma(a_{n})^{\alpha_{n}}=\sigma(a)^{\alpha},~{}\widetilde{a^{\alpha}}=\widetilde{a_{1}}^{\alpha_{1}}\cdots\widetilde{a_{n}}^{\alpha_{n}}=\widetilde{a}^{\alpha}.\end{array}$$

$${\mathbf{LH}}_d(f)=\sum _{d(a^{\alpha (s^{\prime })})=p}{\lambda }_{s^{\prime }}{a}^{\alpha (s^{\prime })},1\le s^{\prime }\le t,$$

$$a^{\alpha}\prec_{gr}a^{\beta}\Leftrightarrow\left\{\begin{array}[]{l}d(a^{\alpha})<d(a^{\beta}),\\ \hbox{or}\\ d(a^{\alpha})=d(a^{\beta})~{}\hbox{and}~{}a^{\alpha}\prec a^{\beta}.\end{array}\right.$$

$$\begin{array}[]{rcl}a^{\alpha}a^{\beta}&=&\lambda_{\alpha,\beta}a^{\alpha+\beta}+f_{\alpha,\beta},\\ &}{}{\hfil&\hbox{where}~{}\lambda_{\alpha,\beta}\in K^{*},~{}a^{\alpha+\beta}=a_{1}^{\alpha_{1}+\beta_{1}}\cdots a_{n}^{\alpha_{n}+\beta_{n}}~{}\hbox{and}~{}f_{\alpha,\beta}\in K\hbox{-span}{\cal B}\\ &}{}{\hfil&\hbox{such that either}~{}f_{\alpha,\beta}=0~{}\hbox{or}~{}{\bf LM}(f_{\alpha,\beta})\prec a^{\alpha+\beta},\end{array}$$

$$a^{\alpha }\prec a^{\beta }impliesd(a^{\alpha })<d(a^{\beta }),$$

$${\cal G}=\left\{g_{ji}=X_{j}X_{i}-\lambda_{ji}X_{i}X_{j}-F_{ji}~{}\left|~{}\begin{array}[]{l}{\bf LM}(g_{ji})=X_{j}X_{i},\\ 1\leq i<j\leq n\end{array}\right.\right\}$$

$$\begin{array}[]{rcl}a_{j}a_{i}&=&\lambda_{ji}a_{i}a_{j}+f_{ji}\\ &}{}{\hfil&\hbox{where}~{}\lambda_{ji}\in K^{*},~{}f_{ji}=\sum\mu_{k}a^{\alpha(k)}\in K\hbox{-span}{\cal B}\\ &}{}{\hfil&\hbox{with}~{}{\bf LM}(f_{ji})\prec a_{i}a_{j}~{}\hbox{if}~{}f_{ji}\neq 0,~{}1\leq j<i\leq n.\end{array}$$

$$a^{\alpha}\prec_{gr}a^{\beta}\Leftrightarrow\left\{\begin{array}[]{l}d(a^{\alpha})<d(a^{\beta})\\ \hbox{or}\\ d(a^{\alpha})=d(a^{\beta})~{}\hbox{and}~{}a^{\alpha}\prec a^{\beta},\end{array}\right.$$

$$\begin{array}[]{rcl}a_{j}a_{i}&=&\lambda_{ji}a_{i}a_{j}+f_{ji}\\ &}{}{\hfil&\hbox{where}~{}\lambda_{ji}\in K^{*},~{}f_{ji}=\sum\mu_{k}a^{\alpha(k)}\in K\hbox{-span}{\cal B}\\ &}{}{\hfil&\hbox{with}~{}{\bf LM}(f_{ji})\prec a_{i}a_{j}~{}\hbox{if}~{}f_{ji}\neq 0,~{}1\leq j<i\leq n.\end{array}$$

$$a^{\alpha}\prec_{gr}a^{\beta}\Leftrightarrow\left\{\begin{array}[]{l}d(a^{\alpha})<d(a^{\beta})\\ \hbox{or}\\ d(a^{\alpha})=d(a^{\beta})~{}\hbox{and}~{}a^{\alpha}\prec a^{\beta},\end{array}\right.$$

$$\sigma (\mathcal{B})=\{\sigma (a{)}^{\alpha }=\sigma (a_1{)}^{{\alpha }_1}\cdots \sigma (a_n{)}^{{\alpha }_n}|\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n\}.$$

$$\sigma (a{)}^{\alpha }\sigma (a{)}^{\beta }={\lambda }_{\alpha ,\beta }\sigma (a{)}^{\alpha +\beta },where\sigma (a{)}^{\alpha +\beta }=\sigma (a_1{)}^{{\alpha }_1+{\beta }_1}\cdots \sigma (a_n{)}^{{\alpha }_n+{\beta }_n};$$

$$\sigma (a{)}^{\alpha }\sigma (a{)}^{\beta }={\lambda }_{\alpha ,\beta }\sigma (a{)}^{\alpha +\beta }+\sum _{d(a^{\alpha (k)})=d(a^{\alpha +\beta })}{\mu }_j^{\alpha ,\beta }\sigma (a{)}^{\alpha (k)}.$$

$$\sigma (a{)}^{\alpha }{\prec }_{{}_{G(A)}}\sigma (a{)}^{\beta }\iff a^{\alpha }{\prec }_{gr}{a}^{\beta },$$

$$\tilde{\mathcal{B}}=\{{\tilde{a}}^{\alpha }{Z}^m={\tilde{a}}_1^{{\alpha }_1}\cdots {\tilde{a}}_n^{{\alpha }_n}{Z}^m|\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n,m\in \mathbb{N}\}.$$

$${\tilde{a}}^{\alpha }{Z}^s\cdot {\tilde{a}}^{\beta }{Z}^t={\lambda }_{\alpha ,\beta }{\tilde{a}}^{\alpha +\beta }{Z}^{s+t},where{\tilde{a}}^{\alpha +\beta }={\tilde{a}}_1^{{\alpha }_1+{\beta }_1}\cdots {\tilde{a}}_n^{{\alpha }_n+{\beta }_n};$$

$$\begin{array}[]{rcl}\widetilde{a}^{\alpha}Z^{s}\cdot\widetilde{a}^{\beta}Z^{t}&=&\lambda_{\alpha,\beta}\widetilde{a}^{\alpha+\beta}Z^{s+t}+\displaystyle\sum_{j}\mu^{\alpha,\beta}_{j}\widetilde{a}^{\alpha(j)}Z^{q-m_{j}},\\ &}{}{\hfil&\hbox{where}~{}q=d(a^{\alpha+\beta})+s+t,~{}m_{j}=d(a^{\alpha(j)}).\end{array}$$

$$\widetilde{a}^{\alpha}Z^{s}\prec_{{}_{\widetilde{A}}}\widetilde{a}^{\beta}Z^{t}\Leftrightarrow\left\{\begin{array}[]{l}a^{\alpha}\prec_{gr}a^{\beta}\\ \hbox{or}\\ a^{\alpha}=a^{\beta}~{}\hbox{and}~{}s<t,\end{array}\right.$$

Full text

Graded Monomial Ordering for $\mathbb{N}$-graded and $\mathbb{N}$-filtered Solvable Polynomial Algebras of $({\cal B},d(~{}))$-type

Huishi Li

Department of Applied Mathematics, College of Information Science and Technology

Hainan University, Haikou 570228, China e-mail: huishipp@yahoo.com

Abstract. Let $K$ be a field, and $A=K[a_{1},\ldots,a_{n}]$ a solvable polynomial algebra in the sense of [K-RW, J. Symbolic Comput., 9(1990), 1–26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type, the $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.

MSC 2010 Primary 13P10; Secondary 16W70, 68W30 (16Z05).

Key words Positive-degree function, $\mathbb{N}$-graded algebra, $\mathbb{N}$-filtered algebra, graded monomial ordering, solvable polynomial algebra.

This note is a complement to [Li6] in which the graded monomial ordering for $\mathbb{N}$-graded and $\mathbb{N}$-filtered solvable polynomial algebras of $({\cal B},d(~{}))$-type is necessarily employed but its existence is not established, and the structures of $\mathbb{N}$-graded and $\mathbb{N}$-filtered solvable polynomial algebras of $({\cal B},d(~{}))$-type are specified but not formally defined.

1. $\mathbb{N}$-graded and $\mathbb{N}$-filtered Algebras of $({\cal B},d(~{}))$-type

Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. We first explore the sufficient and necessary condition that makes $A$ into an $\mathbb{N}$-graded algebra such that

(a) the degree-0 homogeneous part of $A$ is equal to $K$;

(b) every $a^{\alpha}\in{\cal B}$ is a homogeneous element of $A$.

Suppose that $A$ is an $\mathbb{N}$-graded algebra. Then by the definition of a graded algebra we know that $A=\oplus_{p\in\mathbb{N}}A_{p}$ with the degree-$p$ homogeneous part $A_{p}$ which is a subspace of $A$, such that $A_{p}A_{q}\subseteq A_{p+q}$ for all $p,q\in\mathbb{N}$. Furthermore, we assume that $A$ has the above properties (a) and (b). Then $a_{i}\in A_{m_{i}}$ for some $m_{i}>0$, $1\leq i\leq n$. Thus, writing $d_{\rm gr}(h)$ for the graded-degree (abbreviated to gr-degree) of a nonzero homogeneous element $h\in A_{p}$, i.e., $d_{\rm gr}(h)=p$, we have $d_{\rm gr}(a_{i})=m_{i}$, $1\leq i\leq n$. It turns out that if $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}\in{\cal B}$, then $d_{\rm gr}(a^{\alpha})=\sum^{n}_{i=1}\alpha_{i}m_{i}$. This shows that $d_{\rm gr}(~{})$ gives rise to a positive-degree function on $\mathcal{B}$ (or equivalently on $A$) with respect to the $n$-tuple $(m_{1},\ldots,m_{n})\in\mathbb{N}^{n}$, such that

(1) for every $p\in\mathbb{N}$, $A_{p}=K\hbox{-span}\{a^{\alpha}\in\mathcal{B}~{}|~{}d_{\rm gr}(a^{\alpha})=p\}$ (note that this is a finite dimensional subspace of $A$);

(2) for $1\leq i<j\leq n$, if $a_{j}a_{i}\neq 0$ and $a_{j}a_{i}=\sum^{t}_{i=1}\lambda_{i}a^{\alpha(i)}$, then $d_{\rm gr}(a^{\alpha(i)})=m_{j}+m_{i}$, where $\lambda_{i}\in K^{*}$ and $a^{\alpha(i)}\in\mathcal{B}$ .

Conversely, given an $n$-tuple $(m_{1},\ldots,m_{n})\in\mathbb{N}^{n}$ with all $m_{i}>0$, there is a positive-degree function $d(~{})$ on $A$ such that $d(a^{\alpha})=\sum_{i=1}^{n}\alpha_{i}m_{i}$ for all $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}\in{\cal B}$, in particular $d(a_{i})=m_{i}$, $1\leq i\leq n$, and for a nonzero $f=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}$ with $\lambda_{s}\in K^{*}$ and $a^{\alpha(s)}\in{\cal B}$, $d(f)=\max\{d(a^{\alpha(s)})~{}|~{}1\leq s\leq t\}$. With respect to $d(~{})$, the vector space $A$ is equipped with an $\mathbb{N}$-graded structure $A=\oplus_{p\in\mathbb{N}}A_{p}$, where for every $p\in\mathbb{N}$,

[TABLE]

which is a finite dimensional subspace of $A$, in particular, $A_{0}=K$. Clearly, every $a^{\alpha}\in\mathcal{B}$ is a homogeneous element of $A$. However, in general this $\mathbb{N}$-gradation does not necessarily satisfy the condition

[TABLE]

namely $A$ is not necessarily an $\mathbb{N}$-graded algebra with respect to the $\mathbb{N}$-gradation determined by the PBW basis ${\cal B}$ of $A$ and the given positive-degree function $d(~{})$ on $A$ (see Example given after Definition 1.4). To remedy this problem, considering the property (2) presented above, it is straightforward to verify that if

[TABLE]

then the condition ($2^{\prime}$) is satisfied and consequently, $A$ is turned into an $\mathbb{N}$-graded algebra.

Summing up, the above discussion has led to the following result and defination.

1.1. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. The following two statements are equivalent.

(i) $A=\oplus_{p\in\mathbb{N}}A_{p}$ is an $\mathbb{N}$-graded algebra with the degree-$p$ homogeneous part $A_{p}$ which is a subspace of $A$, such that

(a) $A_{0}=K$;

(b) if $a^{\alpha}\in\mathcal{B}$, then $a^{\alpha}\in A_{q}$ for some $q\in\mathbb{N}$.

(ii) With respect to a certain $n$-tuple $(m_{1},\ldots,m_{n})\in\mathbb{N}^{n}$ in which all $m_{i}>0$, there is a positive-degree function $d(~{})$ on $A$ such that $d(a^{\alpha})=\sum_{i=1}^{n}\alpha_{i}m_{i}$ for all $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}\in{\cal B}$, in particular $d(a_{i})=m_{i}$, $1\leq i\leq n$, and such that for $1\leq i<j\leq n$, if $a_{j}a_{i}\neq 0$ and $a_{j}a_{i}=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}$, then $d(a^{\alpha(s)})=m_{j}+m_{i}$, where $\lambda_{i}\in K^{*}$ and $a^{\alpha(s)}\in\mathcal{B}$.

$\Box$

1.2. Definition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. If there is a certain positive-degree function $d(~{})$ on ${\cal B}$ (or equivalently on $A$) such that $A$ is made into an $\mathbb{N}$-graded algebra with respect to the $\mathbb{N}$-gradation $A=\oplus_{p\in\mathbb{N}}A_{p}$ in which each $A_{p}=K\hbox{-span}\{a^{\alpha}\in\mathcal{B}|~{}d(a^{\alpha})=p\}$, then we call $A$ an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type.

Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. We next explore the sufficient and necessary condition that makes $A$ into an $\mathbb{N}$-filtered algebra of type $({\cal B},d(~{}))$, and furthermore, we explore the associated graded structures of $A$ determined by the $\mathbb{N}$-filtration of $A$.

Given an $n$-tuple $(m_{1},\ldots,m_{n})\in\mathbb{N}^{n}$ with all $m_{i}>0$, condider the positive-degree function $d(~{})$ on $A$ such that $d(a^{\alpha})=\sum_{i=1}^{n}\alpha_{i}m_{i}$ for all $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}\in{\cal B}$, in particular $d(a_{i})=m_{i}$, $1\leq i\leq n$, and for a nonzero $f=\sum_{s=1}^{t}\lambda_{s}a^{\alpha(s)}\in A$ with $\lambda_{s}\in K^{*}$ and $a^{\alpha(s)}\in{\cal B}$, $d(f)=\max\{d(a^{\alpha(s)})~{}|~{}1\leq s\leq t\}$. Then, with respect to $d(~{})$, not only an $\mathbb{N}$-gradation may be constructed for the vector space $A$ (as we have seen from the last part), but also an $\mathbb{N}$-filtration may be constructed for the vector space $A$, i.e., $A$ has the $\mathbb{N}$-filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$, where for every $p\in\mathbb{N}$,

[TABLE]

which is a finite dimensional subspace of $A$, in particular $A_{0}=K$, such that $F_{p}A\subseteq F_{p+1}A$ for all $p\in\mathbb{N}$ and $A=\cup_{p\in\mathbb{N}}F_{p}A$. However, in general this $\mathbb{N}$-filtration does not necessarily satisfy the condition

[TABLE]

namely $A$ is not necessarily an $\mathbb{N}$-filtered algebra with respect to the $\mathbb{N}$-filtration determined by the PBW basis ${\cal B}$ of $A$ and the given positive-degree function $d(~{})$ on $A$ (see Example (1) given after Definition 1.4). To remedy this problem, considering the condition (C) presented above by focusing on the representation of the product $a_{j}a_{i}$ by elements in $\mathcal{B}$, where $1\leq i<j\leq n$, we have the following easily verified proposition.

1.3. Proposition With the notation above, $A$ is a filtered algebra with respect to the $\mathbb{N}$-filtration $FA$, or equivalently, the filtration $FA$ satisfies condition (C) presented above, if and only if $A$ satisfies the condition

[TABLE]

$\Box$

In conclusion, we give the formal definition of the $\mathbb{N}$-filtered algebras specified in Proposition 1.3, as follows.

1.4. Definition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. If there is a certain positive-degree function $d(~{})$ on ${\cal B}$ (or equivalently on $A$) such that $A$ is made into an $\mathbb{N}$-filtered algebra with respect to the $\mathbb{N}$-filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ where each $F_{p}A=K\hbox{-span}\{a^{\alpha}\in\mathcal{B}|~{}d(a^{\alpha})\leq p\}$, then we call $A$ an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type.

We now give an example to illustrate Proposition 1.1 and Proposition 1.3.

Example (1) Let $A=K[a_{1},a_{2},a_{3}]$ be the $K$-algebra generated by $\{a_{1},a_{2},a_{3}\}$ subject to the relations

[TABLE]

where $\lambda\in K^{*}$, $\mu\in K$, $f(a_{2})\in K$-span$\{1,a_{2},a_{2}^{2},\ldots,a_{2}^{6}\}$. Then it follows from [Li4] that $A$ has the PBW basis $\mathcal{B}^{\prime}=\{a^{\alpha}=a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}a_{3}^{\alpha_{3}}~{}|~{}\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}\}$. Of course there are many choices for $f(a_{2})$ and $d(~{})$ such that $A$ forms an $\mathbb{N}$-graded or $\mathbb{N}$-filtered algebra of $({\cal B}^{\prime},d(~{}))$-type. For instance, consider the positive-degree function on ${\cal B}^{\prime}$ such that $d(a_{1})=2$, $d(a_{2})=1$ and $d(a_{3})=4$. Then, with $f(a_{2})=a_{2}^{6}$, $A$ is an $\mathbb{N}$-graded algebra of $({\cal B}^{\prime},d(~{}))$-type; noticing that with respect to the given positive-degree function $d(~{})$ on ${\cal B}^{\prime}$ we have $d(f(a_{2}))\leq 6$, it turns out that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B}^{\prime},d(~{}))$-type. But if we consider the positive-degree function such that $d(a_{i})=1$, $1\leq i\leq 3$, then, $A$ is not an $\mathbb{N}$-graded algebra of $({\cal B}^{\prime},d(~{}))$-type with respect to the gradation $A=\oplus_{p\in\mathbb{N}}A_{p}$ where each $A_{p}=K$-span$\{a^{\alpha}\in{\cal B}^{\prime}~{}|~{}d(a^{\alpha})=p\}$, because with $\lambda\neq 0$ and $\mu\neq 0$, $a_{3}a_{1}=\lambda a_{1}a_{3}+\mu a_{2}^{2}a_{3}+f(a_{2})$ in which $d(a_{2}^{2}a_{3})>2$, thereby $A_{1}A_{1}\not\subset A_{2}$; $A$ is also not an $\mathbb{N}$-filtered algebra of $({\cal B}^{\prime},d(~{}))$-type with respect to the filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ where each $F_{p}A=K$-span$\{a^{\alpha}\in{\cal B}^{\prime}~{}|~{}d(a^{\alpha})\leq p\}$, because with $\lambda\neq 0$ and $\mu\neq 0$, $a_{3}a_{1}=\lambda a_{1}a_{3}+\mu a_{2}^{2}a_{3}+f(a_{2})$ in which $d(a_{2}^{2}a_{3})>2$, thereby $F_{1}AF_{1}A\not\subset F_{2}A$.

Let $A$ be an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type with the filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$. Note that $FA$ is constructed with respect to a positive-degree function $d(~{})$ on $A$ such that $F_{p}A=K$-span$\{a^{\alpha}\in{\cal B}~{}|~{}d(a^{\alpha})\leq p\}$, $p\in\mathbb{N}$, in particular $F_{0}A=K$. It turns out that $FA$ is separated in the sense that if $f$ is a nonzero element of $L$, then either $f\in F_{0}A=K$ or $f\in F_{p}A-F_{p-1}A$ for some $p>0$. Actually, this tells us that for a nonzero $f\in A$,

[TABLE]

In order to deal with the associated graded structure of $A$ determined by $FA$ below, we first highlight an intrinsic property of $d(~{})$ with respect to $FA$, as follows.

1.5. Lemma If $f=\sum_{s=1}^{t}\lambda_{s}a^{\alpha(s)}\in A$ with $\lambda_{s}\in K^{*}$ and $a^{\alpha(s)}\in{\cal B}$, then $d(f)=p$ if and only if $d(a^{\alpha(s^{\prime})})=p$ for some $1\leq s^{\prime}\leq t$.

Proof Exercise. $\Box$

Let $A$ be an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type with filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$. Then $A$ has the associated $\mathbb{N}$-graded $K$-algebra $G(A)=\oplus_{p\in\mathbb{N}}G(A)_{p}$ with $G(A)_{0}=F_{0}A=K$ and $G(A)_{p}=F_{p}A/F_{p-1}A$ for $p\geq 1$, where for $\overline{f}=f+F_{p-1}A\in G(A)_{p}$, $\overline{g}=g+F_{q-1}A$, the multiplication is given by $\overline{f}\overline{g}=fg+F_{p+q-1}A\in G(A)_{p+q}$. Another $\mathbb{N}$-graded $K$-algebra determined by $FA$ is the Rees algebra $\widetilde{A}$ of $A$, which is defined as $\widetilde{A}=\oplus_{p\in\mathbb{N}}\widetilde{A}_{p}$ with $\widetilde{A}_{p}=F_{p}A$, where the multiplication of $\widetilde{A}$ is induced by $F_{p}AF_{q}A\subseteq F_{p+q}A$, $p,q\in\mathbb{N}$. For convenience, we fix the following notations once and for all:

$\bullet$ If $h\in G(A)_{p}$ and $h\neq 0$ (i.e., $h$ is a nonzero degree-$p$ homogeneous element of $G(A)$), then, as in the foregoing discussion, we write $d_{\rm gr}(h)$ for the gr-degree of $h$ as a homogeneous element of $G(A)$, i.e., $d_{\rm gr}(h)=p$.

$\bullet$ If $H\in\widetilde{A}_{p}$ and $H\neq 0$, then we write $d_{\rm gr}(H)$ for the gr-degree of the nonzero degree-p homogeneous element $H$ of $\widetilde{A}$, i.e., $d_{\rm gr}(H)=p$.

Concerning the $\mathbb{N}$-graded structure of $G(A)$, if $f\in A$ with $d(f)=p$, then by Lemma 1.5, the coset $f+F_{p-1}A$ represented by $f$ in $G(A)_{p}$ is a nonzero homogeneous element of degree $p$. If we denote this homogeneous element by $\sigma(f)$ (in the literature it is referred to as the principal symbol of $f$), then $d(f)=p=d_{\rm gr}(\sigma(f))$. However, considering the Rees algebra $\widetilde{A}$ of $A$, we note that a nonzero $f\in F_{q}A$ represents a homogeneous element of degree $q$ in $\widetilde{A}_{q}$ on one hand, and on the other hand it represents a homogeneous element of degree $q_{1}$ in $\widetilde{A}_{q_{1}}$, where $q_{1}=d(f)\leq q$. So, for a nonzero $f\in F_{p}A$, we denote the corresponding homogeneous element of degree $p$ in $\widetilde{A}_{p}$ by $h_{p}(f)$, while we use $\widetilde{f}$ to denote the homogeneous element represented by $f$ in $\widetilde{A}_{p_{1}}$ with $p_{1}=d(f)\leq p$. Thus, $d_{\rm gr}(\widetilde{f})=d(f)$, and we see that $h_{p}(f)=\widetilde{f}$ if and only if $d(f)=p$.

Furthermore, if we write $Z$ for the homogeneous element of degree 1 in $\widetilde{A}_{1}$ represented by the multiplicative identity element 1, then $Z$ is a central regular element of $\widetilde{A}$, i.e., $Z$ is not a divisor of zero and is contained in the center of $\widetilde{A}$. Bringing this homogeneous element $Z$ into play, the homogeneous elements of $\widetilde{A}$ are featured as follows:

$\bullet$ If $f\in A$ with $d(f)=p_{1}$ then for all $p\geq p_{1}$, $h_{p}(f)=Z^{p-p_{1}}\widetilde{f}$. In other words, if $H\in\widetilde{A}_{p}$ is a nonzero homogeneous element of degree $p$, then there is a unique $f\in F_{p}A$ such that $H=Z^{p-d(f)}\widetilde{f}=\widetilde{f}+(Z^{p-d(f)}-1)\widetilde{f}$.

With the notation above, it follows that considering the $K$-algebra homomorphism

[TABLE]

we have $G(A)\cong\widetilde{A}/\langle Z\rangle$ where $\langle Z\rangle$ is the ideal generated by $Z$ in $\widetilde{A}$, and considering the $K$-algebra homomorphism

[TABLE]

we have $A\cong\widetilde{A}/\langle 1-Z\rangle$ where $\langle 1-Z\rangle$ is the ideal generated by $1-Z$ in $\widetilde{A}$.

The proposition presented below will very much help us to construct graded monomial orderings (see next section for the definition) on $\mathbb{N}$-graded and $\mathbb{N}$-filtered solvable polynomial algebras of $({\cal B},d(~{}))$-type in the next two sections, so that the results of [Li5, Ch4, Ch5] may be better realized with such graded orderings.

1.6. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the set of generators $\{a_{1},\ldots,a_{n}\}$ and the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$.

(i) Suppose that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type with the filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ where each $F_{p}A=K$-span$\{a^{\alpha}\in{\cal B}~{}|~{}d(a^{\alpha})\leq p\}$. With the notation as before, the following statements hold.

(a) For $f,g\in A$, suppose $d(f)=p_{1}$ and $d(g)=p_{2}$. Then $d(fg)=p_{1}+p_{2}$ if and only if $\sigma(f)\sigma(g)\neq 0$. In the case that $\sigma(f)\sigma(g)\neq 0$ we have $\sigma(f)\sigma(g)=\sigma(fg)$ and $\widetilde{f}\widetilde{g}=\widetilde{fg}$. Also if $p_{1}+p_{2}\leq p$, then $h_{p}(fg)=Z^{p-p_{1}-p_{2}}\widetilde{f}\widetilde{g}$.

(b) For $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}\in{\cal B}$, we have

[TABLE]

(c) $G(A)~{}\hbox{has the PBW basis}~{}\sigma({\cal B})=\{\sigma(a)^{\alpha}~{}|~{}a^{\alpha}\in{\cal B}\}$ and $\widetilde{A}~{}\hbox{has the PBW basis}~{}\widetilde{{\cal B}}=\{\widetilde{a}^{\alpha}Z^{m}~{}|~{}a^{\alpha}\in{\cal B},m\in\mathbb{N}\}$.

(ii) Suppose that $A$ is either an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type or an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type, and let $f=\sum_{s=1}^{t}\lambda_{s}a^{\alpha(s)}\in A$ with $\lambda_{s}\in K^{*}$, $a^{\alpha(s)}\in{\cal B}$ and $d(f)=p$. Then by Lemma 1.5, $f$ is associated to a unique element

[TABLE]

such that $d(f)=p=d({\bf LH}_{d}(f))$. The element ${\bf LH}_{d}(f)$ is usually referred to as the leading homogeneous part of $f$. In the commutative case ${\bf LH}_{d}(f)$ is called the degree form of $f$ (e.g. see [KR2]), and in the noncommutative case the algebra defined by leading homogeneous parts of an ideal is studied in [Li2] and [Li3]. With the notation as fixed, the following holds true.

(a) for any nonzero $f,g\in A$ we have $d(fg)=d(f)+d(g)$ whenever ${\bf LH}_{d}(f){\bf LH}_{d}(g)\neq 0$;

(b) For $a^{\alpha},a^{\beta},a^{\eta}\in{\cal B}$, if ${\bf LH}_{d}(a^{\alpha}a^{\beta}a^{\eta})\not\in\{0,1\}$ and $a^{\beta}\neq{\bf LH}_{d}(a^{\alpha}a^{\beta}a^{\eta})$, then $d(a^{\beta})<d({\bf LH}_{d}(a^{\alpha}a^{\beta}a^{\eta}))$;

(c) For $a^{\gamma},a^{\alpha},a^{\beta},a^{\eta}\in{\cal B}$, if $d(a^{\alpha})<d(a^{\beta})$, ${\bf LH}_{d}(a^{\gamma}a^{\alpha}a^{\eta})\neq 0$ and ${\bf LH}_{d}(a^{\gamma}a^{\beta}a^{\eta})\not\in\{0,1\}$, then $d(a^{\gamma}a^{\alpha}a^{\eta})<d(a^{\gamma}a^{\beta}a^{\eta})$.

Proof All statements may be checked directly by referring to Proposition 1.1, Proposition 1.3 and Lemma 1.5. So we leave the details as an exercise. $\Box$

Remark Let $A$ be either an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type or an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type. Proposition 1.6(ii) may also enable us to have a graded monomial ordering on ${\cal B}$ (see next section for the definition) so that $A$ may have a Gröbner basis theory (though probably theoretical only, i.e., not necessarily realizable in an algorithmic way). More precisely, given a total ordering $\prec$ on ${\cal B}$ (which may not necessarily be a well-ordering), we can define a new ordering $\prec_{gr}$ on ${\cal B}$ as follows: for $a^{\alpha},a^{\beta}\in{\cal B}$,

[TABLE]

Clearly, the obtained $\prec_{gr}$ is now a well-ordering on ${\cal B}$ and this, in turn, determines a unique leading monomial ${\bf LM}(f)$ for every nonzero $f=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}\in A$, where if $d(f)=p$ and ${\bf LH}_{d}(f)=\sum_{d(a^{\alpha(s^{\prime})})=p}\lambda_{s^{\prime}}a^{\alpha(s^{\prime})}$, then ${\bf LM}(f)=a^{\alpha(s^{\prime})}$ for some $s^{\prime}$ and thereby $d({\bf LM}(f))=d({\bf LH}_{d}(F))$. If furthermore the ordering $\prec$ satisfies the conditions (b) and (c) (with ${\bf LH}_{d}(~{})$ replaced by the leading monomial ${\bf LM}_{\prec}(~{})$), then, combining Proposition 1.6(ii), it follows that $\prec_{gr}$ is a graded monomial ordering on ${\cal B}$.

2. Solvable Polynomial algebras

2.1. Definition ([K-RW], [LW]) Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra. Suppose that $A$ has the PBW $K$-basis ${\cal B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$, and that $\prec$ is a (two-sided) monomial ordering on ${\cal B}$. If for all $a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}$, $a^{\beta}=a_{1}^{\beta_{1}}\cdots a^{\beta_{n}}_{n}\in{\cal B}$, the following holds:

[TABLE]

where ${\bf LM}(f_{\alpha,\beta})$ stands for the leading monomial of $f_{\alpha,\beta}$ with respect to $\prec$, then $A$ is called a solvable polynomial algebra.

Usually $({\cal B},\prec)$ is referred to an admissible system of $A$.

2.2. Definition Let $A$ be a solvable polynomial algebra with admissible system $({\cal B},\prec)$. If $d(~{})$ is a positive-degree function $d(~{})$ on ${\cal B}$ such that for all $a^{\alpha},a^{\beta}\in{\cal B}$,

[TABLE]

then we call $\prec$ a graded monomial ordering with respect to $d(~{})$.

The next proposition provides us with a constructive characterization of solvable polynomial algebras.

2.3. Proposition [Li4, Theorem 2.1] Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra, and let $K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle$ be the free $K$-algebras with the standard $K$-basis $\mathbb{B}=\{1\}\cup\{X_{i_{1}}\cdots X_{i_{s}}~{}|~{}X_{i_{j}}\in X,~{}s\geq 1\}$. The following two statements are equivalent:

(i) $A$ is a solvable polynomial algebra in the sense of Definition 2.1.

(ii) $A\cong\overline{A}=K\langle X\rangle/I$ via the $K$-algebra epimorphism $\pi_{1}$: $K\langle X\rangle\rightarrow A$ with $\pi_{1}(X_{i})=a_{i}$, $1\leq i\leq n$, $I=$ Ker$\pi_{1}$, satisfying

(a) with respect to some monomial ordering $\prec_{{}_{X}}$ on $\mathbb{B}$, the ideal $I$ has a finite Gröbner basis $G$ and the reduced Gröbner basis of $I$ is of the form

[TABLE]

where $\lambda_{ji}\in K^{*}$, $\mu^{ji}_{q}\in K$, and $F_{ji}=\sum_{q}\mu^{ji}_{q}X_{1}^{\alpha_{1q}}X_{2}^{\alpha_{2q}}\cdots X_{n}^{\alpha_{nq}}$ with $(\alpha_{1q},\alpha_{2q},\ldots,\alpha_{nq})\in\mathbb{N}^{n}$, thereby ${\cal B}=\{\overline{X}_{1}^{\alpha_{1}}\overline{X}_{2}^{\alpha_{2}}\cdots\overline{X}_{n}^{\alpha_{n}}~{}|~{}$ $\alpha_{j}\in\mathbb{N}\}$ forms a PBW $K$-basis for $\overline{A}$, where each $\overline{X}_{i}$ denotes the coset of $I$ represented by $X_{i}$ in $\overline{A}$; and

(b) there is a (two-sided) monomial ordering $\prec$ on ${\cal B}$ such that ${\bf LM}(\overline{F}_{ji})\prec\overline{X}_{i}\overline{X}_{j}$ whenever $\overline{F}_{ji}\neq 0$, where $\overline{F}_{ji}=\sum_{q}\mu^{ji}_{q}\overline{X}_{1}^{\alpha_{1i}}\overline{X}_{2}^{\alpha_{2i}}\cdots\overline{X}_{n}^{\alpha_{ni}}$, $1\leq i<j\leq n$.

$\Box$

3. Graded Ordering for $\mathbb{N}$-graded Solvable polynomial Algebras of $({\cal B},d(~{}))$-type

Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. Recall from Definition 1.2 that $A$ is called an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type if there is a certain positive-degree function $d(~{})$ on ${\cal B}$ (or equivalently on $A$) such that $A$ is made into an $\mathbb{N}$-graded algebra with respect to the $\mathbb{N}$-gradation $A=\oplus_{p\in\mathbb{N}}A_{p}$ in which each $A_{p}=K\hbox{-span}\{a^{\alpha}\in\mathcal{B}~{}|~{}d(a^{\alpha})=p\}$. By Proposition 1.1(i), this is also equivalent to say that $A=\oplus_{p\in\mathbb{N}}A_{p}$ in which every degree-$p$ homogeneous part $A_{p}$ is a subspace of $A$ such that $A_{p}A_{q}\subset A_{p+q}$ for all $p,q\in\mathbb{N}$, and moreover,

(a) $A_{0}=K$, and

(b) if $a^{\alpha}\in\mathcal{B}$, then $a^{\alpha}\in A_{q}$ for some $q\in\mathbb{N}$.

We recall also from the definition of a solvable polynomial algebra (Definition 2.1) that the condition (S) is equivalent to

[TABLE]

Now, it follows from Proposition 1.1(ii) that we first have the following

3.1. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a solvable polynomial algebra with admissible system $({\cal B},\prec)$. The following statements are equivalent.

(i) $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~{}))$-type (in the sense of Definition 1.2).

(ii) There is a positive-degree function $d(~{})$ on $A$ such that for $1\leq i<j\leq n$, all the relations $a_{j}a_{i}=\lambda_{ji}a_{i}a_{j}+f_{ji}$ with $f_{ji}=\sum\mu_{k}a^{\alpha(k)}$ presented in (S*′*) above satisfy $d(a^{\alpha(k)})=d(a_{i}a_{j})=d(a_{j})+d(a_{i})$ whenever $\mu_{k}\neq 0$.

$\Box$

The proposition (or more precisely its proof) below shows us that every $\mathbb{N}$-graded solvable polynomial algebra of $({\cal B},d(~{}))$-type has a graded monomial ordering $\prec_{gr}$ with respect to the same positive-degree function $d(~{})$ on ${\cal B}$ as described in the remark made at the end of Section 1.

3.2. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. The following statements are equivalent.

(i) $A$ is a solvable polynomial algebra with a graded monomial ordering $\prec_{gr}$ with respect to a certain positive-degree function $d(~{})$ on ${\cal B}$ such that for $1\leq i<j\leq n$, all the relations $a_{j}a_{i}=\lambda_{ji}a_{i}a_{j}+f_{ji}$ with $f_{ji}=\sum\mu_{k}a^{\alpha(k)}$ presented in (S2*′*) above satisfy $d(a^{\alpha(k)})=d(a_{i}a_{j})=d(a_{j})+d(a_{i})$ whenever $\mu_{k}\neq 0$;

(ii) $A$ is an $\mathbb{N}$-graded solvable polynomial algebra of $({\cal B},d(~{}))$-type.

Proof (i) $\Rightarrow$ (ii) This follows immediately from Proposition 3.1.

(ii) $\Rightarrow$ (i) Let $A$ be an $\mathbb{N}$-graded solvable polynomial algebra of $({\cal B},d(~{}))$-type. By Proposition 3.1 it remains to prove that $A$ has a graded monomial ordering with respect to the given positive-degree function $d(~{})$. As $A$ is a solvable polynomial algebra, there is a monomial ordering $\prec$ on ${\cal B}$. If we define a new ordering $\prec_{gr}$ on ${\cal B}$ subject to the rule: for $a^{\alpha}$, $a^{\beta}\in{\cal B}$,

[TABLE]

then, noticing that $A$ is a domain and using Proposition 1.6(ii), it is straightforward to check that $\prec_{gr}$ is a graded monomial ordering on ${\cal B}$ with respect to $d(~{})$, as desired. $\Box$

As we will see in the next section, that every $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type has a graded monomial ordering $\prec_{gr}$. This enables us to show that the associated graded algebra $G(A)$ of an $\mathbb{N}$-filtered solvable polynomial algebra $A$ of $({\cal B},d(~{}))$-type is an $\mathbb{N}$-graded solvable polynomial algebra of $(\sigma({\cal B}),d(~{}))$-type (thereby it has a graded monomial ordering), and the Rees algebra $\widetilde{A}$ of $A$ is an $\mathbb{N}$-graded solvable polynomial algebra of $(\widetilde{{\cal B}},d(~{}))$-type (thereby it has a graded monomial ordering).

Finally, let us also have a review of the algebra $A$ given in Example (1) of Section 1. It follows from [Li4, Proposition 3.1 and Proposition 3.2] that with $f(a_{2})=a_{2}^{6}$ and the positive-degree function such that $d(a_{1})=2$, $d(a_{2})=1$ and $d(a_{3})=4$ , $A$ is turned into an $\mathbb{N}$-graded solvable polynomial algebra of $({\cal B}^{\prime},d(~{}))$-type with the graded monomial ordering $a_{3}\prec_{grlex}a_{2}\prec_{grlex}a_{1}$.

4. Graded Ordering for $\mathbb{N}$-filtered Solvable polynomial Algebras of $({\cal B},d(~{}))$-type

Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. Recall from Definition 1.4 that $A$ is called an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type if there is a certain positive-degree function $d(~{})$ on ${\cal B}$ (or equivalently on $A$) such that $A$ is made into an $\mathbb{N}$-filtered algebra with respect to the $\mathbb{N}$-filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ in which each $F_{p}A=K\hbox{-span}\{a^{\alpha}\in\mathcal{B}~{}|~{}d(a^{\alpha})\leq p\}$. By Proposition 1.3, this is also equivalent to say that $A$ satisfies the condition

(\hbox{C}^{\prime})\quad\quad\begin{array}[]{l}a_{j}a_{i}\neq 0~{}\hbox{and}~{}a_{j}a_{i}=\sum^{t}_{s=1}\lambda_{s}a^{\alpha(s)}~{}\hbox{implies}~{}d(a^{\alpha(s)})\leq m_{j}+m_{i},\\ \hbox{where}~{}\lambda_{s}\in K^{*},~{}a^{\alpha(s)}\in\mathcal{B},~{}1\leq i<j\leq n.\end{array}

Also recall that in the last section we have pointed out that the condition (S) of Definition 2.1 is equivalent to the condition

[TABLE]

Now, it follows from Proposition 1.3 that we first have the following

4.1. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a solvable polynomial algebra with admissible system $({\cal B},\prec)$. The following statements are equivalent.

(i) $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type (in the sense of Definition 1.4).

(ii) There is a positive-degree function $d(~{})$ on $A$ such that for $1\leq i<j\leq n$, all the relations $a_{j}a_{i}=\lambda_{ji}a_{i}a_{j}+f_{ji}$ with $f_{ji}=\sum\mu_{k}a^{\alpha(k)}$ presented in (S*′*) satisfy $d(a^{\alpha(k)})\leq d(a_{i}a_{j})=d(a_{j})+d(a_{i})$ whenever $\mu_{k}\neq 0$.

$\Box$

The proposition (or more precisely its proof) below shows us that every $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type has a graded monomial ordering $\prec_{gr}$ with respect to the same positive-degree function $d(~{})$ on ${\cal B}$ as described in the remark made at the end of Section 1.

4.2. Proposition Let $A=K[a_{1},\ldots,a_{n}]$ be a finitely generated $K$-algebra with the PBW basis $\mathcal{B}=\{a^{\alpha}=a_{1}^{\alpha_{1}}\cdots a_{n}^{\alpha_{n}}~{}|~{}\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}\}$. The following statements are equivalent.

(i) $A$ is a solvable polynomial algebra with a graded monomial ordering $\prec_{gr}$ with respect to a certain positive-degree function $d(~{})$ on ${\cal B}$;

(ii) $A$ is an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type.

Proof (i) $\Rightarrow$ (ii) If $\prec_{gr}$ is a graded monomial ordering with respect to the positive-degree function $d(~{})$ on ${\cal B}$, then by the condition (S*′*) mentioned above we have for $1\leq i<j\leq n$, all the relations $a_{j}a_{i}=\lambda_{ji}a_{i}a_{j}+f_{ji}$ with $f_{ji}=\sum\mu_{k}a^{\alpha(k)}$ satisfy $d(a^{\alpha(k)})\leq d(a_{i}a_{j})=d(a_{j})+d(a_{i})$ whenever $\mu_{k}\neq 0$. Thus it follows from Proposition 4.1 that $A$ is an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type.

(ii) $\Rightarrow$ (i) Let $A$ be an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type, where $d(~{})$ is a positive-degree function on ${\cal B}$. In order to reach (i), we proceed to show that $A$ has a graded monomial ordering with respect to the given positive-degree function $d(~{})$ on ${\cal B}$. To this end, note that $A$ is a solvable polynomial algebra, thereby there is a monomial ordering $\prec$ on ${\cal B}$. If we define a new ordering $\prec_{gr}$ on ${\cal B}$ subject to the rule: for $a^{\alpha}$, $a^{\beta}\in{\cal B}$,

[TABLE]

then, noticing that $A$ is a domain and using Proposition 1.6(ii), it is straightforward to check that $\prec_{gr}$ is a graded monomial ordering on ${\cal B}$ with respect to $d(~{})$, as desired. $\Box$

With the aid of Proposition 4.1, the following examples may be better understood.

Example (1) If $A=K[a_{1},\ldots,a_{n}]$ is an $\mathbb{N}$-graded solvable polynomial algebra of $({\cal B},d(~{}))$-type, i.e., $A=\oplus_{p\in\mathbb{N}}A_{p}$ with the degree-$p$ homogeneous part $A_{p}=K$-span$\{a^{\alpha}\in{\cal B}~{}|~{}d(a^{\alpha})=p\}$, then, $A$ is turned into an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type by the same positive-degree function $d(~{})$ and the $\mathbb{N}$-filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ with each $F_{p}A=\oplus_{q\leq p}A_{q}$.

The next example provides $\mathbb{N}$-filtered solvable polynomial algebras of $({\cal B},d(~{}))$-type in which some generators may have degree $\geq 2$.

Example (2) Consider Example (1) given in Section 1, in which the algebra $A=K[a_{1},a_{2},a_{3}]$ has the PBW basis $\mathcal{B}^{\prime}=\{a^{\alpha}=a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}a_{3}^{\alpha_{3}}~{}|~{}\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{N}^{3}\}$, $a_{1}a_{2}=a_{2}a_{1}$, $a_{3}a_{2}=a_{2}a_{3}$, and $a_{3}a_{1}=\lambda a_{1}a_{3}+\mu a_{2}^{2}a_{3}+f(a_{2})$ with $f(a_{2})\in K$-span$\{1,a_{2},a_{2}^{2},\ldots,a_{2}^{6}\}$. Then by Proposition 3.1, we see that with the positive-degree function such that $d(a_{1})=2$, $d(a_{2})=1$ and $d(a_{3})=4$ , $A$ is turned into an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B}^{\prime},d(~{}))$-type.

Let $A=K[a_{1},\ldots,a_{n}]$ be an $\mathbb{N}$-filtered solvable polynomial algebra of $({\cal B},d(~{}))$-type with the filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ in which each $F_{p}A=K$-span$\{a^{\alpha}\in{\cal B}~{}|~{}d(a^{\alpha})\leq p\}$, and let $\prec_{gr}$ be a graded monomial ordering on ${\cal B}$ with respect to the same positive-degree function $d(~{})$ on ${\cal B}$ (existence guaranteed by Proposition 4.2). Considering the associated graded algebra $G(A)=\oplus_{p\in\mathbb{N}}F_{p}A/F_{p-1}$ of $A$ as well as the Rees algebra $\widetilde{A}=\oplus_{p\in\mathbb{N}}F_{p}A$ of $A$ (see Section 1), we are ready to present the following

4.3. Theorem With the notation as in Section 1 and the notation fixed above, the following statements hold.

(i) The $\mathbb{N}$-graded $K$-algebra $G(A)$ is generated by $\{\sigma(a_{1}),\ldots,\sigma(a_{n})\}$, i.e., $G(A)=K[\sigma(a_{1}),\ldots,\sigma(a_{n})]$, and $G(A)$ has the PBW basis

[TABLE]

By referring to the relation $a^{\alpha}a^{\beta}=\lambda_{\alpha,\beta}a^{\alpha+\beta}+f_{\alpha,\beta}$ in Definition 2.1, where $\lambda_{\alpha,\beta}\in K^{*}$, for $\sigma(a)^{\alpha}$, $\sigma(a)^{\beta}\in\sigma({\cal B})$, if $f_{\alpha,\beta}=0$ then

[TABLE]

in the case where $f_{\alpha,\beta}=\sum_{j}\mu^{\alpha,\beta}_{j}a^{\alpha(j)}\neq 0$ with $\mu^{\alpha,\beta}_{j}\in K$,

[TABLE]

Moreover, the ordering $\prec_{{}_{G(A)}}$ defined on $\sigma({\cal B})$ subject to the rule: for $\sigma(a)^{\alpha},\sigma(a)^{\beta}\in\sigma({\cal B})$,

[TABLE]

is a graded monomial ordering with respect to the positive-degree function $d(~{})$ on $\sigma({\cal B})$ defined by assigning $d(\sigma(a_{i}))=d(a_{i})$ for $1\leq i\leq n$. Thus, with the data $(\sigma({\cal B}),\prec_{{}_{G(A)}},d(~{}))$, $G(A)$ is turned into an $\mathbb{N}$-graded solvable polynomial algebra of $(\sigma({\cal B}),d(~{}))$-type.

(ii) The $\mathbb{N}$-graded $K$-algebra $\widetilde{A}$ is generated by $\{\widetilde{a}_{1},\ldots,\widetilde{a}_{n},Z\}$, i.e., $\widetilde{A}=K[\widetilde{a}_{1},\ldots,\widetilde{a}_{n},Z]$ where $Z$ is the central regular element of degree 1 in $\widetilde{A}_{1}$ represented by 1, and $\widetilde{A}$ has the PBW basis

[TABLE]

By referring to the relation $a^{\alpha}a^{\beta}=\lambda_{\alpha,\beta}a^{\alpha+\beta}+f_{\alpha,\beta}$ in Definition 2.1, where $\lambda_{\alpha,\beta}\in K^{*}$, for $\widetilde{a}^{\alpha}Z^{s}$, $\widetilde{a}^{\beta}Z^{t}\in\widetilde{{\cal B}}$, if $f_{\alpha,\beta}=0$ then

[TABLE]

in the case where $f_{\alpha,\beta}=\sum_{j}\mu^{\alpha,\beta}_{j}a^{\alpha(j)}\neq 0$ with $\mu^{\alpha,\beta}_{j}\in K$,

[TABLE]

Moreover, the ordering $\prec_{{}_{\widetilde{A}}}$ defined on $\widetilde{{\cal B}}$ subject to the rule: for $\widetilde{a}^{\alpha}Z^{s},\widetilde{a}^{\beta}Z^{t}\in\widetilde{{\cal B}}$,

[TABLE]

is a monomial ordering on $\widetilde{{\cal B}}$ (which is not necessarily a graded monomial ordering). Thus, with the data $(\widetilde{{\cal B}},\prec_{{}_{\widetilde{A}}},d(~{}))$, where $d(~{})$ is the positive-degree function on $\widetilde{A}$ defined by assigning $d(Z)=1$ and $d(\widetilde{a_{i}})=d(a_{i})$ for $1\leq i\leq n$, $\widetilde{A}$ is turned into an $\mathbb{N}$-graded solvable polynomial algebra of $(\widetilde{{\cal B}},d(~{}))$-type.

Proof The results stated in the theorem are just analogues of those results concerning quadric solvable polynomial algebras established in ([LW, Section 3], [Li1, CH.IV]), so they may be derived in a similar way as in loc. cit., or else they may also be proved directly by use Proposition 1.6 and thus we leave the detailed argumentation as an exercise. $\Box$

Let $A$ be an $\mathbb{N}$-filtered algebra of $({\cal B},d(~{}))$-type with the filtration $FA=\{F_{p}A\}_{p\in\mathbb{N}}$ in which each $F_{p}A=K$-span$\{a^{\alpha}\in{\cal B}~{}|~{}d(a^{\alpha})\leq p\}$. We end this section by a lemma that may be very helpful in understanding [Li6, Ch5].

4.4. Lemma With the notation as in Theorem 4.3 and Section 1, if $f=\lambda a^{\alpha}+\sum_{j}\mu_{j}a^{\alpha(j)}$ with $d(f)=p$ and ${\bf LM}(f)=a^{\alpha}$, then

(a) $p=d(f)=d_{\rm gr}(\sigma(f))=d_{\rm gr}(\widetilde{f})$;

(b) $\sigma(f)=\lambda\sigma(a)^{\alpha}+\displaystyle\sum_{d(a^{\alpha(j_{k})})=p}\mu_{j_{k}}\sigma(a)^{\alpha(j_{k})}$;

${\bf LM}(\sigma(f))=\sigma(a)^{\alpha}=\sigma({\bf LM}(f))$;

(c) $\widetilde{f}=\lambda\widetilde{a}^{\alpha}+\displaystyle\sum_{j}\mu_{j}\widetilde{a}^{\alpha(j)}Z^{p-d(a^{\alpha(j)})}$;

${\bf LM}(\widetilde{f})=\widetilde{a}^{\alpha}=\widetilde{{\bf LM}(f)},$

where ${\bf LM}(f)$, ${\bf LM}(\sigma(f))$ and ${\bf LM}(\widetilde{f})$ are taken with respect to $\prec_{gr}$, $\prec_{{}_{G(A)}}$ and $\prec_{{}_{\widetilde{A}}}$ respectively.

Proof All properties may be verified by referring to Section 1, in particular, Proposition 1.6. We leave the detailed argumentation as an exercise. $\Box$

References

[K-RW] A. Kandri-Rody, V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput., 9(1990), 1–26.

[Li1] H. Li, Noncommutative Gröbner Bases and Filtered-Graded Transfer. Lecture Notes in Mathematics, Vol. 1795, Springer, 2002.

[Li2] H. Li, $\Gamma$-leading homogeneous algebras and Gröbner bases. In: Recent Developments in Algebra and Related Areas (F. Li and C. Dong eds.), Advanced Lectures in Mathematics, Vol. 8, International Press & Higher Education Press, Boston-Beijing, 2009, 155–200.

[Li3] H. Li, Gröbner Bases in Ring Theory. World Scientific Publishing Company, 2011.

[Li4] H. Li, A Note on solvable polynomial algebras. Computer Science Journal of Moldova, vol.22, 1(64), 2014, 99 – 109. Available at arXiv:1212.5988[math.RA].

[Li5] H. Li, On Computation of minimal free resolutions over solvable polynomial algebras. Commentationes Mathematicae Universitatis Carolinae, 4(56)(2015), 447–503.

[Li6] H. Li, Notes on Gröbner bases and free resolutions of modules over solvable polynomial algebras). arXiv:1510.04381v1[math.RA], 131 pages.

[LW] H. Li. Y. Wu, Filtered-graded transfer of Gröbner basis computation in solvable polynomial algebras. Communications in Algebra, 28(1), 2000, 15–32.