SAGBI Bases in G-Algebras

Abstract

In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm to compute SAGBI bases from a subset of polynomials contained in a subalgebra of a G-algebra.

1. Introduction

Our interest in the topic of this paper was inspired by the work of Levandovskyy [1]. In [1], the author developed the concept and computational criterion for computing Gröbner bases in G-algebra whenever these bases have Poincare-Birkhof-Witt (PBW) bases.

The popular PBW theorem is initially defined in [2] for a special Lie algebra, known as enveloping algebra over a condition of finite dimension. PBW theorem is one of the most important tools to study representation theory, theory of algebra, and rings. The notion, G-algebra developed by Apel [3] and Mora [4]. This algebra is quoted as algebra of solvable types [5,6,7] and PBW algebras [8]. These are also nice generalization of commutative algebras and widely used in non-commutative algebraic geometry [9].

Gordon proposed the idea of Gröbner bases in [10] in 1900 while Gröbner bases for commutative rings of polynomials over a field K were defined and developed by Buchberger [11] in 1965. The theory of Gröbner bases in a free associative algebra was developed by Kandri and Rody [5]. Gröbner bases in G-algebras over a field K were defined by Levandovskyy, he also developed a criterion for the existence of these bases and gave a method to compute them [1].

It is natural to make analogue of Gröbner bases of ideal in K-subalgebra; this work was done independently by Robbiano and Sweedler in [12] and Kapur and Madlener in [13]. These bases are known as SAGBI bases. In [14] Nordbeck developed the concept of SAGBI bases in free associative algebra and gave a method to compute them.

In this paper, we establish the theory of SAGBI bases in a general G-algebra over a field K; and we also develop a computational criterion for its construction.

The sketch of this paper is as follows. In Section 2, we briefly describe the concept of a G-algebra and give some definitions that will be used, including the definition of a SAGBI basis in a G-algebra (Definition 4). In Section 3, we define the process of subalgebra reduction in a G-algebra and introduce the concept SAGBI normal form in a G-algebra. Also, we give an algorithm (Algorithm 1) to compute it and its consequences (Proposition 3.8). Finally, in Section 4, we give a SAGBI bases criterion, (Theorem 1) which determines whether a given set is a SAGBI basis. Based on this criterion, we give an algorithm (Algorithm 2) to compute them.

2. Definitions and Notations

In this section, first, we will introduce G-algebras and then we will review some basic terminologies related with it. G-algebras are significant in the study of non-commutative algebras. It has a wide application area. The theory of Gröbner bases are well developed for G-algebras. The corresponding algorithms are implemented in Singular [15]. Details can be found in [16].

Definition 1.

$T_n=K\langle x_1,\dots ,x_n\rangle$be the free associative K-algebra, generated by$\{x_1,\dots ,x_n\}$over K. Let $c_{ij}\in K\setminus \left\{0\right\}$and$d_{ij}$, denote the standard polynomials in$T_n$, where$1\le i<j\le n$. Consider

$$A=K\langle x_1,\dots ,x_n|x_{ji}=c_{ij}·x_i{x}_j+d_{ij},1\le i<j\le n\rangle$$

A is termed as a G-algebra, if these conditions hold:

1.

There exists a monomial well-ordering < on${\mathbb{N}}^n$such that

$$\mathit{for}\mathit{all}i<j,LM\left(d_{ij}\right)<x_i{x}_j$$

2.

For all$1\le i<j<k\le n$, the polynomial

$$c_{ik}{c}_{jk}·d_{ij}{x}_k-x_k{d}_{ij}+c_{jk}·x_j{d}_{ik}-c_{ij}·d_{ik}{x}_j+d_{jk}{x}_i-c_{ij}{c}_{ik}·x_i{d}_{jk}$$

reduces to 0 with respect to the relations of A.

A K-algebra A has a PBW basis, if $\{x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}\}$, known as a standard word-set, is a K-basis of A.

Proposition 1

([1]).Let A be a G-algebra. Then it is an integral domain and has a PBW basis.

Let $A=K\langle x_1,\dots ,x_n|x_j{x}_i=c_{ij}·x_i{x}_j+d_{ij},1\le i<j\le n\rangle$ be a G-algebra over the field K. As A has a PBW basis, we say standard monomial in A appears as $x^{\alpha }=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}$ of this basis. The set $Mon\left(A\right)$ consist all standard monomials from A, that is,

$$Mon\left(A\right)=\left\{x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}\cdots x_n^{{\alpha }_n}|{\alpha }_k\ge 0\right\}.$$

Now we introduce the notion of a monomial ordering in a G-algebra.

Definition 2.

Let A be a G-algebra in n variables.

1.

A total ordering < on$Mon\left(A\right)$is called a monomial ordering, if it is a well-ordering on$Mon\left(A\right)$and for all$x^{\alpha },x^{\beta },x^{\gamma }\in Mon\left(A\right)$if$x^{\alpha }<x^{\beta }$, then$x^{\alpha +\gamma }<x^{\beta +\gamma }$. If < is a monomial ordering on$Mon\left(A\right)$then < is said to be a monomial ordering on A.

2.

As we know,$Mon\left(A\right)$forms a K-basis of A, therefore any non-zero element f in A could be uniquely written as$f=c_{\alpha }{x}^{\alpha }+g$with$c_{\alpha }\in K\setminus \left\{0\right\}$and$x^{\alpha }$a monomial. Please note that for any non-zero term$c_{\beta }{x}^{\beta }$of g, we have$x^{\beta }<x^{\alpha }$. The monomial$x^{\alpha }\in Mon\left(A\right)$represents the leading monomial of f, denoted by$LM\left(f\right)$. Here$c_{\alpha }\in K\setminus \left\{0\right\}$represents the leading coefficient of f, denoted by$LC\left(f\right)$.

3.

Let$H\subset A$, the notation$K{\langle H\rangle }_A$means the subalgebra S of A generated by H. It is the polynomials set in the H-variables in A.

4.

For$H\subset A$,$m\left(H\right)$denotes a monomial in terms of elements of H, we call it H-monomial. For$m\left(H\right)=h_{i1}{h}_{i2}\cdots h_{it},h_{ij}\in H$we define

$$\overline{LM}m\left(H\right)=LM\left(LM\left(h_{i1}\right)LM\left(h_{i2}\right)\cdots LM\left(h_{it}\right)\right)$$

Also,

$$\overline{LT}m\left(H\right)=LT\left(LT\left(h_{i1}\right)LT\left(h_{i2}\right)\cdots LT\left(h_{it}\right)\right).$$

Example 1.

Consider a subset$H=\{h_1=x^2\partial +1,h_2=2x\partial -\partial ,h_3=x\partial \}$of$A=K\langle x,\partial |\partial x=x\partial +1\rangle$which is the first Weyl algebra. A monomial$m\left(H\right)\in K{\langle H\rangle }_A$is

$$m\left(H\right)=h_1{h}_2=(x^2\partial +1)(2x\partial -\partial )=2x^3{\partial }^2-x^2{\partial }^2+2x^2\partial +2x\partial -\partial .$$

Further

$$\begin{array}{ccc}\hfill LM\left(h_1\right)LM\left(h_2\right)& =& x^3{\partial }^2+x^2\partial \mathit{and},\hfill \\ \hfill \overline{LM}m\left(H\right)& =& LM\left(LM\left(h_1\right)LM\left(h_2\right)\right)=x^3{\partial }^2.\hfill \end{array}$$

3. SAGBI Normal Form in $\mathit{G}$-Algebras

In this section, first, we define the process of reduction together with SAGBI normal form in G-algebras, following which we define the concept of SAGBI bases in G-algebra.

Definition 3.

Let H and s be a subset and a polynomial in a G-algebra A, respectively. If there exists an H-monomial$m\left(H\right)$, and$k\in K$satisfying$\overline{LT}\left(km\left(H\right)\right)=LT\left(s\right)$, then we say that

$$s_o=s-km\left(H\right)$$

(1)

is a one-step s-reduction of s with respect to H. Otherwise, the s-reduction of s with respect to H is s itself.If we apply the one-step s-reduction process iteratively, we can achieve a special form of s with respect to H (which cannot be s-reduced further with respect to H), called SAGBI normal form, and write it as,$s_o:=SNF\left(s\right|H)$. For the reader’s convenience, we give an algorithm for its computation.

Remark 1.

During the reduction process inside the$while$loop,$LM\left(s_0\right)$is strictly smaller than$LM\left(s\right)$(by the choice of k and$m\left(H\right)$). Due to well-ordering of >, Algorithm 1 always terminates after a finite number of sweeps.

Algorithm 1$SNF(s\mid H)$

Require: > a fixed well-ordering on the G-algebra A, $H\subset A$ and $s\in A$   Ensure:$h\in A$ the SAGBI normal form     $s_o:=s$     $H_{s_o}:=\{km\left(H\right)\mid k\in K\mathrm{and}\overline{LT}\left(km\left(H\right)\right)=LT\left(s_o\right)\}$     while $s_o\ne 0\mathrm{and}{H}_{s_o}\ne \varnothing$ do   choose $km\left(H\right)\in H_{s_o}$     $s_o:=s_o-km\left(H\right)$      $H_{s_o}:=\{km\left(H\right)\mid k\in K\mathrm{and}\overline{LT}\left(km\left(H\right)\right)=LT\left(s_o\right)\}$    return $s_o$;

Remark 2.

For different choices “$km\left(H\right)$”in the algorithm above, the output of$SNF$may also be different.

Following is an example of the SAGBI normal form in an enveloping algebra.

Table 1. First possible choice—Example 2.

Turn	${\mathit{h}}_{\mathit{i}}$	${\mathit{H}}_{{\mathit{h}}_{\mathit{i}}}$	Choose	${\mathit{h}}_{\mathit{i}+\mathbf{1}}$
$i=0$	g	$\{q_1{q}_3,q_3{q}_1\}$	$q_1{q}_3$	$-e^{2}f+eh+f$
$i=1$	$h_1$	$\{q_1{q}_2,q_2{q}_1\}$	$q_1{q}_2$	$eh+f$
$i=2$	$h_2$	∅		$SNF\left(g\right|H)=eh+f$

and

Table 2. Second possible choice—Example 2.

Turn	${\mathit{h}}_{\mathit{i}}$	${\mathit{H}}_{{\mathit{h}}_{\mathit{i}}}$	Choose	${\mathit{h}}_{\mathit{i}+\mathbf{1}}$
$i=0$	g	$\{q_1{q}_3,q_3{q}_1\}$	$q_3{q}_1$	$-5e^{2}f+2e{h}^2+13eh+10e+f$
$i=1$	$h_1$	$\{q_1{q}_2,q_2{q}_1\}$	$q_1{q}_2$	$2e{h}^2+13eh+10e+f$
$i=2$	$h_2$	∅		$SNF\left(g\right|H)=2e{h}^2+13eh+10e+f$

in the above example shows that the $SNF$ of different choices are uncommon. In the next example (see

Table 3. First possible choice—Example 3.

Turn	${\mathit{h}}_{\mathit{i}}$	${\mathit{H}}_{{\mathit{h}}_{\mathit{i}}}$	Choose	${\mathit{h}}_{\mathit{i}+\mathbf{1}}$
$i=0$	g	$\{p_1^2{p}_2,p_1{p}_2{p}_1,p_2{p}_1^2\}$	$p_1^2{p}_2$	$-{\partial }_1^2{x}_2^4-2{\partial }_1{x}_2^4{\partial }_2+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2+x_1{x}_2^2+2{\partial }_1{x}_2^2+2x_2^2{\partial }_2-x_1{\partial }_2-1$
$i=1$	$h_1$	$\left\{p_1^2\right\}$	$p_1^2$	$-2{\partial }_1{x}_2^4{\partial }_2+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2+x_1{x}_2^2+2x_2^2{\partial }_2-x_1{\partial }_2$
$i=2$	$h_2$	∅		$SNF\left(g\right|H)=-2{\partial }_1{x}_2^4{\partial }_2+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2+x_1{x}_2^2+2x_2^2{\partial }_2-x_1{\partial }_2$

,

Table 4. Second possible choice—Example 3.

Turn	${\mathit{h}}_{\mathit{i}}$	${\mathit{H}}_{{\mathit{h}}_{\mathit{i}}}$	Choose	${\mathit{h}}_{\mathit{i}+\mathbf{1}}$
$i=0$	g	$\{p_1^2{p}_2,p_1{p}_2{p}_1,p_2{p}_1^2\}$	$p_1{p}_2{p}_1$	$-2x_1{\partial }_1^2{x}_2^3-{\partial }_1^2{x}_2^4-{\partial }_1{x}_2^4{\partial }_2+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2-2{\partial }_1{x}_2^3+2x_1{\partial }_1{x}_2+x_1{x}_2^2+2{\partial }_1{x}_2^2+x_2^2{\partial }_2-x_1{\partial }_2-1$
$i=1$	$h_1$	∅		$-2x_1{\partial }_1^2{x}_2^3-{\partial }_1^2{x}_2^4-{\partial }_1{x}_2^4{\partial }_2+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2-2{\partial }_1{x}_2^3+2x_1{\partial }_1{x}_2+x_1{x}_2^2+2{\partial }_1{x}_2^2+x_2^2{\partial }_2-x_1{\partial }_2-1$

and

Table 5. Third possible choice—Example 3.

Turn	${\mathit{h}}_{\mathit{i}}$	${\mathit{H}}_{{\mathit{h}}_{\mathit{i}}}$	Choose	${\mathit{h}}_{\mathit{i}+\mathbf{1}}$
$i=0$	g	$\{p_1^2{p}_2,p_1{p}_2{p}_1{p}_2{p}_1^2\}$	$p_2{p}_1^2$	$-4x_1{\partial }_1^2{x}_2^3-{\partial }_1^2{x}_2^4+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2+4x_1{\partial }_1{x}_2+x_1{x}_2^2+2{\partial }_1{x}_2^2-x_1{\partial }_2-1$
$i=1$	$h_1$	∅		$SNF\left(g\right|H)=-4x_1{\partial }_1^2{x}_2^3-{\partial }_1^2{x}_2^4+{\partial }_1^3{x}_2^2+2x_1{\partial }_1{x}_2^2{\partial }_2+4x_1{\partial }_1{x}_2+x_1{x}_2^2+2{\partial }_1{x}_2^2-x_1{\partial }_2-1$

) we use second Weyl algebra with all possible choices of the while loop of Algorithm 1.

Example 2.

Let$A=\mathbb{Q}〈e,f,h|fe=ef-h,he=eh+2e,hf=fh-2f〉$. Let S be a subalgebra of A generated by$H=\left\{q_1,q_2,q_3\right\}=\left\{e^2,f,fh+f\right\}$and$g=e^{2}fh+eh+f$, associated with degrevlex ordering(dp). For the computation of$SNF(g\mid H)$, we use Algorithm 1.

Example 3.

Let$A=\mathbb{Q}〈x_1,x_2,{\partial }_1,{\partial }_2\mid {\partial }_i{x}_i=x_i{\partial }_i+1〉$, and the subalgebra S in A generated by$H=\left\{p_1,p_2\right\}=\left\{x_2^2{\partial }_1-1,x_1{\partial }_2+1\right\}$, and a polynomial$g=x_1{\partial }_1^2{x}_2^4{\partial }_2+x_1{x}_2^2+x_2^2{\partial }_1^3$, associated with degrevlex ordering(dp). For the computation of the$SNF(g\mid H)$, we use Algorithm 1.

Let S be a subalgebra of G-algebra A and $H\subset S$. Our interest lies in the case when SAGBI normal form $s_o=0$ for $s\in S$. If there is at least one choice of H-monomials such that $s_o=0$, then we say s reduces weakly over H, and reduces strongly if all possible choices give $s_o=0$.

Definition 4.

Let S be a subalgebra of G-algebra A. A subset$H\subset S$is called a SAGBI basis for S if$\forall s\in S$,$s\ne 0,\exists$a H-monomial,$m\left(H\right)$in$K{\langle H\rangle }_A$such that

$$LM\left(s\right)=\overline{LM}\left(m\left(H\right)\right)$$

(2)

The following proposition illustrates that $s\in K{\langle H\rangle }_A$ reduces strongly to $s_o=0$ if H is a SAGBI basis of S.

Proposition 2.

Let S be subalgebra of A and$H\subseteq S$. We assume H to be a SAGBI basis of S, then

1.

For each$s\in A$,$s\in S$if and only$SNF\left(s\right|H)=0$

2.

H generates the subalgebra S i.e.,$S=K{\langle H\rangle }_A$.

Proof. 

• First assume $SNF\left(s\right|H)=0$, then $s=\sum k_i{m}_i\left(H\right)$ where $k_i\in K$ and hence $s\in S$. Conversely, suppose that $s\in S$ and $SNF\left(s\right|H)\ne 0$ then it cannot be reduced further i.e., $LM\left(SNF\left(s\right|H)\right)\ne \overline{LM}\left(m\left(H\right)\right)$, for any H-monomial $m\left(H\right)$ and this contradicts that H is a SAGBI basis.
• Follows from $\left(1\right)$, $s\in S$ if and only if $SNF\left(s\right|H)=0$, that is, $s=\sum k_i{m}_i\left(H\right)$ with $k_i\in K$, it implies $s\in K{〈H〉}_A$. which shows $S=K{\langle H\rangle }_A$.

☐

4. SAGBI Basis Construction in $\mathit{G}$-Algebras

For the computation of SAGBI bases in G-algebra, we propose an algorithm and explore some ingredients that are necessary for this construction. Throughout this section, let A be a G-algebra over the field K.

Definition 5.

Let$H\subseteq A$and$m\left(H\right)$and$m^{\prime }\left(H\right)$be H-monomials. The pair$(m\left(H\right),m^{\prime }\left(H\right))$is a critical pair of a H if$\overline{LM}\left(m\left(H\right)\right)=\overline{LM}\left(m^{\prime }\left(H\right)\right)$. The T-polynomial of critical pair is defined as$T(m\left(H\right),m^{\prime }\left(H\right))=m\left(H\right)-k{m}^{\prime }\left(H\right)$where$k\in K$such that$\overline{LT}\left(m\left(H\right)\right)=\overline{LT}\left(m^{\prime }\left(H\right)\right)$.

Definition 6.

Let H be a set of polynomials in A and$S=K{〈H〉}_A$be a subalgebra in A. We consider$P\in S$with the representation$P={\sum }_{i=1}^t{k}_i{m}_i\left(H\right)$. Then the height of P with respect to this representation is defined as$ht\left(P\right)={max}_{i=1}^t\left\{\overline{LM}\left(m_i\left(H\right)\right)\right\}$, where the maximum is taken with respect to term ordering in A.

Remark 3.

The height is defined for a specific representation of elements of A, not for the elements itself.

Theorem 1.

(SAGBI Basis Criterion)Assume H generates S as a subalgebra in A, then H is a SAGBI basis of S if every T-polynomial of every critical pair of H gives zero SAGBI normal form.

Proof. 

Assume H is a SAGBI basis of S. Since every T-polynomial is an element of $S=K{\langle H\rangle }_A$, its SAGBI normal form is equal to zero by part (1) of Proposition 2.

Conversely, suppose given $0\ne s\in S$. It is sufficient to prove that it has a representation $s={\sum }_{p=1}^t{k}_p{m}_p\left(H\right)$, where $k_p\in K$ and $m_p\left(H\right)\in K{\langle H\rangle }_A$ with $LM\left(s\right)=ht\left({\sum }_{p=1}^t{k}_p{m}_p\left(H\right)\right)$.

Let $s\in S$ with representation $s={\sum }_{p=1}^t{k}_p{m}_p\left(H\right)$ with smallest possible height X among all possible representations of s in S, that is $X={max}_{p=1}^t\left\{\overline{LM}\left(m_p\left(H\right)\right)\right\}$. Clearly $LM\left(s\right)\leqq X$.

Suppose $LM\left(s\right)\lneqq X$ i.e., cancellation of terms occur then there exist at least two H-monomials such that their leading monomial is equal to X. Assume we have only two H-monomials $m_i\left(H\right),m_j\left(H\right)$ in the representation $s={\sum }_{p=1}^t{k}_p{m}_p\left(H\right)$ such that $\overline{LM}\left(m_i\left(H\right)\right)=\overline{LM}\left(m_j\left(H\right)\right)=X$. If $T(m_i\left(H\right),m_j\left(H\right))=m_i\left(H\right)-k{m}_j\left(H\right)$, we can write

$$\begin{array}{l}s={\displaystyle \sum _{p=1}^t}{k}_p{m}_p\left(H\right)\\ =k_i(m_i\left(H\right)-k{m}_j\left(H\right))+(k_j+k_{i}k)m_j\left(H\right)+{\displaystyle \sum _{p=1,p\ne i,j}^t}{k}_p{m}_p\left(H\right)\\ =k_{i}T(m_i\left(H\right),m_j\left(H\right))+(k_j+k_{i}k)m_j\left(H\right)+{\displaystyle \sum _{p=1,p\ne i,j}^t}{k}_p{m}_p\left(H\right)\end{array}$$

(3)

Since $T(m_i\left(H\right),m_j\left(H\right))$ has a zero SAGBI normal form, then this T-polynomial is either zero or can be written as sum of H-monomials of height $LM(T(m_i\left(H\right),m_j\left(H\right))$ which is less than X. If $k_j+k_{i}k$ is equal to zero, then the right-hand side of Equation (3) is a representation of s that has the height less than X, which contradicts our initial assumption that we had chosen a representation of s that had the smallest possible height. Otherwise, the height is preserved, but on the right-hand side of Equation (3), we have only one H-monomial $m_j\left(H\right)$ such that $\overline{LM}\left(m_j\left(H\right)\right)=X$, which is a contradiction as at least two H-monomials of such type must exist in the representation of s. ☐

Remark 4.

The necessary critical pairs used in SAGBI basis testing are those critical pairs$((m\left(H\right),m^{\prime }\left(H\right))$which cannot be factor as$m\left(H\right)=m_1\left(H\right)\cdots m_t\left(H\right),m^{\prime }\left(H\right)=m_1^{\prime }\left(H\right)\cdots m_t^{\prime }\left(H\right)$with$\overline{LM}{m}_i\left(H\right)=\overline{LM}{m}_i^{\prime }\left(H\right)$for all i. The T-polynomial induced by a necessary critical pair is called the necessary T-polynomial. Since G-algebras are finite factorization domains (Theorem 1.3, [17]), therefore for any critical pair$((m\left(H\right),m^{\prime }\left(H\right))$(possibly not a necessary critical pair), the H-monomials$m\left(H\right)$and$m^{\prime }\left(H\right)$have finite irreducible factors. The necessary critical pairs will be formed by these irreducible factors, therefore the zero SAGBI normal form of T-polynomials induced by necessary critical pairs implies the SAGBI normal form of T-polynomial of a critical pair$((m\left(H\right),m^{\prime }\left(H\right))$, will be zero (for details, see proposition 6 of [14]).

Using Remark 4, Theorem 1 can be restated by replacing every critical pair with necessary critical pairs i.e., a set that generates a subalgebra in a G-algebra is a SAGBI basis if and only if the T-polynomial of all necessary critical pairs of that set gives zero SAGBI normal form.

The following example illustrates Remark 4.

Example 4.

Let$A=\mathbb{Q}〈x,\partial |\partial x=x\partial +1〉$be the first Weyl algebra. Let$S\subseteq A$be the subalgebra generated by$H=\left\{x^2,x\partial ,{\partial }^2\right\}$with$x{>}_{lex}\partial$. Let

$$\begin{array}{ccc}\hfill m\left(H\right)& =& x^3\partial +x^2{\partial }^3+x^2{\partial }^2+x{\partial }^4+x\partial ,\mathit{and}\hfill \\ \hfill m^{\prime }\left(H\right)& =& x^3\partial -x^2{\partial }^3+x{\partial }^4-{\partial }^6+3{\partial }^3.\hfill \end{array}$$

with$\overline{LM}m\left(H\right)=\overline{LM}{m}^{\prime }\left(H\right)$then$(m\left(H\right),m^{\prime }\left(H\right))$is not a necessary critical pair because they can be written in factored form as

$$\begin{array}{ccc}\hfill m\left(H\right)& =& (x^2+x\partial )(x\partial +{\partial }^3):=m_1\left(H\right)m_2\left(H\right),\mathit{and}\hfill \\ \hfill m^{\prime }\left(H\right)& =& (x^2+{\partial }^3)(x\partial -{\partial }^3):=m_1^{\prime }\left(H\right)m_2^{\prime }\left(H\right),\hfill \end{array}$$

with$\overline{LM}{m}_i\left(H\right)=\overline{LM}{m}_i^{\prime }\left(H\right)$for$i=1,2.$Please note that$(m_i\left(H\right),m_i^{\prime }\left(H\right))$are necessary critical pairs. Also, observe that

$$\begin{array}{l}T(m\left(H\right),m^{\prime }\left(H\right))=2x^2{\partial }^3+x^2{\partial }^2+x\partial +{\partial }^6-3{\partial }^3\\ =(x\partial -{\partial }^3)(x\partial +{\partial }^3)+(x^2+{\partial }^3)\left(2{\partial }^3\right)\\ =T(m_1\left(H\right),m_1^{\prime }\left(H\right))m_2\left(H\right)+m_1^{\prime }\left(H\right)T(m_2\left(H\right),m_2^{\prime }\left(H\right)).\end{array}$$

Since SAGBI normal form of T-polynomials on the right-hand side reduces to zero, therefore the SAGBI normal form of$T(m\left(H\right),m^{\prime }\left(H\right))$also vanishes.

Now we give an algorithm based on the SAGBI Basis Criterion to compute SAGBI basis.

Proposition 3.

Let$H_{\infty }=\cup H$, accumulated over a while loop in Algorithm 2. Then$H_{\infty }$is a SAGBI basis for$K{〈H_o〉}_A$. Furthermore, if$K{〈H_o〉}_A$is a finitely generated subalgebra (i.e.,$H_o$is a finite set) and admits a finite SAGBI basis, then Algorithm 2 stops and yields a finite SAGBI basis for$K{〈H_o〉}_A$

Algorithm 2 SAGBI Construction Algorithm

Require: > a fixed well-ordering on the G-algebra A, $H_o\subseteq A$   Ensure: A SAGBI basis H for $K{〈H_o〉}_A$    $H=H_o$ and old $H=\varnothing$    while $H\ne \mathrm{old}{H}_o$ do   Compute $C=$ set of all necessary critical pairs of H     $D=\{T(m\left(H\right),m^{\prime }\left(H\right)):(m\left(H\right),m^{\prime }\left(H\right))\in C\}$     Red$=\left\{SNF\right(p\left|H\right)\mid p\in D\}\setminus \{0\}$     old $H=H$     $H=H\cup \mathrm{Red}$    return H;

Proof. 

First, we will prove the correctness of Algorithm 2, despite its termination.

Correctness: Let $C_{\infty }=\cup C$ (accumulated over a while loop). We will show that for any arbitrary $(m\left(H_{\infty }\right),m^{\prime }\left(H_{\infty }\right))\in C_{\infty }$, for which T-polynomial $p=T(m\left(H_{\infty }\right),m^{\prime }\left(H_{\infty }\right))$, we have $SNF\left(p\right|H_{\infty })=0$.

Since $m\left(H_{\infty }\right)$ and $m^{\prime }\left(H_{\infty }\right)$ can always be written in terms of a finite number of elements, $h_i\in H_{\infty }$. Also, the sets H are nested, therefore these specific $h_i^{\prime }s$ necessarily be in $H_{n_o}$, which is formed during the execution of a finite number, $n_o$, of loops. We can assume that $m\left(H_{\infty }\right)=m\left(H_{n_o}\right)$ and $m^{\prime }\left(H_{\infty }\right)=m^{\prime }\left(H_{n_o}\right)$ which implies $p=T(m\left(H_{n_o}\right),m^{\prime }\left(H_{n_o}\right))$. Clearly, either $SNF\left(p\right|H_{n_o})=0$ or $SNF\left(p\right|H_{n_o+1})=0$. This implies that $SNF\left(p\right|H_{\infty })=0$, thus, by Theorem 1, $H_{\infty }$ is a SAGBI basis for $K{〈H_{\infty }〉}_A=K{〈H_o〉}_A$.

Termination: Now, we suppose that $K{〈H_o〉}_A$ has a finite SAGBI basis S. Because $H_{\infty }$ is also a SAGBI basis for $K{〈H_o〉}_A$, then for each $s\in S$, we have the following expression $LM\left(s\right)=\overline{LM}\left(m\left(H_{\infty }\right)\right)$ for some $H_{\infty }$-monomial $m\left(H_{\infty }\right)$.

These $H_{\infty }$-monomials are in terms of finitely many elements of $H_{\infty }$, we represent this set by $\widehat{H}$. Please note that $\widehat{H}$ is a finite set and $m\left(H_{\infty }\right)=m(\widehat{H})$. Observe that $\overline{LM}(m(\widehat{H}))=\overline{LM}\left(m\left(H_{\infty }\right)\right)=LM\left(s\right)$ which implies $\widehat{H}$ is a SAGBI basis of $K{〈H_o〉}_A$. The finite set $\widehat{H}$ must be a subset of $H_{n_o}$ which is produced after a finite number $n_o$ of loops. Therefore, the set $H_{n_o}$ is a SAGBI basis of $K{〈H_o〉}_A$ and by Theorem 1 the algorithm will terminate after the next pass.

Now we will prove that $H_{n_0}$ is finite for any finite input $H_0$. It follows from Remark 4 that for a finite set $H\subset A$, their exists finitely many irreducible pairs of H-monomials $m\left(H\right)$, $m^{\prime }\left(H\right)$ such that $\overline{LM}\left(m\left(H\right)\right)=\overline{LM}\left(m^{\prime }\left(H\right)\right)$. This implies that there exist finitely many necessary critical pairs at each step in Algorithm 2, i.e., the set C after the while loop is finite at each step, therefore the output of the while loop should necessarily be finite. Hence starting with a finite set $H_0$ in Algorithm 2 and completing a strictly finite number of loops $n_0$, each loop produces a finite output. We finally achieve the output $H_{n_o}$ which is a finite SAGBI basis. ☐

We now give examples of SAGBI bases.

Example 5.

Let$A=\mathbb{Q}〈x,\partial |\partial x=x\partial +1〉$be the first Weyl algebra. Let$S\subseteq A$be the subalgebra generated by$H=\left\{p_1=x^2,p_2=x\partial ,p_3={\partial }^2\right\}$with$x{>}_{lex}\partial$. Then its necessary critical pairs are$(p_1{p}_3,p_2^2)$,$(p_1{p}_3,p_3{p}_1)$and$(p_1{p}_3,p_3{p}_1)$gives T-polynomials that are reduced to zero.

Hence$H=\left\{p_1=x^2,p_2=x\partial ,p_3={\partial }^2\right\}$is a SAGBI basis.

In the next example we o add some elements to the generating set during the construction of SAGBI basis.

Example 6.

Let$A=\mathbb{Q}〈e,f,h|fe=ef-h,he=eh+2e,hf=fh-2f〉$be an enveloping algebra. Let$S\subseteq A$be the subalgebra generated by$H=\left\{e,h^2\right\}$. We construct SAGBI basis of S with respect to the$lex$ordering.

Let$p_1=e$,$p_2=h^2$, then for the necessary critical pair$(m_1\left(H\right),m_2\left(H\right))$where

$$\begin{array}{ccc}\hfill m_1\left(H\right)& =& p_2{p}_1=e{h}^2+4eh+4e,\mathit{and}\hfill \\ \hfill m_2\left(H\right)& =& p_1{p}_2=e{h}^2,\hfill \end{array}$$

the T-polynomial is$T(m_1\left(H\right),m_2\left(H\right))=m_1\left(H\right)-m_2\left(H\right)=4eh+4e$. It is not reduced by elements of H, so$p_3=eh+e$and$H=\left\{p_1,p_2,p_3\right\}$. For the necessary critical pair$(m_3\left(H\right),m_4\left(H\right))$where

$$\begin{array}{ccc}\hfill m_3\left(H\right)& =& p_3^3=e^2{h}^2+4e^{2}h+3e^2,\mathit{and}\hfill \\ \hfill m_4\left(H\right)& =& p_1^2{p}_2=e^2{h}^2,\hfill \end{array}$$

the T-polynomial is

$$T(m_3\left(H\right),m_4\left(H\right))=m_3\left(H\right)-m_4\left(H\right)=4e^{2}h+3e^3=4p_1{p}_3+3p_1^2:=g,$$

and$SNF\left(g\right|H)=0$and all T-polynomials of necessary critical pairs give zero SAGBI normal form. Hence$H=\left\{e,h^2,eh+e\right\}$is a SAGBI basis.

The next example shows that similar to the commutative case, a SAGBI basis of a subalgebra could be infinite.

Example 7.

(Infinite SAGBI basis in the enveloping algebra)

Let$A=\mathbb{Q}〈e,f,h|fe=ef-h,he=eh+2e,hf=fh-2f〉$. Let$S\subseteq A$be the subalgebra generated by$H=\left\{p_1=h,p_2=e^2,p_3=f^2,p_4=efh\right\}$. We construct SAGBI basis of S with respect to the$lex$ordering.

For the necessary critical pair$(m_1\left(H\right),m_2\left(H\right))$where

$$\begin{array}{ccc}\hfill m_1\left(H\right)& =& p_2{p}_3=e^2{f}^2,\mathit{and}\hfill \\ \hfill m_2\left(H\right)& =& p_3{p}_2=e^2{f}^2-4efh+e{h}^2+h^2+2h,\hfill \end{array}$$

the T-polynomial is

$$T(m_1\left(H\right),m_2\left(H\right))=m_1\left(H\right)-m_2\left(H\right)=4efh-e{h}^2+h^2+2h=:g_1,$$

and$SNF\left(g_1\right|H)=e{h}^2-h^2-2h=:p_5$. It is not reduced by elements of H, so$H=\left\{p_1,p_2,p_3,p_4,p_5\right\}$. Continuing in this way we get an infinite SAGBI basis$H=\left\{h,e^2,f^2,efh,e{h}^2,e{f}^2,f{h}^2,\cdots \right\}$.

In this paper, we develop the theory of SAGBI bases in G-Algebras and its corresponding algorithms. It is useful to understand the structure of subalgebras in a given G-algebra. The theory of Gröbner bases of ideals of a subalgebra in a polynomial ring, termed as SAGBI-Gröbner basis was developed by Miller [18]. This work can be evolved into the theory of SAGBI-Gröbner bases in G-algebras, which illustrate a better significance of ideals in a given subalgebra of a G-algebra.