SAGBI Bases in G-Algebras

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.


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.

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 x 1 , ..., x n be the free associative K-algebra, generated by {x 1 , ..., x n } over K. Let c ij ∈ K \ {0} and d ij , denote the standard polynomials in T n , where A is termed as a G-algebra, if these conditions hold: 1.
There exists a monomial well-ordering < on N n such that for all i < j, LM(d ij ) < x i x j

2.
For all reduces to 0 with respect to the relations of A.

Proposition 1 ([1]
).Let A be a G-algebra.Then it is an integral domain and has a PBW basis.
≤ n be a G-algebra over the field K.As A has a PBW basis, we say standard monomial in A appears as n of this basis.The set Mon(A) consist all standard monomials from A, that is, 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(A) is called a monomial ordering, if it is a well-ordering on Mon(A) and for all x α , x β , x γ ∈ Mon(A) if x α < x β , then x α+γ < x β+γ .If < is a monomial ordering on Mon(A) then < is said to be a monomial ordering on A.

2.
As we know, Mon(A) forms a K-basis of A, therefore any non-zero element f in A could be uniquely written as f = c α x α + g with c α ∈ K \ {0} and x α a monomial.Please note that for any non-zero term c β x β of g, we have x β < x α .The monomial x α ∈ Mon(A) represents the leading monomial of f , denoted by LM( f ).Here c α ∈ K \ {0} represents the leading coefficient of f , denoted by LC( f ).

3.
Let H ⊂ A, the notation K H A means the subalgebra S of A generated by H.It is the polynomials set in the H-variables in A.

4.
For H ⊂ A, m(H) denotes a monomial in terms of elements of H, we call it H-monomial.For m(H) =

SAGBI Normal Form in 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(H), and k ∈ K satisfying LT(km(H)) = LT(s), then we say that 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(s|H).
For the reader's convenience, we give an algorithm for its computation.

Remark 1.
During the reduction process inside the while loop, LM(s 0 ) is strictly smaller than LM(s) (by the choice of k and m(H)).Due to well-ordering of >, Algorithm 1 always terminates after a finite number of sweeps.

Algorithm 1 SNF(s | H)
Require: > a fixed well-ordering on the G-algebra A, H ⊂ A and s ∈ A Ensure: h ∈ A the SAGBI normal form Remark 2. For different choices "km(H)"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.Tables 1 and 2 in the above example shows that the SNF of different choices are uncommon.In the next example (see Tables 3-5) we use second Weyl algebra with all possible choices of the while loop of Algorithm 1.

Example 2. Let
Let S be a subalgebra of A generated by H = {q 1 , q 2 , q 3 } = e 2 , f , f h + f and g = e 2 f h + eh + f , associated with degrevlex ordering (dp).For the computation of SNF(g | H), we use Algorithm 1.

Turn h i H h i
Choose

Turn h i H h i
Choose , associated with degrevlex ordering (dp).For the computation of the SNF(g | H), we use Algorithm 1.

Turn h i H h i
Choose

Turn h i H h i
Third possible choice-Example 3.

Turn h i H h i
Let S be a subalgebra of G-algebra A and H ⊂ S. Our interest lies in the case when SAGBI normal form s o = 0 for s ∈ 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.
The following proposition illustrates that s ∈ K H A reduces strongly to Proposition 2. Let S be subalgebra of A and H ⊆ S. We assume H to be a SAGBI basis of S, then 1.
For each s ∈ A, s ∈ S if and only SNF(s|H) = 0 2.
H generates the subalgebra S i.e., S = K H A .
Proof. 1.First assume SNF(s|H) = 0, then s = ∑ k i m i (H) where k i ∈ K and hence s ∈ S. Conversely, suppose that s ∈ S and SNF(s|H) = 0 then it cannot be reduced further i.e., LM(SNF (s|H)) = LM(m(H)), for any H-monomial m(H) and this contradicts that H is a SAGBI basis.

2.
Follows from (1), s ∈ S if and only if SNF(s|H) = 0, that is, s = ∑ k i m i (H) with k i ∈ K, it implies s ∈ K H A . which shows S = K H A .

SAGBI Basis Construction in 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 ⊆ A and m(H) and m (H) be H-monomials. The pair (m(H), m (H)) is a critical pair of a H if LM(m(H)) = LM(m (H)). The T-polynomial of critical pair is defined as T(m(H), m (H)) = m(H) − km (H)
where k ∈ K such that LT(m(H)) = LT(m (H)).Definition 6.Let H be a set of polynomials in A and S = K H A be a subalgebra in A. We consider P ∈ S with the representation P = ∑ t i=1 k i m i (H).Then the height of P with respect to this representation is defined as ht(P) = max t i=1 {LM(m i (H))}, 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.
Since T(m i (H), m j (H)) 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 (H), m j (H)) 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 (H) such that LM(m j (H)) = X, which is a contradiction as at least two H-monomials of such type must exist in the representation of s.
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(H), m (H)) (possibly not a necessary critical pair), the H-monomials m(H) and m (H) 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(H), m (H)), 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
with LMm(H) = LMm (H) then (m(H), m (H)) is not a necessary critical pair because they can be written in factored form as with LMm i (H) = LMm i (H) for i = 1, 2. Please note that (m i (H), m i (H)) are necessary critical pairs.Also, observe that Since SAGBI normal form of T-polynomials on the right-hand side reduces to zero, therefore the SAGBI normal form of T(m(H), m (H)) also vanishes.Proof.First, we will prove the correctness of Algorithm 2, despite its termination.

Now we give an algorithm based on the SAGBI Basis
Correctness: Let C ∞ = ∪C (accumulated over a while loop).We will show that for any arbitrary Since m(H ∞ ) and m (H ∞ ) can always be written in terms of a finite number of elements, h i ∈ H ∞ .Also, the sets H are nested, therefore these specific h i 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(H Termination: Now, we suppose that K H o A has a finite SAGBI basis S. Because H ∞ is also a SAGBI basis for K H o A , then for each s ∈ S, we have the following expression LM(s) = LM(m(H ∞ )) for some H ∞ -monomial m(H ∞ ).
These H ∞ -monomials are in terms of finitely many elements of H ∞ , we represent this set by H.
Please note that H is a finite set and m(H ∞ ) = m( H).
Observe that LM(m( H)) = LM(m(H ∞ )) = LM(s) which implies H is a SAGBI basis of K H o A .The finite set 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 ⊂ A, their exists finitely many irreducible pairs of H-monomials m(H), m (H) such that LM(m(H)) = LM(m (H)).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
In the next example we o add some elements to the generating set during the construction of 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 = Q e, f , h | f e = e f − h, he = eh + 2e, h f = f h − 2 f .Let S ⊆ A be the subalgebra generated by H = p 1 = h, p 2 = e 2 , p 3 = f 2 , p 4 = e f h .We construct SAGBI basis of S with respect to the lex ordering.
For the necessary critical pair (m 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.

Remark 4 .
The necessary critical pairs used in SAGBI basis testing are those critical pairs ((m(H), m (H)) which cannot be factor as m(H
Assume H is a SAGBI basis of S. Since every T-polynomial is an element of S = K H A , its SAGBI normal form is equal to zero by part (1) of Proposition 2.Conversely, suppose given 0 = s ∈ S. It is sufficient to prove that it has a representation s = ∑ t p=1 k p m p (H), where k p ∈ K and m p (H) ∈ K H A with LM(s) = ht(∑ t p=1 k p m p (H)).Let s ∈ S with representation s = ∑ t p=1 k p m p (H) with smallest possible height X among all possible representations of s in S, that is X = max t p=1 {LM(m p (H))}.Clearly LM(s) X. Suppose LM(s) 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 (H), m j (H) in the representation s 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.
Criterion to compute SAGBI basis.Let H ∞ = ∪H, accumulated over a while loop in Algorithm 2. Then H ∞ 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 ⊆ A Ensure: A SAGBI basis H for K H o A H = H o and old H = ∅ while H = old H o do 1 be the first Weyl algebra.Let S ⊆ A be the subalgebra generated by H = p 1 = x 2 , p 2 = x∂, p 3 = ∂ 2 with x > lex ∂.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 be an enveloping algebra.Let S ⊆ A be the subalgebra generated by H = e, h 2 .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 (H), m 2 (H)) where m 1 (H) = p 2 p 1 = eh 2 + 4eh + 4e, andm 2 (H) = p 1 p 2 = eh 2 , the T-polynomial is T(m 1 (H), m 2 (H)) = m 1 (H) − m 2 (H) = 4eh + 4e.It is not reduced by elements of H, so p 3 = eh + e and H = {p 1 , p 2 , p 3 }.For the necessary critical pair (m 3 (H), m 4 (H)) wherem 3 (H) = p 3 3 = e 2 h 2 + 4e 2 h + 3e 2 , and m 4 (H) = p 2 1 p 2 = e 2 h 2 ,the T-polynomial isT(m 3 (H), m 4 (H)) = m 3 (H) − m 4 (H) = 4e 2 h + 3e 3 = 4p 1 p 3 + 3p 2 1 := g,and SNF(g | H) = 0 and all T-polynomials of necessary critical pairs give zero SAGBI normal form.Hence H = e, h 2 , eh + e is a SAGBI basis.