Two Applications of Noncommutative Groebner Bases Li Huishi∗, Wu Yuchun, Zhang Jingliang Department of Mathematics Shaanxi Normal University 710062 Xian, P.R. China

It is well known that the commutative Groebner basis theory has been very successful in many areas, and the noncommutative analogue of this theory has also gained its remarkable applicative prospects (e.g. [AL], [K-RW], [Mor]). In this paper, we give two applications of the noncommutative Groebner bases: • Give an algorithmic description of the defining relations of a quadric algebra with a PBW k-basis, which enables us to use Berger’s q-Jacobi condition in a more general extent. • Give an algorithmic determination of the defining relations of the associated graded algebras of a given algebra with given defining relations. This generalizes the well known result concerning the determination of the defining equations of the projective closure of an affine variety (see [CLO′ ] P.375) to the noncommutative case. More precisely, the contents of this paper are arranged as follows. §1. Quadric Algebras 1.1. Groebner bases in free algebras 1.2. Groebner bases and the PBW bases of quadric algebras 1.3. Quadric algebras satisfying the q-Jacobi condition 1.4. Quadric solvable polynomial algebras §2. The Associated Homogeneous Defining Relations of Algebras 2.1. A description of Ae and G(A) by defining ideals 2.2. Working with standard bases 2.3. Working with Groebner bases Rings (algebras) considered in this paper are associative with 1. Moreover, we refer to [CLO′ ] for a general theory of commutative Groebner bases, [K-RW] for a survey of noncommutative Groebner bases in solvable polynomial algebras, and [Mor] for a general theory of the very noncommutative Groebner bases in free algebras. ∗

Supported by NSFC.

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§1. Quadric Algebras Let k be a field of characteristic 0. By a quadric k-algebra we mean a finitely generated k-algebra A = k[x1 , ..., xn ] subject to the defining relations: (∗)

Rji = xj xi − {xj , xi }, P

n ≥ j > i ≥ 1,

P

λl xl + c with λkh where {xj , xi } = λkh ji xk xh + ji , λl , c ∈ k. And we say that A has a PBW k-basis if the set of standard monomials n



o

xi1 xi2 · · · xin i1 ≤ i2 ≤ · · · ≤ in ∪ {1}

forms a k-basis of A. It is well known that the quadric algebras play very important roles in many areas, e.g. Lie algebras and quantum groups, and many quadric algebras have a PBW k-basis, e.g. Weyl algebras, enveloping algebras of Lie algebras, and q-enveloping algebras in the sense of [Ber]. It is equally well known that if a quadric algebra A has a PBW k-basis, then the structure theory of A, in particular, the representation theory of A will be nicer. However, it seems to the authors that there has been no a general way to know if a quadric algebra has a PBW k-basis. Motivated by the recently developed noncommutative Groebner basis theory [Mor] and Berger’s quantum PBW theorem [Ber], in this part, we give an algorithmic description of the defining relations of a quadric algebra with a PBW k-basis by using the method of [Mor], which enables us to use Berger’s Jacobi condition in a more general extent.

1.1. Groebner bases in free algebras In this section we recall from [Mor] some generalities of the noncommutative Groebner bases in free algebras, and meanwhile we introduce some notation for later use as well. Let k be a field of characteristic 0, X = {Xα }α∈Λ a nonempty set of indeterminates, S = hXi the free semigroup with 1 generated by X, and let khSi be the corresponding free k-algebra (or the noncommutative polynomial k-algebra in variables Xα , α ∈ Λ). By a monomial ordering on S we mean a well-ordering > which is compatible with the product: for each l, r, t1 , t2 ∈ S, t1 < t2 implies lt1 r < lt2 r. For example, the graded lexicographical order on S, denoted >grlex , is a monomial ordering: For u, v ∈ S, u >grlex v if and only if either d(v) < d(u) or d(u) = d(v) and v is lexicographically less than u, where we say that v is lexicographically less than u if either there is r ∈ S such that u = vr or there are l, r1 , r2 ∈ S, Xj1 , Xj2 with j1 < j2 such that v = lXj1 r1 , u = lXj2 r2 . 2

Given a monomial ordering > on S, each element f ∈ khSi has a unique ordered representation as a linear combination of elements of S: f=

s X

ci ti ,

ci ∈ k − {0}, ti ∈ S, t1 > t2 > · · · > ts .

i=1

So to each nonzero element f ∈ khSi we can associate LM(f ) = t1 , the leading monomial of f , and LC(f ) = c1 , the leading coefficient of f . If I ⊂ khSi is a two-sided ideal, the set

n

o

LM(I) = LM(f ) ∈ S f ∈ I ⊂ S

is a two-sided semigroup ideal of S and the set

O(I) = S − LM(I) is a two-sided order ideal of S. 1.1.1. Theorem ([Mor] Theorem 1.3) The following holds: (i) khSi = I ⊕ Spank (O(I)). (ii) There is a k-vector space isomorphism between khSi/I and Spank (O(I)). (iii) For each f ∈ khSi there is a unique g = Can(f, I) ∈ Spank (O(I)) such that f − g ∈ I. Moreover, (a) Can(f, I) = Can(g, I) if and only if f − g ∈ I, (b) Can(f, I) = 0 if and only if f ∈ I. 2 For each f ∈ khSi, the unique Can(f, I) determined by I above is called the canonical form of f . It is known from [Mor] that Can(f, I) can be algorithmically computed. 1.1.2. Definition With notation as above, a set G = {gi }i∈J ⊂ I is called a Groebner basis of I if LM(G) = LM(I) where LM(G) is the two-sided semigroup ideal generated by {LM(g) | g ∈ G}. 1.1.3. Theorem ([Mor] Theorem 1.8) With notation as above, the following conditions are equivalent: (i) G is a Groebner basis of I; (ii) For each f ∈ khSi: P

f = Can(f, I) + ti=1 ci ui gi vi , ci ∈ k − {0}, ui , vi ∈ S, gi ∈ G, LM(f ) ≥grlex u1 LM(g1 )v1 >grlex · · · >grlex ui LM(gi )vi >grlex ui+1 LM(gi+1 )vi+1 >grlex · · · ;

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(iii) For each f ∈ khSi, f ∈ I if and only if P

t f = i=1 ci ui gi vi , ci ∈ k − {0}, ui , vi ∈ S, gi ∈ G, LM(f ) = u1 LM(g1 )v1 >grlex · · · >grlex ui LM(gi )vi >grlex ui+1 LM(gi+1 )vi+1 >grlex · · · .

Such a presentation is called a Groebner representation. 2 Given a generating set G = {gi }i∈J of an ideal I ⊂ khSi, it is generally difficult to know if G is a Groebner basis of I. However, if khSi is the free k-algebra generated by a finite set of indeterminates X = {X1 , ..., Xn }, and if G = {g1 , ..., gs } is also finite, then from [Mor] we know that the noncommutative version of Buchberger’s algorithm does exist and it can be used to produce a Groebner basis of I from G (provided the procedure halts), though the obtained basis is usually no longer finite. The existence of such an algorithm in khSi is based on a technical analysis for the noncommutative analogue of the S-elements (used in the commutative Buchberger’s algorithm) which we are going to recall in some detail below. Let X = {X1 , ..., Xn }, khSi be as before and G = {g1 , ..., gs }. Let S × S be the Cartesian product of the free semigroup S and > a monomial ordering on S. If (l, r), (λ, ρ) ∈ S × S are such that lLM(gj )r = λLM(gi )ρ, then the S-element of gj and gi is defined as S(i, j; l, r; λ, ρ) = lgj r − λgi ρ. We say that S(i, j; l, r; λ, ρ) has a weak Groebner representation if S(i, j; l, r; λ, ρ) =

X

ckµ lkµ gk rkµ , and

k,µ

for each k, µ, lku LM(gk )rkµ < lLM(gj )r. The product S × S has a natural S-bimodule structure in the sense that for each t ∈ S, for each (l, r) ∈ S × S, t(l, r) = (tl, r), (l, r)t = (l, rt). To be convenient, we denote this algebraic structure on S × S by S ⊗ S. An ideal of S ⊗ S is a subset J ⊂ S ⊗ S such that if (l, r) ∈ J , t ∈ S, then (tl, r) ∈ J , (l, rt) ∈ J ; a set of generators for J is a (not necessarily finite) set G ⊂ J such that for each (l, r) ∈ J there are l1 , r1 ∈ S, (wl , wr ) ∈ G such that l = l1 wl and r = wr r1 . For s ≥ j ≥ 1, we write ST (LM(gj )) for the ideal of S ⊗ S generated by the set SOB(LM(gj )) = and we put

) r 6= 1 and there is (l, 1) ∈ S ⊗ S (1, r) ∈ S ⊗ S , such that lLM(gj ) = LM(gj )r

(

n



o

Tj (G) = (l, r) ∈ S ⊗ S lLM(gj )r ∈ Ij ∪ ST (LM(gj )), 4

where Ij stands for the ideal of S generated by {LM(g1 ), LM(g2 ), ..., LM(gj−1 )}. (One may see that Tj (G) is indeed an ideal of S ⊗ S.) Let B be a minimal generating set of the ideal Tj (G). For each σ = (lσ , rσ ) ∈ B, choose iσ , λσ , ρσ such that lσ LM(gj )rσ = λσ LM(giσ )ρσ , iσ < j or iσ = j and there is w ∈ S such that rσ = wρσ , and we let M IN (j) = {(iσ , j; lσ , rσ ; λσ , ρσ )}. An element (i, j; l, r; λ, ρ) ∈ M IN (j) is said to be trivial if there is w ∈ S such that either l = λLM(gi )w (and so ρ = wLM(gj )r) or λ = lLM(gj )w (and so r = wLM(gi )ρ). 1.1.4. Theorem ([Mor] Corollary 5.8, Theorem 5.9) Let G = {g1 , ..., gs } be a generating set of the ideal I ⊂ khSi where LC(gi ) = 1 for each i. (i) The set OBS(j) = {(i, j; l, r; λ, ρ) ∈ M IN (j) and nontrivial} is finite. (ii) G is a Groebner basis of I if and only if for each j, for each nontrivial (i, j; l, r; λ, ρ) ∈ M IN (j), the S-element S(i, j; l, r; λ, ρ) has a weak Groebner representation. 2

1.2. Groebner bases and the PBW k-bases of quadric algebras In this section, we give a Groebner basis description of the defining relations of a quadric algebra with a PBW k-basis by using the method of [Mor]. Let khSi be the free k-algebra generated by X = {X1 , ..., Xn } over k, where S = hX1 , ..., Xn i is the free semigroup generated by X with 1. Let A = k[x1 , ..., xn ] be a finitely generated k-algebra subject to the defining relations: Rji = Xj Xi − {Xj , Xi }, n ≥ j > i ≥ 1, P P where {Xj , Xi } = λkh λl Xl + c, λkh ji Xk Xh + ji , λl , c ∈ k.

Writing I = hRji in≥j>i≥1 for the ideal of khSi generated by G = {Rji }n≥j>i≥1 , then A = khSi/I. From now on we let > be a monomial ordering on S such that (∗)

LM(Rji ) = Xj Xi ,

n ≥ j > i ≥ 1.

1.2.1. Lemma With notation as in section 1.1, if we put gk = Rkj , gj = Rji for n ≥ k, j ≥ 1, then OBS(k) = {(h, k; l, r; λ, ρ) ∈ M IN (k) and nontrivial} = {(j, k; 1, Xi ; Xk , 1) | k > j > i} . 5

Proof Recall from the last section that

n

o

Tk (G) = (l, r) ∈ S ⊗ S lLM(gk )r ∈ Ik ∪ ST (LM(gk )),

where Ik stands for the ideal of S generated by {LM(g1 ), LM(g2 ), ..., LM(gk−1 )}, and ST (LM(gk )) is the ideal of S ⊗ S generated by the set SOB(LM(gk )) =

(

) r 6= 1 and there is (l, 1) ∈ S ⊗ S (1, r) ∈ S ⊗ S . such that lLM(gk ) = LM(gk )r

It is easy to see that SOB(LM(gk )) = {(1, wXk Xj ) | w ∈ S} and SOB(LM(gk )) is a minimal generating set of ST (LM(gk )). It is also not hard to check that for each (1, wXk Xj ) ∈ SOB(LM(gk )) every (j, k; 1, wXk Xj ; λ, ρ) ∈ M IN (j) is trivial. Furthermore if (l, r) ∈ S ⊗ S is such that lXk Xj r = λXt Xi ρ for some λ, ρ ∈ S ⊗ S, where k > t, then one sees that (t, k; l, r; λ, ρ) is trivial in case j 6= t; In the case where j = t, one may also easily sees that (1, Xi ) generates all nontrivial elements in M IN (k), or more precisely, OBS(k) = {(h, k; l, r; λ, ρ) ∈ M IN (k) and nontrivial} = {(j, k; 1, Xi ; Xk , 1) | k > j > i}, as desired. 2 Now we are able to mention the Groebner basis description of the defining relations of a quadric algebra with a PBW k-basis. 1.2.2. Theorem With notation as before, let > be a monomial ordering on S such that the assumption (∗) above is satisfied. The following are equivalent: (i) The k-algebra A = khSi/I = k[x1 , ..., xn ] has a PBW k-basis, where each xi is the image of Xi in khSi/I. (ii) G = {Rji }n≥j>i≥1 is a Groebner basis of the ideal I = hRji in≥j>i≥1 ⊂ khSi. (iii) For n ≥ k > j > i ≥ 1, every Rkj Xi − Xk Rji has a weak Groebner representation. Proof (i) ⇔ (ii) Since Xj Xi = LM(Rji ) for n ≥ j > i ≥ 1, this follows from Theorem 1.1.1. (ii) ⇒ (iii) since every Rkj Xi − Xk Rji is in I, this follows from Theorem 1.1.3. (iii) ⇒ (ii) By Lemma 1.2.1 we have OBS(k) = {(h, k; l, r; λ, ρ) ∈ M IN (k) and nontrivial} = {(j, k; 1, Xi ; Xk , 1) | k > j > i}. And for each (j, k; 1, Xi ; Xk , 1) ∈ OBS(k) the corresponding S-element is nothing but Rkj Xi − Xk Rji which has a weak Groebner representation by (iii). It follows from Theorem 1.1.4 that (ii) holds. 2 To realize Theorem 1.2.2, one may, of course, use the very noncommutative division algorithm [Mor] to check if every Rkj Xi − Xk Rji has a weak Groebner representation or not. However, in the next section we will see that sometimes there may be other way to reduce the division procedure.

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1.3. Quadric algberas satisfying the q-Jacobi condition In this section we show that Berger’s q-Jacobi condition can be used to realize Theorem 1.2.2(iii) in a more general extent. All notions and notation are maintained as before. Let khSi, A = khSi/I be as in section 1.2, where I = hRji in≥j>i≥1 and {Rji }n≥j>i≥1 satisfies the assumption (∗) with respect to the monomial ordering > on S. We start by rewriting the defining relations A as follows. Rji = Xj Xi − qji Xi Xj − {Xj , Xi }, qji ∈ k, n ≥ j > i ≥ 1, X X where {Xj , Xi } = αkl αh Xh + c, αkl ji Xk Xl + ji , αh , c ∈ k. k,l

h

For n ≥ k > j > i ≥ 1, the q-Jacobi sum J(Xk , Xj , Xi ) is defined as J(Xk , Xj , Xi ) = {Xk , Xj }Xi − qki qji Xi {Xk , Xj } −qji {Xk , Xi }Xj + qkj Xj {Xk , Xi } +qkj qki {Xj , Xi }Xk − Xk {Xj , Xi }. Furthermore, we define two k-subspaces of khSi as in [Ber]: n



o

E1 = k-Span Rji n ≥ j > i ≥ 1 , n



o

E2 = k-Span Xi Rji , Rji Xi , Xj Rji , Rji Xj n ≥ j > i ≥ 1 ,

1.3.1. Definitoin For n ≥ k > j > i ≥ 1, if every J(Xk , Xj , Xi ) is contained in E1 + E2 , then A is said to satisfy the q-Jacobi condition. 1.3.2. Proposition If A satisfies the q-Jacobi condition, then {Rji }n≥j>i≥1 forms a Groebner basis for I. Proof We claim that if A satisfies the q-Jacobi condition, then it satisfies Theorem 1.2.2(iii). To see this, from the defining relations we read out {Xk , Xj } = Xk Xj − qkj Xj Xk − Rkj , {Xk , Xi } = Xk Xi − qki Xi Xk − Rki , {Xj , Xi } = Xj Xi − qji Xi Xj − Rji , and then we obtain J(Xk , Xj , Xi ) = Xk Rji − Rkj Xi + qji Rki Xj − qkj Xj Rki − qkj qki Rji Xk + qki qji Xi Rkj , which is obviously contained in I. Now we see that Rkj Xi − Xk Rji = qji Rki Xj − qkj Xj Rki − qkj qki Rji Xk + qki qji Xi Rkj − J(Xk , Xj , Xi ). 7

If J(Xk , Xj , Xi ) ⊂ E1 + E2 , then from the structure of E1 + E2 it is clear that J(Xk , Xj , Xi ) has a Groebner representation, and from the above Rkj Xi − Xk Rji has a Groebner representation. So we are done. 2 Recall from [Ber] that a k-algebra A = k[x1 , ..., xn ] is said to be a q-algebra (note that in [Ber] k is generally a commutative ring and we are working on a field) if A ∼ = khSi/I and I is generated by Rji = Xj Xi − qji Xi Xj − {Xj , Xi }, n ≥ j > i ≥ 1, X X αkl αh Xh + c, αkl where {Xj , Xi } = ji Xk Xl + ji , αh , c ∈ k, k,l

satisfying if

αkl ji

h

6= 0, then i < k ≤ l < j, and k − i = j − l,

and a q-algebra satisfying the q-Jacobi condition defined above is called a q-enveloping algebra. If we use the monomial ordering >grlex on S, then it is clear that LM(Rji ) = Xj Xi ,

n ≥ j > i ≥ 1,

satisfying the assumption (∗) of the last section. It follows from Proposition 1.3.2 that we have the following 1.3.3. Corollary If A is a q-enveloping algebra over a field k in the sense of [Ber], then the defining relations of A form a Groebner basis in khSi. 2 In what follows, we give examples to show that (a) There are q-algebras which do not satisfy the Jacobi condition but have PBW k-bases. (b) There are quadric algebras which are not q-algebras but satisfy the q-Jacobi condition, and hence have PBW k-bases. 1.3.4. Example A = k[x1 , ..., x5 ] subject to the defining relations: X2 X1 X3 X1 X3 X2 X4 X1 X4 X2 X4 X3

= = = = = =

X1 X2 , X1 X3 , X2 X3 , X1 X4 , X2 X4 , X3 X4 ,

X5 X1 X5 X2 X5 X3 X5 X4

= = = =

X1 X5 , X2 X5 + X3 X4 , X3 X5 , X4 X5 .

It is clear that A is a q-algebra. Using >grlex such that X5 >grlex X4 >grlex · · · >grlex X1 , one may check by Theorem 1.2.2 that the defining relations of A form a Groebner basis in khX1 , ..., X5 i, and hence A has a PBW k-basis. But A does not satisfy the q-Jacobi condition, as a calculation shows: J(X5 , X2 , X1 ) = X3 X4 X1 − X1 X3 X4 . 8

1.3.5. Example A = k[x1 , x2 , x3 ] subject to the defining relations R21 = X2 X1 − X1 X2 − X12 − 1, R31 = X3 X1 − X1 X3 − X12 − 1, R32 = X3 X2 − X2 X3 − 1. It is clear that A is not a q-algebra. Using >grlex such that X3 >grlex X2 >grlex X1 , one may check by Theorem 1.2.2 that the defining relations of A form a Groebner basis in khX1 , ..., X5 i, and hence A has a PBW k-basis, and A also satisfies the q-Jacobi condition since a calculation shows: J(X3 , X2 , X1 ) = −X3 X12 + X2 X12 + X12 X3 − X12 X2 = −R31 X1 + R21 X1 − X1 R31 + X1 R21 .

1.4. Quadric solvable polynomial algebras Another class of quadric algebras having PBW k-bases comes from the solvable polynomial algebras in the sense of [K-RW]. Recall from [K-RW] that if A = k[x1 , ..., xn ] is a solvable polynomial algebra over a field k with respect to >grlex such that xn >grlex xn−1 >grlex · · · >grlex x1 , then by the definition, (S1) the set {xα1 1 xα2 2 · · · xαnn | αi ∈ ZZ ≥0 } forms a k-basis of A, i.e., A has a PBW k-basis, and the k-algebra generators of A satisfy (S2) for n ≥ j > i ≥ 1, xj xi = λji xi xj +

X

xi xj >grlex xk xh

λkh xk xh +

X

λl xl + c,

where λji , λkh , λl , c ∈ k, and λji 6= 0. It follows from Theorem 1.2.2 that 1.4.1. Conclusion The defining relations of a quadric solvable polynomial algebra A form a Groebner basis in khX1 , ..., Xn i with respect to >grlex. In what follows, we give one example to show that there are quadric solvable polynomial algebras which do not satisfy the q-Jacobi condition, and we give another example to show that there are quadric algebras which are neither q-algebras nor solvable algebras but have PBW k-bases.

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1.4.2. Example A = k[x1 , x2 , x3 , x4 ] subject to the defining relations: X2 X1 X3 X1 X3 X2 X4 X1 X4 X2 X4 X3

= = = = = =

X1 X2 , X1 X3 + X1 X2 , X2 X3 , X1 X4 , X2 X4 , X3 X4 ,

Using >grlex such that X4 >grlex X3 >grlex X2 >grlex X1 , one may check by Theorem 1.2.2 that the defining relations of A form a Groebner basis in khX1 , X2 , X3 , X4 i, and hence A has a PBW k-basis. Moreover, from the defining relations we also see that A is a solvable polynomial algebra with respect to >grlex . But A does not satisfy the q-Jacobi condition, as a calculation shows that J(X4 , X3 , X1 ) = −X4 X1 X2 + X1 X2 X4 . 1.4.3. Example A = k[x1 , x2 , x3 , x4 ] subject to the defining relations R21 R31 R32 R41 R42 R43

= = = = = =

X2 X1 − X1 X2 − 1, X3 X1 − X1 X3 − 1, X3 X2 − X2 X3 − 1, X4 X1 − X1 X4 − 2X32 + 1, X4 X2 − X2 X4 − 2X32 − 1, X4 X3 − X3 X4 − 2.

From the definition it is clear that A is not a q-algebra. Using >grlex such that X3 >grlex X2 >grlex X1 , one may check by Theorem 1.2.2 that the defining relations of A form a Groebner basis in khX1 , X2 , X3 , X4 i, and hence A has a PBW k-basis, and A also satisfies the q-Jacobi condition since a calculation shows that J(X3 , X2 , X1 ) = J(X4 , X3 , X1 ) = J(X4 , X3 , X2 ) = 0, J(X4 , X2 , X1 ) = −2X32 X2 + 2X32 X1 + 2X2 X32 − 2X1 X32 = −2X3 R32 + 2X3 R31 − 2R32 X3 + 2R31 X3 . But from the defining relation we see that A is not a solvable polynomial algebra with respect to >grlex . Finally, we point out that generally a q-enveloping algebra over a field k in the sense of [Ber] is not solvable with respect to >grlex (this may be seen from the definition of a q-algebra), though the q-PBW theorem holds for such algebras.

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§2. The Associated Homogeneous Defining Relations of Algebras Let k be a field of characteristic 0, and k[x1 , ..., xn ] the commutative polynomial k-algebra in n variables. Let I be an ideal of k[x1 , ..., xn ], and let I ∗ denote the homogenization ideal of I in k[x0 , x1 , ..., xn ] with respect to x0 . It is well known that the projective algebraic set V (I ∗ ) defined by I ∗ in the projective n-space Pnk is the projective closure of the affine algebraic set V (I) defined by I in the affine n-space Ank . The following remarkable result tells us that, using the Groebner basis method, the defining equations of the projective closure V (I ∗ ) of the affine algebraic set V (I) can be determined from that of V (I), or equivalently, the defining relations of the graded k-algebra k[x0 , x1 , ..., xn ]/I ∗ can be determined from that of k[x1 , ..., xn ]/I. (•) (cf. [CLO′ ] P.375, Theorem 4) If G = {g1 , ..., gs } is a Groebner basis for I with respect to a graded monomial order in k[x1 , ..., xn ], then G∗ = {g1∗ , ..., gs∗ } is a Groebner basis for I ∗ ⊂ k[x0 , x1 , ..., xn ], where gi∗ is the homogenization of gi in k[x0 , x1 , ..., xn ]. It is natural to ask: Question Is there any analogue of the above (•) in the noncommutative algebraic structure theory? In this part, we will give a positive answer to the above question. To this end, in section 1, by applying the homogenization and dehomogenization of graded rings to a free algebra, for any (not necessarily finitely generated) k-algebra A with the standard filtration F A on A, we give a clear description of the Rees algebra Ae and the associated graded algebra G(A) of A by defining ideals, and from this we also clearly see the geometric meaning of Ae and G(A) in the commutative case corresponding to the above (•). In section 2, we show that if the defining relations of A form a standard basis in the sense of (e.g. [Gol]), then the defining relations of G(A) and Ae can be determined from that of A. Based on the result of section 2, we derive the main result of this part in section 3, namely, if the defining relations of A form a Groebner basis in the sense of [Mor], then the defining relations of Ae and G(A) can be determined from that of A. 2.1. A description of Ae and G(A) by defining ideals

As in part 1 of this paper, we let k be a fixed field of characteristic 0, and X = {Xα }α∈Λ a nonempty set of indeterminates, S = hXi the free semigroup with 1 generated by X, and let khSi be the corresponding free associative k-algebra. If w is a word of S, then the length of w is called the degree of w and is denoted by d(w). We write d(1) = 0. The positive gradation defined on khSi is given by the k-subspaces: khSip =

  X 

d(w)=p

  cw w cw ∈ k, w ∈ S , 

11

p ≥ 0,

i.e., khSi = ⊕p≥0 khSip . If f ∈ khSi, we let d(f ) denote the highest degree appearing in a homogeneous decomposition for f . Now let A = k[xα ]α∈Λ be an associative k-algebra generated by {xα }α∈Λ over k. Then the naturally defined standard filtration F A on A consists of k-subspaces of A: Fp A =

  

X

cα xiα11 · · · xiαnn

i1 +···in ≤p

  cα ∈ k, ij ≥ 0 , 

p ≥ 0.

This filtration yields two graded k-algebra structures, namely, the associated graded k-algebra of A which is by definition the graded ring G(A) = ⊕p≥0 (Fp A/Fp−1 A), and the Rees k-algebra of A which is by definition the graded ring Ae = ⊕p≥0 Fp A. For each a ∈ Fp A − Fp−1 A, we write ep = Fp A represented by a. If X denotes the ep for the homogeneous element of degree p in A a homogeneous element of degree 1 in Ae1 = F1 A represented by 1A , then X is in the centre of Ae and it is not a divisor of zero. Moreover, each homogeneous element of degree p ≥ 1 can e e eh where h ≤ p. Thus, it is easy to see that A ∼ − XiA, be uniquely written as X p−h a = A/h1 e e The importance of studying G(A) for A and A e is well known in the literature G(A) ∼ A. = A/X (e.g. [M-R], [LVO1]). To obtain the main result of this part, in this section we first give a clear description of Ae and G(A) by defining ideals. The description we will give mainly use the homogenization and dehomogenization trick on a free algebra. For some generalities of the homogenization and dehomogenization for graded rings we refer to (e.g. [NVO]). With notation as before, let khSi be the free k-algebra with natural gradation, where S = hXi is the free semigroup generated by X = {Xα }α∈Λ . Considering the polynomial ring khSi[t] over khSi in commuting variable t, there is a ring epimorphism φ: khSi[t] → khSi with φ(t) = 1. Hence kerφ = h1 − ti where the latter is the ideal of khSi[t] generated by 1 − t. If furthermore we consider the “mixed” gradation on khSi[t] which is defined by putting khSi[t]p =

  X 

i+j=p

then we have the following observation:

  Fi tj Fi ∈ khSii , 

p ≥ 0,

• For every p ≥ 0 khSi[t]p + h1 − ti khSi[t]p+1 + h1 − ti ⊂ , h1 − ti h1 − ti and

[ khSi[t]p + h1 − ti

p≥0

h1 − ti

=

khSi[t] ∼ = khSi. h1 − ti

• For every f ∈ khSi, there exists a homogeneous element F ∈ khSi[t]p , for some p, such that φ(F ) = f . More precisely, if f = F0 + F1 + · · · + Fp where Fi ∈ khSii , then

12

F = tp F0 + tp−1 F1 + · · · + tFp−1 + Fp is a homogeneous element in khSi[t]p satisfying φ(F ) = f . 2.1.1. Definition (i) For any F ∈ khSi[t], we write F∗ = φ(F ). F∗ is called the dehomogenization of F with respect to t. (ii) For any f ∈ khSi, if f = F0 + F1 + · · · + Fp , then the homogeneous element f ∗ = tp F0 + tp−1 F1 + · · · + tFp−1 + Fp in khSi[t]p is called the homogenization of f with respect to t. (iii) If I is a two-sided ideal of khSi, then we let I ∗ stand for the graded two-sided ideal of khSi[t] generated by {f ∗ | f ∈ I}. I ∗ is called the homogenization ideal of I with respect to t. Note that since t is a commuting variable, the following lemma can be easily verified as in the commutative case. 2.1.2. Lemma (i) For F, G ∈ khSi[t], (F + G)∗ = F∗ + G∗ , (F G)∗ = F∗ G∗ . (ii) For f, g ∈ khSi, (f g)∗ = f ∗ g∗ , ts (f + g)∗ = tr f ∗ + th g∗ , where r = d(g), h = d(f ), and s = r + h − d(f + g). (iii) For any f ∈ khSi, (f ∗ )∗ = f . (iv) If F is a homogeneous element in khSi[t], then tr (F∗ )∗ = F , where r = d(F ) − d((F∗ )∗ ). (v) If I is a two-sided ideal of khSi, then each homogeneous element F ∈ I ∗ is of the form tr f ∗ for some f ∈ I. 2 2.1.3. Proposition Let I be a proper two-sided ideal of khSi. Then there is a ring epimorphism α: khSi[t]/I ∗ → khSi/I with Kerα = h1 − ti, where t denotes the coset of t in khSi[t]/I ∗ . Moreover, t is a regular element in khSi[t]/I ∗ (i.e. t is not a divisor of zero), and hence h1 − ti does not contain any nonzero homogeneous element of khSi[t]/I ∗ . Proof If we define α by putting α

khSi[t]/I ∗ −→ khSi/I F + I ∗ 7→ F∗ + I,

F ∈ khSi[t],

then by Lemma 2.1.2 we easily see that α is a ring epimorphism with Kerα = h1 − ti. For any homogeneous element F ∈ khSi[t], if tF ∈ I ∗ , then F∗ = (tF )∗ ∈ (I ∗ )∗ ⊂ I by Lemma 2.1.2. Again by Lemma 2.1.2 we have F = tr (F∗ )∗ ∈ I ∗ . Hence t is a regular element of khSi[t]/I ∗ . The fact that h1−ti does not contain any nonzero homogeneous element of khSi[t]/I ∗ easily follows from the regularity of t. 2 Now let us consider the standard filtration F khSi on khSi which is by definition given by the k-subspaces: Fp khSi = ⊕i≤p khSii , p ≥ 0.

13

If I is any two-sided ideal of khSi and we put A = khSi/I, then F khSi gives a filtration F A on A: Fp A = (Fp khSi + I)/I, p ≥ 0. Indeed, F A coincides with the standard filtration on the k-algebra A = khSi/I = k[X α ]α∈Λ where X α is the coset of Xα in khSi/I. If we consider the associated graded ring G(A) = ⊕p≥0 (Fp A/Fp−1 A) of A and the Rees ring Ae = ⊕p≥0 Fp A of A, then the following proposition shows that G(A) and Ae are determined by I ∗. 2.1.4. Proposition With notation as above, there are graded k-algebra isomorphisms: ∼ khSi[t]/I ∗ , and (i) Ae = ∼ khSi[t]/(hti + I ∗ ), where hti denotes the ideal of khSi[t] generated by t. (ii) G(A) = Proof Using the ring homomorphism α of Proposition 2.1.3 and the regularity of t in khSi[t]/I ∗ , e we have an easily verified graded ring isomorphism α: khSi[t]/I ∗ =

M khSi[t]p + I ∗ p≥0

I∗

F + I∗

α e

−→ 7→

M Fp khSi + I p≥0

F∗ + I,

I

= Ae

F ∈ khSi[t]p

e sends the central regular element t of degree 1 to the canonical central regular Note that since α element of Ae which is by definition the image of 1A in F1 A via the inclusion map F0 A ⊂ F1 A, it follows from [LVO] that (i) and (ii) hold. 2

Remark If we go back to the commutative case and put A = k[x1 , ..., xn ]/I, where I is an ideal of the polynomial algebra k[x1 , ..., xn ], then it is clear that with respect to the standard filtration F A on A, the defining relations of the Rees algebra Ae of A correspond to the defining equations of the projective closure V (I ∗ ) of the affine algebraic set V (I) and the defining relations of the associated graded ring G(A) correspond to the defining equations of the part of the projective closure V (I ∗ ) at infinity.

2.2. Working with standard basis With notation as we have fixed in section 2.1, let A = khSi/I be a k-algebra with the set of defining relations {fi = 0}i∈J , i.e., the two-sided ideal I is generated by {fi }i∈J . Let F A be the standard filtration on A, and let G(A) and Ae be the associated graded algebra and Rees algebra of A with respect to F A, respectively. In view of Proposition 2.1.4 we further to consider the following question. Question Can we determine the defining relations of G(A) and Ae from the defining relations of A? 14

Before studying the above question in detail, we first look at some well known examples. Example (i) Let g = kx1 ⊕ · · · ⊕ kxn be an n-dimensional Lie algebra over k with [xi , xj ] = Pn h=1 cij,h xh , and let A = U (g) be the enveloping algebra of g with the standard filtration F U (g). Then by the famous PBW theorem we know that G(U (g)) is, as a graded k-algebra, isomorphic to the polynomial k-algebra in n variables. (ii) Let A = An (k) = k[x1 , ..., xn , y1 , ..., yn ] be the n-th Weyl algebra over k with [xi , yj ] = δij , [xi , xj ] = [yi , yj ] = 0. Then it is well known that, with respect to the standard filtration (or Bernstein filtration) on An (k), G(An (k)) is, as a graded k-algebra, isomorphic to the polynomial k-algebra in 2n variables. Note that in both examples (i) and (ii) the proof of the fact about G(A) is nontrivial in the literature. (iii) Let g and A = U (g) be as in (i). In [LeS] and [LeV], the authors have constructed a regular algebra H(g) in the sense of Artin and Schelter, which is called the homogenized enveloping algebra and is generated by x0 , x1 , ..., xn where x0 is taken to be central and the remaining P defining relations are [xi , xj ] = nh=1 cij,h xh x0 . This new algebra looks very like the Rees algebra of U (g), namely, we also have H(g)/h1 − x0 iH(g) ∼ = U (g), H(g)/x0 H(g) ∼ = G(U (g)). (We will see in the finall section that H(g) is exactly the Rees algebra of U (g).) From the above examples one might expect that for a k-algebra A with standard filtration F A, the defining relations of G(A) resp. Ae may be given by simply taking the homogeneous part of highest degree from the defining relations of A resp. by simply taking the homogenization of the defining relations of A. However, as shown by the following examples (even in the commutative case) the question we posed above is not so trivial to answer in general. Example (i) Consider I = hf1 , f2 i = hx2 − x21 , x3 − x31 i, the ideal of the affine twisted cubic in IR3 . If we homogenize f1 , f2 , then we get the ideal J = hx2 x0 −x21 , x3 x20 −x31 i in IR[x0 , x1 , x2 , x3 ]. One may directly check that for f3 = f2 − x1 f1 = x3 − x1 x2 ∈ I, f3∗ = x3 x0 − x1 x2 6∈ J, i.e., J 6= I ∗ . (ii) Let khSi be the free k-algebra generated by {X1 , X2 , X3 }, and let f = 2X3 X2 X1 − 3X1 X32 , g12 = X2 X1 − X1 X2 , g13 = X3 X1 − X1 X3 , g23 = X3 X2 − X2 X3 . Considering the two-sided ideal I = hf, g12 , g13 , g23 i, then it can be directly verified that h = −3X1 X32 + 2X1 X2 X3 = ∗ , g ∗ , g ∗ i (note that the latter is equal to f − 2X3 g12 + 2g13 x2 + 2X1 g23 ∈ I ∗ , but h 6∈ hf ∗ , g12 13 23 hf, g12 , g13 , g23 i in khSi[t]). Remark In the above examples (i) and (ii) we did not say anything about G(A). However, it will be clear from the below Lemma 3.3 and Proposition 3.5 that generally the defining relations of G(A) cannot be simply taken to be the homogeneous part of the highest degree from the

15

defining relations of A. Nevertheless, based on Proposition 2.1.4. the result (•) we mentioned in §1 still gives us the light, i.e., we may ask Question If the defining relations of A form a Groebner basis in the sense of [Mor], what will e happen to the defining relations of G(A) and A?

As a preliminary result of our answer to the above question we show that if {fi }i∈J is a standard basis of I in the sense of (e.g. [Gol]), then the defining relations of Ae and G(A) can be completely determined. To see why the standard basis is the first choice in our discussion, we first strengthen Proposition 2.1.4(ii) as follows. For any f ∈ khSi we denote by LH(f ) the highest homogeneous part of f , i.e., if f = F0 + F1 + · · · + Fp with Fi ∈ khSii , then LH(f ) = Fp . If I is an ideal of khSi, we denote by hLH(I)i the ideal generated by {LH(f ) | f ∈ I} in khSi. 2.2.1. Proposition Let A = khSi/I and F A the standard filtration on A. Then G(A) ∼ = khSi/hLH(I)i. Proof From Proposition 2.1.4 we have known that G(A) ∼ = khSi[t]/(hti + I ∗ ). To prove the theorem, we first recall that if f = F0 + F1 + · · · + Fp ∈ khSi, then LH(f ) = Fp and f ∗ = LH(f ) + tFp−1 + · · ·. Hence the inclusion map khSi ֒→ khSi[t] yields the inclusion hLH(f )i ⊂ hti + I ∗ . This, in turn, gives a graded ring homomorphism khSi khSi[t] ϕ −→ hLH(f )i (hti + I ∗ ) g + hLH(f )i 7→ g + (hti + I ∗ ) Obviously, ϕ is surjective. On the other hand, each element F ∈ khSi[t] has a unique presentation F = F0 + F ′ where F0 ∈ khSi, F ′ ∈ hti. Moreover, from §2 we know that each homogeneous element in I ∗ is of the form tr f ∗ for some f ∈ I. If f = Fp + Fp−1 + · · · + F0 with LH(f ) = Fp , then f ∗ = LH(f ) + tFp−1 + · · · + tp F0 . Therefore, each element of hti + I ∗ can be written as a sum u + v, where u ∈ hLH(f )i, v ∈ hti. Thus we can define a ring homomorphism khSi[t] khSi ψ −→ ∗ (hti + I ) hLH(I)i F + (hti + I ∗ ) 7→ F0 + hLH(I)i Since ψ ◦ ϕ = 1, it follows that ϕ is also injective and hence an isomorphism. (Indeed, ψ is the ring homomorphism induced by the canonical homomorphism khSi[t] → khSi which sends t to 0.) 2

16

Suppose I = hfi ii∈J . From the above proposition we certainly expect that hLH(I)i = hLH(fi )ii∈J . This leads to the use of standard bases. 2.2.2. Definition The set {fi }i∈J is called a standard basis of I if each element f ∈ I with P p = d(f ) has a presentation as a finite sum f = i gi fi hi , where gi , hi ∈ khSi, and d(gi ) + d(fi ) + d(hi ) ≤ p for all i. Let I be a graded ideal of khSi, i.e., I = ⊕p≥0 (I ∩ khSip ). If {fi }i∈J is a generating set of I consisting of homogeneous elements, then it is easy to see that {fi }i∈J is a standard basis of I. But generally it is not so easy to check if a generating set of an ideal is a standard basis. We refer to [Gol] for a homological criterion of standard basis. The first easy but important property of a standard basis is the following 2.2.3. Lemma If {fi }i∈J is a standard basis of I, then for any f ∈ I, LH(f ) = LH(gi )LH(fi )LH(hi ) for some gi , hi ∈ khSi, fi ∈ {fi }i∈J . Indeed, we have the more stronger result: {fi }i∈J is a standard basis if and only if hLH(fi )ii∈J = hLH(I)i. P

Proof If {fi }i∈J is a standard basis of I, then by the definition it is easy to see that for P any f ∈ I, LH(f ) = LH(gi )LH(fi )LH(hi ) for some gi , hi ∈ khSi, fi ∈ {fi }i∈J . Hence hLH(fi )ii∈J = hLH(I)i. Conversely, if hLH(fi )ii∈J = hLH(I)i, then for any f ∈ I with d(f ) = p, say f = fp + fp−1 + · · · P with fi ∈ khSii , we have LH(f ) = fp = gj LH(fj )hj for some gj , hj ∈ khSi, fj ∈ {fi }i∈J , and P d(gj ) + d(LH(fj )) + d(hj ) = d(gj ) + d(fj ) + d(hj ) = p. Now the element f ′ = f − gj fj hj ∈ I has d(f ′ ) < p, we may repeat the above argumentation and after a finite number of steps we will P reach a presentation f = gi fi hi where gi , hi ∈ khSi, fi ∈ {fi }i∈J and d(gi ) + d(fi ) + d(hi ) ≤ p for all i. It follows that {fi }i∈J is a standard basis of I. 2 2.2.4. Proposition With notation as before, if {fi }i∈J is a standard basis of I, then (i) G(A) has defining relations LH(fi ) = 0, i ∈ J; and moreover (ii) {LH(fi )}i∈J is a standard basis of hLH(I)i. Proof This follows immediately from Proposition 2.2.1 and Lemma 2.2.3.

2

2.2.5. Proposition With notation as before, if {fi }i∈J is a standard basis of I, then (i) I ∗ is generated by {fi∗ }i∈J , or in other words, Ae has defining relations tXα − Xα t = 0, α ∈ Λ, fi∗ = 0, i ∈ J; and moreover (ii) {fi∗ }i∈J is a standard basis of I ∗ . Proof By §2, each homogeneous element in I ∗ is of the form tr f ∗ for some f ∈ I. Suppose P f = i hi fi gi . Since {fi }i∈J is a standard basis of I, it follows from Lemma 2.2.3 and the

17

definition of homogenization of f that d(h∗i ) + d(fi∗ ) + d(gi∗ ) ≤ d(f ∗ ) and P f ∗ − i h∗i fi∗ gi∗ = tr1 m∗1 + tr2 m∗2 + · · · with rj > 0, mj ∈ I, and d(trj m∗j ) ≤ d(f ∗ ) for all mj . Similarly, for each m∗j ∈ I ∗ where mj =

P

i

hij fij gij , we have

d(h∗ij ) + d(fi∗j ) + d(gi∗j ) ≤ d(m∗j ) and m∗j −

P

∗ ∗ ∗ i hij fij gij

= tr1j m∗1j + tr2j m∗2j + · · · with

rkj > 0, mkj ∈ I, and d(t

r kj

m∗kj ) ≤ d(m∗j ) for all mkj .

Since d(f ∗ ) is finite, after a finite number of steps we will reach f ∗ ∈ hfi∗ ii∈J , in particular, P f ∗ = j h∗j fj∗ gj∗ with d(h∗j ) + d(fj∗ ) + d(gj∗ ) ≤ d(f ∗ ) for all j. (Note that I is a proper ideal, the P final step of the reduction procedure cannot reach the form l tl .) This proves the conclusions of (i) and (ii). 2 Remark One may also obtain Proposition 2.2.4 from Proposition 2.2.5. To see this, suppose P I ∗ = hfi∗ ii∈J . Then since each element F of I ∗ is of the form F = i Hi fi∗ Gi , Hi , Gi ∈ khSi[t], we can define the ring homomorphism ψ and complete the argumentation as in the proof of Proposition 2.2.1. We finish this section with a class of examples. 2.2.6. Lemma Let A = k[x1 , ..., xn ] be a k-algebra generated by {x1 , ..., xn }. If A satisfies (a) the set of ordered monomials n



o

{1} ∪ xi1 · · · xin i1 ≤ i2 ≤ · · · ≤ in , n ≥ 1

forms a k-basis of A, and (b) the associated graded k-algebra G(A) of A with respect to the standard filtration of A is a domain, then the defining relations of A is a standard basis for the defining ideal of A. Proof Suppose A ∼ = khX1 , ..., Xn i/I with I = hfi ii∈J . Considering the standard filtration F A on A, it is easily seen from the conditions (a) and (b) that the set

n

o

{1} ∪ σ(xi1 )σ(xi1 ) · · · σ(xin ) i1 ≤ · · · ≤ in , n ≥ 1

forms a k-basis of G(A), where each σ(xij ) denotes the image of xij in F1 A/F0 A = G(A)1 . Then it follows from Proposition 2.1.4 and Proposition 2.2.1 that the natural graded k-algebra homomorphism khX1 , ..., Xn i/hLH(fi )ii∈J → khX1 , ..., Xn i/hLH(I)i is an isomorphism. Hence hLH(fi )ii∈J = hLH(I)i, i.e., {fi }i∈J is a standard basis for I. 2 18

2.2.7. Corollary (i) every q-enveloping k-algebra A in the sense of [Ber] satisfies the conditions (a) and (b) of Lemma 2.2.6. (ii) Any sovable polynomial algebra A in the sense of [K-RW] satisfies the conditions (a) and (b) of Lemma 2.2.6.

2.3. Working with Groebner basis In this section we retain the notation fixed in the preceding sections, but we restrict to a free k-algebra with finite generating set, i.e., we let X = {X1 , ..., Xn } and khSi the free k-algebra where S = hX1 , ..., Xn i is the free semigroup generated by X. Moreover, we let >grlex denote the graded lexicographical order on S (see section 1.1 of the first part of the paper). Under the restriction fixed above, we aim to show that if G = {fi }i∈J ⊂ khSi is a Groebner basis in the sense of [Mor], then for the k-algebra A = khSi/I with the standard filtration F A, where I = hGi is the two-sided ideal of khSi generated by G, the defining relations of Ae and G(A) can be determined from that of A. Note that since we are using >grlex on S, it follows immediately from Theorem 1.1.3(iii) of the first part that any Groebner basis G for I is a standard basis of I in the sense ofsection 2.2. Thus we have reached the following 2.3.1. Theorem With notation as before, let G = {fi }i∈J be a Groebner basis of I and A = khSi/I. Then G(A) has the defining relations LH(fi ) = 0, i ∈ J; and Ae has the defining relations tXj − Xj t = 0, j = 1, ..., n, fi∗ = 0, i ∈ J. 2 Furthermore, let us consider the k-basis n



o

B(S, t) = wtr w ∈ S, r ≥ 0

for khSi[t]. Then the order >grlex on S induces an order on B(S, t) which is a well-ordering and is compatible with the multiplication of khSi[t]: w1 tr1 ≻ w2 tr2 if and only if d(w1 ) + r1 > d(w2 ) + r2 or d(w1 ) + r1 = d(w2 ) + r2 but w1 >grlex w2 . With the definition as abobe, we still let >grlex denote this order on B(S, t). Thus, we have X1 >grlex X2 >grlex · · · >grlex Xn >grlex t, and we can discuss the Groebner basis in khSi[t] exactly as in khSi. Before giving the next theorem we also note that for any nonzero f ∈ khSi, the degree-compatible order >grlex gives us the following equalities: LM(f ) = LM(LH(f )) LM(f ∗ ) = LM(f ). 19

2.3.2. Theorem With notation as before, let G = {fi }i∈J ⊂ I where I is a two-sided ideal of khSi. The following are equivalent: (i) G is a Groebner basis of I; (ii) {fi∗ }i∈J is a Groebner basis of I ∗ in khSi[t]; (iii) {LH(fi )}i∈J is a Groebner basis of the two-sided ideal hLH(I)i in khSi. Proof (i) ⇒ (ii). By the above remark, this can be proved exactly as we did in the proof of Proposition 2.2.5. (ii) ⇒ (i). From §2 we have known that (f ∗ )∗ = f holds for any f ∈ I. So the assertion follows again from the above remark. (i) ⇔ (iii). Using the above remark, this can be directly checked. 2 Suppose that G = {f1 , ..., fs } is a finite subset in khSi and that I is the two-sided ideal generated by G. From [Mor] we know that the noncommutative Buchberger’s algorithm possibly works for producing a Groebner basis for I from G, in particular, it gives us some effective criterion of checking Groebner basis in case we are considering quadric algebras, as shown in the first part of the paper. Therefore, the advantage of using Groebner basis in answering our question concerning the defining relations of Ae and G(A) is not only theoretical but also practical. We finish this paper with two typical examples.

(i) Let A = An (k) = k[x1 , ..., xn , y1 , ..., yn ], the n-th Weyl algebra over k with defining relations Yj Xi − Xi Yj = δij , Xj Xi − Xi Xj = Yj Yi − Xi Yj = 0. As in part 1 of the paper, it can be easily verified that G = {Yj Xi − Xi Yj − δij , Xj Xi − Xi Xj , Yj Yi − Yi Yj }ni,j=1 is a Groebner basis. Hence Theorem 2.3.1 can be used to give the defining relations of Ae and G(A).

(ii) Let g = kx1 ⊕ · · · ⊕ kxn be the n-dimensional Lie algebra over k with defining relations P [Xi , Xj ] = nh=1 λij,h xh where j > i. A = U (g), the enveloping algebra of g. As in part 1 of the paper, it is also easy to verify that G = {Xj Xi − Xi Xj − [Xi , Xj ]}j>i is a Groebner basis. Hence Theorem 2.3.1 can be used to give the defining relations of Ae and G(A). In particular, the so called homogenized enveloping algebra defined in [LeV] and [LeS] is nothing but the Rees algebra of U (g) with respect to the standard filtration on U (g).

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[Gol] E.S. Golod, Standard bases and homology, Algebra, in “Some current Trends, Proceedings, Varna, 1986”, pp. 88–95, Lecture Notes in Mathematics, Vol. 1352, SpringerVerlag, Berlin/New York, 1988. [K-RW] A. Kandri-Rody and V. Weispfenning, Non-commutative Groebner bases in algebras of solvable type, J. Symbolic Computation, 9(1990), 1–26. [LeV] L. Le Bruyn and M. Van den Bergh, On quantum spaces of Lie algebras, Proc. Amer. Math. Soc., Vol. 119, 2(1993), 407–414. [LeS] L. Le Bruyn and S.P. Smith, Homogenized sl(2), Proc. Amer. Math. Soc., Vol. 118, 3(1993), 725–730. [LVO] Li Huishi and F. Van Oystaeyen, Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals, Proc. Amer. Math. soc., Vol. 115, 1(1992), 1–11. [LVO1] Li Huishi and F. Van Oystaeyen, Zariskian Filtrations, Kluwer Academic Publishers, 1996. [LW] Li Huishi and Wu Yihong, Filtered-graded transfer of Groebner basis computation in solvable polynomial algebras, Comm. Alg. 1(28)(2000), to appear. [Mor] T. Mora, An introduction to commutative and noncommutative Groebner bases, Theoretical Computer Science, 134(1994), 131–173. [M-R] J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, 1987. [NVO] C. Nˇ astˇ asescu and F. Van Oystaeyen, Graded ring theoey, Math. Library 28, North Holland, Amsterdam, 1982.

21