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 noncommu-

tative 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 al-

gebras 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 A 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.

1

§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 = xjxi − {xj , xi}, n ≥ j > i ≥ 1,

where {xj , xi} =∑λkhji xkxh +

∑λlxl + c with λkhji , λl, c ∈ k. And we say that A has a PBW

k-basis if the set of standard monomials

{xi1xi2 · · · 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 = 〈X〉

the free semigroup with 1 generated by X, and let k〈S〉 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 lt1r < lt2r.

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 = lXj1r1, u = lXj2r2.

2

Given a monomial ordering > on S, each element f ∈ k〈S〉 has a unique ordered representation

as a linear combination of elements of S:

f =s∑

i=1

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

So to each nonzero element f ∈ k〈S〉 we can associate LM(f) = t1, the leading monomial of f ,

and LC(f) = c1, the leading coefficient of f .

If I ⊂ k〈S〉 is a two-sided ideal, the set

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) k〈S〉 = I ⊕ Spank(O(I)).

(ii) There is a k-vector space isomorphism between k〈S〉/I and Spank(O(I)).

(iii) For each f ∈ k〈S〉 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 ∈ k〈S〉, 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 setG = {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 ∈ k〈S〉:

f = Can(f, I) +∑ti=1 ciuigivi, ci ∈ k − {0}, ui, vi ∈ S, gi ∈ G,

LM(f) ≥grlex u1LM(g1)v1 >grlex · · ·

>grlex uiLM(gi)vi >grlex ui+1LM(gi+1)vi+1 >grlex · · · ;

3

(iii) For each f ∈ k〈S〉, f ∈ I if and only if

f =∑ti=1 ciuigivi, ci ∈ k − {0}, ui, vi ∈ S, gi ∈ G,

LM(f) = u1LM(g1)v1 >grlex · · ·

>grlex uiLM(gi)vi >grlex ui+1LM(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 ⊂ k〈S〉, it is generally difficult to know if

G is a Groebner basis of I. However, if k〈S〉 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 k〈S〉 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}, k〈S〉 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) = lgjr − λgiρ.

We say that S(i, j; l, r;λ, ρ) has a weak Groebner representation if

S(i, j; l, r;λ, ρ) =∑

k,µ

ckµlkµgkrkµ, and

for each k, µ, lkuLM(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 = l1wl and r = wrr1.

For s ≥ j ≥ 1, we write ST (LM(gj)) for the ideal of S ⊗ S generated by the set

SOB(LM(gj)) =

{(1, r) ∈ S ⊗ S

∣∣∣∣∣r 6= 1 and there is (l, 1) ∈ S ⊗ S

such that lLM(gj) = LM(gj)r

},

and we put

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 thatlσLM(gj)rσ = λσLM(giσ )ρσ, iσ < j or iσ = j and

there is w ∈ S such that rσ = wρσ,

and we let MIN(j) = {(iσ , j; lσ , rσ ;λσ, ρσ)}.

An element (i, j; l, r;λ, ρ) ∈ MIN(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 ⊂ k〈S〉 where LC(gi) = 1 for each i.

(i) The set

OBS(j) = {(i, j; l, r;λ, ρ) ∈MIN(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) ∈

MIN(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 k〈S〉 be the free k-algebra generated by X = {X1, ...,Xn} over k, where S = 〈X1, ...,Xn〉 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 = XjXi − {Xj ,Xi}, n ≥ j > i ≥ 1,

where {Xj ,Xi} =∑λkhji XkXh +

∑λlXl + c, λkhji , λl, c ∈ k.

Writing I = 〈Rji〉n≥j>i≥1 for the ideal of k〈S〉 generated byG = {Rji}n≥j>i≥1, then A = k〈S〉/I.

From now on we let > be a monomial ordering on S such that

(∗) LM(Rji) = XjXi, 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,

thenOBS(k) = {(h, k; l, r;λ, ρ) ∈MIN(k) and nontrivial}

= {(j, k; 1,Xi ;Xk, 1) | k > j > i} .

5

Proof Recall from the last section that

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)) =

{(1, r) ∈ S ⊗ S

∣∣∣∣∣r 6= 1 and there is (l, 1) ∈ S ⊗ S

such that lLM(gk) = LM(gk)r

}.

It is easy to see that SOB(LM(gk)) = {(1, wXkXj) | w ∈ S} and SOB(LM(gk)) is a mini-

mal generating set of ST (LM(gk)). It is also not hard to check that for each (1, wXkXj) ∈

SOB(LM(gk)) every (j, k; 1, wXkXj ;λ, ρ) ∈MIN(j) is trivial. Furthermore if (l, r) ∈ S ⊗ S is

such that

lXkXjr = λXtXiρ 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 MIN(k), or more precisely,

OBS(k) = {(h, k; l, r;λ, ρ) ∈ MIN(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 = k〈S〉/I = k[x1, ..., xn] has a PBW k-basis, where each xi is the image of

Xi in k〈S〉/I.

(ii) G = {Rji}n≥j>i≥1 is a Groebner basis of the ideal I = 〈Rji〉n≥j>i≥1 ⊂ k〈S〉.

(iii) For n ≥ k > j > i ≥ 1, every RkjXi −XkRji has a weak Groebner representation.

Proof (i) ⇔ (ii) Since XjXi = LM(Rji) for n ≥ j > i ≥ 1, this follows from Theorem 1.1.1.

(ii) ⇒ (iii) since every RkjXi −XkRji 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) ∈ MIN(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 RkjXi −XkRji 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 RkjXi − XkRji 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.

6

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 k〈S〉, A = k〈S〉/I be as in section 1.2, where I = 〈Rji〉n≥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 = XjXi − qjiXiXj − {Xj ,Xi}, qji ∈ k, n ≥ j > i ≥ 1,

where {Xj ,Xi} =∑

k,l

αkljiXkXl +∑

h

αhXh + c, αklji , αh, c ∈ k.

For n ≥ k > j > i ≥ 1, the q-Jacobi sum J(Xk,Xj ,Xi) is defined as

J(Xk,Xj ,Xi) = {Xk,Xj}Xi − qkiqjiXi{Xk,Xj}

−qji{Xk,Xi}Xj + qkjXj{Xk,Xi}

+qkjqki{Xj ,Xi}Xk −Xk{Xj ,Xi}.

Furthermore, we define two k-subspaces of k〈S〉 as in [Ber]:

E1 = k-Span{Rji

∣∣∣ n ≥ j > i ≥ 1},

E2 = k-Span{XiRji, RjiXi, XjRji, RjiXj

∣∣∣ 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} = XkXj − qkjXjXk −Rkj,

{Xk,Xi} = XkXi − qkiXiXk −Rki,

{Xj ,Xi} = XjXi − qjiXiXj −Rji,

and then we obtain

J(Xk,Xj ,Xi) = XkRji −RkjXi

+ qjiRkiXj − qkjXjRki − qkjqkiRjiXk + qkiqjiXiRkj,

which is obviously contained in I. Now we see that

RkjXi −XkRji = qjiRkiXj − qkjXjRki − qkjqkiRjiXk + qkiqjiXiRkj

− 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 RkjXi −XkRji 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 ∼= k〈S〉/I and I is generated

by

Rji = XjXi − qjiXiXj − {Xj ,Xi}, n ≥ j > i ≥ 1,

where {Xj ,Xi} =∑

k,l

αkljiXkXl +∑

h

αhXh + c, αklji , αh, c ∈ k,

satisfying if αklji 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) = XjXi, 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 k〈S〉.

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:

X2X1 = X1X2,

X3X1 = X1X3,

X3X2 = X2X3,

X4X1 = X1X4,

X4X2 = X2X4,

X4X3 = X3X4,

X5X1 = X1X5,

X5X2 = X2X5 +X3X4,

X5X3 = X3X5,

X5X4 = X4X5.

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

k〈X1, ...,X5〉, 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) = X3X4X1 −X1X3X4.

8

1.3.5. Example A = k[x1, x2, x3] subject to the defining relations

R21 = X2X1 −X1X2 −X21 − 1,

R31 = X3X1 −X1X3 −X21 − 1,

R32 = X3X2 −X2X3 − 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 k〈X1, ...,X5〉,

and hence A has a PBW k-basis, and A also satisfies the q-Jacobi condition since a calculation

shows:J(X3,X2,X1) = −X3X

21 +X2X

21 +X2

1X3 −X21X2

= −R31X1 +R21X1 −X1R31 +X1R21.

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,

xjxi = λjixixj +∑

xixj>grlexxkxh

λkhxkxh +∑

λlxl + 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 k〈X1, ...,Xn〉 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.

9

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

X2X1 = X1X2,

X3X1 = X1X3 +X1X2,

X3X2 = X2X3,

X4X1 = X1X4,

X4X2 = X2X4,

X4X3 = X3X4,

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 k〈X1,X2,X3,X4〉, 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) = −X4X1X2 +X1X2X4.

1.4.3. Example A = k[x1, x2, x3, x4] subject to the defining relations

R21 = X2X1 −X1X2 − 1,

R31 = X3X1 −X1X3 − 1,

R32 = X3X2 −X2X3 − 1,

R41 = X4X1 −X1X4 − 2X23 + 1,

R42 = X4X2 −X2X4 − 2X23 − 1,

R43 = X4X3 −X3X4 − 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 k〈X1,X2,X3,X4〉, 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) = −2X23X2 + 2X2

3X1 + 2X2X23 − 2X1X

23

= −2X3R32 + 2X3R31 − 2R32X3 + 2R31X3.

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.

10

§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∗ = {g∗1 , ..., g∗s} is a Groebner basis

for I∗ ⊂ k[x0, x1, ..., xn], where g∗i 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 FA on A, we

give a clear description of the Rees algebra A and the associated graded algebra G(A) of A by

defining ideals, and from this we also clearly see the geometric meaning of A 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 A 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 A and G(A) can be determined from that of

A.

2.1. A description of A 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 = 〈X〉 the free semigroup with 1 generated by X, and let

k〈S〉 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 k〈S〉 is given by the k-subspaces:

k〈S〉p =

∑

d(w)=p

cww

∣∣∣∣ cw ∈ k, w ∈ S

, p ≥ 0,

11

i.e., k〈S〉 = ⊕p≥0k〈S〉p. If f ∈ k〈S〉, 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 FA on A consists of k-subspaces of A:

FpA =

∑

i1+···in≤p

cαxi1α1

· · · xinαn

∣∣∣∣ 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(FpA/Fp−1A), and the Rees k-algebra

of A which is by definition the graded ring A = ⊕p≥0FpA. For each a ∈ FpA− Fp−1A, we write

ap for the homogeneous element of degree p in Ap = FpA represented by a. If X denotes the

homogeneous element of degree 1 in A1 = F1A represented by 1A, then X is in the centre of

A and it is not a divisor of zero. Moreover, each homogeneous element of degree p ≥ 1 can

be uniquely written as Xp−hah where h ≤ p. Thus, it is easy to see that A ∼= A/〈1 − X〉A,

G(A) ∼= A/XA. The importance of studying G(A) for A and A is well known in the literature

(e.g. [M-R], [LVO1]).

To obtain the main result of this part, in this section we first give a clear description of A

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 k〈S〉 be the free k-algebra with natural gradation, where S = 〈X〉

is the free semigroup generated by X = {Xα}α∈Λ. Considering the polynomial ring k〈S〉[t] over

k〈S〉 in commuting variable t, there is a ring epimorphism φ: k〈S〉[t] → k〈S〉 with φ(t) = 1.

Hence kerφ = 〈1 − t〉 where the latter is the ideal of k〈S〉[t] generated by 1 − t. If furthermore

we consider the “mixed” gradation on k〈S〉[t] which is defined by putting

k〈S〉[t]p =

∑

i+j=p

Fitj

∣∣∣∣ Fi ∈ k〈S〉i

, p ≥ 0,

then we have the following observation:

• For every p ≥ 0k〈S〉[t]p + 〈1 − t〉

〈1 − t〉⊂k〈S〉[t]p+1 + 〈1 − t〉

〈1 − t〉,

and ⋃

p≥0

k〈S〉[t]p + 〈1 − t〉

〈1 − t〉=k〈S〉[t]

〈1 − t〉∼= k〈S〉.

• For every f ∈ k〈S〉, there exists a homogeneous element F ∈ k〈S〉[t]p, for some p, such

that φ(F ) = f . More precisely, if f = F0 + F1 + · · · + Fp where Fi ∈ k〈S〉i, then

12

F = tpF0 + tp−1F1 + · · · + tFp−1 + Fp is a homogeneous element in k〈S〉[t]p satisfying

φ(F ) = f .

2.1.1. Definition (i) For any F ∈ k〈S〉[t], we write F∗ = φ(F ). F∗ is called the dehomogeniza-

tion of F with respect to t.

(ii) For any f ∈ k〈S〉, if f = F0 + F1 + · · · + Fp, then the homogeneous element f∗ = tpF0 +

tp−1F1 + · · · + tFp−1 + Fp in k〈S〉[t]p is called the homogenization of f with respect to t.

(iii) If I is a two-sided ideal of k〈S〉, then we let I∗ stand for the graded two-sided ideal of k〈S〉[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 ∈ k〈S〉[t], (F +G)∗ = F∗ +G∗, (FG)∗ = F∗G∗.

(ii) For f, g ∈ k〈S〉, (fg)∗ = f∗g∗, ts(f + g)∗ = trf∗ + thg∗, where r = d(g), h = d(f), and

s = r + h− d(f + g).

(iii) For any f ∈ k〈S〉, (f∗)∗ = f .

(iv) If F is a homogeneous element in k〈S〉[t], then tr(F∗)∗ = F , where r = d(F ) − d((F∗)

∗).

(v) If I is a two-sided ideal of k〈S〉, then each homogeneous element F ∈ I∗ is of the form trf∗

for some f ∈ I.

2

2.1.3. Proposition Let I be a proper two-sided ideal of k〈S〉. Then there is a ring epimorphism

α: k〈S〉[t]/I∗ → k〈S〉/I with Kerα = 〈1 − t〉, where t denotes the coset of t in k〈S〉[t]/I∗.

Moreover, t is a regular element in k〈S〉[t]/I∗ (i.e. t is not a divisor of zero), and hence 〈1 − t〉

does not contain any nonzero homogeneous element of k〈S〉[t]/I∗.

Proof If we define α by putting

k〈S〉[t]/I∗α

−→ k〈S〉/I

F + I∗ 7→ F∗ + I, F ∈ k〈S〉[t],

then by Lemma 2.1.2 we easily see that α is a ring epimorphism with Kerα = 〈1 − t〉.

For any homogeneous element F ∈ k〈S〉[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

k〈S〉[t]/I∗. The fact that 〈1−t〉 does not contain any nonzero homogeneous element of k〈S〉[t]/I∗

easily follows from the regularity of t. 2

Now let us consider the standard filtration Fk〈S〉 on k〈S〉 which is by definition given by the

k-subspaces:

Fpk〈S〉 = ⊕i≤pk〈S〉i, p ≥ 0.

13

If I is any two-sided ideal of k〈S〉 and we put A = k〈S〉/I, then Fk〈S〉 gives a filtration FA on

A:

FpA = (Fpk〈S〉 + I)/I, p ≥ 0.

Indeed, FA coincides with the standard filtration on the k-algebra A = k〈S〉/I = k[Xα]α∈Λ

where Xα is the coset of Xα in k〈S〉/I.

If we consider the associated graded ring G(A) = ⊕p≥0(FpA/Fp−1A) of A and the Rees ring

A = ⊕p≥0FpA of A, then the following proposition shows that G(A) and A are determined by

I∗.

2.1.4. Proposition With notation as above, there are graded k-algebra isomorphisms:

(i) A ∼= k〈S〉[t]/I∗, and

(ii) G(A) ∼= k〈S〉[t]/(〈t〉 + I∗), where 〈t〉 denotes the ideal of k〈S〉[t] generated by t.

Proof Using the ring homomorphism α of Proposition 2.1.3 and the regularity of t in k〈S〉[t]/I∗,

we have an easily verified graded ring isomorphism α:

k〈S〉[t]/I∗ =⊕

p≥0

k〈S〉[t]p + I∗

I∗α

−→⊕

p≥0

Fpk〈S〉 + I

I= A

F + I∗ 7→ F∗ + I, F ∈ k〈S〉[t]p

Note that since α sends the central regular element t of degree 1 to the canonical central regular

element of A which is by definition the image of 1A in F1A via the inclusion map F0A ⊂ F1A,

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

FA on A, the defining relations of the Rees algebra A 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 = k〈S〉/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 FA be the

standard filtration on A, and let G(A) and A be the associated graded algebra and Rees algebra

of A with respect to FA, 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 A 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 ] =∑nh=1 cij,hxh, and let A = U(g) be the enveloping algebra of g with the standard filtration

FU(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

defining relations are [xi, xj ] =∑nh=1 cij,hxhx0. This new algebra looks very like the Rees

algebra of U(g), namely, we also have H(g)/〈1 − x0〉H(g) ∼= U(g), H(g)/x0H(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 FA,

the defining relations of G(A) resp. A 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 = 〈f1, f2〉 = 〈x2 − x21, x3 − x3

1〉, the ideal of the affine twisted cubic in

IR3. If we homogenize f1, f2, then we get the ideal J = 〈x2x0−x21, x3x

20−x

31〉 in IR[x0, x1, x2, x3].

One may directly check that for f3 = f2 − x1f1 = x3 − x1x2 ∈ I, f∗3 = x3x0 − x1x2 6∈ J , i.e.,

J 6= I∗.

(ii) Let k〈S〉 be the free k-algebra generated by {X1,X2,X3}, and let f = 2X3X2X1 − 3X1X23 ,

g12 = X2X1 − X1X2, g13 = X3X1 − X1X3, g23 = X3X2 − X2X3. Considering the two-sided

ideal I = 〈f, g12, g13, g23〉, then it can be directly verified that h = −3X1X23 + 2X1X2X3 =

f − 2X3g12 + 2g13x2 + 2X1g23 ∈ I∗, but h 6∈ 〈f∗, g∗12, g∗13, g

∗23〉 (note that the latter is equal to

〈f, g12, g13, g23〉 in k〈S〉[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

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 A 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 ∈ k〈S〉 we denote by LH(f) the highest homogeneous part of f ,

i.e., if f = F0 + F1 + · · · + Fp with Fi ∈ k〈S〉i, then LH(f) = Fp. If I is an ideal of k〈S〉, we

denote by 〈LH(I)〉 the ideal generated by {LH(f) | f ∈ I} in k〈S〉.

2.2.1. Proposition Let A = k〈S〉/I and FA the standard filtration on A. Then G(A) ∼=

k〈S〉/〈LH(I)〉.

Proof From Proposition 2.1.4 we have known that G(A) ∼= k〈S〉[t]/(〈t〉 + I∗). To prove the

theorem, we first recall that if f = F0 + F1 + · · · + Fp ∈ k〈S〉, then LH(f) = Fp and f∗ =

LH(f) + tFp−1 + · · ·. Hence the inclusion map k〈S〉 → k〈S〉[t] yields the inclusion 〈LH(f)〉 ⊂

〈t〉 + I∗. This, in turn, gives a graded ring homomorphism

k〈S〉

〈LH(f)〉

ϕ−→

k〈S〉[t]

(〈t〉 + I∗)

g + 〈LH(f)〉 7→ g + (〈t〉 + I∗)

Obviously, ϕ is surjective. On the other hand, each element F ∈ k〈S〉[t] has a unique presentation

F = F0 + F ′ where F0 ∈ k〈S〉, F ′ ∈ 〈t〉. Moreover, from §2 we know that each homogeneous

element in I∗ is of the form trf∗ for some f ∈ I. If f = Fp +Fp−1 + · · ·+F0 with LH(f) = Fp,

then f∗ = LH(f) + tFp−1 + · · · + tpF0. Therefore, each element of 〈t〉 + I∗ can be written as a

sum u+ v, where u ∈ 〈LH(f)〉, v ∈ 〈t〉. Thus we can define a ring homomorphism

k〈S〉[t]

(〈t〉 + I∗)

ψ−→

k〈S〉

〈LH(I)〉

F + (〈t〉 + I∗) 7→ F0 + 〈LH(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 k〈S〉[t] → k〈S〉 which sends t to

0.) 2

16

Suppose I = 〈fi〉i∈J . From the above proposition we certainly expect that 〈LH(I)〉 =

〈LH(fi)〉i∈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 = d(f) has a presentation as a finite sum f =∑i gifihi, where gi, hi ∈ k〈S〉, and d(gi) +

d(fi) + d(hi) ≤ p for all i.

Let I be a graded ideal of k〈S〉, i.e., I = ⊕p≥0(I ∩ k〈S〉p). 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 ∈ k〈S〉, fi ∈ {fi}i∈J . Indeed, we have the more stronger

result: {fi}i∈J is a standard basis if and only if 〈LH(fi)〉i∈J = 〈LH(I)〉.

Proof If {fi}i∈J is a standard basis of I, then by the definition it is easy to see that for

any f ∈ I, LH(f) =∑

LH(gi)LH(fi)LH(hi) for some gi, hi ∈ k〈S〉, fi ∈ {fi}i∈J . Hence

〈LH(fi)〉i∈J = 〈LH(I)〉.

Conversely, if 〈LH(fi)〉i∈J = 〈LH(I)〉, then for any f ∈ I with d(f) = p, say f = fp+ fp−1 + · · ·

with fi ∈ k〈S〉i, we have LH(f) = fp =∑gjLH(fj)hj for some gj, hj ∈ k〈S〉, fj ∈ {fi}i∈J , and

d(gj) + d(LH(fj)) + d(hj) = d(gj) + d(fj) + d(hj) = p. Now the element f ′ = f −∑gjfjhj ∈ I

has d(f ′) < p, we may repeat the above argumentation and after a finite number of steps we will

reach a presentation f =∑gifihi where gi, hi ∈ k〈S〉, 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 〈LH(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 {f∗i }i∈J , or in other words, A has defining relations tXα−Xαt = 0, α ∈ Λ,

f∗i = 0, i ∈ J ; and moreover

(ii) {f∗i }i∈J is a standard basis of I∗.

Proof By §2, each homogeneous element in I∗ is of the form trf∗ for some f ∈ I. Suppose

f =∑i hifigi. 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(f∗i ) + d(g∗i ) ≤ d(f∗) and

f∗ −∑i h

∗i f

∗i g

∗i = tr1m∗

1 + tr2m∗2 + · · · with

rj > 0, mj ∈ I, and d(trjm∗j) ≤ d(f∗) for all mj.

Similarly, for each m∗j ∈ I∗ where mj =

∑i hijfijgij , we have

d(h∗ij ) + d(f∗ij ) + d(g∗ij ) ≤ d(m∗j ) and

m∗j −

∑i h

∗ijf∗ijg

∗ij

= tr1jm∗1j

+ tr2jm∗2j

+ · · · with

rkj> 0, mkj

∈ I, and d(trkjm∗

kj) ≤ d(m∗

j ) for all mkj.

Since d(f∗) is finite, after a finite number of steps we will reach f∗ ∈ 〈f∗i 〉i∈J , in particular,

f∗ =∑j h

∗jf

∗j g

∗j with d(h∗j ) + d(f∗j ) + d(g∗j ) ≤ d(f∗) for all j. (Note that I is a proper ideal, the

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

I∗ = 〈f∗i 〉i∈J . Then since each element F of I∗ is of the form F =∑iHif

∗i Gi, Hi, Gi ∈ k〈S〉[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

{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 ∼= k〈X1, ...,Xn〉/I with I = 〈fi〉i∈J . Considering the standard filtration FA

on A, it is easily seen from the conditions (a) and (b) that the set

{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 F1A/F0A = G(A)1.

Then it follows from Proposition 2.1.4 and Proposition 2.2.1 that the natural graded k-algebra

homomorphism k〈X1, ...,Xn〉/〈LH(fi)〉i∈J → k〈X1, ...,Xn〉/〈LH(I)〉 is an isomorphism. Hence

〈LH(fi)〉i∈J = 〈LH(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 k〈S〉 the free k-algebra

where S = 〈X1, ...,Xn〉 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 ⊂ k〈S〉 is a Groebner basis in

the sense of [Mor], then for the k-algebra A = k〈S〉/I with the standard filtration FA, where

I = 〈G〉 is the two-sided ideal of k〈S〉 generated by G, the defining relations of A 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 = k〈S〉/I. Then G(A) has the defining relations LH(fi) = 0, i ∈ J ; and A has the defining

relations tXj −Xjt = 0, j = 1, ..., n, f∗i = 0, i ∈ J .

2

Furthermore, let us consider the k-basis

B(S, t) ={wtr

∣∣∣ w ∈ S, r ≥ 0}

for k〈S〉[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 k〈S〉[t]: w1tr1 ≻ w2t

r2 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 k〈S〉[t]

exactly as in k〈S〉.

Before giving the next theorem we also note that for any nonzero f ∈ k〈S〉, 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

k〈S〉. The following are equivalent:

(i) G is a Groebner basis of I;

(ii) {f∗i }i∈J is a Groebner basis of I∗ in k〈S〉[t];

(iii) {LH(fi)}i∈J is a Groebner basis of the two-sided ideal 〈LH(I)〉 in k〈S〉.

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 k〈S〉 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 A 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

YjXi −XiYj = δij , XjXi −XiXj = YjYi −XiYj = 0. As in part 1 of the paper, it can be easily

verified that G = {YjXi−XiYj−δij , XjXi−XiXj , YjYi−YiYj}ni,j=1 is a Groebner basis. Hence

Theorem 2.3.1 can be used to give the defining relations of A and G(A).

(ii) Let g = kx1 ⊕ · · · ⊕ kxn be the n-dimensional Lie algebra over k with defining relations

[Xi,Xj ] =∑nh=1 λij,hxh 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 = {XjXi −XiXj − [Xi,Xj ]}j>i is a Groebner basis.

Hence Theorem 2.3.1 can be used to give the defining relations of A 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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