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
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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,
(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 ,