Bull. Sci. math. 125, 8 (2001) 689–715  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0007-4497(01)01090-9/FLA RE-FILTERING AND EXACTNESS OF THE GELFAND–KIRILLOV DIMENSION BY J.L. BUESO, J. GÓMEZ-TORRECILLAS, F.J. LOBILLO 1 Dept. Algebra, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain Manuscript presented by M.P. M ALLIAVIN, received in February 2001 A BSTRACT . – We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Our methods apply to algebras of current interest like the quantized enveloping algebras, iterated differential operators algebras, quantum matrices or quantum Weyl algebras.  2001 Éditions scientifiques et médicales Elsevier SAS Keywords: Multi-filtration; Gelfand–Kirillov dimension; Exactness; Finitely partitive algebra; Semi-commutative algebra Introduction Let k be a field. It is still an open problem whether the Gelfand– Kirillov dimension is exact for finitely generated modules over any noetherian k-algebra. Some partial positive answers were given by T.H. Lenagan [13] for noetherian PI-algebras and by P. Tauvel [27, Théorème 4.4] for finitely filtered algebras with finitely generated and (left) noetherian associated graded algebra. For the case of the quantum enveloping algebras associated to Cartan matrices of type AN McConnell [19] found an infinite dimensional N2 -filtration with E-mail addresses: jbueso@ugr.es (J.L. Bueso), torrecil@ugr.es (J. Gómez-Torrecillas), jlobillo@ugr.es (F.J. Lobillo). 1 Research supported by the grant PB97-0806 from DGES. 690 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 semi-commutative associated graded algebra for which some results from [21] could be applied. In [8] this idea was systematized in order to endow some algebras R with an Nn -filtration is such a way that the corresponding generalization of Tauvel’s result holds, namely, if the associated Nn -algebra G(R) is left noetherian and finitely generated, then the Gelfand–Kirillov dimension is exact, whenever the filtering vector subspaces are finite-dimensional. In particular, this applies when G(R) is semi-commutative. This is the case for algebras like quantum matrices, quantum Weyl algebras, iterated differential operator algebras and the quantum enveloping algebra Uq (C) associated to a Cartan matrix C of type AN , but the author was unable to cover the case when C is not of type AN , due to the fact that the arguments developed there do not work when the filtering vector subspaces are not finite-dimensional. Here, we prove that any multi-filtered algebra R with semi-commutative associated graded algebra G(R) can be re-filtered by finite-dimensional subspaces keeping up the semi-commutativity on the new associated graded algebra. Therefore, the theory developed in [8,9] and [27], can be applied to get the exactness of Gelfand–Kirillov dimension and that the algebra is left and right finitely partitive in the sense of [20, 8.13.17]. This applies in particular to the quantized enveloping algebras because they were multifiltered (by infinite dimensional subspaces) in [6]. Therefore, Uq (C) enjoys the exactness property for the Gelfand–Kirillov dimension, and it is finitely partitive, as conjectured J.C. McConnell in [19]. The scope of our methods is explored in Section 3, where we point out further properties of multi-filtered algebras with semi-commutative associated multi-graded algebra. During our re-filtering process we discover that multi-filtered algebras R such that G(R) is semi-commutative are just those algebras generated by finitely many elements x1 , . . . , xs that satisfy relations of the type xj xi = qj i xi xj + pj i , where the qj i ’s are non-zero scalars and the pj i ’s are polynomials of total weighted degree strictly less than the corresponding degree of xi xj . Moreover, all generators have strictly positive degree. The simplex algorithm is used in the Appendix to compute explicitly these degrees for Uq (B2 ). For the notion and properties of Gelfand–Kirillov dimension, the reader is referred to [12,15,20]. J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 691 1. Multi-filtrations and bounded quantum algebras Throughout this paper, K will denote a commutative ring, and R, A and B will be associative and unitary K-algebras. By k we shall denote a (commutative) field. Let n be a positive integer and consider Nn the additive monoid consisting of all n-tuples α = (α1 , . . . , αn ), where αi ∈ N, for every i = 1, . . . , n. For n = 0, N0 = {0} is the trivial monoid. D EFINITION 1.1. – An admissible order  on Nn is a total order such that, 1. 0 = (0, . . . , 0)  α for every α ∈ Nn . 2. For all α, β, γ ∈ Nn with α  β it follows α + γ  β + γ . Remark 1.2. – By Dickson’s Lemma (see, e.g., [1, Corollary 4.48]), admissible orders on Nn are well orderings (i.e., any nonempty subset of Nn has a first element). Let  be an admissible order on Nn . Observe that, for n > 1 there are uncountably infinitely many admissible orders. A basic example is the lexicographical order
lex on Nn with ε 1 <lex · · · <lex ε n , where ε i denotes the ith vector in the canonical basis of Rn i.e., ε i = (0, . . . , 1 , . . . , 0). (i) We shall recall the notion of multi-filtration, as used in [8]. D EFINITION 1.3. – An (Nn , )-filtration on R is a family F = Fα (R) | α ∈ Nn
 of K-submodules of R satisfying the following axioms. 1. Fα (R) ⊆ Fβ (R) for all α  β ∈ Nn . 2. F (R)Fβ (R) ⊆ Fα+β (R) for all α, β ∈ Nn . α 3. α∈Nn Fα (R) = R. 4. 1 ∈ F0 (R). We will say that R is an (Nn , )-filtered K-algebra or, sometimes, that R is a multi-filtered K-algebra. For n = 1 the foregoing definition recovers the usual notion of N-filtered algebra, because the only admissible order on N is the usual one. 692 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 The multi-filtered algebra R has an associated Nn -graded K-algebra given as follows. For each α ∈ Nn , define Fα− (R) =  Fβ (R) β≺α and GF α (R) = Fα (R) Fα− (R) (it is understood that F0− (R) = {0}). Consider the Nn -graded K-module  n F G (R) = α∈Nn GF α (R). For every r ∈ R, define exp(r) ∈ N as the n least α ∈ N such that r ∈ Fα (R). Every homogeneous element in GF (R) − (R). The product of homogeneous elements in is of the form r + Fexp(r) GF (R) is given by − − − r + Fexp(r) (R) s + Fexp(s) (R) = rs + Fexp(r)+exp(s) (R)      for r, s ∈ R. With this product, GF (R) becomes an Nn -graded algebra over K called the associated graded algebra of R with respect to F . We will need the following easy lemma. L EMMA 1.4. – Let r1 , . . . , rt be elements in an (Nn , )-filtered algebra R and suppose that α 1 , . . . , α t ∈ Nn are such that ri ∈ Fα i (R) for i = 1, . . . , t. The product in GF (R) of the elements ri + Fα− (R) is computed as follows r1 + Fα−1 (R) · · · rt + Fα−t (R) = r1 · · · rt + Fα−1 +···+α t (R).     Proof. – A straightforward argument by induction on t reduces the proof to the case t = 2. Now, if α i = exp(ri ) for i = 1, 2 then the desired equality is just the definition of the product in GF (R). If α i = exp(ri ) for some i = 1, 2, then ri ∈ Fα−i (R) for such index i. It follows easily that the equality is just 0 = 0. ✷ We are interested in multi-filtered algebras over a field for which the associated graded algebra is semi-commutative in the sense of [19]. D EFINITION 1.5. – Let A be an algebra over a field k. We will say that two elements x, y of an algebra A are semi-commuting if there is J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 693 0 = q ∈ k such that yx = qxy. An Nn -graded k-algebra A is called semi-commutative if A is generated as a k-algebra by a finite set of homogeneous semi-commuting elements. The following is one of our motivating examples. Example 1.6. – Let U = Uq (C) be the quantum enveloping algebra in the sense of [16,7] associated to a Cartan matrix C. This is an algebra over C(q), where q is an indeterminate. Following [6], U is endowed with a multi-filtration, say F , with semi-commutative associated graded algebra [7, Proposition 10.1]. In this case the first filtering subspace F0 (U ) is not finite-dimensional over the ground field and, therefore, no filtering subspace is finite-dimensional. This is a serious obstruction to the lift of good properties of the Gelfand–Kirillov dimension from GF (U ) to U . However, we will show that U can be N-filtered (in the usual sense) by finite-dimensional vector subspaces in such a way that the new associated graded algebra still being semi-commutative (see Theorem 2.3). The aim of this paper is to prove the following T HEOREM 1.7. – Let R be any (Nn , )-filtered k-algebra with semicommutative associated Nn -graded algebra. Then the Gelfand–Kirillov dimension is exact on R and R is left and right finitely partitive. This theorem has been recently proved in [8, Corollary 2.12] and [9, Theorem 2.9] under the additional conditions that the multi-filtration is pointed (i.e., F0 (R) = k) and locally finite in the sense that Fα (R) is finite-dimensional over k for every α ∈ Nn . Our strategy will be to refilter R in such a way that the referred results are applicable. Our first results are of technical nature, but they are fundamental for our theoretical development and for providing new examples of multifiltered algebras. The main notion here is that of bounded quantum algebra over a field, but we will start with a more general definition which, we think, is interesting. Let B be a K-algebra extension of a given K-algebra A and let x1 , . . . , xs be elements in B. The monomials of the form xα = x1α1 · · · xsαs , where α = (α1 , . . . , αs ) ∈ Ns will be called standard monomials in x1 , . . . , xs . Consider 1 and 2 two admissible orders in Nm and Ns , respectively, and suppose that A is an (Nm , 1 )-filtered K-algebra with multi-filtration F = {Fα (A) | α ∈ Nm }. 694 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 D EFINITION 1.8. – The extension A ⊆ B is said to be a (left) 2 -bounded extension of the multi-filtered algebra A with respect to x1 , . . . , xs if 1. B is generated as a ring by x1 , . . . , xs and A. 2. For every α ∈ Nm and every i = 1, . . . , s, (1) xi Fα (A) ⊆ Fα (A)xi +  γ ≺2 Axγ . εi 3. For every 1
i < j
s, there exists qj i ∈ F0 (A) such that (2) xj xi − qj i xi xj ∈  Axγ . γ ≺2 ε i +ε j Example 1.9. – Following [20, 1.6.10] the extension A ⊆ B is said to be almost normalizing if the generators x1 , . . . , xs satisfy 1. xi A + A = Axi + A, 2. xj xi − xi xj ∈ sf =1 Axf + A. Let
deglex denote the degree lexicographical order on Ns , i.e., first order by total degree and then lexicographically, if needed. It is clear that B is a
deglex -bounded extension, where A is assumed to be trivially N0 -filtered. Example 1.10. – Following [20, 8.6.6] the finitely generated extension A ⊆ B is said to be almost centralizing extension if for every r ∈ A and for every 1
i, j
s, 1. rxi − xi r ∈ A, 2. xj xi − xi xj ∈ sf =1 Axf + A. It is clear that B is a
deglex -bounded extension of A for any given multifiltration over A. A typical example of this situation is the crossed product A ∗ U (g), where g is a finite-dimensional Lie algebra over a field (see [20, 1.7.12] for the definition and examples of crossed products). Example 1.11. – We say that B is an iterated differential operator K-algebra over A, if we have a sequence of K-algebra extensions A = B0 ⊆ B1 ⊆ · · · ⊆ Bs = B and for each i = 1, . . . , s we have a K-linear derivation δi over Bi−1 such that Bi = Bi−1 [xi ; δi ] is an Ore extension. We will use the notation B = A[x1 ; δ1 ] · · · [xs ; δs ]. J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 695 The commutation rules in B are 1. xi r = rxi + δi (r) for every i = 1, . . . , x and every r ∈ A. 2. xj xi = xi xj + δj (xi ) for 1
i < j
s. Then B is a
lex -bounded extension for any given multi-filtration on A. For K = Z we obtain the iterated differential operator rings of [26]. Every multi-filtered algebra is (trivially) a bounded extension. Our next objective is to prove that every bounded extension A ⊆ B can be multifiltered in some useful way. Let us define the following admissible order 3 on Nm+s : α 1 , α 2 3 β 1 , β 2 ⇐⇒      2 2   α ≺2 β   or α 2 = β 2 and α 1 1 β 1 . At this point we have to be extremely careful with the notation. The letters q, r, s, t with subscripts and/or superscripts represent elements in A. The notation r α , where α ∈ Nm , means r α ∈ Fα (A), while rα just an element in A not necessarily belonging to Fα (A). However, xβ still represents the β standard monomial x1 1 · · · xsβs . Let F(α 1 ,α 2 ) (B) be the additive subgroup 1 2 1 of B generated by the elements of the form r β xβ where r β ∈ Fβ 1 (A) and (β 1 , β 2 ) 3 (α 1 , α 2 ). Notice that F(α 1 ,α 2 ) (B) is a K-submodule of B. L EMMA 1.12. – Every f ∈ F(α 1 ,α 2 ) (B) can be written as 1 2 f = r α xα +  2 rβ 2 xβ . β 2 ≺2 α 2 1 − α ∈ / Fα−1 (A). Moreover, if f ∈ / F(α 1 ,α 2 ) (B) then r Proof. – An element f in F(α 1 ,α 2 ) (B) is, by definition, a sum of 2 1 ‘monomials’ of the form r β xβ , for (β 1 , β 2 ) 3 (α 1 , α 2). For such an element, either β 2 ≺2 α 2 or β 2 = α 2 . In the first case, the monomial 2 1 r β xβ contributes to the second summand in the desired expression of f . In the second case, β 1 1 α 1 and it is clear that the corresponding 1 coefficient r β ∈ Fα 1 (R), so that it contributes to the first summand in the 1 2 2 1 just proved expression f = r α xα + β 2 ≺2 α 2 rβ 2 xβ . If r α ∈ Fβ 1 (A) for − some β 1 ≺1 α 1 then f ∈ F(β 1 ,α 2 ) (B) ⊆ F(α 1 ,α 2 ) (B), so the second part in the lemma follows. ✷ 696 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 P ROPOSITION 1.13. – With the above notations, F = F(α 1 ,α 2 ) (B) | α 1 , α 2 ∈ Nm+s
   is an (Nm+s , 3 )-filtration on B. In particular, every element f ∈ B can be represented as 2 1  f = r α xα + 2 rβ 2 xβ , β 2 ≺2 α 2 where (α 1 , α 2 ) = exp(f ). Proof. – Clearly 1 ∈ F0 (B) and (α 1 , α 2 )  (β 1 , β 2 ) implies F(α 1 ,α 2 ) (B) ⊆ F(β 1 ,β 2 ) (B). We shall prove that for every (α 1 , α 2 ), (β 1 , β 2 ) ∈ Nm+s , it follows (3) F(α 1 ,α 2 ) (B) · F(β 1 ,β 2 ) (B) ⊆ F(α 1 +β 1 ,α 2 +β 2 ) (B). We proceed by induction on α 2 + β 2 . The case α 2 = 0 is clear, which gives in particular the first step of the induction for α 2 + β 2 = 0. So, assume α 2 = 0. Let us introduce some new notation: Let α ∈ Ns be a nonzero vector; by f↓α we shall denote an element in F(γ ,δ)(B), where δ ≺2 α and γ ∈ Nm (we define f↓0 = 0). The meaning of the ′′ ′ , g↓α , . . . is the same. Thus, Lemma 1.12 tell us that , f↓α notations f↓α any element in F(α 1 ,α 2 ) (B) (resp. in F(β 1 ,β 2 ) (B)) can be represented as 2 1 2 1 f = r α xα + f↓α 2 (resp. g = r β xβ + g↓β 2 ). Now, by induction, 1 2 1 2 1 2 1 2 1 1 2 2 f g = r α xα r β xβ + r α xα g↓β 2 + f↓α 2 r β xβ + f↓α 2 g↓β 2 = r α xα r β xβ + h↓α 2 +β 2 . 1 2 Therefore, to prove f g ∈ F(α 1 +α 2 ,β 1 +β 2 ) (B) we can assume f = r α xα 2 i 2 2 1 and g = r β xβ without loss of generality. We have xα = xi xα −ε for some 1
i
s. Then 2 1 xα r β = xi xα 2 −ε i 1 rβ = xi s β xα   1 2 −ε i β1 + f↓α 2 −εi  α 2 −ε i = si xi + f↓εi x β1 = si xi xα β1 2 2 −ε i + f↓εi xα ′ = si xα + f↓α 2. (induction)  + xi f↓α 2 −εi 2 −ε i by (1) + xi f↓α 2 −εi (induction) J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 697 Therefore, by using (1) and induction, 1 (4) 2 1 2 1 β1 2 ′ β r α xα r β xβ = r α si xα + f↓α 2 x 1  β1 2  2 2 = r α si xα xβ + f↓α 2 +β 2 . 2 2 Let us see now what happens with xα xβ . We call h (resp. i) the index corresponding to the first non zero component of α 2 (resp. β 2 ). Analogously, let j (resp. k) be the index corresponding to the last non zero component of α 2 (resp. β 2 ). We are going to study several cases. Case h
i: Then 2 2 xα xβ = xh xα 2 −ε h = xh r 0 xα xβ 2 2 −ε h +β 2 + xh g↓α 2 −εh +β 2 (induction) α 2 −ε h +β 2 by (1) = rh0 xh + g↓εh x  α 2 +β 2 = rh0 x  + xh g↓α 2 −εh +β 2 (induction) + g↓α 2 +β 2 . 2 2 2 2 k Case j
k: Write xα xβ = xα xβ −ε xk and proceed similarly to case h
i. Case h > i and j > k: Note that in this case i < j . We have 2 −ε i xα xβ = xα 2 −ε j xj xi xβ = xα 2 −ε j (qj i xi xj + g↓εi +εj )xβ 2 2 = q 0 xα  2 −ε j 2 −ε i + xα = q0x = q 0 xα 2 −ε j + g↓α 2 −εj xi xj xβ g↓εi +εj xβ α 2 −ε j  2 −ε i  2 −ε j 2 −ε j  (induction) β 2 −ε j + g↓α 2 −εj xi xj x xi xj xβ by (2) 2 −ε i + g↓α 2 +β 2 + g↓α 2 −εj xi xj xβ (induction) 2 −ε i + g↓α 2 +β 2 = q 0 t 0 xα  2 −ε j +ε i β ′ + g↓α 2 −ε j +ε i xj x + g↓α 2 −εj +εi xj xβ = q 0 t 0 xα 2 −ε j +ε i 2 −ε i xj xβ β ′′ + g↓α 2 −ε i +ε j xj x  + g↓α 2 +β 2 2 −ε i 2 −ε i + g↓α 2 +β 2 2 −ε i (∗) 698 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 = q 0 t 0 xi xα 2 −ε j xj xβ 2 −ε i ′′ + g↓α 2 −ε j +ε i g↓β 2 −ε i +ε j + g↓α 2 +β 2 = q 0 t 0 xi xα xβ 2 2 −ε i = q 0 t 0 xi s 0 xα 2 +β 2 −ε i  α 2 +β 2 −ε i ′′ + g↓α 2 +β 2 −εi + g↓α 2 +β 2 2 +β 2  ′′′ + g↓α 2 +β 2 α 2 +β 2 −ε i = q 0 t si0 xi + g↓εi x = q 0 t 0 si0 xα (induction) ′ + g↓α 2 +β 2 + g↓α 2 +β 2 = q 0 t 0 xi s 0 x  0 (∗∗)  (induction) (induction) by (1) ′′′ + g↓α 2 +β 2 + h↓α 2 +β 2 . (∗ ∗ ∗) Equality (∗) holds by induction, since 0 2 β 2 − ε i and, thus, α 2 − ε j + ε i ≺2 α 2 + ε i 2 α 2 + β 2 . To see equality (∗∗), consider that 2 j 2 j i xα −ε +ε = xi xα −ε because i < h, and that β 2 − ε i + ε j ≺2 α 2 + β 2 allows the use of the induction hypothesis. Equality (∗ ∗ ∗) follows by 2 2 2 2 i induction and from the fact that xi xα +β −ε = xα +β because i < h. In all cases we have 2 2 xα xβ = r 0 xα (5) 2 +β 2 + g↓α 2 +β 2 . By replacing (5) into (4), we obtain 1 2 1 2 1 β 1  0 α 2 +β 2 r α xα r β xβ = r α si r x  + g↓α 2 +β 2 + f↓α 2 +β 2 which belongs to F(α 1 +β 1 ,α 2 +β 2 ) (S). This finishes the induction and (3) is proved. Finally, we have to prove that B=  F(α,β) (B). (α,β)∈Nm+s Since B is generated by A and x1 , . . . , xs , every element in B is a sum of ‘monomials’ of the form r1 xi1 r2 · · · xit rt +1 . Now, let α j ∈ Nm such that rj ∈ Fα j (A) for j = 1, . . . , t. As a consequence of (3) we get r1 xi1 r2 · · · xit rt +1 ∈ F(α 1 +···+α j ,εi1 +···+εit ) (B).  The mapping from GF (A) into GF (B) given on homogeneous − − (B) is an injective K-algebra (A) → r + F(exp(r),0) elements by r + Fexp(r) J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 699 homomorphism. Thus, we can identify GF (A) as the K-subalgebra F α∈Nm G(α,0) (B) of G (B).  − P ROPOSITION 1.14. – Let yi = xi + F(0,ε i ) (B) for i = 1, . . . , s. The K-algebra GF (B) is generated by GF (A) and y1 , . . . , ys . Moreover, 1. For every α ∈ Nm and every i = 1, . . . , s, F yi GF α (A) ⊆ Gα (A)yi . (6) 2. For every 1
i < j
s, (7) yj yi = qj i yi yj . Proof. – Every homogeneous element h of GF (B) is of the form − 1 2 m+s h = f + F(α . By 1 ,α 2 ) (B), where f ∈ B with exp(f ) = (α , α ) ∈ N 1 2 Proposition 1.13, f = r α xα + − F(α 1 ,α 2 ) (B) and, by Lemma 1.4, 1 β2 β 2 ≺2 α 2 rβ 2 x . 2 2 1 Therefore, h = r α xα + 1 2 − − − α α + F(α xα + F(0,α h = r α + F(α 1 ,0) (B) y . 2 ) (B) = r 1 ,0) (B)     The statements in the proposition follow now without difficulty.  ✷ We are interested in special classes of bounded extensions of algebras over fields which give semi-commutative associated graded algebras. The following is the fundamental example. D EFINITION 1.15. – Let R be an algebra generated by x1 , . . . , xs over the field k and  an admissible order on Ns . We say that R is a -bounded quantum algebra if for every 1
i < j
s there exists 0 = qj i ∈ k satisfying (8) xj xi − qj i xi xj ∈  kxγ . γ ≺ε i +ε j Remark 1.16. – A -bounded quantum algebra R with the additional condition that the standard monomials xα α ∈ Ns form a k-basis of R is just a polynomial ring of solvable type in the sense of [11]. We also refer them as Poincaré–Birkhoff–Witt (or PBW) algebras (cf. [5]). Notice that Uq (C) is not in general a polynomial ring of solvable type. 700 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 C OROLLARY 1.17. – Every -bounded quantum k-algebra R is filtered by the (Ns , )-filtration F given by Fα (R) = γ α kxγ for every α ∈ Ns . The associated Ns -graded algebra GF (R) is semi-commutative. Proof. – It is clear that every -bounded quantum algebra R is a -bounded extension of the ground field k (which is trivially N0 -filtered). By Proposition 1.13 R is endowed with the (Ns , )-filtration F given by Fα (R) = γ α kxγ for every α ∈ Ns . By Proposition 1.14, GF (R) is semi-commutative. ✷ Example 1.18. – Let R = k[x1 ][x2 ; σ2 , δ2 ] · · · [xs ; σs , δs ] be an iterated Ore extension of the polynomial ring k[x1 ]. Assume that the k-linear automorphisms σi satisfy that (9) σj (xi ) = qj i xi + fj i for every i < j , where qj i ∈ k \ {0} and fj i ∈ k[x1 ][x2 ; σ2 , δ2 ] · · · [xi−1 ; σi−1 , δi−1 ] for every i = 1, . . . , s. Then R is a
lex -bounded quantum extension of the ground field k (in fact, R is a PBW algebra). We call these algebras triangular iterated Ore extensions. Examples of triangular iterated Ore extensions are the quantum coordinate algebras of matrices; (k) introduced in [18] the multiparameter quantized Weyl algebras AQ,Γ n (see also [10]); the quantum symplectic spaces (see [23]), the positive part Uq (C)+ of quantized enveloping algebras (see [24] and [17]) and the iterated differential operator algebras of [26]. Remark 1.19. – Propositions 1.13 and 1.14 can be used to see that every almost centralizing extension or every iterated differential operator k-algebra of a multi-filtered k-algebra A with (semi-)commutative associated graded algebra can be endowed with a multi-filtration having a semi-commutative associated graded algebra. They also are bounded quantum algebras, as the following proposition shows. P ROPOSITION 1.20. – Let R be an algebra over the field k. If there is an (Nn , )-filtration F on R such that GF (R) is semi-commutative, generated by s semi-commuting homogeneous elements, then there is an admissible order ′ on Ns such that R is a ′ -bounded quantum algebra. J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 701 Proof. – Let y1 , . . . , ys be homogeneous semi-commuting generators of GF (R). By hypothesis there are nonzero scalars qj i ∈ k, 1
i < j
s such that yj yi = qj i yi yj for 1
i < j
s. It is clear that G(R) is spanned as k-vector space by all standard monomials yγ = γ y1 1 · · · ysγs , where γ ∈ Ns . For i = 1, . . . , s let α i ∈ Nn be the degree of the homogeneous element yi . Consider the matrix M of order s × n whose rows are α 1 , . . . , α s . Given any element r ∈ R, the homogeneous element − (R) ∈ GF (R) can be represented as a standard polynomial r + Fexp(r) in y1 , . . . , ys and, being these elements homogeneous, all monomials − (R) can be chosen to be appearing in the expression of r + Fexp(r) homogeneous of degree exp(r). Therefore (10)  − (R) = r + Fexp(r) cγ yγ , γ M=exp(r) where the cγ ’s are scalars in k. Let yi = xi + Fα i (R) for i = 1, . . . , s. By (10) and Lemma 1.4, − (R) = r + Fexp(r)  − (R). cγ xγ + Fexp(r) γ M=exp(r) As a consequence, we can prove by induction on exp(r) that (11) r=  aγ xγ γ Mexp(r) for every r ∈ R (the aγ ’s are scalars). For 1
i < j
s do the following computation in GF (R) (12) 0 = yj yi − qj i yi yj = xj + Fα−j (R) xi + Fα−i (R) − qj i xi + Fα−i (R) xj + Fα−j (R)    = (xj xi − qj i xi xj ) + Fα−i +α j (R).   Thus, exp(xj xi − qj i yi yj ) ≺ α i + α j and, by (11), (13) xj xi − qj i xi xj =  γ M≺α i +α j aγ xγ  702 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 for some scalars aγ . Let ′ be the admissible order on Ns given by γ ≺′ µ ⇐⇒    γ M ≺ µM   or γ M = µM and γ <lex µ. Notice that α i + α j = (ε i + ε j )M. By (11), R is generated as a k-algebra by x1 , . . . , xs and, by (13), these elements satisfy xj xi − qj i xi xj =  aγ xγ γ ≺′ ε i +ε j for 1
i < j
s, where the aγ ’s are scalars. Thus, R is a ′ -bounded quantum algebra. ✷ In order to further illustrate the potential of Propositions 1.13 and 1.14 to construct examples of multi-filtered algebras with semi-commutative associated graded algebra which are, by Proposition 1.20, bounded quantum algebras, let us consider the following construction. Example 1.21. – Let B be a k-algebra which is a 2 -bounded extension of an (Nm , 1 )-filtered algebra A. In order to obtain that GF (B) is semi-commutative it is reasonable, after Proposition 1.14, to require that GF (A) is itself semi-commutative. Clearly, we should also suppose that the qj i ’s in (6) are non-zero scalars in the ground field k. So, let z1 , . . . , zr be the homogeneous semi-commuting generators of GF (A) with degrees α 1 , . . . , α r ∈ Nm . Let zk = tk + Fα−k (A), for some t1 , . . . , tr ∈ A. Condition (7) says that for every i = 1, . . . , s, k = 1, . . . , r, there exists rik ∈ Fα j (A) such that xi tk − rik xi ∈  Axγ . γ ≺2 ε i If we assume in addition that for every i, k, there are a nonzero scalar q ik ∈ k and an element sik ∈ Fα−k (A) such that rik = q ik tk + sik , then, by Proposition 1.14, GF (B) is semi-commutative. By Proposition 1.20, B becomes a ′ -bounded quantum algebra for certain admissible order ′ on Nr+s . J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 703 2. Re-filtering by finite-dimensional subspaces In this section we will complete the proof of Theorem 1.7. We know from Proposition 1.20 that every multi-filtered algebra R over a field k with semi-commutative associated multi-graded algebra is a ′ -bounded quantum k-algebra for some admissible order ′ on Ns . Our next aim is to show that we can replace the order ′ by a locally finite admissible order ∗ , in the sense that for every α ∈ Ns , the set {β ∈ Ns | β ∗ α} is finite. To do this we will use [22, Corollary 2.2] or [28, Proposition 2.1]. For the convenience of the reader we shall prove this result by elementary methods (see Corollary 2.2). The first remark is that the admissible orders in Ns are just the restrictions of the total orders  on Qs compatible with the additive group structure (that is, the group orders) such that Ns is included in the non-negative cone (see [25] for details). This last condition is equivalent to require 0 ≺ ε i for every i = 1, . . . , s. In what follows, every admissible order on Ns will be considered automatically as the restriction of its unique extension to Qs . The order to be found is of some special type, namely, for every u = (u1 , . . . , us ) ∈ Zs , define the group order on Qs by    u, λ < u, µ λ
u µ ⇐⇒   or u, λ = u, µ and λ <lex µ. The restriction to Ns of this order is admissible and locally finite if and only if ui > 0 for every i = 1, . . . , s. We recall some facts about convex sets that we will need. Let E be a d–dimensional real affine space. Given C ⊆ E, we denote E(C) the convex hull of C. By Carathéodory theorem (see [2, 11.1.8.6 Theorem]) we have E(C) = d+1  i=1  d+1    i ri α  α ∈ C, ri  0, and ri = 1 . i i=1 Moreover, as a consequence of [2, 11.1.3.3 Corollary] and [2, 11.4.5 Corollary], two disjoint compact convex sets in E can be strictly separated (in the sense of [2, 11.4.3 Definition]) by an hyperplane. This fact is used in the following proposition. 704 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 P ROPOSITION 2.1. – Let C be any nonempty finite subset of Qs . An element α ∈ C is maximal in C with respect to some group order in Qs if and only if there exists u ∈ Zs such that u, β < u, α for every β ∈ C \ {α}. Proof. – Assume that α ∈ C is maximal with respect to some group order  in Qs . Let C ′ = {β − α | β ∈ C \ {α}}. Then γ ≺ 0 for all γ ∈ C ′ . Assume for a contradiction 0 ∈ E(C ′ ). Then there is a subset {γ 1 , . . . , γ n } ⊆ C ′ with n minimal such that 0 = ni=1 ri γ i , ri  0 and ni=1 ri = 1. The minimality of n implies that ri > 0 for every i = 1, . . . , n and so γ 1 = ni=2 − rr1i γ i . We claim that the points γ 1, . . . , γ n are affinely independent. In fact, let E be the affine subspace of Rs spanned by these elements, and let t be its dimension. It is clear that E(γ1 , . . . , γn ) is contained in E and that 0 ∈ E(γ1 , . . . , γn ), whence, by Carathéodory’s theorem and minimality of n, n
t + 1. This clearly implies that n = t + 1, i.e., the points {γ 1, . . . , γ n} are affinely independent and, thus, the scalars − rr1i are unique. Therefore, − rr1i ∈ Q. As γ i ≺ 0 and − rr1i < 0 we have − rr1i γ i ≻ 0. So γ 1 = ni=2 − rr1i γ i ≻ 0, a contradiction. So the compact convex sets {0} and E(C ′ ) are disjoint and strictly separated by an hyperplane, i.e., there exists v ∈ Rs such that v, β − α < 0 for all β ∈ C \ {α}. Finally, write C = {α, β 1 , . . . , β m } and let v ∈ Rs such that v, β j  < v, α for every j = 1, . . . , m. Let ε > 0 be a real number such that v, α − v, β j  > ε for every j = 1, . . . , m and j choose a number N > 0 such that |αi − βi | < N for every j = 1, . . . , m and every i = 1, . . . , s. Now, if v = (v1 , . . . , vs ) ∈ Rs then for every i = 1, . . . , s there exists a rational number wi such that |vi − wi | < ε/sN . A straightforward computation shows that w = (w1 , . . . , ws ) ∈ Qs satisfies that w, β j  < w, α for every j = 1, . . . m. By multiplying w by an adequate positive integer, we get a vector u in Zs as required. Conversely, assume u ∈ Zs given such that u, β < u, α for every β ∈ C \ {α}. Consider the group order
u in Qs . Obviously, α is the maximal element of C. ✷ C OROLLARY 2.2 ([22,28]). – Let C be a nonempty finite subset of Qs . An element α ∈ C is maximal in C with respect to (the extension of) some admissible order in Ns if and only if there exists u ∈ Ns+ such that u, β < u, α for every β ∈ C \ {α}. J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 705 Proof. – Notice that an admissible order  in Qs is the extension of an admissible order in Ns if and only if −ε i ≺ 0 for every i = 1, . . . , s. Apply Proposition 2.1 to the set C ∪ {α − ε 1 , . . . , α − ε s } in order to get this corollary. ✷ We are ready to give the main result in this section. It completes our refiltering process in a somewhat surprising form: The implication (i) ⇒ (v) in the following theorem tells us that every multi-filtered algebra with semi-commutative associated graded algebra can be re-filtered (in the usual setting) by finite-dimensional vector subspaces in such a way that the new associated graded algebra is semi-commutative. Therefore, the algebras satisfying the conditions of Theorem 2.3 (which are already the -bounded quantum algebras for some order ) should be called somewhat semi-commutative algebras (cf. [20, 8.6.9]). T HEOREM 2.3. – Let R be an algebra over a field k. The following conditions are equivalent. (i) There exists a positive integer n, an admissible order  on Nn and an (Nn , )-filtration F on R such that GF (R) is semicommutative. (ii) There exists a positive integer s and an admissible order ′ on Ns such that R is a ′ -bounded quantum algebra generated by s elements x1 , . . . , xs . (iii) There exists a positive integer s and a vector u ∈ Ns+ such that R is a
u -bounded quantum algebra generated by s elements x1 , . . . , xs . (iv) There exist a positive integer s, a vector u ∈ Ns+ and a locally finite and pointed (Ns ,
u )-filtration H on R such that GH (R) is semi-commutative. (v) There exists a locally finite and pointed N-filtration L on R such that GL (R) is semi-commutative. Proof. – The equivalence between (i) and (ii) is given by Corollary 1.17 and Proposition 1.20. Corollary 1.17 gives also (iii) ⇒ (iv). Let us prove that (ii) does imply (iii) and (v). For every 1
i < j
s there is a nonzero scalar qj i such that xj xi − qj i xi xj ∈  α≺′ ε i +ε j kxα . 706 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 Thus, there is a finite subset Aj i of Ns satisfying that α ≺′ ε i + ε j for every α ∈ Aj i and (14) xj xi − qj i xi xj ∈  kxα . α∈Aji Write  C = {0} ∪ Cj i , 1
i<j
s where Cj i = Aj i − ε i − ε j . It is clear that C is a subset of Zs such that 0 is maximal in C with respect to ′ . By Corollary 2.2, there is u = (u1 , . . . , us ) ∈ Ns+ such that u, α < 0 for every 0 = α ∈ C. This means that, for every 1
i < j
s, (15) for every α ∈ Aj i . u, α < ui + uj The admissible order
u is locally finite and, by (14), we have xj xi − qj i xi xj ∈  kxα . α<u ε i +ε j Therefore, R is a
u -bounded quantum algebra with generators x1 , . . . , xs and (iii) is proved. By Corollary 1.17, R is endowed with the (Ns ,
u )filtration H given by Hα (R) = γ
u α kxγ . Since
u is locally finite, the vector subspace Hα (R) is finite-dimensional for every α ∈ Ns . For each n ∈ N, define Rn =  Hα (R). u,α
n A straightforward verification proves that L = {Rn | n ∈ N} is an N-filtration on R. Since u has all its components strictly positive, it is clear that R0 = k and that, for a given non-negative integer n, there are finitely many α ∈ Ns such that u, α
n. Therefore, Rn is finitedimensional for every n ∈ N and we have proved that the filtration L is pointed and locally finite. Let us prove that GL (R) is semi-commutative. For every i = 1, . . . , s, let yi = xi + Rui −1 , where u = (u1 , . . . , us ). Every homogeneous element of GL (R) is of the form u,α=n aα xα + Rn−1 , where the aα ’s are in k. By Lemma 1.4, xα + Rn−1 = yα for every J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 707 α satisfying u, α = n. Therefore, y1 , . . . , ys generate GL (R) as a k-algebra. ✷ C OROLLARY 2.4 ([21]). – Let R be an N-filtered k-algebra with filtration F such that GF (R) is commutative and finitely generated as a k-algebra. Then there exists a locally finite N-filtration H over R such that GH (R) is commutative and finitely generated. We are now ready to give a proof for Theorem 1.7. Proof of Theorem 1.7. – Let R be a multi-filtered algebra with semicommutative associated graded algebra. By Theorem 2.3, R can be endowed with a pointed and locally finite multi-filtration keeping up the semi-commutativity of the new associated graded algebra. The exactness of Gelfand–Kirillov dimension is obtained now from [8, Corollary 2.12]. By [9, Theorem 2.9], R is left and right finitely partitive. ✷ Remark 2.5. – Theorem 1.7 applies to every triangular iterated Ore extension (see Example 1.18) and, thus, to a plenty of quantum algebras that can be expressed as a triangular iterated Ore extensions. Some classical examples are also covered. This is the case of every almost centralizing extension of a somewhat commutative algebra (see Example 1.10). An interesting consequence of Theorem 1.7 is the following. It answers to an open problem left by McConnell in [19]. C OROLLARY 2.6. – The Gelfand–Kirillov dimension is exact for the quantized enveloping algebra Uq (C) of any Cartan matrix C and Uq (C) is a left and right finitely partitive algebra. 3. Final remarks Let R be a bounded quantum algebra over k with generators x1 , . . . , xs (see Definition 1.17). Equivalently, R is a multi-filtered algebra with semi-commutative associated graded algebra or R has a filtration R =  vector spaces and the n∈N Rn such that all Rn are finite-dimensional  associated graded algebra gr(R) = n∈N Rn /Rn−1 is semi-commutative (Theorem 2.3). These characterizations allow to get further interesting information for this wide class of algebras, which include Uq (C), Uq (C)+ , quantum matrices, iterated differential operators algebras, quantum Weyl 708 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 algebras. . . , and it is closed by almost centralizing extensions (by Proposition 1.14). Thus, by [19, Theorem 3.8], R satisfies the Nullstellensatz over k, i.e., endomorphisms division rings of simple R-modules are algebraic over k and each prime ideal of R is an intersection of primitive ideals. Moreover, it follows from [3] that R is Auslander– Gorenstein (resp. Auslander-regular) whenever G(R) is. This is the case for the Poincaré–Birkhoff–Witt algebras: Following [5], R is said to be a Poincaré–Birkhoff–Witt algebra if the standard monomials xα are linearly independent over k and, thus, form a k-basis of R. By Theorem 2.3, the original admissible order  can be replaced by the order
u associated to certain vector u = (u 1 , . . . , us ) of strictly positive integer components. Filter the algebra R = n∈N Rn , where Rn = u,α
n kxα is a finite dimensional vector space for every n ∈ N. It can be easily proved that the associated graded algebra gr(R) is a quantum affine space over k, which is known to be Auslander-regular. Thus, every PBW algebra over k is Auslander-regular. This applies in particular that every triangular iterated Ore extension of k (see Example 1.18). However, the re-filtering process destroys in general part of the structure of the original semi-commutative multi-graded algebra GF (R) in the sense that the new graded algebra gr(R) could enjoy different properties. This is the case of the integral domain GF (Uq (C)) which turns into an algebra gr(Uq (C)) with zero divisors. Finally, one could conclude that the GK-dimension theory of multifiltered algebras can be reduced to the theory of filtered algebras. But it seems that this is not, for the moment, the case: If R is multi-filtered with a pointed and locally finite filtration F and GF (R) such that GF (R) is a left noetherian finitely generated algebra, then the Gelfand–Kirillov dimension is exact for R ([8, Theorem 2.10]). But we do not know any general re-filtering process in the non semi-commutative case. Even for N-filtrations, the known results in this direction require rather strong conditions on GF (R) (see [21]). 4. Appendix: Computing weights Let R be a k-algebra generated by finitely many elements x1 , . . . , xs . An element p in R is a standard polynomial if p is a k-linear combination of standard monomials (see Section 1). We shall say that the generators x1 , . . . , xs satisfy a set of quantum relations if there are non-zero scalars J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 709 qj i ∈ k and standard polynomials pj i ∈ R for every 1
i < j
s such that (16) xj xi − qj i xi xj − pj i = 0 (1
i < j
n). The polynomials pj i will be called the tails of the quantum relations. By Proposition 1.20, every multi-filtered algebra with semi-commutative associated graded algebra satisfies a set of quantum relations. Moreover, the size of the tails of these relations are controlled by some admissible order in Ns . This leads to the following definition. D EFINITION 4.1. – Let i < j be numbers in {1, . . . , s} and fix a standard representation pj i = α∈Ns aα xα , where aα ∈ k for every α. The Newton diagram of pj i (with respect to the given representation) is defined as the finite subset N (pj i ) of Ns given by those α ∈ Ns such that aα = 0. We shall say that the quantum relations (16) are bounded by an admissible order ′ on Ns (shortly, they are ′ -bounded) if max′ N (pj i ) ≺′ ε i + ε j for every 1
i < j
s. It is clear that the algebras generated by finitely many elements x1 , . . . , xs that satisfy a set of ′ -bounded quantum relations are just the ′ -bounded quantum algebras. Some questions arise now: Given generators satisfying a set of quantum relations, how to decide if these relations are bounded by some order? If such an order does exist, can we compute it? We will show that the existence of ′ can be effectively decided by linear programming methods. In addition, when the answer is positive, a bounding locally finite order can be explicitly computed. This is important from the point of view of the effective computation of the Gelfand–Kirillov dimension for finitely generated modules over R developed in [4] (see also [14, Capítulo 5]). For every 1
i < j
s, let Cj i = N (pj i ) − εi − εj and write C=  1
i<j
s Cj i \ {0} = α 1 , . . . , α m .
 710 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 Consider the following linear programming problem (17)  minimize f (u) = u1 + · · · + us     with the constraints  u 1 (i = 1, . . . , s)  i   Φ ≡ u, α j 
−1 (j = 1, . . . , m). P ROPOSITION 4.2. – The set of relations (16) is ′ -bounded for some admissible order ′ on Ns if and only if the linear programming problem (17) has a solution. Moreover, each solution u of (17) gives a  filtration R = n∈N Rn , with Rn =  kxα u,α
n such that the associated graded algebra gr(R) = commutative.  n∈N Rn /Rn−1 is semi- Proof. – The linear programming problem (17) has a solution if and only if the feasible region Φ is not empty (notice that the linear functional f (u) is bounded under below whenever the feasible region is not empty). But the proof of Theorem 2.3 says that this region is not empty if and only if a bounding admissible order does exist for the quantum relations (16). The filtration given for each solution u is that in the proof of Theorem 2.3. ✷ In the following examples, the simplex algorithm implemented in MAPLE has been used to solve the linear programming problems. Example 4.3. – In this example we deal with Uq (A2 ). Yamane’s basis for this algebra is
f γ k β eα | α, γ ∈ N3 , β ∈ Z2 , β β γ  γ γ α1 α2 α3 where eα = e12 e13 e23 , k β = k1 1 k2 2 and f γ = f121 f132 f233 . Let u = (f12 , f13, f23 , k1 , k2 , l1 , l2 , e12, e13 , e23) be one of the vectors provided by Theorem 2.3(iii). The relations of this algebra (see [29, §3]) say that u can be obtained as a solution of the J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 711 following linear programming problem (17). minimize f (u) = f12 + f13 + f23 + k1 + k2 + l1 + l2 + e12 + e13 + e23 with the constraints f12  1 k1  1 e12  1 f13  1 k2  1 e13  1 f23  1 l1  1 e23  1 l2  1 −f12 + f13 − f23
−1 −e12 + e13 − e23
−1 2k1 + f23 − e12 − f13
−1 2k2 + e12 − e13 − f23
−1 2l1 + e23 − e13 − f12
−1 f12 + 2l2 − e23 − f13
−1 2k1 − e12 − f12
−1 2l1 − e12 − f12
−1 2k1 + 2k2 − e13 − f13
−1 2l1 + 2l2 − e13 − f13
−1 2k2 − e23 − f23
−1 2l2 − e23 − f23
−1. Minimum is at: f23 = 3, e13 = 1, k1 = 1, e12 = 1, e23 = 1,
l2 = 1, f12 = 3, l1 = 1, f13 = 5, k2 = 1 . Therefore, we obtain the weight vector  u = (3, 5, 3, 1, 1, 1, 1, 1, 1, 1). Example 4.4. – We will use again the simplex algorithm implemented in MAPLE to compute one of the vectors u = (e1 , e12 , e122, e2 , k1 , k2 , l1 , l2 , f2 , f122, f12, f1 ) prescribed in Theorem 2.3(iii) for the algebra Uq (B2 ) with respect to the Lusztig’s generators E1 , E12 , E122 , E2 , K1 , K2 , K1−1 , K2−1 , F2 , F122 , F12 , F1 . The commutation relations among them can be computed (with enough amount of time and paper) from [7, Sections §9, §12 and Appendix] (see also [16]). They give rise the following linear programming problem (17). 712 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 minimize f (u) = e1 + e12 + e122 + e2 + k1 + k2 + l1 + l2 + f2 + f122 + f12 + f1 with the constraints e1  1, k1  1, f2  1 e12  1, k2  1, f122  1 e122  1, l1  1, f12  1 e2  1, l2  1, f1  1 −e1 − e2 + e12
−1, −f1 − f2 + f12
−1 −e12 − e2 + e122
−1, −f12 − f2 + f122
−1 −e1 − e122 + 2e12
−1, −f1 − f122 + 2f12
−1 −e1 − f1 + k1
−1, −e1 − f1 + l1
−1 −e1 − f12 + k1 + f2
−1, −e1 − f12 + l1 + f2
−1 −e1 − f122 + k1 + 2f2
−1, −e1 − f122 + l1 + 2f2
−1 −e12 − f1 + e2 + k1
−1 −e12 − f1 + e2 + l1
−1 −e12 − f12 + k1 + k2
−1 −e12 − f12 + l1 + l2
−1 −e12 − f122 + e1 + k2 + f2 + f1
−1 −e12 − f122 + k1 + k2 + f2
−1 −e12 − f122 + k2 + l1 + f2
−1 −e12 − f122 + e1 + k2 + f12
−1 −e12 − f122 + e1 + k2 + 2l1 + f2 + f1
−1 −e12 − f122 + k2 + l1 + f2
−1 −e12 − f122 + k2 + 3l1 + f2
−1 −e12 − f122 + e1 + k2 + 2l1 + f12
−1 −e12 − f2 + e1 + k2
−1 −e12 − f2 + e1 + l2
−1 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 −e122 − f1 + 2e2 + k1
−1 −e122 − f1 + 2e2 + l1
−1 −e122 − f12 + e1 + e2 + 2k1 + k2 + f1
−1 −e122 − f12 + e12 + 2k1 + k2 + f1
−1 −e122 − f12 + e2 + 3k1 + k2
−1 −e122 − f12 + e2 + k1 + k2
−1 −e122 − f12 + e1 + e2 + l2 + f1
−1 −e122 − f12 + e12 + l2 + f1
−1 −e122 − f12 + e2 + k1 + l2
−1 −e122 − f12 + e2 + l1 + l2
−1 −e122 − f122 + k1 + 2k2
−1 −e122 − f122 + l1 + 2l2
−1 −e122 − f2 + e12 + k2
−1 −e122 − f2 + e1 + e2 + k2
−1 −e122 − f2 + e12 + l2
−1 −e122 − f2 + e1 + e2 + l2
−1 −e2 − f12 + k2 + f1
−1, −e2 − f122 + k2 + f12
−1 −e2 − f12 + l2 + f1
−1, −e2 − f122 + l2 + f12
−1 −e2 − f2 + k2
−1, −e2 − f122 + k2 + f2 + f1
−1 −e2 − f2 + l2
−1, −e2 − f122 + l2 + f2 + f1
−1. Minimum is at: e1 = 4, e12 = 9, e122 = 15, e2 = 7, k1 = 1, k2 = 1, l1 = 1, l2 = 1, f2 = 2, f122 = 2, f12 = 1, f1 = 1. Therefore, we obtain the weight vector u = (4, 9, 15, 7, 1, 1, 1, 1, 2, 2, 1, 1). 713 714 J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 REFERENCES [1] Becker T., Weispfenning V., Gröbner Bases. A Computational Approach to Commutative Algebra, Springer-Verlag, 1993. [2] Berger M., Geometry, Vol. I, Springer-Verlag, 1987. [3] Björk J.-E., The Auslander condition on noetherian rings, in: Malliavin M.P. (Ed.), Sém. d’Algèbre P. Dubreil et M.P. Malliavin 1987–1988, Lecture Notes in Mathematics, Vol. 1404, Springer-Verlag, 1989, pp. 137–173. [4] Bueso J.L., Gómez-Torrecillas J., Lobillo F.J., Effective computation of the Gelfand–Kirillov-dimension for modules over Poincaré–Birkhoff–Witt algebras with non quadratic relations, in: Proceedings of EACA-98, Universidad Complutense and Universidad de Alcalá, 1998. [5] Bueso J.L., Castro F.J., Gómez-Torrecillas J., Lobillo F.J., An introduction to effective calculus in quantum groups, in: Caenepeel S., Verschoren A. (Eds.), Rings, Hopf Algebras and Brauer Groups, Lect. Notes in Pure and Appl. Math., Vol. 197, Marcel-Dekker, 1998, pp. 55–83. [6] De Concini C., Kac V., Representations of quantum groups at roots of 1, in: Progress in Math., Vol. 92, Birkhäuser, 1990, pp. 471–506. [7] De Concini C., Procesi C., Quantum groups, in: Zampieri G., D’Agnolo A. (Eds.), D-Modules, Representation Theory and Quantum Groups, Lecture Notes in Math., Vol. 1565, Springer, 1993, pp. 31–140. [8] Gómez-Torrecillas J., Gelfand–Kirillov dimension of multi-filtered algebras, P. Edinburgh Math. Soc. 42 (1999) 155–168. [9] Gómez-Torrecillas J., Lenagan T.H., Poincaré series of multi-filtered algebras and partitivity, J. London Math. Soc. 62 (2000) 370–380. [10] Goodearl K.R., Lenagan T.H., Catenarity in quantum algebras, J. Pure Appl. Algebra 111 (1996) 123–142. [11] Kandri-Rody A., Weispfenning V., Non-commutative Gröbner bases in algebras of solvable type, J. Symb. Comput. 9/1 (1990) 1–26. [12] Krause G.R., Lenagan T.H., Growth of Algebras and Gelfand–Kirillov Dimension, Research Notes in Mathematics, Vol. 116, Pitman, London, 1985. [13] Lenagan T.H., Gelfand–Kirillov dimension is exact for noetherian PI-algebras, Canadian Math. Bull. 27 (1984) 247–250. [14] Lobillo F.J., Métodos algebraicos y efectivos en grupos cuánticos. Ph.D. Thesis, Universidad de Granada, 1998. [15] Lorenz M., Gelfand–Kirillov dimension and Poincaré series, Cuadernos de Algebra, Vol. 7, Universidad de Granada, 1988. [16] Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447–498. [17] Malliavin M.P., La catenarité de la partie positive de l’algèbre envaloppante quantifiée de l’algèbre de Lie simple de type B2 , Beiträge zur Algebra und Geometrie 35 (1994) 73–83. [18] Maltsiniotis G., Calcul Différentiel Quantique, 1992. J.L. BUESO ET AL. / Bull. Sci. math. 125 (2001) 689–715 715 [19] McConnell J.C., Quantum groups, filtered rings and Gelfand–Kirillov dimension, in: Noncommutative Ring Theory, Lecture Notes in Math., Vol. 1448, Springer, 1990, pp. 139–149. [20] McConnell J.C., Robson J.C., Noncommutative Noetherian Rings, Wiley, Chichester, 1987. [21] McConnell J.C., Stafford J.T., Gelfand–Kirillov dimension and associated graded modules, J. Algebra 125 (1989) 197–214. [22] Mora T., Robbiano L., The Gröbner fan of an ideal, J. Symbolic Comput. 6 (2–3) (1988) 183–208. [23] Oh S.Q., Primitive ideals of the coordinate ring of quantum symplectic space, J. Algebra 174 (1995) 531–552. [24] Ringel C.M., PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996) 51–88. [25] Robbiano L., Term orderings on the polynomial ring, in: Caviness B. (Ed.), EUROCAL ’85, Vol. 2, Lecture Notes in Comput. Sci., Vol. 204, Springer, 1985, pp. 513–517. [26] Sigurdsson G., Ideals in universal enveloping algebras of solvable Lie algebras, Comm. Algebra 15 (1987) 813–826. [27] Tauvel P., Sur la dimension de Gelfand–Kirillov, Comm. Algebra 10 (1982) 939– 963. [28] Weispfenning V., Constructing universal Gröbner bases, in: Proceedings of AAECC 5, Lecture Notes in Comput. Sci., Vol. 356, Springer, 1987, pp. 408–417. [29] Yamane H., A Poincaré–Birkhoff–Witt theorem for quantized universal enveloping algebras of type AN , Publ. RIMS Kyoto Univ. 25 (1989) 503–520.