First-order characterization of noncommutative birational equivalence
aa r X i v : . [ m a t h . R A ] S e p FIRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVEBIRATIONAL EQUIVALENCE
HUGO LUIZ MARIANO, JO ˜AO SCHWARZ
Abstract.
Let Σ be a root system with Weyl group W . Let k be an al-gebraically closed field of zero characteristic, and consider the correspondingsemisimple Lie algebra g k , Σ . Then there is a first-order sentence φ Σ in thelanguage L = (1 , , + , ∗ ) of rings sucht that, for any algebraically closed field k of char = 0, the validity of the Gelfand-Kirillov Conjecture for g k , Σ is equiv-alent to ACF ⊢ φ Σ . By the same method, we can show that the validity ofNoncommutative Noether’s Problem for A n ( k ) W , k any algebraically closedfield of char = 0 is equivalent to ACF ⊢ φ W , φ W a formula in the samelanguage. As consequences, we obtain results on the modular Gelfand-KirillovConjecture and we show that, for F algebraically closed with characteristic p >> A n ( F ) W is a case of positive solution of modular NoncommutativeNoether’s Problem. Introduction
Connections between Algebra and Logic are well known (cf. [10], [34]). In thespecific topic of Algebraic Geometry, this line of inquiry began with the work ofAlfred Tarski on the decidability via quantifier elimination in the theory of alge-braically closed fields, and have achieved a remarkable development through theyears using more sophisticated methods of Model Theory in Algebraic Geometry,such as the work of Ax, Kochen and Ershov on Artin’s Conjecture, or the celebratedproof of Mordell-Lang Conjecture by Hrushovski (cf. [27], [8], [24], [35]).Let’s fix some conventions. All our rings and fields will be algebras over a basisfield k .One of the main problems of algebraic geometry is the birational classification ofvarieties ([25]). A particular important example for us is Noether’s Problem ([39]),which asks:Let G be a finite group acting linearly on k ( x , . . . , x n ) G . When is k ( x , . . . , x n ) G isomorphic to k ( x , . . . , x n )? In algebraic-geometric terms: when is the va-riety k n /G rational?This is a very important area of research, given its connection to the inverseGalois problem (cf. [31]), PI-algebras (cf. [14]), study of moduli spaces (cf. [11]).In the 1966 the study of birational geometry of noncommutative objects began.In his adress at the 1966 ICM in Moscow, A. A. Kirillov proposed to classify, up tobirational equivalence, the enveloping algebras U ( g ) of finite dimensional algebraicLie algebras g when k is algebraically closed of zero characteristic. This means to MSC 2020 Primary: 03C60; Secundary: 16S85, 16W22, 17B35Keywords: First-order characterization, noncommutative birational equivalence, Gelfand-Kirillov Conjecture find canonical division rings among the equivalence classes of
F rac U ( g ), the skewfield of fractions of the enveloping algebra, which is an Ore domain ([37]).The idea became mature in the groundbreaking paper [22], where A. A. Kirillovand I. M. Gelfand formulated the celebrated Gelfand-Kirillov Conjecture. Beforewe formulate it, lets recall some definitions: Definition 1.1.
The n − th Weyl algebra is the algebra with generators x , . . . , x n and y , . . . , y n and relations [ x i , x j ] = [ y i , y j ] = 0; [ y i , x j ] = δ ij , i, j = 1 , . . . , n .We denote by A n,s ( k ) the algebra A n ( k ( t , . . . , t s )) . We denote the Weyl fields D n,s ( k ) , D n ( k ) the skew field of fractions of A n,s ( k ) , A n ( k ) , respectively. The nature of the Weyl algebra changes a lot if k has 0 characteristic (cf. [37])or prime characteristic (cf. [6], [42]). Nonetheless, it is always an Ore domain. Conjecture. (Gelfand-Kirilov Conjecture) : consider the enveloping algebra U ( g ) , g a noncommutative finite dimensional algebraic Lie algebra over k algebraicallyclosed of zero characteristic. When its skew field of fractions, F rac U ( g ) , is of thethe form D n,s ( k ) , for some n > , s ≥ ? The Conjecture was shown to be true for gl n and sl n , as well for nilpotent Liealgebras in [22]. This later fact was generalized for solvable Lie algebras in [36],[32], [7]; other cases considered in [38] [5], [40]; and certain modifications of itconsidered in [23] and [12]. The Conjecture was very influential in the developmentof Lie theory and, recently, quantum group theory (cf. [13, Problem 3]; [9, I.2.11,II.10.4]); it also became a paradigmatic example on the study of skew field offractions of many Ore domains (cf. [17], [2], [18], [3]). However, it was eventuallyshown that the Conjecture is false in general ([4]). Regarding simple Lie algebras,it was known to be true in the seminal work of Gelfand and Kirillov for type A simple Lie algebras. This question was revisited by Premet in [41], where usingreduction module prime techniques, he showed the Conjecture to be false for types B, D, F, G . About the types
C, G , nothing is known at this moment.An influential work in this line of inquiry is a noncommutative analogue ofNoether’s problem, considered by Jacques Alev and Fran¸cois Dumas in [1] andsystematically studied in [2], under the name
Noncommutative Noether’s Problem .The question is as follows:
Problem. (Noncommutative Noether’s Problem) : Let k be a field of zero charac-teristic, and G a finite group acting linearly on the Weyl algebra A n ( k ) . When wehave D n ( k ) G ∼ = D n ( k ) ? Noncommutative Noether’s Problem was shown to be true whenever n = 1 , G decomposes as a direct sum of one dimensional representations, with connectionswith rings of differential operators on Kleinian singularities (cf. [2]). Later it wasshown to be true for algebraically closed k and G = S n in [20], with applications tothe Gelfand-Kirillov Conjecture for finite W -algebras of type A (such as the classicalcase of U ( gl n )). This was later generalized for k = C and G is any complex reflec-tion group ([16]), obtaining a vast generalization of the Gelfand-Kirillov Conjecturefor a class of Galois algebras (discussed in [18]) and the Gelfand-Kirillov Conjec-ture for rational Cherednik algebras. The most recent work shows that a linear In general, it is not the case that a noncommutative domain can be embedded in a divisionring, as shown by Malcev. Ore domains are an example of when a quotient division ring exists ina particularly nice form, cf. [33, Chapter 4].
IRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVE BIRATIONAL EQUIVALENCE3 action that gives a positive solution to Noether’s Problem gives a positive solutionto noncommutative Noether’s Problem as well ([21]), with connections to construc-tive aspects of noncommutative invariant theory, and the modified Gelfand-KirillovConjecture discussed in [23]. Other noncommutative analogues of Noether’s Prob-lem have shown interesting applications in the study of skew field of fractions ofquantum groups ([19], [26]).The pourpose of this paper is two-fold. The first is to show that, surprisingly,given a pair (Σ , W ) of root system/Weyl group (cf. [13] 11.1) and any algebraicallyclosed field k with char k = 0, for the finite dimensional Lie algebra g k , Σ , and forthe algebra of invariants A n ( k ) W (with the canonical crystallographic action of W ),Gelfand-Kirillov Conjecture and Noncommutative Noether’s Problem respectively,are equivalent to the validity of a certain first-order sentence in the language ofrings L (0 , , + , ∗ ) in the theory of algebraically closed fields of zero characteristic— ACF (cf. [27]).Being more precise, we have our main Theorem: Theorem 1.2.
Given a root system Σ and Weyl group W , there are first-ordersentences φ Σ , φ W in the language of rings L (0 , , + , ∗ ) such that itens (1), (2),(3) are equivalent, as well itens (4), (5), (6). (1) For some k algebraically closed of zero characteristic, the Gelfand-KirillovConjecture holds for g k , Σ . (2) For any k algebraically closed of zero characteristic, the Gelfand-KirillovConjecture holds for g k , Σ . (3) ACF ⊢ φ Σ . (4) For some k algebraically closed of zero characteristic, NoncommutativeNoether’s Problem holds for A n ( k ) W . (5) For any k algebraically closed of zero characteristic, Noncommutative Noether’sProblem holds for A n ( k ) W . (6) ACF ⊢ φ W .Moreover, φ Σ and φ W are naturally constructed as existential closures of booleancombinations of atomic formulas in the language. We remark that the expression of a statement as a first-order sentence in
ACF is a very important question. One of the main applications of this idea is Lefchetz’sPrinciple (cf. [43]), frequently used in algebraic-geometry, using the fact that ACF is a complete theory ([27]): in order to prove a statement for a variety over analgebraically closed field of zero characteristic, it suffices to show it for k = C ,where transcendental methods are appliable.In fact, we will prove a stronger form of this Theorem (Theorem 2.12, cf. [41,Remark 1]). From it, Theorem 1.2 will follow as a Corollary.We remark that Gelfand-Kirillov Conjecture makes perfect sense, with the samedefinition of objects, when k is algebraically closed of prime characteristic (cf. [41,Introduction]). The same remark applies to the Noncommutative Noether’s Prob-lem, since the action of W on its canonical representation is crystallographic.Our methods allows a diferent proof of [41, Theorem 2]: Obviously, since
ACF admits quantifier elimination, every sentence in the language of ringsis equivalent to a boolean combination of atomic sentences. HUGO LUIZ MARIANO, JO˜AO SCHWARZ
Theorem 1.3.
Let Σ be any root system. If the Gelfand-Kirillov Conjecture holdsfor g k , Σ , then it holds in its modular version for all algebraically closed fields F with char F = p >> . They also allow a different proof a less general statement found in [6, Theorem2.2.3].Finally, it allows us to make a unnoticed observation about the Gelfand-KirillovConjecture for non-algebraically closed fields (cf. Proposition 3.2).Our last main theorem is:
Theorem 1.4.
Let Σ be any root system, W the Weyl group. Modular Noncommu-tative Noether’s Problem holds for A n ( F ) , for all algebraically closed fields F with char F = p >> . Using similar techniques, one can show:
Theorem 1.5.
Modular Noncommutative Noether’s Problem has a positive solutionfor the permutation action of the alternating groups A n , in A n ( F ) , n = 3 , , ,whenever char F = p >> . The structure of the paper is as follows. In section 2 we introduce the notionof Z -compatible family of algebras, which is the technical heart of the paper. InSection 3 we show that U ( g k , Σ ) and A n ( k ) W belong to a Z -compatible family ofalgebras, obtaining the above results as a consequence. Acknowledgments
J. S. is supported by FAPESP grant 2018/18146-5. The second author is gratefulto Vyacheslav Futorny and Ivan Shestakov for very fruitful discussions regardingthis paper. He is also grateful to M. Soares and M. C. Cardoso.2. Z -Compatible family of algebras Let
T deg denote the Gelfand-Kirillov transcendence degree of a k algebra (cf.[22], [44]); tdeg will denote the usual transcendence degree of commutative fields.Over a field k of characteristic 0, T deg D n,l ( k ) = 2 n + l , as computed in [22, Thm.2]. The same computation hold if the characteristic is prime ([44, Theorem1.1]).We are interested in a family of algebras with satisfy the following properties: Definition 2.1.
A family of algebras K = { A k } , where k ranges over all alge-braically closed fields and A k is a k -algebra; is called Z -compatible if: (1) For each algebraically closed field k of any characteristic, we have an algebra A k defined over this field. There exists an Z -algebra A Z sucht that for allfields A k = A Z ⊗ Z k . (2) For each algebraically closed field k , A k is an Ore domain. A Z is an Oredomain. (3) A Z has a filtration F = { A Z i } i ≥ such that A = Z and A i = A i , i ≥ is a finitely generated free module over Z ; and each A k admits a finitedimensional filtration { A k i } i ≥ , A k i = A Z i ⊗ Z k , for each k algebraically closed. (4) For k of characteristic , Z ( F rac A k ) ∼ = F rac Z ( A k ) . (5) For each algebraically closed field k , T deg F rac A k = dim k A k . IRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVE BIRATIONAL EQUIVALENCE5 (6) Z ( A k ) , when k algebraically closed of zero characteristic, is a polynomialalgebra k [ x , . . . , x s ] , where x , . . . , x s belong to A Z ; the images of x , . . . , x s in A F , F of prime characteristic, belongs to Z ( A F ) . Remark 1.
Strictly speaking, K as above is not a set but a proper class, but weindulge in this slight abuse of terminology. There is somewhat simpler way to show that a family of algebras K = { A k } is Z -compatible. Lemma 2.2.
Let K = { A k } be a family of algebras that satisfy conditions (1), (3),(4), (6) above, and such that for the given filtration in (3), both gr A k and gr A Z are affine domains; tdeg gr A k = dim k A k in the case of algebraically closed fields.Then K is a Z -compatible family of algebras.Proof. Item (2) follows from the usual filtered graded techniques in ring theory([37, Chapter 1]). (5) follows from [44, Corollary 6.9]. (cid:3)
Let us denote by Q the field of algebraic numbers. The following lemma isobvious, but crucial in what follows. Lemma 2.3. A Z ⊆ A Q ⊆ A k , where k is any algebraically closed field of zerocharacteristic. Also, for each i ≥ , A Z i ⊆ A Q i ⊆ A k i . Lemma 2.4.
Let k be an algebraically closed field of zero characteristic, and A k be an algebra of a Z -compatible family of algebras. If F rac A k ∼ = D n,l ( k ) , then F rac Z ( A k ) ∼ = k ( x , . . . , x l ) .Proof. Obvious, since Z ( D n,l ( k )) = k ( x , . . . , x l ), cf. [22, Thm. 2]. (cid:3) Definition 2.5.
Let A k , for k an algebraically closed field of zero characteristic, bepart of family of Z -compatible algebras. Then we say that Gelfand-Kirillov Conjec-ture holds for A k if and only if F rac A k ∼ = D n,l ( k ) , where T deg F rac A k = 2 n + l , l = tdegF rac Z ( A k ) . The following Proposition is a broad generalization of [6, Lemma 1.2.3], of inde-pendent interest.
Proposition 2.6.
Let A be an Ore domain over k (arbitrary characteristic) suchthat T deg A = 2 n + l . Suppose we have elements x , . . . , x n , y , . . . , y n , z , . . . , z l of F rac A such that [ x i , y j ] = δ i,j and all other commutators between these elementsvanish. Let B be the subalgebra of F rac A generated by these elements. If
F rac B = F rac A then B ∼ = A n,l ( k ) and F rac A ∼ = D n,l ( k ) .Proof. By the presentation of the Weyl algebra by generators and relations, we havesurjective map θ : A n,l ( k ) → B with kernel I . By [44, Theorem 1.1, Proposition3.1], 2 n + l = GK A n,l ( k ) ≥ GK B ≥ T deg B ≥ T deg F rac B = 2 n + l . Since I = { } implies GK B < GK A n,l ( k ) ([37, Corollary 8.3.6]), which is absurd, θ isan isomorphism. Hence the claim follows. (cid:3) Proposition 2.7.
Let A k be an algebra over k algebraically closed of zero char-acteristic, part of a Z -compatible family of algebras K , and let x , . . . , x n be abasis of A k contained in A Z . Then F rac A k ∼ = D m,l ( k ) if and only if we have ele-ments w , . . . , w m , w m +1 , . . . , w m in F rac A k , such that [ w i , w j ] = [ w m + i , w m + j ] =0 , [ w i , w m + j ] = δ ij , i, j = 1 , . . . , m , and polynomials p i , q i in w ’s and coefficients in Z ( A k ) such that q i x i = p i , i = 1 , . . . , n . HUGO LUIZ MARIANO, JO˜AO SCHWARZ
Proof.
Necessity is clear; the only point worth a remark is about the polynomials p i , q i . They exist because the algebra generated inside F rac A k by w , . . . , w m ,w m +1 , . . . , w m has as basis monomials on these variables with non-negative integerexponents. Sufficiency follows from Proposition 2.6. (cid:3) The next result is a kind of “Nullstellesatz”; its proof follows closely [41, Theorem2.1].
Theorem 2.8.
Let A k be an algebra over an algebraically closed field of zero char-acteristic belonging to a class K of Z -compatible algebras. Then there is a locallyclosed subvariety X k of k N , defined by polynomials with integer coefficients and N >> ; such that the Gelfand-Kirillov hods for A k if and only if X k = ∅ .Proof. Suppose
F rac A k ∼ = D m,l ( k ). By Proposition 2.7, we can find elements x , . . . , x n ∈ A Z , a basis of A k , w , . . . , w m , w m +1 , . . . , w m in F rac A k , such that:( † ) [ w i , w j ] = [ w m + i , w m + j ] = 0 , [ w i , w m + j ] = δ ij , i, j = 1 , . . . , m, and polynomials p i , q i in w ’s and coefficients in Z ( A k ) such that( ‡ ) q i x i = p i , i = 1 , . . . , n. Call Z ( A k ) = k [ φ , . . . φ l ], φ i ∈ A Z .Using the fact that A k \ { } satisfy the Ore condition, we can find a i , b j ∈ A k d ( i ) , i = 1 , . . . , m , for a certain d ( i ) ∈ N , such that w i = b − i a i ; and similarly a i,j , b i,j ∈ A k d ( i,j ) , i, j = 1 , . . . , n , such that(1) b i,j a i = a i,j b j , i, j = 1 , . . . , m. We can then introduce c i,j , d i,j ∈ A k d ( i,j ) such that(2) c i,j b i,j b i = d i,j b j,i b j , i, j = 1 , . . . , m ;and hence, because of ( † ),(3) c i,j a i,j a j = d i,j a j,i a i , i, j = 1 , . . . , m or i, j = m, . . . , m ;(4) c i,m + j a i,m + j a m + j = δ i,j c i,m + j b i,m + j b i + d i,m + j a m + j,i a i , i, j = 1 , . . . , m. When m ≥ m -uple i = ( i (1) , . . . , i ( m )) with 1 ≤ i (1) ≤ . . . ≤ i ( m ) ≤ m , we can inductively find elements a i (1) ...i ( k ) , b i (1) ...i ( k ) in A k d ( i ) , where3 ≤ k ≤ m and such that(5) b i (1) ...i ( k ) a i (1) ...i ( k − a i ( k − = a i (1) ...i ( k ) b i ( k ) . Write b i = Q m k =1 b i (1) ...i ( m − k +1) and a i = a i (1) ...i ( m ) u i ( m ) . We can considerthe tuples i as above such that, calling M = max { deg p i , deg q i | i = 1 , . . . , n } , P m ℓ =1 i ( ℓ ) ≤ M . Call them { i (1) , . . . , i ( r ) } . We have p k = P rj =1 λ j,k b − i ( j ) a i ( j ) and q k = P rj =1 µ j,k b − i ( j ) a i ( j ) .We can write λ j,k = X λ j,k ( n , . . . , n l ) φ n . . . φ n l l ,µ j,k = X µ j,k ( n , . . . , n l ) φ n . . . φ n l l , IRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVE BIRATIONAL EQUIVALENCE7 where summation is over finitely many l -tuples ( n , . . . , n l ) ∈ N l and λ j,k ( n , . . . , n l ) ,µ j,k ( n , . . . , n l ) ∈ k .There exists 0 = e i ( j ); k , f i ( j ); k ∈ A k d ( i ( j ); k ) with(6) a i ( j ) x k f i ( j ); k = b i ( j ) e i ( j ); k , j = 1 , . . . , r, k = 1 , . . . n, and p k = q k x k implies r X j =1 λ j,k b − i ( j ) a i ( j ) = r X j =1 µ j,k e i ( j ); k f − i ( j ); k , k = 1 , . . . , n. Let us now introduce a i ( j ) (0) = a i ( j ) , b i ( j ) (0) = b i ( j ) ; e i ( j ); k (0) = e i ( j ); k ; f i ( j ); k (0) = f i ( j ); k ; and inductively, for each 0 < s < j ≤ r , a i ( j ) ( s ) , b i ( j ) ( s ), e i ( j ); k ( s ) , f i ( j ); k ( s ),all in A k d ( i ( j ) ,k,s ) such that the following holds:(7) b i ( j ) ( s ) b i ( s ) ( s −
1) = a i ( j ) ( s ) b i ( j ) ( s − , (8) f i ( j ); k ( s − e i ( j ); k ( s ) = f i ( s ); k ( s − f i ( j ); k ( s )Finally, the following algebraic relation holds:(9) ( r X j =1 λj, k r − j Y ℓ =1 b i ( r − ℓ +1) ( r − ℓ ) j Y ℓ =1 a i ( j − ℓ +1) ( j − ℓ )) r Y ℓ =1 f i ( ℓ ); k ( ℓ −
1) =( r Y ℓ =1 b i ( r − ℓ +1) ( r − ℓ ) × ( r X j =1 µ j,k j Y ℓ =1 e i ( ℓ ); k ( ℓ − r Y ℓ = j +1 f i ( ℓ ); k ( ℓ − . It follows from the above that we have elements a i , b i , a i,j , b i,j , c i,j , d i,j , a i (1) ...i ( s ) , b i ( i ) ...i ( s ) , a i ( j ) ( ℓ ) , b i ( j ) ( ℓ ) , e i ( j ); k ( ℓ ) , f i ( j ); k ( ℓ ) , where i, j, k, s, l, i ( j ) ranges through a finite set of indices and belong to A k M for M >>
0, as well two finite collections of elements of k , λ j,k ( a , . . . , a l ) , µ j,k ( a , . . . , a l ),linked be algebraic equations(1) (2) (3) (4) (5) (6) (7) (8) (9) . This data can be parametrized by a locally closed subset defined by the simultaneousvanishing of polynomials { f , . . . , f r } and non-vanishing of some g ∈ { g , . . . , g s } in k N , a sufficiently big affine space; and all these polynomials have integer coefficients.Call this variety X k . Then the Gelfand-Kirillov Conjecture holding for A k implies X k = ∅ . This process is reversible — hence we have an equivalence of of X k = ∅ and the Gelfand-Kirillov Conjecture for A k . (cid:3) Corollary 2.9.
Let K be a family of Z -compatible algebras. If the Gelfand-KirillovConjecture holds for one of them, A k ∈ K , k algebraically closed of zero character-istic, then it holds for A Q . HUGO LUIZ MARIANO, JO˜AO SCHWARZ
Proof.
By the above Theorem (the notation which we use here), X k = ∅ . Hence X k ( Q ) = ∅ , because X is defined by polynomials with integer coefficients. Theprocess of Theorem 2.8 is reversible, and relation (9) of it reffers to the groundfield; hence, we can assume that elements in ( † ) , ( ‡ ) belongs to F rac A Q . Sincethe elements x , . . . , x n also belong to A Q and, by definition, are a basis of thisvector space, then we can apply Proposition 2 . A Q . (cid:3) Proposition 2.10.
Let K = { A k } be a family of Z -compatible algebras. If theGelfand-Kirillov Conjecture holds for A Q , then it holds for all A k , k any alge-braically closed field of zero characteristic.Proof. By Lemma 2.3, if Gelfand-Kirillov Conjecture holds for A Q and we can findelements x ′ s ∈ A Q and w ′ s ∈ F rac A Q as in Proposition 2.7, then these elementslie, respectively, also in A k and F rac A k . Then by applying Proposition 2.7 in theother direction, we conclude that Gelfand-Kirillov Conjecture also holds for A k . (cid:3) Theorem 2.11.
Let K = { A k } be a family of Z -compatible algebras. The Gelfand-Kirillov Conjecture holds for A Q if and only if ACF ⊢ φ K , φ K a formula in thelanguage of rings L = (0 , , + , − ) ; and φ K is the existential closure of a booleancombination of atomic formulas.Proof. As in the proof Theorem 2.8, there is locally closed subvariety X Q of acertain Q N such that Gelfand-Kirillov Conjecture holds if and only if X Q = ∅ .Recall how X Q is defined: the simultaneous vanishing of polynomials { f , . . . , f r } and non-vanishing of some g ∈ { g , . . . , g s } in Q N . Let ψ K := ( T f i = 0) ∩ ( S g j =0), and denote by φ the existential closure of ψ . Clearly X Q = ∅ if and only if Q (cid:15) φ . Since the theory of algebraically closed fields of a fixed characteristic iscomplete ([27]), this is the case if and only if ACF ⊢ φ . (cid:3) Theorem 2.12.
Given a Z -compatible family of algebras K = { A k } there is a first-order sentence φ K in the language of rings L (0 , , + , ∗ ) such that itens (1), (2), (3)are equivalent. (1) For some k algebraically closed of zero characteristic, the Gelfand-KirillovConjecture holds for A k . (2) For any k algebraically closed of zero characteristic, the Gelfand-KirillovConjecture holds for A k . (3) ACF ⊢ φ K .Moreover, φ K is naturally presented as the existential closure of a boolean combi-nation of atomic formulas in the language.Proof. (1) implies (2) by Corollary 2.9 and Proposition 2.10. (2) implies (1) trivially.Let φ K be as in Theorem 2.11. Then Proposition 2.10 and Theorem 2.11 show that(2) and (3) are equivalent. (cid:3) Definition 2.13.
Let F be an algebraically closed field of prime characteristic. Let K = { A k } be a Z -compatible family of algebras. Then we say that the modularGelfand-Kirillov Conjecture holds for A F if its skew field of fractions is a Weyl fieldover a purely transcendental extension of F . Part of the reasoning in the next result is similar to [41, Theorem 2].
IRST-ORDER CHARACTERIZATION OF NONCOMMUTATIVE BIRATIONAL EQUIVALENCE9
Theorem 2.14.
If the Gelfand-Kirillov Conjecture holds for some (any) A k , k algebraically field o zero characteristics, then its modular version holds for all A F ,whenever char F >> , F an arbitrary algebraically closed field.Proof. Since
ACF ⊢ φ K , by compactness ([27]), ACF p ⊢ φ K , p >>
0. Hence, if char F >> X Q ( F ) = ∅ . Reasoning like Theorem 2.8, we obtain by reductionmodulo prime the equivalents of(1) , (2) , (3) , (4) , (5) , (6) , (7) , (8) , (9);the process that gives this algebraic relations is reversible, and hence we haveanalogues of ( † ) and ( ‡ ) in prime chracteristic. Using Proposition 2.6, we concludethat the modular Gelfand-Kirillov Conjecture holds for A F . (cid:3) Applications
Lets fix Σ be a root system, with a fixed basis of simple roots ∆, Weyl group W , rank n = | ∆ | and N positive roots. Consider the Lie ring g Z , Σ obtained byChevalley basis in the complex Lie algebra (cf. [28, 25.2].For every algebraically closed field k , we have g k , Σ = g Z , Σ ⊗ Z k , and similarly U ( g k , Σ ) = U ( g Z , Σ ) ⊗ Z k (cf. [13], [29], [30]). Proposition 3.1. K = { U ( g k , Σ ) } is a Z -compatible family of algebras.Proof. We check the items of Definition 2.1. (1) was discussed above. By thePBW theorem ([13, 2.1]), U ( g k , Σ ) and U ( g Z , Σ ) have a finite dimensional filtrationsuch that the graded associated algebra is (Ξ) the polynomial algebra S ( g k , Σ ) orpolynomial ring S ( g Z , Σ ) respectively. We hence have (3). (4) is [13, 4.2.3]. (6)follows from [13, 7.4.6] and [30, 9.6]. Finally, (2) and (5) holds because (Ξ) allowus to apply Lemma 2.2. (cid:3) This proves the first part of Theorem 1.2. As an immediate application of The-orem 2.14, we recover Theorem 1.3.
Proposition 3.2.
Let g k , Σ , h k , Σ be a pair of semisimple Lie algebra and Cartansubalgebra such that g is split-semisimple (cf. [29] , [13] ), over a non-necessarelyclosed field of zero characteristic. If Σ is irreducible of type A , Gelfand-KirillovConjecture is true for all fields of zero characteristic. If Σ is irreducible of types B, D, E, F , then Gelfand-Kirillov Conjecture is false for all fields for which thealgebra is split-semisimple.Proof.
The remark about type A follows by noticing that the proof in [22] works for h k , Σ the standard Cartan subalgebra, which is split-semisimple over any field of zerocharacteristic. By Theorem 1.2, the negation of the Gelfand-Kirillov Conjectureis the samething as the validity of a Q formula. Since they are preserved bysubstructures and any field can be embbeded in its algebraic closure, the secondstatement follows. (cid:3) Introduce the Z -Weyl algebra given by the same relations as in Definition 1.1, A n ( Z ). When we have a field k , A n ( k ) can be seen as the subalgebra of End k k [ x , . . . , x n ]generated by the polynomial algebra (identified as a subalgebra of the endomor-phism algebra by left multiplication), and the derivations Der k k [ x , . . . , x n ], whichconsists of linear maps δ : k [ x , . . . , x n ] → k [ x , . . . , x n ] such that δ ( λ ) = 0; δ ( f g ) = δ ( f ) g + f δ ( g ), f, g, ∈ k [ x , . . . , y n ], λ ∈ k . A n ( Z ) has a similar description (cf. [37,15.1], [15, 2.1]). Lemma 3.3.
Each Weyl algebra (over Z or a field k ) has the Bernstein filtration B = { B j } j ≥ , B j = span h x a . . . x a n n y b . . . y b n n i with a ′ i s, b ′ i s ∈ N and P ni =1 a i + b i ≤ j . The graded associated algebra is Z [ x , . . . , x n , y , . . . , y n ] ; respectively k [ x , . . . , x n , y , . . . , y n ] .Proof. [15, 2.1.2]. (cid:3) Since the action of W is crystallographic on Z [∆] ∼ = Z [ x , . . . , x n ] (cf. [28,Section10]), we can make it acts on A n ( Z ) by conjugation: g ∈ W acts on D ∈ A n ( Z ) by : g ( D ( g − )). Similarly, it acts on A n ( k ) when k is any algebraicallyclosed field — reducing module a prime if necessary. We clearly have A n ( k ) W = A n ( Z ) W ⊗ Z k . A n ( Z ) W and A n ( k ) W inherit a filtration from the Bernstein filtration by consid-ering W -invariant elements. The graded associated algebras, in view of Lemma 3.3,are (Ψ) Z [ x , . . . , x n , y , . . . , y n ] W ; respectively k [ x , . . . , x n , y , . . . , y n ] W (cf. [15,3.2.3]).We shall need Theorem 3.4.
For an algebraically closed field of zero characteristic k , F rac A n ( k ) W ∼ = D n ( k ) .Proof. It follows from [21, Theorem 1], in view of Chevalley-Shephard-Todd Theo-rem (cf. [11]). (cid:3)
Proposition 3.5. K = { A n ( k ) W } is a Z -compatible family of algebras.Proof. Items (1) and (3) of Definition 2.1 were discussed above — cf. (Ψ). k = Z ( F rac A n ( k ) W ) ⊇ F rac Z ( A n ( k ) W ) ⊇ k for any algebraically closed field of zerocharacteristics by [22, Thm. 2] and Theorem 3.4; so item (4) follows. Item (6) istrivial as we’ve just seen that the center restrict to the scalars. Items (2) and (5)follow from Lemma 2.2 in view of remark (Ψ). So we are done. (cid:3) This proves the second part of Theorem 1.2. With this, Theorem 1.4 followsfrom Theorem 2.14. Theorem 1.5 follows from [21, Corollary 3.5] with a similarreasoning as above.
Remark 2.
The technical notion herein introduced of Z -compatible family of al-gebras was very useful to establish algebraic/model-theoretical results in the twoinstances treated in the present work. We are considering other classes of inter-esting algebras (but not only algebras over a field) were this device can be used toprovide other new results. References [1] J. Alev, F. Dumas. Sur les invariants des alg`ebres de Weyl et de leurs corps de fractions.(French) [Invariants of Weyl algebras and of their fields of fractions] Rings, Hopf algebras, andBrauer groups (Antwerp/Brussels, 1996), 1-10, Lecture Notes in Pure and Appl. Math., 197,Dekker, New York, 1998.[2] J. Alev and F. Dumas, Operateurs differentiels invariants et probl`eme de Noether, Studiesin Lie theory, 21-50, Progr. Math., 243, Birkh¨auser Boston, Boston, MA, 2006. Birkhauser,Boston, 2006.
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Instituto de Matematica e Estatistica, Universidade de S˜ao Paulo,S˜ao Paulo, Brasil
E-mail address : [email protected] J. Schwarz
Instituto de Matematica e Estatistica, Universidade de S˜ao Paulo,S˜ao Paulo, Brasil
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