Independence of the B-KK Isomorphism of Infinite Prime
aa r X i v : . [ m a t h . AG ] F e b Independence of the B-KK Isomorphism ofInfinite Prime
Alexei Kanel-Belov ∗ , Andrey Elishev † , and Jie-Tai Yu ‡ College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518061, China Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics andTechnology, Dolgoprudny, Moscow Region, 141700, Russia Department of Innovations and High Technology, Moscow Institute of Physics and Technology,Dolgoprudny, Moscow Region, 141700, Russia
Abstract
We investigate a certain class of Ind-scheme morphisms corresponding to homomorphismsbetween the automorphism group of the n -th complex Weyl algebra and the group of Poissonstructure-preserving automorphisms of the commutative complex polynomial algebra in 2 n variables. An open conjecture of Kanel-Belov and Kontsevich states that these automorphismgroups are canonically isomorphic in characteristic zero, with the mapping discussed herebeing the candidate for the isomorphism. The objective of the present paper is to establishthe independence of the said mapping of the choice of infinite prime - that is, the class [ p ] ofprime number sequences modulo fixed non-principal ultrafilter U on the index set of positiveintegers. To that end, we introduce the augmented versions of algebras in question andstudy the augmented Ind-morphism between the normalized automorphism Ind-schemes in thecontext of tame automorphism approximation. In order to correctly implement approximationin our proof, we study singularities of curves in augmented automorphism Ind-schemes andtheir images under Ind-scheme morphisms. The Belov – Kontsevich conjecture [1], sometimes Kanel-Belov – Kontsevich con-jecture, dubbed B − KKC n for positive integer n , seeks to establish a canonicalisomorphism between algebra automorphism groups Conjecture 1.1.
Aut( A n, C ) ≃ Aut( P n, C ) . Here A n, C is the n -th Weyl algebra over the field of complex numbers A n, C = C h x , . . . , x n , y , . . . , y n i / ( x i x j − x j x i , y i y j − y j y i , y i x j − x j y i − δ ij ) , and P n, C ≃ C [ z , . . . , z n ] is the commutative polynomial ring viewed as a C -algebraand equipped with the standard Poisson bracket: { z i , z j } = ω ij ≡ δ i,n + j − δ i + n,j Key words:
Weyl algebra automorphisms, polynomial symplectomorphisms, deformation quantization, infiniteprime number. ∗ [email protected] † [email protected] ‡ [email protected] he automorphisms from Aut( P n, C ) preserve the Poisson bracket.Let ζ i , i = 1 , . . . , n denote the standard generators of the Weyl algebra (theimages of x j , y i under the canonical projection). The filtration by total degree on A n, C induces a filtration on the automorphism group:Aut ≤ N A n, C := { f ∈ Aut( A n, C ) | deg f ( ζ i ) , deg f − ( ζ i ) ≤ N, ∀ i = 1 , . . . , n } . The obvious maps Aut ≤ N A n, C → Aut ≤ N +1 A n, C are Zariski-closed embeddings, the entire group Aut( A n, C ) is a direct limit of theinductive system formed by Aut ≤ N together with these maps. The same can be saidfor the symplectomorphism group Aut( P n, C ).The Belov – Kontsevich conjecture admits a stronger form, with C being replacedby the field of rational numbers.Since Makar-Limanov [8], [9], Jung [6] and van der Kulk [7], the Belov-Kontsevichconjecture is known to be true for n = 1. The proof is essentially a direct descriptionof the automorphism groups. Such a direct approach for n > . Nevertheless, at least one known candidate forisomorphism may be constructed in a rather straightforward fashion. The idea is tostart with an arbitrary Weyl algebra automorphism, lift it after a shift by a certainautomorphism of C to an automorphism of a larger algebra (of formal power serieswith powers taking values in the ring ∗ Z of hyperintegers) and then restrict to asubset of its center isomorphic to C [ z , . . . , z n ].This construction goes back to Tsuchimoto [12], who obtained a morphismAut( A n, C ) → Aut( P n, C )in order to prove the stable equivalence between the Jacobian and the Dixmier con-jectures. It was independently considered by Kontsevich and Kanel-Belov [2], whooffered a shorter proof of the Poisson structure preservation which does not employ p -curvatures. It should be noted, however, that Tsuchimoto’s thorough inquiry into p -curvatures has exposed a multitude of problems of independent interest, in whichcertain statements from the present paper might appear.The construction we describe in detail in the following sections differs from thatof Tsuchimoto in one aspect: a Weyl algebra automorphism f may in effect undergoa shift by an automorphism of the base field γ : C → C prior to being lifted, and thisextra procedure is homomorphic. Taking γ to be the inverse nonstandard Frobeniusautomorphism (see below), we manage to get rid of the coefficients of the form a [ p ] ,with [ p ] an infinite prime, in the resulting symplectomorphism. The key result hereis that for a large subgroup of automorphisms, the so-called tame automorphisms,one manages to completely eliminate the dependence of the whole construction onthe choice of the infinite prime [ p ]. Also, the resulting Ind-morphism φ [ p ] is anisomorphism of the tame subgroups. In particular, for n = 1 all automorphismsof A , C are tame (Makar-Limanov’s theorem), and the map φ [ p ] is the conjecturedcanonical isomorphism.These observations motivate the question whether for any n the group homomor-phism φ [ p ] is independent of infinite prime. In particular, we do not know whether all elements of Aut( P n, C ) are tame when n > he main result of the present paper is formulated as follows. Given two distinctinfinite primes [ p ] and [ p ′ ] (cf. the definition and construction below), one mayimplement Tsuchimoto’s construction and obtain two group homomorphisms φ [ p ] , φ p ′ : Aut( A n, C ) → Aut( P n, C ) . We assume a stronger version of Conjecture 1.1, namely that the homomorphismsof the form φ [ p ] are bijective. Main Theorem.
Assuming Conjecture 1.1, the homomorphism φ [ p ] ◦ φ − p ′ : Aut( P n, C ) → Aut( P n, C ) is the identity map for every pair [ p ] , [ p ′ ] . The proof of Main Theorem is based around the study of Ind-scheme morphisms φ [ p ] ◦ φ − p ′ and its Weyl algebra counterpart by means of approximation by tame au-tomorphism sequences, which is introduced in the next subsection. Approximationof arbitrary automorphisms by convergent sequences of tame automorphisms is, ofcourse, a natural idea. However, as will be made clear in the sequel, tame approxi-mation in the non-augmented case differs significantly from that of the augmentedalgebras. The point in introducing augmentation by central variables lies in thefact that the tame approximation is canonical with respect to the Ind-morphismof Conjecture 1.1: distinct tame sequences converging to the same automorphism(in the irreducible component of the identity automorphism) will have images con-verging to the same automorphism as well. This canonical property is establishedby investigating the singularities of one-parametric families (curves) of augmentedautomorphisms and their images under the Ind-morphisms. A similar argument al-lows one to construct the canonical lifting from Aut( P hn, C ) to Aut( A hn, C ) and proveConjecture 1.1 (see [27]).As will be seen later in the course of the paper, the augmentation and approxima-tion approach requires (Proposition 3.2 and Lemma 3.11) the Ind-map in questionto be a morphism of normalized schemes. On the other hand, the Tsuchimoto con-struction leading to φ [ p ] works equally well [12] for the semigroup End A n, C of Weylalgebra endomorphisms. If one were to assume the analogue of Conjecture 1.1 forthe semigroups End and assume that the semigroup homomorphism φ [ p ] induces amorphism of normalized varieties, then one could use approximation and singu-larities’ technique to repeat the steps of the main proof of independence of infiniteprime. Thus, one would have the following Theorem 1.2.
Suppose that φ [ p ] : End A n, C → End P n, C is a semigroup isomorphism which induces a morphism of the associated (normal-ized) varieties. Then φ [ p ] is independent of the choice of the infinite prime [ p ] . Remark 1.3.
Positivity of the Jacobian Conjecture, together with the results of thepresent paper, would render the latter theorem trivial.
The main body of the paper is divided into two parts. In the first part weconsider the construction of Tsuchimoto and define map φ [ p ] . In the second part Something which does not follow from the proof of Proposition 3.2, for there we do require invertibility ofendomorphisms. e demonstrate that the map φ [ p ] possesses several important properties that willbe used in the subsequent proof of the Main Theorem. At that point, a certaindeformation (or rather, augmentation by central variables) of algebras A n and P n are introduced, and the critical proposition (Proposition 3.9) is established. TheMain Theorem will be a consequence of Proposition 3.9. Ind -schemes, morphisms, approximation, and singularities technique
The Main Theorem is proved under the assumption of Conjecture 1.1 – or rather, itsstronger form with ϕ [ p ] as the group isomorphism. Recently [27] we have managedto prove this form of Conjecture 1.1. Our proof relies upon the augmentation of thealgebras in question by central variables (introduced in the subsection below).Dodd [4], on the other hand, has established a number of far-reaching resultsof homological nature which imply Conjecture 1.1. His argument is based in thetheory of holonomic D -modules and properties of the so-called p -support, definedby Kontsevich in [3]. Various subtle properties of p -support are due to Bitoun [29]and Van den Bergh [30].Our approach is entirely different from that of Dodd and focuses mainly of thetopological properties of automorphism Ind-schemes, such as Aut A n . In this respect,our method adheres to the school of thought of Shafarevich (the reader is referredparticularly to [20]).More precisely, the schemes we consider in the paper are given by normalizationsof schemes corresponding to varieties Aut ≤ N ; in effect, we will later understandby Aut ≤ N the normalizations of the automorphism varieties defined previously. Aswe will observe (Proposition 3.2, Corollary 3.4, and Proposition 3.10), transitionto normal schemes is required for the establishment of the tame approximationproperty. The insufficiency of naive approximation approach, as well as the generalcontext regarding automorphism Ind-schemes, specifically the study of singularitiesof curves in those, is explored in [25] in greater detail.In fact, it is the investigation of the behavior of singularities of curves of aug-mented automorphisms that allows us to establish the canonicity of approximationin the augmented setting. This critical property is not present in the non-augmentedcase; thus the introduction of augmented algebras is justified.The theory of Ind-schemes is an actively studied area that contains a number ofopen problems, apart from the ones formulated here. See for instance [13] or otherpapers of J.-P. Furter and references therein.The proof of the Main Theorem, as well as the strengthened version of Conjecture1.1 (cf. our newest paper [27]), relies on the phenomenon of approximation by tameautomorphisms. The definition is as follows.Suppose first that K [ x , . . . , x n ] is the polynomial algebra over a field K , and let ϕ be an automorphism of this algebra. Definition 1.4. ϕ is an elementary automorphism if it is of the form ϕ = ( x , . . . , x k − , ax k + f ( x , . . . , x k − , x k +1 , . . . , x n ) , x k +1 , . . . , x n ) with a ∈ K × . Observe that linear invertible changes of variables – that is, transformations ofthe form ( x , . . . , x n ) ( x , . . . , x n ) A, A ∈ GL( n, K )are realized as compositions of elementary automorphisms. efinition 1.5. A tame automorphism is an element of the subgroup TAut K [ x , . . . , x n ] generated by all elementary automorphisms. Automorphisms thatare not tame are called wild . All automorphisms of K [ x, y ] are tame [6, 7]; there are examples of wild automor-phisms of K [ x, y, z ] (Nagata automorphism). It is unknown whether there are wildautomorphisms in the case of 2 n ( n >
1) generators. An open conjecture asserts thatevery polynomial automorphism becomes tame after adjunction of a finite numberof variables on which its action is extended by the identity map.A tame polynomial symplectomorphism is a tame automorphism that preservesthe Poisson structure – in other words, the group TAut P n is the intersection of thetame automorphism group with Aut P n . With the standard Poisson structure fixed,it is easy to see the necessary and sufficient conditions for elementary automorphismsto be symplectic.Similarly, tame automorphisms of the Weyl algebra are defined.One of the crucial properties of the homomorphism φ [ p ] , proved in [1], is as follows. Proposition 1.6.
The restriction of φ [ p ] to the subgroup TAut A n, C is an isomor-phism TAut A n, C → TAut P n, C Another important result is the symplectic version of a classical theorem of D.Anick ( [10]), which states that the tame automorphism subgroup is dense in theautomorphism group in the formal power series topology. The proof is in [24].
Theorem 1.7.
The subgroup
TAut P n, K ( char K = 0 ) is dense in Aut P n, K in thepower series topology. Approximation by tame automorphisms is essentially Anick’s theorem and, inour context, its symplectic analogue (Theorem 1.7). Given a symplectomorphism,one may choose a sequence of tame symplectomorphism which converges to it in thepower series topology. Then, using the tame isomorphism of Proposition 1.6, onemay form a sequence of tame automorphisms of the corresponding Weyl algebra;one can then take its formal limit (whose action is given by power series in the Weylgenerators) and ask whether this limit is well defined (independent of the choice oftame symplectomorphism sequence) and exists in Aut A n . If one manages to provethe correctness and power series truncation, then the inverse to the homomorphism φ [ p ] is constructed and Conjecture 1.1 is proved. The tame approximation methodis the main subject of [27]. As it turns out, the simplest way to construct theinverse (the lifting map) is to introduce an augmentation of the Poisson (and Weylcommutator) structure by adjoining a central variable, which is analogous to thedeformation parameter (Planck’s constant) in the Kontsevich’s quantization recipeand which distorts the power series topology, construct a well defined lifting map forthe new algebras and then show that a specialization of the augmentation variables(required to return to the non-augmented case of Conjecture 1.1) is valid.The proof of the Main Theorem is also based on the tame approximation tech-nique, with parts of the specialization argument essentially copying that of [27]. Themorphism in question is Φ = φ [ p ] ◦ φ − p ′ for a fixed choice of [ p ] = [ p ′ ]. Nevertheless, some properties of Φ can be extractedusing methods of combinatorial nature, without the introduction of power seriestopology. The first half of Section 3 provides several examples. The techniqueutilized in the proofs is not dissimilar from those described in our prior work [25] n Ind-schemes. While these properties are not essential to the proof of the mainresult, we believe the corresponding proofs to be of independent interest and in factto be quite illustrative of the nature of the problems in question.In particular, the proof techniques we implement (Proposition 3.7) will enable usto obtain the following result. Theorem 1.8.
Let Φ be an Ind -automorphism of the
Ind -scheme
Aut P n, C (thatis, Φ is a collection of maps Φ N of the Ind -scheme strata, such that all Φ N aremorphisms). Suppose that Φ preserves asymptotic expansions of curves: if L ( t ) is acurve in Aut P n, C parameterized by t and at t = t it has a pole of order k , such thatthe coordinate functions a i on the curve admit, as functions of t , a decomposition a i ( t ) = a − k ( t − t ) − k + · · · then we say that Φ preserves asymptotic expansions if the image Φ( L ( t )) has a poleat t of the same order with the same asymptotic expansion as L ( t ) . Then, if Φ preserves asymptotic expansions of curves, then there is a positive integer m suchthat the composition Φ m = Φ ◦ . . . ◦ Φ is the identity morphism. In [25], we have obtained several results on the geometry (and composition) ofInd-scheme morphisms of Aut K [ x , . . . , x n ], the Ind-scheme of automorphisms of thepolynomial algebra. In analogy with the main results of that study, we formulatethe following conjecture. Conjecture 1.9.
Every
Ind -automorphism of the
Ind -scheme
Aut P n, K is inner(that is, a conjugation by an element of Aut P n ), whenever the ground field K isof characteristic zero. An identical statement may be formulated for the Weyl algebra.In the subsection below, we introduce the augmented versions of the Weyl andPoisson algebras. The images of these algebras under Aut are Ind-schemes, thereforean analogue of Conjecture 1.9 may be stated for these Ind-schemes as well.Finally, the construction of the morphism φ [ p ] , as well as the notion of infiniteprime in the context of algebraic geometry, has its roots in model theory. Applica-tion of model theory to problems in algebraic geometry has been developed by E.Hrushovski and B. Zilber, cf. for instance [18] and [19].The study of Ind-schemes using tools from model theory is also our objective,with Conjecture 1.1 being one of the more significant examples. It is our hope thatthe investigation of such instances will yield new insights into the theory itself. In the proof of our main result we make use of the augmented (quantized) versionsof A n and P n , which we now define. In order to quantize the Weyl algebra A n , weintroduce additional generators h , . . . , h n and modify the commutator between y and x by setting [ y i , x j ] = h i δ ij . Alternatively, one can start with the free algebra K h a , . . . , a n , b , . . . , b n , c , . . . , c n i and take the quotient with respect to the following set of identities: b i a j − a i b j − δ ij c i , a i c j − a j c j , b i c j − b j c i . he quotient algebra, which we denote by B n (or B n, K to indicate the base field) isthe augmented Weyl algebra.Similarly, we may distort the Poisson bracket of P n : { p i , x j } = h i δ ij to reflect the augmentation of A n into B n in the classical counterpart. (Here we haverenamed the generators z i into x i and p i , according to their behavior with respect tothe Poisson bracket. The notation is standard.) The resulting polynomial algebrawill be denoted by R n .The algebras B n and R n are connected by an analogue of the Kontsevich homo-morphism (which is constructed for the undeformed case below); the main pointis that the independence of this quantized homomorphism of infinite prime impliesthe independence of infinite prime of the non-augmented morphism. Also, the au-tomorphism groups of algebras B n and R n admit nice approximation propertieswith respect to their formal power series topologies (a direct analogue of Theorem1.7 holds) as well as generally behave in a way best suited for some of our prooftechniques. For instance, unlike the Poisson algebra P n , the augmented version R n admits linear changes of generators of the form x Λ x, p Λ p, h Λ h (here we rename the generators z i and partition them into two groups x i and p i –coordinates and momenta – according to the Poisson bracket; the introduction ofauxiliary generators h i made it possible for both x and p to be simultaneously actedupon by a single matrix; in particular, both x and p can now be dilated). A versionof these augmented algebras appeared in the work of Myung and Oh [28]. Let
U ⊂ N be an arbitrary non-principal ultrafilter on the set of all positive numbers( N will almost always be regarded as the index set in this note) . Let P be the setof all prime numbers, and let P N denote the set of all sequences p = ( p m ) m ∈ N ofprime numbers. We refer to a generic set A ∈ U as an index subset in situationsinvolving the restriction p | A : A → P . We will call a sequence p of prime numbers U -stationary if there is an index subset A ∈ U such that its image p ( A ) consists ofone point.A sequence p : N → P is bounded if the image p ( N ) is a finite set. Thanksto the ultrafilter finite intersection property, bounded sequences are necessarily U -stationary.Any non-principal ultrafilter U generates a congruence ∼ U ⊆ P N × P N in the following way. Two sequences p and p are U -congruent iff there is an indexsubset A ∈ U such that for all m ∈ A the following equality holds: p m = p m . The corresponding quotient ∗ P ≡ P N / ∼ U ontains as a proper subset the set of all primes P (naturally identified with classesof U -stationary sequences), as well as classes of unbounded sequences. The latterare referred to as nonstandard, or infinitely large, primes. We will use both namesand normally denote such elements by [ p ], mirroring the convention for equivalenceclasses. The terminology is justified, as the set of nonstandard primes is in one-to-one correspondence with the set of prime elements in the ring ∗ Z of nonstandardintegers in the sense of Robinson [22].Indeed, one may utilize the following construction, which was thoroughly studied in [23]. Consider the ring Z ω = Q m ∈ N Z - the product of countably many copies of Z indexed by N . The minimal prime ideals of Z ω are in bijection with the set of allultrafilters on N (perhaps it is opportune to remind that the latter is precisely theStone-Cech compactification β N of N as a discrete space). Explicitly, if for every a = ( a m ) ∈ Z ω one defines the support complement as θ ( a ) = { m ∈ N | a m = 0 } and for an arbitrary ultrafilter U ∈ N sets( U ) = { a ∈ Z ω | θ ( a ) ∈ U } , then one obtains a minimal prime ideal of Z ω . It is easily shown that every minimalprime ideal is of such a form. Of course, the index set N may be replaced by any set I , after which one easily gets the description of minimal primes of Z I (since thosecorrespond to ultrafilters, there are exactly 2 | I | of them if I is infinite and | I | when I is a finite set). Note that in the case of finite index set all ultrafilters are principal,and the corresponding ( U ) are of the form Z × · · · × (0) × · · · × Z - a textbookexample.Similarly, one may replace each copy of Z by an arbitrary integral domain andrepeat the construction above. If for instance all the rings in the product happen tobe fields, then, since the product of any number of fields is von Neumann regular,the ideal ( U ) will also be maximal.The ring of nonstandard integers may be viewed as a quotient (an ultrapower) Z ω / ( U ) = ∗ Z . The class of U -congruent sequences [ p ] corresponds to an element (also an equiv-alence class) in ∗ Z , which may as well as [ p ] be represented by a prime numbersequence p = ( p m ), only in the latter case some but not too many of the primes p m may be replaced by arbitrary integers. For all intents and purposes, this differenceis insignificant.Also, observe that [ p ] indeed generates a maximal prime ideal in ∗ Z : if one for(any) p ∈ [ p ] defines an ideal in Z ω as( p, U ) = { a ∈ Z ω | { m | a m ∈ p m Z } ∈ U } , then, taking the quotient Z ω / ( p, U ) in two different ways, one arrives at an isomor-phism ∗ Z / ([ p ]) ≃ Y m Z p m ! / ( U ) , and the right-hand side is a field by the preceding remark. For a fixed non-principal also cf. [16] and an infinite prime [ p ], we will call the quotient Z [ p ] ≡ ∗ Z / ([ p ])the nonstandard residue field of [ p ]. Under our assumptions this field has character-istic zero. We have seen that the objects [ p ] - the infinite prime - behaves similarly to the usualprime number in the sense that a version of a residue field corresponding to this ob-ject may be constructed. Note that the standard residue fields are contained as adegenerate case in this construction, namely if we drop the condition of unbound-edness and instead consider U -stationary sequences, we will arrive at a residue fieldisomorphic to Z p , with p being the image of the stationary sequence in the chosenclass. The fields of the form Z [ p ] are a realization of what is known as pseudofinitefield, cf. [18].The nonstandard case is surely more interesting. While the algebraic closure of astandard residue field is countable, the nonstandard one itself has the cardinality ofthe continuum. Its algebraic closure is also of that cardinality and has characteristiczero, which implies that it is isomorphic to the field of complex numbers. We proceedby demonstrating these facts. Proposition 2.1.
For any infinite prime [ p ] the residue field Z [ p ] has the cardinalityof the continuum .Proof. It suffices to show there is a surjection h ∗ : Z [ p ] → P , where P = { , } ω is the Cantor set given as the set of all countable strings of bitswith the 2-adic metric d ( x, y ) = 1 /k, k = min { m | x m = y m } . The map h ∗ is constructed as follows. If Z ⊂ P is the subset of all strings with finitenumber of ones in them, and e : Z + → Z , e X k
The algebraic closure Z [ p ] of Z [ p ] is isomorphic to the field of complexnumbers. We now fix the notation for the aforementioned isomorphisms in order to employit in the next section.For any nonstandard prime [ p ] ∈ ∗ P fix an isomorphism α [ p ] : C → Z [ p ] comingfrom the preceding theorem. Denote by Θ [ p ] : Z [ p ] → Z [ p ] the nonstandard Frobeniusautomorphism - that is, a well-defined field automorphism that sends a sequence ofelements to a sequence of their p m -th powers:( x m ) ( x p m m ) . The automorphism Θ [ p ] is identical on Z [ p ] ; conjugated by α [ p ] , it yields a wildautomorphism of complex numbers, as by assumption no finite power of it (as always,in the sense of index subsets A ∈ U ) is the identity homomorphism. The n -th Weyl algebra A n, C ≃ A n, Z [ p ] can be realized as a proper subalgebra of thefollowing ultraproduct of algebras A n ( U , [ p ]) = Y m ∈ N A n, F pm ! / U . Here for any m the field F p m = Z p m is the algebraic closure of the residue field Z p m .This larger algebra contains elements of the form ( ζ I m ) m ∈ N with unbounded | I m | -something which is not present in A n, Z [ p ] , hence the proper embedding. Note thatfor the exact same reason (with degrees | I m | of differential operators having beenreplaced by degrees of minimal polynomials of algebraic elements) the inclusion Z [ p ] ⊆ Y m ∈ N F p m ! / U is also proper.It turns out that, unlike its standard counterpart A n, C , the algebra A n ( U , [ p ]) hasa huge center described in this proposition: Proposition 2.3.
The center of the ultraproduct of Weyl algebras over the se-quence of algebraically closed fields { F p m } coincides with the ultraproduct of centersof A n, F pm : C ( A n ( U , [ p ])) = Y m C ( A n, F pm ) ! / U . he proof is elementary and is left to the reader. As in positive characteristicthe center C ( A n, F p ) is given by the polynomial algebra F p [ x p , . . . , x pn , y p , . . . , y pn ] ≃ F p [ ξ , . . . , ξ n ] , there is an injective C -algebra homomorphism C [ ξ , . . . ξ n ] → Y m F p m [ ξ ( m )1 , . . . ξ ( m )2 n ] ! / U from the algebra of regular functions on A n C to the center of A n ( U , [ p ]), evaluatedon the generators in a straightforward way: ξ i [( ξ ( m ) i ) m ∈ N ] . Just as before, this injection is proper.Furthermore, the image of this monomorphism (the set which we will simply referto as the polynomial algebra) may be endowed with the canonical Poisson bracket.Recall that in positive characteristic case for any a, b ∈ Z p [ ξ , . . . , ξ n ] one can define { a, b } = − π (cid:18) [ a , b ] p (cid:19) . Here π : A n, Z → A n, Z p is the modulo p reduction of the Weyl algebra, and a , b arearbitrary lifts of a, b with respect to π . The operation is well defined, takes values inthe center and satisfies the Leibnitz rule and the Jacobi identity. On the generatorsone has { ξ i , ξ j } = ω ij . The Poisson bracket is trivially extended to the entire center F p [ ξ , . . . , ξ n ] and thento the ultraproduct of centers. Observe that the Poisson bracket of two elementsof bounded degree is again of bounded degree, hence one has the bracket on thepolynomial algebra. The point of this construction lies in the fact that thus defined Poisson structure onthe (injective image of) polynomial algebra is preserved under all endomorphisms of A n ( U , [ p ]) of bounded degree. Every endomorphism of the standard Weyl algebra isspecified by an array of coefficients ( a i,I ) (which form the images of the generatorsin the standard basis); these coefficients are algebraically dependent, but with onlya finite number of bounded-order constraints. Hence the endomorphism of the stan-dard Weyl algebra can be extended to the larger algebra A n ( U , [ p ]). The restrictionof any such obtained endomorphism on the polynomial algebra C [ ξ , . . . , ξ n ] pre-serves the Poisson structure. In this setup the automorphisms of the Weyl algebracorrespond to symplectomorphisms of A n C . Example . If x i and y i are standard generators, then one may perform a linearsymplectic change of variables: f ( x i ) = n X j =1 a ij x j + n X j =1 a i,n + j y j , i = 1 , . . . , n,f ( d i ) = n X j =1 a i + n,j x j + n X j =1 a i + n,n + j y j , a ij ∈ C . n this case the corresponding polynomial automorphism f c of C [ ξ , . . . , ξ n ] ≃ C [ x [ p ]1 , . . . , x [ p ] n , y [ p ]1 , . . . , y [ p ] n ]acts on the generators ξ as f c ( ξ i ) = n X j =1 ( a ij ) [ p ] ξ j , where the notation ( a ij ) [ p ] means taking the base field automorphism that is conju-gate to the nonstandard Frobenius via the Steinitz isomorphism.Let γ : C → C be an arbitrary automorphism of the field of complex numbers.Then, given an automorphism f of the Weyl algebra A n, C with coordinates ( a i,I ), onecan build another algebra automorphism using the map γ . Namely, the coefficients γ ( a i,I ) define a new automorphism γ ∗ ( f ) of the Weyl algebra, which is of the samedegree as the original one. In other words, every automorphism of the base fieldinduces a map γ ∗ : A n, C → A n, C which preserves the structure of the ind-object. Itobviously is a group homomorphism.Now, if P n, C denotes the commutative polynomial algebra with Poisson bracket,we may define an ind-group homomorphism φ : Aut( A n, C ) → Aut( P n C ) as follows.Previously we had a morphism f f c , however as the example has shown it ex-plicitly depends on the choice of the infinite prime [ p ]. We may eliminate thisdependence by pushing the whole domain Aut( A n, C ) forward with a specific basefield automorphism γ , namely γ = Θ − p ] - the field automorphism which is Steinitz-conjugate with the inverse nonstandard Frobenius, and only then constructing thesymplectomorphism f c Θ as the restriction to the (nonstandard) center. For the sub-group of tame automorphisms such as linear changes of variables this procedure hasa simple meaning: just take the [ p ]-th root of all coefficients ( a i,I ) first. We thusobtain a group homomorphism which preserves the filtration by degree and is infact well-behaved with respect to the Zariski topology on Aut (indeed, the filtra-tion Aut N ⊂ Aut N +1 is given by Zariski-closed embeddings). Formally, we have aproposition: Proposition 2.4.
There is a system of morphisms φ [ p ] ,N : Aut ≤ N ( A n, C ) → Aut ≤ N ( P n, C ) . such that the following diagram commutes for all N ≤ N ′ : Aut ≤ N ( A n, C ) Aut ≤ N ( P n, C )Aut ≤ N ′ ( A n, C ) Aut ≤ N ′ ( P n, C ) φ [ p ] ,N µ NN ′ ν NN ′ φ [ p ] ,N ′ The corresponding direct limit of this system is given by φ [ p ] , which maps a Weylalgebra automorphism f to a symplectomorphism f c Θ . The strengthened form of the Belov – Kontsevich conjecture (Conjecture 1.1)then states:
Conjecture 2.5. φ [ p ] is a group isomorphism. njectivity may be established right away. Theorem 2.6. φ [ p ] is an injective homomorphism. (See [1] for the fairly elementary proof). Let us at first assume that the Belov – Kontsevich conjecture holds, with φ [ p ] fur-nishing the isomorphism between the automorphism groups. (This would be thecase if all automorphisms in Aut( A n, C ) were tame, which is unknown at the momentfor n >
1. See also [27] for the justification of the above assumption.)Let [ p ] and [ p ′ ] be two distinct classes of U -congruent prime number sequences -that is, two distinct infinite primes. We then have the following diagram:Aut( A n, C ) Aut( P n, C )Aut( A n, C ) Aut( P n, C ) φ [ p ] isom isomφ [ p ′ ] with all arrows being isomorphisms. Vertical isomorphisms answer to different pre-sentations of C as Z [ p ] and Z [ p ′ ] . The corresponding automorphism C → Z [ p ] isdenoted by α [ p ] for any [ p ].The fact that all the arrows in the diagram are isomorphisms allows one insteadto consider a loop morphism of the formΦ : Aut( A n, C ) → Aut( A n, C ) . Furthermore, as it was noted in the previous section, the morphism Φ belongs toAut(Aut( A n, C )). The main result of the paper is as follows: Theorem 3.1.
If one assumes that φ [ p ] ,N is surjective for any infinite prime [ p ] ,then Φ is necessarily the identity morphism. In order to prove the above theorem, we are going to establish several impor-tant properties of Φ and related maps, the first of which is given by the followingproposition.
Proposition 3.2.
For an arbitrary Weyl algebra automorphism f ∈ Aut ≤ N ( A n, C ) ,the coordinates of Φ N ( f ) are given by algebraic functions of the coordinates of f .Proof. The statement of this Proposition is contained in Theorem 2 of [1]; for theconvenience of the reader, we reproduce the proof with technical modifications inaccordance with the context of this section. The idea of utilizing local ad-nilpotencyis originally due to O. Gabber.It suffices to demonstrate the Proposition as a property of φ [ p ] . More precisely, itis enough to show that, given an automorphism f p of the Weyl algebra in positivecharacteristic p with coordinates ( a i,I ), its restriction to the center (a symplectomor-phism) f cp has coordinates which are algebraic in ( a pi,I ), with the additional propertyof being implicitly defined by polunomial identities of degree independent of p (sothat one can return to characteristic zero). he switch to positive characteristic and back is performed for a fixed f ∈ Aut( A n, C ) on an index subset A f ∈ U .Let f be an automorphism of A n, C and let N = deg f be its degree. The auto-morphism f is specified by its coordinates a i,I ∈ C , i = 1 , . . . , n , I = { i , . . . , i n } ,obtained from the decomposition of algebra generators ζ i in the standard basis ofthe free module: f ( ζ i ) = X i,I a i,I ζ I , ζ I = ζ i · · · ζ i n n . Let ( a i,I,p ) denote the class α [ p ] ( a i,I ), p = ( p m ), and let { R k ( a i,I | i, I ) = 0 } k =1 ,...,M be a finite set of algebraic constraints for coefficients a i,I . Let us denote by A , . . . , A M the index subsets from the ultrafilter U , such that A k is precisely the subset on whoseindices the constraint R k is valid for ( a i,I,p ). Take A f = A ∩ . . . ∩ A M ∈ U and for p m , m ∈ A f , define an automorphism f p m of the Weyl algebra in positive characteristic A n, F pm by setting f p m ( ζ i ) = X i,I a i,I,p m ζ I . All of the constraints are valid on A f , so that f corresponds to a class [ f p ] moduloultrafilter U of automorphisms in positive characteristic. The degree of every f p m ( m ∈ A f ) is obviously less than or equal to N = deg f .Now consider f ∈ Aut ≤ N ( A n, C ) with the index subset A f over which its definingconstraints are valid. The automorphisms f p m = f p : A n, F p → A n, F p defined for m ∈ A f ∈ U provide arrays of coordinates a i,I,p . Let us fix any valid p m = p anddenote by F p k a finite subfield of F p which contains the respective coordinates a i,I,p (one may take k to be equal to the maximum degree of all minimal polynomials ofelements a i,I,p which are algebraic over Z p ).Let a , . . . , a s be the transcendence basis of the set of coordinates a i,I,p and let t , . . . , t s denote s independent (commuting) variables. Consider the field of rationalfunctions: F p k ( t , . . . , t s ) . The vector space Der Z p ( F p k ( t , . . . , t s ) , F p k ( t , . . . , t s ))of all Z p -linear derivations of the field F p k ( t , . . . , t s ) is finite-dimensional with Z p -dimension equal to ks ; a basis of this vector space is given by elements { e a D t b | a = 1 , . . . , k, b = 1 , . . . , s } where e a are basis vectors of the Z p -vector space F p k , and D t b is the partial derivativewith respect to the variable t b .Set a , . . . , a s = t , . . . , t s (i.e. consider an s -parametric family of automor-phisms), so that the rest of the coefficients a i,I,p are algebraic functions of s variables t , . . . , t s . We need to show that the coordinates of the corresponding symplecto-morphism f cp are annihilated by all derivations e a D t b .Let δ denote a derivation of the Weyl algebra induced by an arbitrary basisderivation e a D t b of the coefficient field. For a given i , let us introduce the short-hand notation a = f p ( ζ i ) , b = δ ( a ) . We need to prove that ( f c ( ξ i )) = δ ( f p ( ζ pi )) = 0 . In our notation δ ( f p ( ζ pi )) = δ ( a p ), so by Leibnitz rule we have: δ ( f p ( ζ pi )) = ba p − + aba p − + · · · + a p − b. Let ad x : A n, F p → A n, F p denote a Z p -derivation of the Weyl algebra correspondingto the adjoint action (recall that all Weyl algebra derivations are inner):ad x ( y ) = [ x, y ] . We will call an element x ∈ A n, F p locally ad-nilpotent if for any y ∈ A n, F p there isan integer D = D ( y ) such that ad Dx ( y ) = 0 . All algebra generators ζ i are locally ad-nilpotent. Indeed, one could take D ( y ) =deg y + 1 for every ζ i .If f is an automorphism of the Weyl algebra, then f ( ζ i ) is also a locally ad-nilpotent element for all i = 1 , . . . , n . That means that for any i = 1 , . . . , n thereis an integer D ≥ N + 1 such thatad Df p ( ζ i ) ( δ ( f p ( ζ i ))) = ad Da ( b ) = 0 . Now, for p ≥ D + 1 the previous expression may be rewritten as0 = ad p − a ( b ) = p − X l =0 ( − l (cid:18) p − l (cid:19) a l ba p − − l ≡ p − X l =0 a l ba p − − l (mod p ) , and this is exactly what we wanted.We have thus demonstrated that for an arbitrary automorphism f p of the Weylalgebra in characteristic p the coordinates of the corresponding symplectomorphism f cp are algebraic functions in the p -th powers of the coordinates of f p , provided that p is greater than deg f p + 1.As the sequence (deg f p m ) is bounded from above by N for all m ∈ A f , we see thatthere is an index subset A ∗ f ∈ U such that the coordinates of the symplectomorphism f cp m for m ∈ A ∗ f are algebraic functions in p m -th powers of a i,I,p m . Furthermore, thedegree of these functions’ defining polynomials (i.e. polynomials whose roots arethese algebraic functions) is also bounded from above, as f p is in Aut ≤ N . There-fore, the image Φ N ( f ) of the characteristic zero automorphism will have coordinatesalgebraic in the coefficients of f . Remark 3.3.
The nature of local ad-nilponency is not to be considered trivial. Thisis especially so in the context of ultrafilter decomposition (modulo infinite primereduction), for various propositions involving the Weyl algebra in positive charac-teristic may well contain elements whose degree is dependent on the prime p . Thisproblem presents certain difficulties, sometimes of quite substantial nature (one no-table example being the Dixmier conjecture itself, or rather the impossibility of itsnegation by a naive example in positive characteristic with subsequent ultrafilter re-construction). or a more immediate illustration, consider the following elementary automor-phism of A , F p ϕ : x x + ay p − , y y with a constant. The rank of this automorphism grows as p , which prevents it frombeing reconstructed in characteristic zero. Moreover, the problem becomes even moreinvolved if one looks at its image under the Kontsevich homomorphism: the actionupon the central element x p will result in the appearance of coefficients which willbe (even after the Frobenius untwisting) explicitly dependent on p . Also, thanks tothe Weyl algebra commutation relations, the term ay p − x p − in the image of x p willcontribute (in the standard ordering of generators) a constant term a , so that theimage looks as this: ϕ ( x p ) = x p + a p y p ( p − + a. This would have a prohibitive effect on the theorems on symplectomorphism ap-proximation (as developed in [24]), were the latter to be applied naively.
Proposition 3.2 has the following important corollary.
Corollary 3.4.
Let (Aut ≤ N ( A n, C )) ν denote the normalization of the scheme Aut ≤ N ( A n, C ) (corresponding to the variety defined previously). Then the maps Φ N induce monomorphisms of normal schemes Φ N : (Aut ≤ N ( A n, C )) ν → (Aut ≤ N ( A n, C )) ν . Proof.
This corollary is immediate since the coefficients of Φ N ( f ) are algebraic incoordinates of f and are therefore in the coordinate ring of the normalized variety.Injectivity follows from the assumption on φ [ p ] at the beginning of the section.In view of Corollary 3.4, it is sensible to amend the definitions of automorphismInd-schemes we have given in the previous sections. From this point forward, byAut ≤ N we mean normalized versions of varieties defined in the introduction (thusdropping the superscript ν for the sake of notational convenience). The correspond-ing direct limits will therefore be normal Ind-schemes. Accordingly, the Ind-mapΦ (corresponding to the group homomorphism Φ) will always mean the system ofmorphisms of Corollary 3.4. Remark 3.5.
It is possible to establish a stronger version of Proposition 3.2 (orrather, the property of φ [ p ] demonstrated in the proof ) for the subschemes TAut of tame automorphisms: the map φ [ p ] is polynomial and indeed is an isomorphism(independent of the choice of infinite prime, as proved in [1]). We now proceed with our investigation of properties of the Ind-morphism Φ.The automorphism Φ acting on elements f ∈ Aut( A n, C ), takes the set of coor-dinates ( a i,I ) and returns a set ( G i,I ( a k,K )) of the same size. All functions G i,I arealgebraic by the above proposition. It is convenient to introduce a partial order onthe set of coordinates. We say that a i,I ′ is higher than a i,I (for the same generator i )if | I | < | I ′ | and we leave pairs with i = j or with | I | = | I ′ | unconnected. We definethe dominant elements a i,I (or rather, dominant places ( i, I )) to be the maximalelements with respect to this partial order, and subdominant elements to be theelements covered by maximal ones (in other words, for fixed i , subdominant placesare the ones with | I | = | I max | − emma 3.6. Functions G i,I corresponding to dominant places ( i, I ) are identities: G i,I ( a k,K ) = a i,I . Proof.
Indeed, it follows from the commutation relations that for any i = 1 , . . . , n and f p , p = p m , m ∈ A f ∈ U , the highest-order term in f cp ( ξ i ) = f p ( ζ pi ) = f p ( ζ i ) p has the coefficient a pi,I,p . The shift by the inverse Frobenius then acts as the p -throot on the dominant place, so that we deduce that the latter is independent of thechoice of [ p ].Let us now fix N ≥ N : Aut ≤ N A n, C → Aut ≤ N A n, C – the restriction of Φ to the subvariety Aut ≤ N A n, C , which is well defined by the abovelemma. The morphism corresponds to an endomorphism of the ring of functionsΦ ∗ N : O (Aut ≤ N A n, C ) → O (Aut ≤ N A n, C )Let us take a closer look at the behavior of Φ N (and of Φ ∗ N , which is essentially thesame up to an inversion), specifically at how Φ N affects one-dimensional subvarietiesof automorphisms. Let X N be the set of all algebraic curves of automorphisms inAut ≤ N A n, C ; by virtue of Lemma 3.6 we may without loss of generality consider thesubset of all curves with fixed dominant places – we denote such a subset by X ′ N ,and, for that same matter, the subsets X ( k ) N of curves with fixed places of the form( i, I ′ ), which are away from a dominant place by a path of length at most ( k − X ′ N = X (1) N .The morphism Φ N yields a map˜Φ N : X N → X N and its restrictions ˜Φ ( k ) N : X ( k ) N → X N . Our immediate goal is to prove that for all attainable k we have˜Φ ( k ) N : X ( k ) N → X ( k ) N , i.e. the map Φ N preserves the terms corresponding to non-trivial differential mono-mials.In spite of minor abuse of language, we will call the highest non-constant termsof a curve in X ( k ) N dominant, although they cease to be so when that same curve isregarded as an element of X N .Let A ∈ X N be an algebraic curve in general position. Coordinate-wise A an-swers to a set ( a i,I ( τ )) of coefficients parameterized by an indeterminate. By Lemma3.6, Φ N leaves the (coefficients corresponding to) dominant places of this curve un-changed, so we may well set A ∈ X (1) N . In fact, it is easily seen that the subdominantterms are not affected by Φ N either, thanks to the commutation relations that definethe Weyl algebra: for every p participating in the ultraproduct decomposition, afterone raises to the p -th power one should perform a reordering within the monomials– a procedure which decreases the cardinality | I | by an even number. Therefore, othing contributes to the image of any subdominant term other than that subdom-inant term itself, which therefore is fixed under Φ N . We are then to consider theimage ˜Φ (2) N ( A ) ∈ X (2) N . Again, given a positive characteristic p within the ultraproduct decomposition,suppose the curve A (or rather its component answering to the chosen element p ) has a number of poles attained on dominant terms. Let us pick among thesepoles the one of the highest order k , and let ( i , I ) be its place. By definition ofan automorphism of Weyl algebra as a set of coefficients, the number i does notactually carry any meaningful data, so that we are left with a pair ( k, | I | ). Aswe can see, this pair is maximal from two different viewpoints; in fact, the pairrepresents a vertex of a Newton polygon taken over the appropriate field, with thediscrete valuation given by | I | . The coordinate function a i ,I corresponding to thispole admits a decomposition a i ,I = a − k t k + · · · , with t a local parameter. Acting upon this curve by the morphism Φ N amounts totwo steps: first, we raise everything to the p -th power and then assemble the com-ponents within the ultraproduct decomposition, then we take the preimage, whichis essentially the same as taking the p ′ -root, with respect to a different ultraprod-uct decomposition. The order of the maximal pole is then multiplied by an integerduring the first step and divided by the same integer during the second one. Bymaximality, there are no other terms that might contribute to the resulting place in˜Φ (2) N ( A ). It therefore does not change under Φ N .We may process the rest of the dominant (with respect to X (2) N ) terms similarly:indeed, it suffices to pick a different curve in general position. We then move downto X ( k ) N with higher k and argue similarly.After we have exhausted the possibilities with non-constant terms, we arriveat the conclusion that all that Φ N does is permute the irreducible components ofAut ≤ N A n, C . We have thus arrived at the following proposition. Proposition 3.7.
Let Φ ∗ N,M : O ( M ) (Aut ≤ N A n, C ) → O ( M ) (Aut ≤ N A n, C ) denote the linear map of finite-dimensional vector spaces O ( M ) (Aut ≤ N A n, C ) obtainedby restricting Φ ∗ N to regular functions of total degree less than or equal to M . If λ is an eigenvalue of Φ ∗ N,M , then there exists a natural number k such that λ k = 1 . Our further exploration of the morphism Φ will be divided into two steps. In thefirst step we are going to show that the linear maps Φ ∗ N,M induced by Φ are givenby diagonal matrices in the standard basis; more precisely, we will argue that thesematrices do not have non-trivial Jordan blocks. This strengthened version of theprevious proposition is of independent interest. The second step is the conclusion ofthe proof of the main theorem, which will, however, proceed along a different line ofreasoning than the first one. At the moment there seems to be no way of extendingthe fairly simple argument concerning the absence of Jordan blocks to process themain diagonal as well (i.e. to guarantee k = 1 in the proposition above). With respect to X (2) N , i.e. the highest terms that actually change - see above where we specify this convention. .1 Matrices Φ ∗ N,M – no non-diagonal blocks
Suppose that Φ ∗ N,M carries a Jordan block in the standard basis which, after a re-parametrization of the corresponding variety, amounts to a change of variables u λu + v, v v. Let us assume first that λ = 1, or rather that k = 1 in the previous proposition.Suppose given an automorphism f ∈ Aut ≤ N , then ˜ f = Φ( f ) is also an automor-phism, and we can look at the image Φ( ˜ f ◦ f − ). The coefficients (coordinates) of f and ˜ f are functions analytic in u and v , with the action of Φ upon f given by thechange of local variables (as above); this means that the coefficients ˜ a i ( u, v ) of ˜ f areequal to a i ( u + v, v ) with a i being the coordinates of f . Then, the coordinates ofΦ( ˜ f ◦ f − ) will be given by a i ( u + 2 v, v ) − a i ( u + v, v ). These are analytic, so that a i are power series. But then the expressions of the form( u + 2 v ) l − ( u + v ) l would provide a non-zero contribution to the term with u l − , which contradicts theinitial assumption on the spectrum of Φ ∗ N,M .If k = 1 in the proposition, the argument can be easily modified by consideringthe iterated loop Φ k ( ˜ f ◦ f − )and thus getting rid of λ . We therefore arrive at the stronger proposition. Proposition 3.8.
The linear maps Φ ∗ N,M cannot have non-trivial Jordan blocks. Φ is the identity morphism In order to conclude the proof of the main theorem, we proceed along the way of acertain topological argument. To do so, we modify our setting, passing from the ini-tial Weyl algebra A n, C to another associative unital algebra B n, C , which is defined asthe quotient of the free algebra on 3 n indeterminates C h a , . . . , a n , b , . . . , b n , c , . . . , c n i by the ideal generated by elements of the form b i a j − a i b j − δ ij c i , a i c j − a j c j , b i c j − b j c i . We will denote the images of a i , b i and c i in the quotient algebra by x i , d i and h i , respectively. The generators of B n are subject to the commutation relationswhich, at least with respect to the first two sets of generators ( x i and d i ), aresimilar in structure to those of the Weyl algebra A n , however new generators h i are also involved. One could view, if one so desired, such a peculiar construct as ageneralization of the canonical commutation relations[ d i , x j ] = ~ δ ij arising in the setting of deformation quantization.Accordingly, the commutative Poisson algebra P n, C is similarly modified, byadding new (commuting) variables h , . . . , h n and setting { p i , x j } = h i δ ij . Thisis the classical counterpart of B n , which we denote by R n .The idea of passing to augmented algebras plays a key role in the proof of Con-jecture 1.1 in our upcoming paper [27]. t can be verified that these new algebras behave in a way almost identical tothe one we described in the previous sections; in particular, the notions of tameautomorphism, tame (modified) symplectomorphism and homomorphisms φ h [ p ] ,N : Aut ≤ N ( B n, C ) → Aut ≤ N ( R n, C ) . which is identical on the tame points, are present. Note also that in the algebra B n, C , the variables h i are central.The loop morphism Φ h : Aut( B n, C ) → Aut( B n, C )(or rather, its normalized version) is then constructed, and a similar problem ofverifying it to be the identity map is posed; namely, one has Proposition 3.9. Φ h is the identity map. Obviously (by means of a specialization argument), Φ h = Id implies the MainTheorem.The point of the construction lies in the fact that the Ind-varieties Aut( B n, C ) andAut( R n, C ) admit a certain (formal power series) topology, which is preserved underΦ h . Moreover, the subgroups TAut( B n, C ) and TAut( R n, C ) of tame automorphismsare dense in this topology (this is a slight generalization of our more recent result [24]on tame symplectomorphisms).The power series topology is introduced by defining the augmentation ideal I in R n, C : I = ( x , . . . , x n , p , . . . , p n , h , . . . , h n );The system of neighborhoods { H N } of the identity automorphism in Aut R n, C isdefined by setting H N = { g ∈ Aut R n, C : g ( ξ ) ≡ ξ (mod I N ) } (here ξ denotes any generator in the set { x , . . . , x n , p , . . . , p n , h , . . . , h n } , so thatelements of H N are precisely those automorphisms which are identity modulo termswhich lie in I N ). Alternatively, one could define the power series topology by in-troducing the degree of generators (and setting the degree of every generator in thechosen set to 1) and defining the distance between pointsDist( f, g ) = exp( − inf ξ { ht( f ( ξ ) − g ( ξ )) } )with the height ht( f ( ξ )) of a polynomial being the degree of its smallest-degreemonomial with a non-zero coefficient. The functionht( f ) = inf ξ { ht( f ( ξ )) } is called the height of an automorphism. Similar notions are valid for the non-commutative algebra B n (once the proper ordering of the generators in the chosenset is fixed).We also define the subgroupsˆ H N = { g ∈ Aut R n, C : g ( ξ ) ≡ cξ (mod I N ) , c ∈ C } f deformed symplectomorphisms which are homothety modulo higher-degree terms.Obviously one has H N ⊂ ˆ H N .The establishment of tame approximation results (the fact that the tame de-formed symplectomorphism subgroup is dense in the power series topology of Aut R n, C )proceeds along the lines of our recent developments in [24].Once the tame approximation and lifting results are established, the main theo-rem of this paper will follow from the continuity of the loop morphism with respectto the augmentation topology. It is somewhat easier to prove the continuity of theautomorphism of the symplectomorphism groupsΦ hs : Aut R n, C → Aut R n, C rather than that of Φ h ; the morphism Φ hs is constructed in the same way as Φ h (withthe different starting point in the commutative diagram) and possesses essentiallythe same properties as Φ (explored in the previous section). Proposition 3.10.
Let Φ hs be the automorphism of the augmented symplectomor-phism groups as above. Then Φ hs ( H N ) ⊆ H N . The proof of the Proposition is based on the following straightforward observa-tion.
Lemma 3.11.
Let ϕ : X → Y be a morphism of affine algebraic sets, and let Λ( t ) be a curve (more simply, a one-parameter family of points) in X . Suppose that Λ( t ) does not tend to infinity as t → . Then the image ϕ Λ( t ) under ϕ also does not tend to infinity as t → . The proof of the lemma is an easy exercise and is left to the reader.In our proof, the one-parameter family of points Λ( t ) will be given by linearchanges of generators x i and p i of R n, C . The linear changes will be represented byinvertible matrices Λ which in addition are required to be skew-symmetric (in orderto preserve the commutation relations); the action is given by x Λ x, p Λ p, h Λ h (here x , p and h are vectors of the corresponding generators; we will not make use ofthis notation any further). Such linear changes correspond to tame automorphismsof R n, C ; in particular, they are preserved by ˜Φ h . Note also that the undeformed Pois-son algebra does not admit automorphisms of such form, unless mighty restrictionsare imposed on the matrix Λ. If g is an automorphism of the deformed Poisson algebra R n, C , one can conjugateit with a fixed one-parameter family Λ( t ) of linear substitutions to produce anotherone-parameter family of automorphismsΛ( t ) g Λ( t ) − . We are going to examine the behavior of such one-parameter families near singular-ities of Λ( t ). To be sure, the deformed version has its restrictions – namely those coming from the commutation relations;they are far less prohibitive, however, and in fact allow for a sufficiently broad set of matrices to be considered inorder for the proof below to work. This is the point in introducing the deformation via h i . uppose that, as t tends to zero, the i -th eigenvalue of Λ( t ) also tends to zero as t k i , k i ∈ N . Such a family will always exist.Let { k i , i = 1 , . . . n } be the set of degrees of singularity of eigenvalues of Λ( t ) atzero. Suppose that for every pair ( i, j ) the following holds: if k i = k j , then thereexists a positive integer m such thateither k i m ≤ k j or k j m ≤ k i . We will call the largest such m the order of Λ( t ) at t = 0. As k i are all set to bepositive integer, the order equals k max k min .We now formulate the main technical statement in our proof. Lemma 3.12.
Let g be a fixed deformed symplectomorphism. The curve Λ( t ) g Λ( t ) − has no singularity at zero for any Λ( t ) of order ≤ N if and only if g ∈ ˆ H N , where ˆ H N is the subgroup of automorphisms which are homothety modulo the N -th powerof the augmentation ideal.Proof. Suppose g ∈ ˆ H N . Then the action of Λ( t ) g Λ( t ) − upon any generator ξ ( x i , p i or h i ) is given by the expressionΛ( t ) g Λ( t ) − ( ξ ) = cξ + t − k i X l + ··· + l n = N a l ...l n t k l + ··· + k n l n ) P N ( ξ ) + S i ( t, ξ ) , where c denotes the homothety ratio of the linear part of g , P N is a homogeneouspolynomial of degree N , and S i is a polynomial in x i , p i and h i of height greaterthan N . One sees that for any choice of l , . . . , l n in the sum, the expression k l + · · · + k n l n − k i ≥ k min X l j − k i = k min N − k i ≥ i , so whenever t goes to zero, the coefficient will not go to infinity. Thesame argument applies to higher-degree monomials within S i . The If part of thelemma is proved.The other direction is established by contraposition: assuming g / ∈ ˆ H N , we mustshow the existence of a curve Λ( t ) such that the conjugation of g by Λ( t ) producesa singularity at zero.Suppose first that the linear part ¯ g of g is not a scalar matrix. Then, withoutloss of generality (modulo basis change) it is not a diagonal matrix; as such it has anon-zero entry in position ( i, j ). Consider a diagonal matrix Λ( t ) = D ( t ) such thatit has t k i at entry ( r, r ) for every r = j , while at the remaining main diagonal entry( j, j ) it has t k j . Then D ( t )¯ gD − ( t ) has entry ( i, j ) with the coefficient t k i − k j and if k j > k i it has a singularity at t = 0.Let also k i < k j . Then the non-linear part of g does not produce singularitiesand cannot compensate the singularity of the linear part. This case is thus processed.Now suppose that the linear part ¯ g is scalar. Then the linear part of Λ( t ) g Λ( t ) − does not possess a singularity, therefore one needs to look at the smallest non-linearterm. Let g ∈ ˆ H N \ ˆ H N +1 . Changing variables if necessary, we may assume that g ( x ) = αx + βx N + S, with ht( S ) > N .Let Λ( t ) = D ( t ) be a diagonal matrix of the form ( t k , t k , t k , . . . , t k ) and let( N + 1) · k > k > N · k . Then in Λ( t ) − g Λ( t ) the term βx N will be transformed nto βx N t Nk − k , and all other terms are multiplied by t lk + sk − k with ( l, s ) = (1 , l, s >
0. In this case lk + sk − k >
0. This concludes the second case, andwith it the proof of the lemma.We can now complete the proof of Proposition 3.10. Suppose that for some N ,the image Φ hs ( H N ) is not contained in H N . Then there exists a deformed symplec-tomorphism g which is identity modulo I N whose image Φ hs ( g ) has terms of degreestrictly between 1 and N . Then, by Lemma 3.12, there exists a family of linearsubstitutions Λ( t ), such that the curveΛ( t )Φ hs ( g )Λ( t ) − admits a singularity of order N at t = 0. On the other hand, as Φ hs preserves tameautomorphisms, the curve Λ( t )Φ hs ( g )Λ( t ) − is the imageΦ hs (Λ( t ) g Λ( t ) − )of the curve Λ( t ) g Λ( t ) − which, again by Lemma 3.12, has no singularity of order N at t = 0. This yields a contradiction with Lemma 3.11, as Φ hs is a morphism.Proposition 3.10 is proved. Corollary 3.13. Φ hs is the identity map.Proof. Proposition 3.10, along with tame approximation (the fact that the subgroupof tame automorphisms is dense in power series topology), implies that Φ hs is theidentity on the union of all Zariski-irreducible subspaces which contain the point ofAut R n, C corresponding to the identity automorphism. Let, on the other hand, ϕ bean automorphism outside that union. We need to show that Φ hs ( ϕ ) = ϕ . In orderto do so, we introduce a Poisson pair of auxiliary variables u, v and extend Φ hs tothe new automorphism variety. We now consider the tame map ψ t : u u + tx i , p i p i + tv where t ∈ C and i is fixed.The curve γ t = ϕ − ◦ ψ t ◦ ϕ lies in an irreducible component of the identity (the identity automorphism is thepoint γ ) and is therefore point-wise stable under Φ hs . It follows that ϕ ◦ Φ hs ( ϕ ) − = Idas desired.Finally, the Main Theorem follows from these observations by a specialization to h i = 1. Remark 3.14.
The proof of the main theorem requires the assumption of the Kont-sevich conjecture: indeed, we did assume φ [ p ] to be bijective, which of course impliesthe isomorphism between the automorphism groups. We have recently [26] put for-ward a proof of this conjecture that uses approximation by tame symplectomorphismsdeveloped in [24] and lifting to automorphisms which preserve the (class of ) associa-tive product parameterized by ~ that arises in the context of deformation quantiza-tion. The proof is somewhat cumbersome in that it relies on a procedure (lifting ofapproximating tame sequences) that, in the case of the non-deformed Poisson algebra s not canonical. As it turns out, deforming the algebra by adding auxiliary variables h i and then lifting directly to the (augmented) Weyl algebra B n resolves this issue(because the augmentation topology will then be sensitive to the central variables h i ,which in a way become the substitute for the deformation quantization and ~ ). Thenew proof is the subject of our recent paper [27]. Acknowledgments
We thank Roman Karasev, David Kazhdan, V. O. Manturov, Eugene Plotkin,Eliyahu Rips and Boris Zilber for numerous helpful remarks. We also thank Alexan-der Zheglov and Georgy Sharygin, as well as Ilya Karzhemanov and Ilya Zhdanovskiiand the participants of their respective seminars who provided for a most engagingand fruitful discussion of our findings.This work is supported by the Russian Science Foundation grant No. 17-11-01377.
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