Citation: Zehra Velioǧlu, Mukaddes Balçik. On the Tame automorphisms of differential polynomial algebras[J]. AIMS Mathematics, 2020, 5(4): 3547-3555. doi: 10.3934/math.2020230
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P. Cohn [3] proved that the automorphisms of a free Lie algebra with a finite set of generators are tame. The tameness of the automorphisms of polynomial algebras and free associative algebras in two variables is well known [2,6,9,10]. It was proved [13,15] that polynomial algebras and free associative algebras in three variables in the case of characteristic zero have wild automorphisms. For example the Nagata automorphism [11,13] is a wild automorphism of a free Poisson algebra in three variables. Let Δ={δ1,…,δm} be a basic set of derivation operators and R{x1,x2,…,xn} be the polynomial algebra in the variables x1,x2,…,xn over a differential ring R. The basic concepts of differential algebras can be found in [1,5,7,8,12,14]. The tameness of automorphisms of differential polynomial algebras is studied by B. A. Duisengaliyeva, A. S. Naurazbekova and U. U. Umirbaev [4]. They have proved that the tame automorphism group of a differential polynomial algebra over a field of characteristic 0 in two variables with m commuting derivations is a free product with amalgamation. In this paper we give some important subetaoups of the group of differential automorphisms of R{x,y} with one derivation operator. Furthermore using the method in Essen [16], we prove that the Tame subetaoup of automorphism of R{x,y} is the amalgamated free product of the Triangular and the Affine subetaoups over their intersection.
Let R be any commutative ring with unity. A mapping d:R→R is called a derivation if
d(s+t)=d(s)+d(t) |
d(st)=d(s)t+sd(t) |
holds for all s,t∈R.
Let Δ={δ1,…,δm} be a basic set of derivation operators.
A ring R is said to be a differential ring or Δ-ring if all elements of Δ act on R as a commuting set of derivations, i.e., the derivations δi:R→R are defined for all i and δiδj=δjδi for all i,j. If Δ={δ} (that is, if Δ consists of only one derivation), then R is called an ordinary differential ring and will be denoted as δ-ring, unless R is called a partial differential ring. In this work, we study on ordinary differential rings.
Let Θ be the free commutative monoid generated by a derivation operator δ. For n=0,1,2,..., the elements
θ=(δδ…δ⏟(n−1)−times)δ=δn, |
of the monoid Θ are called derivative operators. The order of θ is defined as |θ|=n. If R is a δ-ring and R is a field (resp. integral domain), then R is called a differential field (resp. integral domain). Let x be a differential indeterminate and let Θx={θx|θ∈Θ} be the set of symbols enumerated by the elements of Θ. Consider the polynomial algebra R[Θx] over a δ-ring R generated by the set of (algebraically) independent indeterminates Θx. It is easy to check that the derivation δ can be uniquely extended to a derivation of R[Θx] by δ(θx)=(δθ)x. Denote this differential ring by R{x}; it is called the ring of differential polynomials in x over R.
By adjoining more differential indeterminates, we obtain the differential ring R{x1,x2, …,xn} of the differential polynomials in x1,x2, …,xn over R. The ring R{x1,x2, …,xn} coincides with the polynomial algebra
R[θxi:θ∈Θ,1≤i≤n]. |
Consequently, the set of differential monomials
M=θ1xk1⋯θsxks |
where 1≤ki≤n, θi∈Θ, 1≤i≤s, form a linear basis of R{x1,x2,…,xn} over R.
The degree deg(M) for a monomial M=∏si=1(θixki), 1≤ki≤n, θi∈Θ, 1≤i≤s is defined as in the algebraic case: deg(M) = s. It is clear that deg(θxi)=1, for each i, θ∈Θ. The elements of the ring R{x1,x2, …,xn} are called differential polynomials. The degree of a differential polynomial f is deg(f) = maxM∈fdeg(M), where M∈f means that M is a differential monomials occurring in f. If each term of a differential polynomial f has the same degree, then f is a homogeneous differential polynomial. By degxif we denote the degree of f with respect to xi and its derivatives. We have degxi(θxj)=δij where 1≤i≤n, 1≤j≤n, θ∈Θ and δij is the Kronecker delta function.
On the other hand one defines the weight of the monomial M as wt(M)=∑si=1|θi|, where θi∈Θ and the weight of a differential polynomial f as wt(f) = maxM∈f wt(M). The terms occurring in f that have the same weight are called isobaric component of f.
Let R be a δ-ring, an ideal a∈R is a differential ideal (δ-ideal) if a∈a, we have δ(a)∈a. Additionally if a is a radical ideal, then it is called as radical δ-ideal.
A subset V⊆R2 is Kolchin-closed if it is the set of all common zeros in R2 of a radical differential ideal a⊆R{x,y}.
Let R be a δ-field, we say that R is differentially closed if every consistent system of differential polynomial equations with coefficients in R has a solution in R.
Definition 1. Let R be a δ-ring with a derivation operator δ. A differential ring homomorphism, or simply a δ-homomorphism, of R is a ring automorphism φ of R such that φ(δ(a))=δ(φ(a)), for all a∈R. If the δ-homomorphism φ is a ring automorphism of R, then φ is a δ-automorphism.
Definition 2. Let K be a differential ring extension of a δ-ring R. A δ-automorphism φ of K is called an R-δ-automorphism, provided φ(a)=a, for all a∈R. The set of all R-δ-automorphism of K is a group under composition, denoted by Autδ(K|R).
From now, let R be an integral domain with only derivation operator δ and R{x,y} be the differential polynomial ring in x,y over R. Notice that since R{x,y} is the free object on the set {x,y} in the category of differential R-δ-algebras and hence has the universal mapping property.
Definition 3. Let F1=F1(x,y) and F2=F2(x,y) be two differential polynomials in R{x,y}. A tuple F=(F1,F2) in R{x,y}2 defines uniquely a R-δ-homomorphism σF:R{x,y}→R{x,y} such that σF(x)=F1, σF(y)=F2. For any P∈R{x,y}, σF(P)=P(F1,F2) (that is, P acts like a bivariate differential operator on pair (F1,F2)). Conversely, every R-δ-homomorphism σ of R{x,y}(in particular, the inverse of an R-δ-automorphism σ∈ Autδ(R{x,y}|R)) is of form σF for some tuple F=(F1,F2)∈R{x,y}2.
Definition 4. A differential polynomial map is a polynomial map φF:R2→R2 defined by a tuple F=(F1,F2) in R{x,y}2 such that φF(a,b)=(F1(a,b) ,F2(a,b)) for any (a,b)∈R2.
Definition 5. A differential polynomial map φF said to be invertible if there exists G=(G1(x,y),G2(x,y))∈R{x,y}2 such that φF∘φG=ı, where ı is the identity map on the set R2.
The following theorem is a simplified version of a theorem of Kolchin [8, Theorem 4, p. 105] and is analogous to the algebraic fact that in characteristic zero, any two transcendence bases of a field extension K over k have the same cardinality. Kolchin proved the analog of for arbitrary characteristic in terms of differential inseparability basis of a partial differential field extension K over a partial differential field k.
Theorem 1. Let K be a differential field extension of an ordinary differential field k of characteristic zero. Then every set Σ⊂K that (differentially) generates K over k (that is, K=k⟨Σ⟩) contains a differential transcendence basis K of over k and any two differential transcendence bases of K over k have the same cardinality (called the differential dimension of K over k).
The next theorem is known as differential analogue of Hilbert's Nullstellensatz, but is a much deeper result as it involves the notion of a differential closed field.
Theorem 2. Let R be differentially closed field of characteristic zero. The correspondence between the set of radical differential ideals a of R{x,y} and the set of Kolchin-closed subset V⊆R2, given by
a↦V(a)={(a,b)∈R2|P(a,b)=0 for all P∈a} |
is bijective with the inverse given by
V⊆R2↦a(V)={P∈R{x,y}|P(a,b)=0 for all (a,b)∈V}. |
Remark 1. It follows that V((0))=R2 and a(R2)=(0). In particular, we have this property : "For any P∈R{x,y}, if P vanishes for every (a,b)∈R2, then P=0".
From now, for simplicity, we will use P instead of P(x,y) for a differential polynomial P(x,y) of R{x,y}.
In the next theorem, we show the relation between differential polynomial maps and differential automorphisms. Here when we say that "for some G" we mean "for some G=(G1,G2)∈R{x,y}2".
Theorem 3. Let R be a differential integral domain. Let F=(F1,F2)∈R{x,y}2.
I. The following are equivalent:
(a) σF is a differential automorphism with an inverse σG for some G.
(b) σF is surjective (equivalently, R{x,y}=R{F1,F2}).
(c) x=G1(F1,F2) and y=G2(F1,F2) for some G.
II. If F satisfies any (and all) of the conditions in I., then
(d) φF is a invertible differential polynomial map (that is, φF∘φG=ı for some G).
(e) x=F1(G1,G2) and y=F2(G1,G2) for some G.
(f) φF has an inverse φG for some G.
III. If R is a differentially closed field of characteristic zero, then (a)–(f) are all equivalent, and the tuple G∈R{x,y}2 involved are unique and the same.
Proof. Ⅰ. The implications (a)⇒(b)⇒(c) are trivial. We now show the converses: (c)⇒(b)⇒(a). From (c), we have x,y∈R{F1,F2}, hence σF(R{x,y})=R{F1,F2}=R{x,y}, which is (b). From (b), let K be the quotient field of R (R is a domain). Then K is a differential field and the two quotient fields K⟨x,y⟩ of R{x,y} and K⟨F1,F2⟩ of R{F1,F2} coincide. By Theorem 1, the set {F1,F2} forms a differential transcendence basis of K⟨x,y⟩=K⟨F1,F2⟩ over K. So F1,F2 are differentially algebraic independent over K and a fortiori over R, which means σF is injective and hence (a).
Ⅱ. We now assume F satisfies (a), (b), and (c) and prove the second part by proving several implications. In this part of the proof, let G be as in (c).
(c)⇒(d):
By (c), for all (a,b)∈R2,
φG∘φF(a,b)=φG(F1(a,b),F2(a,b))=(G1(F1(a,b),F2(a,b)),G2(F1(a,b),F2(a,b)))=(a,b). | (3.1) |
Hence φG∘φF=ı, the identity map on R2 and (d) holds.
(c)⇒(e):
From (c), we have σF(F1(G1,G2))=F1(G1(F1,F2),G2(F1,F2))=F1(x,y)=σF(x). Since (c)⇒(a), σF is injective and so x=F1(G1,G2). Similarly, y=F2(G1,G2) and (e) holds.
(c)⇒(f):
By (c), σF∘σG=I, where I is the identity automorphism of R{x,y} and from equation (3.1) in (c)⇒(d), φG∘φF=ı. Since (c)⇒(e), σG∘σF=I and φF∘φG=ı. Hence (f). In fact, we also prove σG is the inverse of σF.
Ⅲ. Let R be a differential closed field of characteristic zero. In Ⅰ. we have proved that the G in (c) works for (a) and vice versa. In Ⅱ. we have proved that the G from (c) works for (d), (e), and (f). We now prove the converses and that the G from any of (d), (e), or (f) works for (c), too and hence G is unique (because G defines the inverse of σF) and all G in all the equivalent conditions are the same.
(d)⇒(c):
Suppose there exists G=(G1,G2)∈R{x,y}2 such that φG∘φF=ı. Then from equation (3.1), the differential polynomials P=G1(F1,F2)−x and Q=G2(F1,F2)−y each vanishes at all (a,b)∈R2. By Remark 1, P=Q=0, which proves (c).
(e)⇒(c):
Let G=(G1,G2) be as given in (e). We now apply the already proven first two parts of this theorem to G (taking the place of F). If we temporarily ornate the corresponding item labels in the theorem for G with †, then (c)† holds using the F=(F1,F2) from (e). Since we have proved (c)†⇒(e)†, using the F of (c)†, it follows that (e)⇒(c), using the G from (e).
(f)⇒(c):
Let G be such that φG is the inverse of φF. Then of course φG∘φF=ı, which is (d), and which implies(c).
From now let R be a differentially closed field of characteristic zero. Now, for each subset S⊆R{x,y}2 of interest, we want to find any necessary and sufficient conditions CS on F that will make σF an R-δ-automorphism of R{x,y} for every F∈S, that is, for σF to have an inverse σ−1F∈Autδ(R{x,y}/R). Thus CS is not only the set of necessary and sufficient conditions for σF to be both injective and surjective, but also becomes part of defining properties of the set S. Then, it is valid to identify S as a subset Σ(S) of Autδ(R{x,y}/R) via F→σF and not merely as a set of R-δ-automorphisms. Moreover, we want to identify those S⊆R{x,y}2 for which Σ(S)={σF|F∈S} are subetaoups of Autδ(R{x,y}/R).
Notice that there are two properties Σ(S) (or S) must satisfy to ensure that Σ(S) is a subetaoup. The first of these properties is after showing that σ−1F exists and hence σF, σ−1F ∈Autδ(R{x,y}/R), it needs to belong to Σ(S), that is, σ−1F=σG for some G=(G1,G2)∈S. The second is that for any σF and σG in Σ(S), their composition σF∘σG must also belong to Σ(S). That means σF∘σG=σH for some H=(H1,H2)∈S.
Some important subetaoups of Autδ(R{x,y}/R) are defined as follows:
(1). The Affine differential subetaoup: We determine it as Affδ(R{x,y}/R).
Consider Fi∈R{x,y}, deg(Fi)=1, wt(Fi)=0, (i=1,2), then for a,b,c,d,e,f∈R put
F1=ax+by+c, F2=dx+ey+f and det([abde])=|JF|.
After calculating the conditions CS for S=Affδ(R{x,y}/R) and F=(F1,F2)∈S, we define Affine-δ subetaoup as follows:
Affδ(R{x,y}/R)={(ax+by+c,dx+ey+f)|a,b,c,d,e,f∈R,|JF|∈R∗}.
(2). The Triangular differential subetaoup: We determine this subetaoup as Jδ(R{x,y}/R) and similarly we calculate it as follows:
Jδ(R{x,y}/R)={(ax+f(y),by+c)|a,b∈R∗,c∈R and f(y)∈R{y}}.
Here R∗ denotes nonzero elements of R.
(3). The Elementary differential subetaoup: We determine this subetaoup as Eδ(R{x,y}/R) and we calculate it as:
Eδ(R{x,y}/R)={(ax+f(y),y)|a∈R∗ and f(y)∈R{y}}.
Definition 6. The subetaoup of Autδ(R{x,y}|R) generated by the Affine R-δ-automorphisms and the Triangular R-δ-automorphisms is called Tame δ-subetaoup and is denoted by Tδ(R{x,y}|R).
Now we will show that Tδ(R{x,y}|R) is the amalgamated free product of Affδ(R{x,y}|R) and Jδ(R{x,y}|R) over their intersection.
To prove the results announced above we need the following lemmas.
Lemma 1. Let G be a group generated by two subetaoups H and K. Then every element g of G can be written as
g=h0k1h1⋯kℓhℓ |
for some ℓ≥1, where hi∈H∖ K for all 1≤i≤ℓ−1 and ki∈K∖ H for all 1≤i≤ℓ and h0∈H.
Proof. See, [16], Lemma 5.1.1, p.86.
Let F=(F1,F2),G=(G1,G2)∈R{x,y}2 and let σF and σF be automorphisms defined by F and G respectively. We define composition of σF and σG as follows: there exists H=(H1,H2)∈R{x,y}2 such that
σH(x)=(σF∘σG)(x)=G1(F1(x,y),F2(x,y))=H1 |
σH(y)=(σF∘σG)(y)=G2(F1(x,y),F2(x,y))=H2. |
To formulate Lemma 2 and Corollary 1 below, we need following notations: let F=(F1,F2)∈R{x,y}2. Then
degF=max{deg(F1),deg(F2)}, |
bidegF=(deg(F1),deg(F2)), |
wtF=max{wt(F1),wt(F2)}, |
biwtF=(wt(F1),wt(F2)). |
Now let F∈Tδ(R{x,y}|R). Then applying Lemma 1 to G=Tδ(R{x,y}|R), H=Affδ(R{x,y}|R), K=Jδ(R{x,y}|R) and g=F we can write
F=λ0τ1λ1⋯τℓλℓ |
with λi∈Affδ(R{x,y}|R)∖Jδ(R{x,y}|R) for all 1≤i≤ℓ−1 and τi∈Jδ(R{x,y}|R)∖Affδ(R{x,y}|R) for all 1≤i≤ℓ. Write
λi=(aix+biy+ci,dix+eiy+fi). |
Note that for an integer j, if λj∈Affδ(R{x,y}|R)∖Jδ(R{x,y}|R), then dj=0.
Remark 2. Observe that if τ∈Jδ(R{x,y}|R)∖Affδ(R{x,y}|R) then either wt(τ)≥1 or wt(τ)=0 and deg(τ)≥2.
Lemma 2. For any positive integer ℓ we have
bideg(λ1∘τ1∘⋯∘λℓ∘τℓ)=(∏ℓj=1deg(τj(x)),∏ℓ−1j=1deg(τj(x))) |
where for all 1≤i≤ℓ, wt(τi)=0 and the second product is by definition 1 if ℓ=1.
Proof. We prove the lemma by induction on ℓ. The case ℓ=1 is obvious. So let us assume that the statement is true for some n and consider bideg(λ1∘τ1∘⋯∘λn∘τn∘λn+1). Since λn+1∉Jδ(R{x,y}|R), we have that dn+1≠0. Therefore
bideg(λ1∘τ1∘⋯∘λn∘τn∘λn+1)=(pi,∏nj=1deg(τj(x))), |
where pi≤∏nj=1deg(τj(x)). Finally since τn+1∉Affδ(R{x,y}|R), we have that
bideg(λ1∘τ1∘⋯∘λn∘τn∘λn+1∘τn+1)=(n∏j=1deg(τj(x)).deg(τn+1(x)),n∏j=1deg(τj(x)))=(n+1∏j=1deg(τj(x)),n∏j=1deg(τj(x))) |
which completes the proof.
Corollary 1. Tδ(R{x,y}|R) is the amalgamated free product of Affδ(R{x,y}|R) and Jδ(R{x,y}|R) over their intersection, i.e., Tδ(R{x,y}|R) is generated by these two groups and if τi∈Jδ(R{x,y}|R)∖Affδ(R{x,y}|R) and λi∈Affδ(R{x,y}|R)∖Jδ(R{x,y}|R) then τ1∘λ2⋯∘τn−1∘λn∘τn does not belong to Affδ(R{x,y}|R).
Proof. By the definition, Tδ(R{x,y}|R) is generated by Affδ(R{x,y}|R) and Jδ(R{x,y}|R). So suppose that
τ1∘λ2⋯∘τn−1∘λn∘τn=λ∈Affδ(R{x,y}|R), |
with τi∈Jδ(R{x,y}|R)∖Affδ(R{x,y}|R) and λi∈Affδ(R{x,y}|R)∖Jδ(R{x,y}|R) for all i. Then
λ−1∘τ1∘λ2⋯∘τn−1∘λn∘τn(x,y)=(x,y). |
Therefore we have
bideg(λ−1∘τ1∘λ2⋯∘τn−1∘λn∘τn)=(1,1) | (4.1) |
biwt(λ−1∘τ1∘λ2⋯∘τn−1∘λn∘τn)=(0,0) | (4.2) |
Since τi∈Jδ(R{x,y}|R)∖ Affδ(R{x,y}|R), then
wt(τi)≥1 or wt(τi)=0. |
Now let us suppose for all i, wt(τi)=0 so deg(τi)≥2. By the Lemma 2, we have
bideg(λ−1∘τ1∘λ2⋯∘τn−1∘λn∘τn)=(∏n−1j=1deg(τj(x)).deg(τn(x)),∏n−1j=1deg(τj(x))) |
From the equation (4.1), deg(τn)=1 which is a contradiction. Let us suppose that for all i, wt(τi)≥1, so
biwt(λ−1∘τ1∘λ2⋯∘τn−1∘λn∘τn)>(0,0) |
which is a contradiction by (4.2).
The Theorem 3 is by, and included with permission from William Sit through private communications. The authors would like to thank Professor William Sit and Professor Ualbai Umirbaev for their supports and valuable comments and suggestions to improve the quality of the paper.
The authors declare no conflict of interest.
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