The group of automorphisms of the Lie algebra of derivations of a polynomial algebra
aa r X i v : . [ m a t h . R A ] A p r The group of automorphisms of the Lie algebra of derivationsof a polynomial algebra
V. V. Bavula
Abstract
We prove that the group of automorphisms of the Lie algebra Der K ( P n ) of derivationsof a polynomial algebra P n = K [ x , . . . , x n ] over a field of characteristic zero is canonicallyisomorphic to the the group of automorphisms of the polynomial algebra P n . Key Words: Group of automorphisms, monomorphism, Lie algebra, automorphism, locallynilpotent derivation.Mathematics subject classification 2010: 17B40, 17B20, 17B66, 17B65, 17B30.
In this paper, module means a left module, K is a field of characteristic zero and K ∗ is its groupof units, and the following notation is fixed: • P n := K [ x , . . . , x n ] = L α ∈ N n Kx α is a polynomial algebra over K where x α := x α · · · x α n n , • G n := Aut K ( P n ) is the group of automorphisms of the polynomial algebra P n , • ∂ := ∂∂x , . . . , ∂ n := ∂∂x n are the partial derivatives ( K -linear derivations) of P n , • D n := Der K ( P n ) = L ni =1 P n ∂ i is the Lie algebra of K -derivations of P n where [ ∂, δ ] := ∂δ − δ∂ , • δ := ad( ∂ ) , . . . , δ n := ad( ∂ n ) are the inner derivations of the Lie algebra D n determinedby the elements ∂ , . . . , ∂ n (where ad( a )( b ) := [ a, b ]), • G n := Aut Lie ( D n ) is the group of automorphisms of the Lie algebra D n , • D n := L ni =1 K∂ i , • H n := L ni =1 KH i where H := x ∂ , . . . , H n := x n ∂ n , • A n := K h x , . . . , x n , ∂ , . . . , ∂ n i = L α,β ∈ N n Kx α ∂ β is the n ’th Weyl algebra , • for each natural number n ≥ u n := K∂ + P ∂ + · · · + P n − ∂ n is the Lie algebra of triangularpolynomial derivations (it is a Lie subalgebra of the Lie algebra D n ) and Aut K ( u n ) is itsgroup of automorphisms.The aim of the paper is to prove the following theorem. Theorem 1.1 G n = G n .Structure of the proof . (i) G n ⊆ G n via the group monomorphism (Lemma 2.3.(3)) G n → G n , σ σ : ∂ σ ( ∂ ) := σ∂σ − . (ii) Let σ ∈ G n . Then ∂ ′ := σ ( ∂ ) , . . . , ∂ ′ n := σ ( ∂ n ) are commuting, locally nilpotent deriva-tions of the polynomial algebra P n (Lemma 2.6.(1)).1iii) T ni =1 ker P n ( ∂ ′ i ) = K (Lemma 2.6.(2)).(iv)(crux) There exists a polynomial automorphism τ ∈ G n such that τ σ ∈ Fix G n ( ∂ , . . . , ∂ n )(Corollary 2.9).(v) Fix G n ( ∂ , . . . , ∂ n ) = Sh n (Proposition 2.5.(3)) whereSh n := { s λ ∈ G n | s λ ( x ) = x + λ , . . . , s λ ( x n ) = x n + λ n } is the shift group of automorphisms of the polynomial algebra P n and λ = ( λ , . . . , λ n ) ∈ K n .(vi) By (iv) and (v), σ ∈ G n , i.e. G n = G n . (cid:3) An analogue of the Jacobian Conjecture is true for D n . The Jacobian Conjecture claimsthat certain monomorphisms of the polynomial algebra P n are isomorphisms: Every algebra endo-morphism σ of the polynomial algebra P n such that J ( σ ) := det( ∂σ ( x i ) ∂x j ) ∈ K ∗ is an automorphism. The condition that J ( σ ) ∈ K ∗ implies that the endomorphism σ is a monomorphism. Conjecture . Every homomorphism of the Lie algebra D n is an automorphism. Theorem 1.2 [4] Every monomorphism of the Lie algebra u n is an automorphism.Remark . Not every epimorphism of the Lie algebra u n is an automorphism. Moreover, thereare countably many distinct ideals { I iω n − | i ≥ } such that I = { } ⊂ I ω n − ⊂ I ω n − ⊂ · · · ⊂ I iω n − ⊂ · · · and the Lie algebras u n /I iω n − and u n are isomorphic (Theorem 5.1.(1), [5]).Theorems 1.2 and Conjecture have bearing of the Jacobian Conjecture and the Conjectureof Dixmier [8] for the Weyl algebra A n over a field of characteristic zero that claims: everyhomomorphism of the Weyl algebra is an automorphism . The Weyl algebra A n is a simple algebra,so every algebra endomorphism of A n is a monomorphism. This conjecture is open since 1968 forall n ≥
1. It is stably equivalent to the Jacobian Conjecture for the polynomial algebras as wasshown by Tsuchimoto [9], Belov-Kanel and Kontsevich [7], (see also [2] for a short proof which isbased on the author’s new inversion formula for polynomial automorphisms [1]).
An analogue of the Conjecture of Dixmier is true for the algebra I := K h x, ddx , R i of polynomial integro-differential operators . Theorem 1.3 (Theorem 1.1, [3])
Each algebra endomorphism of I is an automorphism. In contrast to the Weyl algebra A = K h x, ddx i , the algebra of polynomial differential operators,the algebra I is neither a left/right Noetherian algebra nor a simple algebra. The left localizations, A ,∂ and I ,∂ , of the algebras A and I at the powers of the element ∂ = ddx are isomorphic. Forthe simple algebra A ,∂ ≃ I ,∂ , there are algebra endomorphisms that are not automorphisms [3]. The group of automorphisms of the Lie algebra u n . In [6], the group of automor-phisms Aut K ( u n ) of the Lie algebra u n of triangular polynomial derivations is found ( n ≥ T n ⋉ (UAut K ( P n ) n ⋊ ( F ′ n × E n ))where T n is an algebraic n -dimensional torus, UAut K ( P n ) n is an explicit factor group of thegroup UAut K ( P n ) of unitriangular polynomial automorphisms, F ′ n and E n are explicit groupsthat are isomorphic respectively to the groups I and J n − where I := (1 + t K [[ t ]] , · ) ≃ K N J := ( tK [[ t ]] , +) ≃ K N . Comparing the groups G n and Aut K ( u n ) we see that the group(UAut K ( P n ) n of polynomial automorphisms is a tiny part of the group Aut K ( u n ) but in contrast G n = Aut K ( P n ). It is shown that the adjoint group of automorphisms A ( u n ) of the Lie algebra u n is equal to the group UAut K ( P n ) n (Theorem 7.1, [6]). Recall that the adjoint group A ( G ) of aLie algebra G is generated by the elements e ad( g ) := P i ≥ g ) i i ! ∈ Aut K ( G ) where g runs throughall the locally nilpotent elements of the Lie algebra G (an element g is a locally nilpotent element if the inner derivation ad( g ) := [ g, · ] of the Lie algebra G is a locally nilpotent derivation). This section can be seen as a proof of Theorem 1.1. The proof is split into several statements thatreflect ‘Structure of the proof of Theorem 1.1’ given in the Introduction.
The Lie algebra D n is Z n -graded . The Lie algebra D n = M α ∈ N n n M i =1 Kx α ∂ i (1)is a Z n -graded Lie algebra D n = M β ∈ Z n D n,β where D n,β = M α − e i = β Kx α ∂ i , i.e. [ D n,α , D n,β ] ⊆ D n,α + β for all α, β ∈ N n where e := (1 , , . . . , , . . . , e n := (0 , . . . , ,
1) is thecanonical free basis for the free abelian group Z n . This follows from the commutation relations[ x α ∂ i , x β ∂ j ] = β i x α + β − e i ∂ j − α j x α + β − e j ∂ i . (2)Clearly, for all i, j = 1 , . . . , n and α ∈ N n ,[ H j , x α ∂ i ] = ( α j x α ∂ i if j = i, ( α i − x α ∂ i if j = i, (3)[ ∂ j , x α ∂ i ] = α j x α − e j ∂ i . (4)The support Supp( D n ) := { β ∈ Z n | D n,β = 0 } is a submonoid of Z n . Let us find the supportSupp( D n ), the graded components D n,β and their dimensions dim K D n,β . For each i = 1 , . . . , n ,let N n,i := { α ∈ N n | α i = 0 } and P ∂ i n := ker P n ( ∂ i ). It follows from the decompositions P n = P ∂ i n ⊕ P n x i for i = 1 , . . . , n that D n = n M i =1 ( P ∂ i n ⊕ P n x i ) ∂ i = n M i =1 P ∂ i n ∂ i ⊕ n M i =1 P n H i ,D n = n M i =1 P ∂ i n ∂ i ⊕ M α ∈ N n x α H n . (5)Hence, Supp( D n ) = n a i =1 ( N n,i − e i ) a N n . (6) D n,β = ( x α ∂ i if β = α − e i ∈ N n,i − e i ,x β H n if β ∈ N n . (7)dim K D n,β = ( β = α − e i ∈ N n,i − e i ,n if β ∈ N n . G be a Lie algebra and H be its Lie subalgebra. The centralizer C G ( H ) := { x ∈ G | [ x, H ] =0 } of H in G is a Lie subalgebra of G . In particular, Z ( G ) := C G ( G ) is the centre of the Lie algebra G . The normalizer N G ( H ) := { x ∈ G | [ x, H ] ⊆ H} of H in G is a Lie subalgebra of G , it is thelargest Lie subalgebra of G that contains H as an ideal.Let V be a vector space over K . A K -linear map δ : V → V is called a locally nilpotent map if V = S i ≥ ker( δ i ) or, equivalently, for every v ∈ V , δ i ( v ) = 0 for all i ≫
1. When δ is a locallynilpotent map in V we also say that δ acts locally nilpotently on V . Every nilpotent linear map δ , that is δ n = 0 for some n ≥
1, is a locally nilpotent map but not vice versa, in general. Let G be a Lie algebra. Each element a ∈ G determines the derivation of the Lie algebra G by the rulead( a ) : G → G , b [ a, b ], which is called the inner derivation associated with a . The set Inn( G ) ofall the inner derivations of the Lie algebra G is a Lie subalgebra of the Lie algebra (End K ( G ) , [ · , · ])where [ f, g ] := f g − gf . There is the short exact sequence of Lie algebras0 → Z ( G ) → G ad → Inn( G ) → , that is Inn( G ) ≃ G /Z ( G ) where Z ( G ) is the centre of the Lie algebra G and ad([ a, b ]) = [ad( a ) , ad( b )]for all elements a, b ∈ G . An element a ∈ G is called a locally nilpotent element (respectively, a nilpotent element ) if so is the inner derivation ad( a ) of the Lie algebra G . The Cartan subalgebra H n of D n . A nilpotent Lie subalgebra C of a Lie algebra G is calleda Cartan subalgebra of G if it coincides with its normalizer. We use often the following obviousobservation: An abelian Lie subalgebra that coincides with its centralizer is a maximal abelian Liesubalgebra . Lemma 2.1 H n is a Cartan subalgebra of D n .2. H n = C D n ( H n ) is a maximal abelian subalgebra of D n .Proof . Statements 1 and 2 follows from (6) and (7). (cid:3) P n is a D n -module . The polynomial algebra P n is a (left) D n -module: D n × P n → P n ,( ∂, p ) ∂ ∗ p . In more detail, if ∂ = P ni =1 a i ∂ i where a i ∈ P n then ∂ ∗ p = n X i =1 a i ∂p∂x i . The field K is a D n -submodule of P n and n \ i =1 ker P n ( ∂ i ) = K. (8) Lemma 2.2
The D n -module P n /K is simple with End D n ( P n /K ) = K id where id is the identitymap.Proof . Let M be a nonzero submodule of P n /K and 0 = p ∈ M . Using the actions of ∂ , . . . , ∂ n on p we obtain an element of M of the form λx i for some λ ∈ K ∗ . Hence, x i ∈ M and x α = x α ∂ i ∗ x i ∈ M for all 0 = α ∈ N n . Therefore, M = P n /K . Let f ∈ End D n ( P n /K ). Thenapplying f to the equalities ∂ i ∗ ( x + K ) = δ i for i = 1 , . . . , n , we obtain the equalities ∂ i ∗ f ( x + K ) = δ i for i = 1 , . . . , n. Hence, f ( x + K ) ∈ T ni =2 ker P n /K ( ∂ i ) ∩ ker P n /K ( ∂ i ) = ( K [ x ] /K ) ∩ ker P n /K ( ∂ i ) = K ( x + K ).So, f ( x + K ) = λ ( x + K ) and so f = λ id, by the simplicity of the D n -module P n /K . (cid:3) he G n -module D n . The Lie algebra D n is a G n -module, G n × D n → D n , ( σ, ∂ ) σ ( ∂ ) := σ∂σ − . Every automorphism σ ∈ G n is uniquely determined by the elements x ′ := σ ( x ) , . . . , x ′ n := σ ( x n ) . Let M n ( P n ) be the algebra of n × n matrices over P n . The matrix J ( σ ) := ( J ( σ ) ij ) ∈ M n ( P n ),where J ( σ ) ij = ∂x ′ j ∂x i , is called the Jacobian matrix of the automorphism (endomorphism) σ andits determinant J ( σ ) := det J ( σ ) is called the Jacobian of σ . So, the j ’th column of J ( σ ) is the gradient grad x ′ j := ( ∂x ′ j ∂x , . . . , ∂x ′ j ∂x n ) T of the polynomial x ′ j . Then the derivations ∂ ′ := σ∂ σ − , . . . , ∂ ′ n := σ∂ n σ − are the partial derivatives of P n with respect to the variables x ′ , . . . , x ′ n , ∂ ′ = ∂∂x ′ , . . . , ∂ ′ n = ∂∂x ′ n . (9)Every derivation ∂ ∈ D n is a unique sum ∂ = P ni =1 a i ∂ i where a i = ∂ ∗ x i ∈ P n . Let ∂ :=( ∂ , . . . , ∂ n ) T and ∂ ′ := ( ∂ ′ , . . . , ∂ ′ n ) T where T stands for the transposition. Then ∂ ′ = J ( σ ) − ∂, i . e . ∂ ′ i = n X j =1 ( J ( σ ) − ) ij ∂ j for i = 1 , . . . , n. (10)In more detail, if ∂ ′ = A∂ where A = ( a ij ) ∈ M n ( P n ), i.e. ∂ i = P nj =1 a ij ∂ j . Then for all i, j = 1 , . . . , n , δ ij = ∂ ′ i ∗ x ′ j = n X k =1 a ik ∂x ′ j ∂x k where δ ij is the Kronecker delta function. The equalities above can be written in the matrix formas AJ ( σ ) = 1 where 1 is the identity matrix. Therefore, A = J ( σ ) − .Suppose that a group G acts on a set S . For a nonempty subset T of S , St G ( T ) := { g ∈ G | gT = T } is the stabilizer of the set T in G and Fix G ( T ) := { g ∈ G | gt = t for all t ∈ T } is the fixator of the set T in G . Clearly, Fix G ( T ) is a normal subgroup of St G ( T ). The maximal abelian Lie subalgebra D n of D n . Lemma 2.3 C D n ( D n ) = D n and so D n is a maximal abelian Lie subalgebra of D n .2. Fix G n ( D n ) = Fix G n ( ∂ , . . . , ∂ n ) = Sh n .3. D n is a faithful G n -module, i.e. the group homomorphism G n → G n , σ σ : ∂ σ∂σ − ,is a monomorphism.4. Fix G n ( ∂ , . . . , ∂ n , H , . . . , H n ) = { e } .Proof . 1. Statement 1 follows from (2).2. Let σ ∈ Fix G n ( D n ) and J ( σ ) = ( J ij ). By (10), ∂ = J ( σ ) ∂ , and so, for all i, j = 1 , . . . , n , δ ij = ∂ i ∗ x j = J ij , i.e. J ( σ ) = 1, or equivalently, by (8), x ′ = x + λ , . . . , x ′ n = x n + λ n for some scalars λ i ∈ K , and so σ ∈ Sh n .3 and 4. Let σ ∈ Fix G n = ( ∂ , . . . , ∂ n , H , . . . , H n ). Then σ ∈ Fix G n ( ∂ , . . . , ∂ n ) = Sh n , bystatement 2. So, σ ( x ) = x + λ , . . . , σ ( x n ) = x n + λ n where λ i ∈ K . Then x i ∂ i = σ ( x i ∂ i ) =( x i + λ i ) ∂ i for i = 1 , . . . , n , and so λ = · · · = λ n = 0. This means that σ = e . So, Fix G n =( ∂ , . . . , ∂ n , H , . . . , H n ) = { e } and D n is a faithful G n -module. (cid:3) By Lemma 2.3.(3), we identify the group G n with its image in G n .5 emma 2.4 D n is a simple Lie algebra.2. Z ( D n ) = { } .3. [ D n , D n ] = D n .Proof . 1. Let 0 = a ∈ D n and a = ( a ) be the ideal of the Lie algebra D n generated by theelement a . We have to show that a = D n . Using the inner derivations δ , . . . , δ n we see that ∂ i ∈ a for some i . Then a = D n since x α ∂ j = ( α i + 1) − [ ∂ i , x α + e i ∂ j ] ∈ a for all α and j .2 and 3. Statements 2 and 3 follow from statement 1. (cid:3) Proposition 2.5 Fix G n ( ∂ , . . . , ∂ n , H , . . . , H n ) = { e } .2. Let σ, τ ∈ G n . Then σ = τ iff σ ( ∂ i ) = τ ( ∂ i ) and σ ( H i ) = τ ( H i ) for i = 1 , . . . , n .3. Fix G n ( ∂ , . . . , ∂ n ) = Sh n .Proof . 1. Let σ ∈ F := Fix G n ( ∂ , . . . , ∂ n , H , . . . , H n ). We have to show that σ = e . Since σ ∈ Fix G n ( H , . . . , H n ), the automorphism σ respects the weight decomposition of D n . By (7), σ ( x α ∂ i ) = λ α,i x α ∂ i for all α ∈ N n,i and i = 1 , . . . , n where λ α,i ∈ K . Clearly, λ ,i = 1 for i = 1 , . . . , n . Since σ ∈ Fix G n ( ∂ , . . . , ∂ n ), by applying σ to the relations α j x α − e j ∂ i = [ ∂ j , x α ∂ i ],we get the relations α j λ α − e j ,i x α − e j ∂ i = [ ∂ j , λ α,i x α ∂ i ] = α j λ α,i x α − e j ∂ i . Hence λ α,i = λ α − e j ,i provided α j = 0. We conclude that all the coefficients λ α,i are equal toone of the coefficients λ e i ,j where i, j = 1 , . . . , n and i = j . The relations ∂ j = [ ∂ i , x i ∂ j ] impliesthe relations ∂ j = [ ∂ i , λ e i ,j x i ∂ j ] = λ e i ,j ∂ j , hence all the coefficients λ e i ,j are equal to 1. So, ⊕ ni =1 P ∂ i n ∂ i ⊆ F := Fix D n ( σ ) := { ∂ ∈ D n | σ ( ∂ ) = ∂ } . To finish the proof of statement 1 it sufficesto show that x α H i ∈ F for all α ∈ N n and i = 1 , . . . , n , see (5) and (6). We use induction on | α | := α + · · · + α n . If | α | = 0 the statement is obvious as σ ∈ F . Suppose that | α | >
0. Usingthe commutation relations [ ∂ j , x α H i ] = ( α j x α − e j H i if j = i, ( α i + 1) x α ∂ i if j = i, (11)the induction and the previous case, we see that[ ∂ j , σ ( x α H i ) − x α H i ] = 0 for i = 1 , . . . , n. Therefore, σ ( x α H i ) − x α H i ∈ C D n ( D n ) = D n . Since the automorphism σ respects the weightdecomposition of D n , we must have σ ( x α H i ) − x α H i ∈ x α H n ∩ D n = { } . Hence, x α H i ∈ F , asrequired.2. Statement 2 follows from statement 1.3. Clearly, Sh n ⊆ F = Fix G n ( ∂ , . . . , ∂ n ). Let σ ∈ F and H ′ i := σ ( H i ) , . . . , H ′ n := σ ( H n ).Applying the automorphism σ to the commutation relations [ ∂ i , H j ] = δ ij ∂ i gives the relations[ ∂ i , H ′ j ] = δ ij ∂ i . By taking the difference, we see that [ ∂ i , H ′ j − H j ] = 0 for all i and j . Therefore, H ′ i = H i + d i for some elements d i ∈ C D n ( D n ) = D n (Lemma 2.3.(1)), and so d i = P nj =1 λ ij ∂ j forsome elements λ ij ∈ K . The elements H ′ , . . . , H ′ n commute, hence[ H j , ∂ i ] = [ H i , ∂ j ] for all i, j,
6r equivalently, λ ij ∂ j = λ ji ∂ i for all i, j. This means that λ ij = 0 for all i = j , i.e. H ′ i = H i + λ ii ∂ i = ( x i + λ ii ) ∂ i = s λ ( H i )where s λ ∈ Sh n , s λ ( x i ) = x i + λ ii for all i . Then s − λ σ ∈ Fix G n ( ∂ , . . . , ∂ n , H , . . . , H n ) = { e } (statement 2), and so σ = s λ ∈ Sh n . (cid:3) Lemma 2.6
Let σ ∈ G n and ∂ ′ := σ ( ∂ ) , . . . , ∂ ′ n := σ ( ∂ n ) . Then1. ∂ ′ , . . . , ∂ ′ n are commuting, locally nilpotent derivations of P n .2. T ni =1 ker D n ( ∂ ′ i ) = K .Proof . 1. The derivations ∂ ′ , . . . , ∂ ′ n commute since ∂ , . . . , ∂ n are commute. The inner deriva-tions δ , . . . , δ n of the Lie algebra D n are commuting and locally nilpotent. Hence, inner derivations δ ′ := ad( ∂ ′ ) , . . . , δ ′ n := ad( ∂ ′ n )of the Lie algebra D n are commuting and locally nilpotent. The vector space P n ∂ ′ i is closed underthe derivations δ ′ j since δ ′ j ( P n ∂ ′ i ) = [ ∂ ′ j , P n ∂ ′ i ] = ( ∂ ′ j ∗ P n ) · ∂ ′ i ⊆ P n ∂ ′ i . Therefore, ∂ ′ , . . . , ∂ ′ n are locally nilpotent derivations of the polynomial algebra P n .2. Let λ ∈ T ni =1 ker P n ( ∂ ′ i ). Then λ∂ ′ ∈ C D n ( ∂ ′ , . . . , ∂ ′ n ) = σ ( C D n ( ∂ , . . . , ∂ n )) = σ ( C D n ( D n )) = σ ( D n ) = σ ( n M i =1 K∂ i ) = n M i =1 K∂ ′ i , since C D n ( D n ) = D n , Lemma 2.3.(1). Then λ ∈ K since otherwise the infinite dimensional space L i ≥ Kλ i ∂ ′ would be a subspace of a finite dimensional space σ ( D n ). (cid:3) The following lemma is well-known and it is easy to prove.
Lemma 2.7
Let ∂ be a locally nilpotent derivation of a commutative K -algebra A such that ∂ ( x ) =1 for some element x ∈ A . Then A = A ∂ [ x ] is a polynomial algebra over the ring A ∂ := ker( ∂ ) ofconstants of the derivation ∂ in the variable x . The next theorem is the most important point in the proof of Theorem 1.1 and, roughlyspeaking, the main reason why Theorem 1.1 holds.
Theorem 2.8
Let ∂ ′ , . . . , ∂ ′ n be commuting, locally nilpotent derivations of the polynomial algebra P n such that T ni =1 ker P n ( ∂ ′ i ) = K . Then there exist polynomials x ′ , . . . , x ′ n ∈ P n such that ∂ ′ i ∗ x ′ j = δ ij . (12) Moreover, the algebra homomorphism σ : P n → P n , x x ′ , . . . , x n x ′ n is an automorphism such that ∂ ′ i = σ∂ i σ − = ∂∂x ′ i for i = 1 , . . . , n . roof . Case n = 1: By Lemma 2.6, the derivation ∂ ′ of the polynomial algebra P is a locallynilpotent derivation with K ′ := ker P ( ∂ ′ ) = K . Hence, ∂ ′ ∗ x ′ = 1 for some polynomial x ′ ∈ P .By Lemma 2.7, K [ x ] = K ′ [ x ′ ] = K [ x ′ ], and so σ : K [ x ] → K [ x ], x x ′ , is an automorphismsuch that ∂ ′ = ddx ′ = σ ddx σ − .Case n ≥
2. Let K ′ i := ker P n ( ∂ ′ i ) for i = 1 , . . . , n . Clearly, K ⊆ K ′ i .(i) K ′ i = K for i = 1 , . . . , n : If K ′ i = K for some i then by the same argument as in the case n = 1 there exists a polynomial x ′ i ∈ P n such that ∂ ′ i ∗ x ′ i = 1, and so P n = K ′ i [ x ′ i ] = K [ x i ], acontradiction.(ii) Let m be the maximum of card( I ) such ∅ 6 = I ⊆ { , . . . , n − } and T i ∈ I K ′ i = K . By (i),2 ≤ m ≤ n −
1. Changing (if necessary) the order of the derivations ∂ ′ , . . . , ∂ ′ n we may assumethat A := T mi =1 K ′ i = K . Then the algebra A is infinite dimensional (since K = A ⊆ P n ) andinvariant under the action of the derivations ∂ ′ j for j = m + 1 , . . . , n . By the choice of m , A ∂ ′ j = K ′ j ∩ m \ i =1 K ′ i = K for j = m + 1 , . . . , n and the derivations ∂ ′ j acts locally nilpotently on the algebra A ∂ ′ j . Therefore, for each index j = m + 1 , . . . , n , there exists an element x ′ j ∈ A such that ∂ ′ j ∗ x ′ j = 1, and so (Lemma 2.7) A = A ∂ ′ j [ x ′ j ] = K [ x ′ j ] for j = m + 1 , . . . , n. (13)(ii)(a) Suppose that m = n −
1, i.e. ∂ ′ i ∗ x ′ n = δ in for all i = 1 , . . . , n . By Lemma 2.7, P n = K ′ n [ x ′ n ].The algebra K ′ n admits the set of commuting, locally nilpotent derivations ∂ ′′ := ∂ ′ | K ′ n , . . . , ∂ ′′ n − := ∂ ′ n − | K ′ n with T n − i =1 ker K ′ n ( ∂ ′′ i ) = K ′ n ∩ T n − i =1 K ′ i = K. (ii)(b) Suppose that m < n −
1. By (13), K ∗ x ′ m +1 + K = K ∗ x ′ m +2 + K = · · · = K ∗ x ′ n + K, and so λ j := ∂ ′ j ∗ x ′ n ∈ K for j = m + 1 , . . . , n −
1. Hence, ( ∂ ′ j − λ j ∂ ′ n ) ∗ x ′ n = 0 for j = m + 1 , . . . , n −
1. A linear combination of commuting, locally nilpotent derivations is a locallynilpotent derivation (the proof boils down to the case ∂ + δ of two commuting, locally nilpotentderivations, then the result follows from ( ∂ + δ ) m = P mi =0 (cid:0) mi (cid:1) ∂ i δ m − i and ∂ i δ m − i = δ m − i ∂ i ).Using the set of commuting, locally nilpotent derivations ∂ ′ , . . . , ∂ ′ n that satisfy (12) we obtainthe set of commuting, locally nilpotent derivations δ ′ := ∂ ′ , . . . , δ ′ m := ∂ ′ m , δ ′ m +1 := ∂ ′ m +1 − λ m +1 ∂ ′ n , . . . , δ ′ n − := ∂ ′ n − − λ n − ∂ ′ n , δ ′ n := ∂ n that satisfy (12) with δ ′ i ∗ x ′ n = δ in for i = 1 , . . . , n. Then repeating the arguments of (ii)(a), we see that P n = K ′ n [ x ′ n ]. The algebra K ′ n admits theset of commuting, locally nilpotent derivations ∂ ′′ := δ ′ | K ′ n , . . . , ∂ ′′ n − := δ ′ n − | K ′ n with n − \ i =1 ker K ′ n ( ∂ ′′ i ) = K ′ n ∩ n − \ i =1 ker P n ( δ ′ i ) = K ′ n ∩ n − \ i =1 ker P n ( ∂ ′ i ) = n \ i =1 K ′ i = K. (iii) Using the cases (ii)(a) and (ii)(b) n − x ′ , . . . , x ′ n andcommuting set of locally nilpotent derivations of P n , say, ∆ , . . . , ∆ n that satisfy (12) and suchthat( α ) ∆ i ∗ x ′ j = δ ij for all i, j = 1 , . . . , n ; 8 β ) the n -tuple of derivations ∆ = (∆ , . . . , ∆ n ) T is obtained from the n -tuple of derivations ∂ ′ = ( ∂ ′ , . . . , ∂ ′ n ) T by unitriangular (hence invertible) scalar matrix Λ = ( λ ij ) ∈ M n ( K ) such that∆ = Λ ∂ ′ ; and( γ ) (where K ′′ := ker P n (∆ ) , . . . , K ′′ n := ker P n (∆ n )) P n = K ′′ n [ x ′ n ] = ( K ′′ n − ∩ K ′′ n )[ x ′ n − , x ′ n ] = · · · = ( n \ i = s K ′′ i )[ x ′ s , . . . , x ′ n ] = · · · = ( n \ i =1 K ′′ i )[ x ′ , . . . , x ′ n ] = K [ x ′ , . . . , x ′ n ] . (iv) Replacing the row x ′ = ( x ′ , . . . , x ′ n ) by the row x ′ Λ gives the required elements of the theorem.Indeed, by ( α ), Λ · ( ∂ ′ i ∗ x ′ j ) = 1, the identity n × n matrix. Hence, ( ∂ ′ i ∗ x ′ j ) · Λ = 1, as required.(v) Let x ′ , . . . , x ′ n be the set of polynomials as in the theorem. Then σ is an algebra automor-phism (see ( γ ) and (iv)) such that ∂ ′ i = σ∂ i σ − = ∂∂x ′ i for i = 1 , . . . , n . (cid:3) Corollary 2.9
Let σ ∈ G n . Then τ σ ∈ Fix G n ( ∂ , . . . , ∂ n ) for some τ ∈ G n .Proof . By Lemma 2.6, the elements ∂ ′ := σ ( ∂ ) , . . . , ∂ ′ n := σ ( ∂ n ) satisfy the assumptions ofTheorem 2.8. By Theorem 2.8, ∂ ′ := τ − ( ∂ ) , . . . , ∂ ′ n := τ − ( ∂ n ) for some τ ∈ G n . Therefore, τ σ ∈ Fix G n ( ∂ , . . . , ∂ n ). (cid:3) Proof of Theorem 1.1 . Let σ ∈ G n . By Corollary 2.9, τ σ ∈ Fix G n ( ∂ , . . . , ∂ n ) = Sh n (Proposition 2.5.(3)). Therefore, σ ∈ G n , i.e. G n = G n . (cid:3) Acknowledgements
The work is partly supported by the Royal Society and EPSRC.
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