The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture

Belov-Kanel, Alexei; Kontsevich, Maxim

doi:10.17323/1609-4514-2007-7-2-209-218

Cited by 93 publications

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References 6 publications

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“…The Jacobian conjecture in dimension 2n implies the generalized Dixmier conjecture in dimension n. This was proved independently by Tsuchimoto [9] and by Belov and Kontsevich [4]. A shorter proof can be found in [2].…”

Section: An Analogue Results For the Jacobian Conjecturementioning

confidence: 91%

The Starred Dixmier Conjecture forA₁

Moskowicz

Valqui²

2015

Communications in Algebra

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Let A 1 K = K X Y YX − XY = 1 be the first Weyl algebra over a characteristic zero field K, and let be the exchange involution on A 1 K given by X = Y and Y = X. The Dixmier conjecture of Dixmier (1968) asks the following question: Is every algebra endomorphism of the Weyl algebra A 1 K an automorphism? The aim of this paper is to prove that each -endomorphism of A 1 K is an automorphism. Here an -endomorphism of A 1 K is an endomorphism which preserves the involution . We also prove an analogue result for the Jacobian conjecture in dimension 2, called − JC 2 .

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Section: An Analogue Results For the Jacobian Conjecturementioning

confidence: 91%

The Starred Dixmier Conjecture forA₁

Moskowicz

Valqui²

2015

Communications in Algebra

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“…. , ∂ s (see (5) and (6)) are locally nilpotent (iff (∂ i ) (deg σ ) s−1 +1 (x j ) = 0 for all i, j, by Theorem 2.3).…”

Section: The Algebra a Is An Iterated Ore Extensionmentioning

confidence: 99%

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Bavula

2007

Journal of Pure and Applied Algebra

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Let A n be the nth Weyl algebra and P m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A n ⊗ P m } is proved: an algebra A admits a finite set δ 1 , . . . , δ s of commuting locally nilpotent derivations with generic kernels and ∩ s i=1 ker(δ i ) = K iff A A n ⊗ P m for some n and m with 2n + m = s, and vice versa. The inversion formula for automorphisms of the algebra A n ⊗ P m (and for P m :has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given σ ∈ Aut K (P m ), then deg σ −1 ≤ (deg σ ) m−1 (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given σ ∈ Aut K (A n ⊗ P m ), then deg σ −1 ≤ (deg σ ) 2n+m−1 . Any automorphism σ ∈ Aut K (P m ) is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for σ ∈ Aut K (A n ⊗ P m ). One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242.[6]] problem 1) into a single question, (JD): is a K -algebra endomorphism σ : A n ⊗ P m → A n ⊗ P m an algebra automorphism provided σ (P m ) ⊆ P m and det( ∂σ (x i )

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“…Dixmier's problem has been also referred to as the Dixmier conjecture. It was proved in [7,27] that the Dixmier conjecture is stably equivalent to the Jacobian conjecture [14]. Recently, there has been some interest in establishing a quantum analogue of the Dixmier conjecture.…”

Section: Introductionmentioning

confidence: 99%

The automorphism groups for a family of generalized Weyl algebras

Tang

2018

J. Algebra Appl.

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Abstract. In this paper, we study a family of generalized Weyl algebras {A p (λ, µ, K q [s, t])} and their polynomial extensions. We will show that the algebra) S when none of p and q is a root of unity. As an application, we determine all the height-one prime ideals and the center foris cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras A p (λ, µ, K q [s, t]) and their polynomial extensions in the case where none of p and q is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization (A p (1, 1, K q [s, t])) S . Moreover, we will completely determine the automorphism group for the algebra A p (1, 1, K q [s, t]) and its polynomial extension when p = 1 and q = 1.

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The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture

Cited by 93 publications

References 6 publications

The Starred Dixmier Conjecture forA₁

The Starred Dixmier Conjecture forA₁

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

The automorphism groups for a family of generalized Weyl algebras

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The Jacobian Conjecture is Stably Equivalent to the Dixmier Conjecture

Cited by 93 publications

References 6 publications

The Starred Dixmier Conjecture forA1

The Starred Dixmier Conjecture forA1

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

The automorphism groups for a family of generalized Weyl algebras

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The Starred Dixmier Conjecture forA₁

The Starred Dixmier Conjecture forA₁