On the augmentation topology of automorphism groups of affine spaces and algebras

Kanel-Belov, Alexei; Yu, Jie-Tai; Elishev, Andrey

doi:10.1142/s0218196718400040

Cited by 11 publications

(12 citation statements)

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“…The automorphisms of the group of polynomial automorphisms on free associative algebras and commutative algebras at the level of Ind-schemes were obtained by Belov, Elishev and J.-T.Yu in Reference [30]. Let K[x 1 , .…”

Section: Plotkin's Problem: Automorphisms Of Endomorphism Semigroups mentioning

confidence: 99%

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Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

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Section: Plotkin's Problem: Automorphisms Of Endomorphism Semigroups mentioning

confidence: 99%

“…A group of automorphisms of ind-schemes was computed in Reference [30]. In investigating the Jacobian conjecture and automorphisms of the Weyl algebra, Plotkin's problem for symplectomorphisms is also extremely important.…”

Section: Introductionmentioning

confidence: 99%

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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“…The singularity trick is rather useful when dealing with direct systems of varieties equipped with the power series topology. It was first introduced in our prior paper [26] (Theorem 3.2, Lemma 3.5, 3.6 and 3.7). In our proof, this technique is employed in an essentially the same manner, and is contained in Lemma 3.6 and Proposition 3.4.…”

Section: Approximation and The Singularity Trickmentioning

confidence: 99%

“…The point of introducing the skew algebras W h n,C [k ij ] and P h n,C [k ij ] is that a certain singularity analysis procedure (the singularity tricks mentioned in the introduction) can be implemented for these algebras in full analogy with the case of the commutative polynomial algebra processed in our preceding study [26], while on the other hand there seems to be no straightforward way to execute the singularity trick for the algebras W h n and P h n . Furthermore, after adjunction of k −1 ij (together with the entries of the inverse matrix) and extension of scalars (the localization procedure) one can embed the h-augmented C-algebras W h n and P h n in the skew augmented algebras over the larger coefficient ring, thus connecting the h-augmented automorphisms with the skew augmented ones which are linear on k ij , as we shall see below.…”

Section: Continuity Of φ Hk and The Singularity Trickmentioning

confidence: 99%

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Lifting of polynomial symplectomorphisms and deformation quantization

Kanel-Belov

Grigoriev

Elishev

et al. 2018

Communications in Algebra

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The objective of this paper is the proof of a conjecture of Kontsevich [1] on the isomorphism between groups of polynomial symplectomorphisms and automorphisms of the corresponding Weyl algebra in characteristic zero. The proof is based on the study of topological properties of automorphism Ind-varieties of the so-called augmented and skew augmented versions of Poisson and Weyl algebras. Approximation by tame automorphisms as well as a certain singularity analysis procedure is utilized in the construction of the lifting of augmented polynomial symplectomorphisms, after which specialization of the augmentation parameter is performed in order to obtain the main result.

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Algebraic overshear density property

Andrist,

Kutzschebauch

2024

Beitr Algebra Geom

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We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.

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On the augmentation topology of automorphism groups of affine spaces and algebras

Abstract: We study topological properties of Ind-

Cited by 11 publications

References 28 publications

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

Lifting of polynomial symplectomorphisms and deformation quantization

Algebraic overshear density property

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