Nonstandard analysis, deformation quantization and some logical aspects of (Non)commutative algebraic geometry

Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie Tai Yu, Wenchao Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Article number1694
Pages (from-to)1-33
Number of pages33
JournalMathematics
Volume8
Issue number10
DOIs
StatePublished - Oct 2020

Bibliographical note

Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

Keywords

  • Affine algebraic geometry
  • Affine spaces
  • Algorithmic unsolvability
  • Automorphisms
  • Deformation quantization
  • Elementary equivalence
  • Embeddability of varieties
  • Finitely presented algebraic systems
  • First order rigidity
  • Free associative algebras
  • Ind-group
  • Infinite prime number
  • Isotypic algebras
  • Noncommutative Gröbner-Shirshov basis
  • Polynomial symplectomorphisms
  • Semi-inner automorphism
  • Turing machine
  • Undecidability
  • Universal algebraic geometry
  • Weyl algebra automorphisms

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