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 and Wenchao Zhang
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Alexei Kanel-Belov: Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
Alexei Chilikov: Department of Information Security, Bauman Moscow State Technical University, ul. Baumanskaya 2-ya, 5, 105005 Moscow, Russia
Ilya Ivanov-Pogodaev: Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudnyi, Institutskiy Pereulok, 141700 Moscow Oblast, Russia
Sergey Malev: Department of Mathematics, Ariel University of Samaria, Ariel 40700, Israel
Eugeny Plotkin: Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
Jie-Tai Yu: College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, China
Wenchao Zhang: Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel

Mathematics, 2020, vol. 8, issue 10, 1-33

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.

Keywords: universal algebraic geometry; affine algebraic geometry; elementary equivalence; isotypic algebras; first order rigidity; Ind-group; affine spaces; automorphisms; free associative algebras; Weyl algebra automorphisms; polynomial symplectomorphisms; deformation quantization; infinite prime number; semi-inner automorphism; embeddability of varieties; undecidability; noncommutative Gröbner-Shirshov basis; finitely presented algebraic systems; algorithmic unsolvability; turing machine (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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