TY - JOUR

T1 - Nonstandard analysis, deformation quantization and some logical aspects of (Non)commutative algebraic geometry

AU - Kanel-Belov, Alexei

AU - Chilikov, Alexei

AU - Ivanov-Pogodaev, Ilya

AU - Malev, Sergey

AU - Plotkin, Eugeny

AU - Yu, Jie Tai

AU - Zhang, Wenchao

N1 - Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

KW - Affine algebraic geometry

KW - Affine spaces

KW - Algorithmic unsolvability

KW - Automorphisms

KW - Deformation quantization

KW - Elementary equivalence

KW - Embeddability of varieties

KW - Finitely presented algebraic systems

KW - First order rigidity

KW - Free associative algebras

KW - Ind-group

KW - Infinite prime number

KW - Isotypic algebras

KW - Noncommutative Gröbner-Shirshov basis

KW - Polynomial symplectomorphisms

KW - Semi-inner automorphism

KW - Turing machine

KW - Undecidability

KW - Universal algebraic geometry

KW - Weyl algebra automorphisms

UR - http://www.scopus.com/inward/record.url?scp=85092891388&partnerID=8YFLogxK

U2 - 10.3390/math8101694

DO - 10.3390/math8101694

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AN - SCOPUS:85092891388

SN - 2227-7390

VL - 8

SP - 1

EP - 33

JO - Mathematics

JF - Mathematics

IS - 10

M1 - 1694

ER -