Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry
-
Published:2020-10-02
Issue:10
Volume:8
Page:1694
-
ISSN:2227-7390
-
Container-title:Mathematics
-
language:en
-
Short-container-title:Mathematics
Author:
Kanel-Belov AlexeiORCID,
Chilikov AlexeiORCID,
Ivanov-Pogodaev Ilya,
Malev SergeyORCID,
Plotkin Eugeny,
Yu Jie-Tai,
Zhang WenchaoORCID
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.
Funder
Russian Science Support Foundation
Israel Science Foundation
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference111 articles.
1. Universal Algebra, Algebraic Logic and Databases;Plotkin,1994
2. Algebra, categories, and databases;Plotkin,2000
3. Groups, Algebras and Identities
4. Varieties of algebras and algebraic varieties;Plotkin;Israel J. Math.,1996
5. Varieties of algebras and algebraic varieties. Categories of algebraic varieties;Plotkin;Sib. Adv. Math.,1997