72 research outputs found
Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
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 Section 2.1 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. Section 2.2 is dedicated to
particular cases of Plotkin's problem. Section 2.3 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 two
sections deal with algorithmic problems for noncommutative and commutative
algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in
non-commutative situation. Despite the existence of an algorithm for checking
equalities, the zero divisors and nilpotency problems are algorithmically
unsolvable. Section 3.2 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.Comment: In this review we partially used results of arXiv:1512.06533,
arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a
new contex
Recommended from our members
Topologie
The Oberwolfach conference “Topologie” is one of only a few opportunities for researchers from many different areas in algebraic and geometric topology to meet and exchange ideas. The program covered new developments in fields such as automorphisms of manifolds, applications of algebraic topology to differential geometry, quantum field theories, combinatorial methods in low-dimensional topology, abstract and applied homotopy theory and applications of L2-cohomology. We heard about new results describing the cohomology of the automorphism spaces of some smooth manifolds, progress on spaces of positive scalar curvature metrics, a variant of the Segal conjecture without completion, advances in classifying topological quantum field theories, and a new undecidability result in combinatorial group theory, to mention some examples. As a special attraction, the conference featured a series of three talks by Dani Wise on the combinatorics of CAT(0)-cube complexes and applications to 3-manifold topology
Admissible orders on quotients of the free associative algebra
An admissible order on a multiplicative basis of a noncommutative algebra A is a term order satisfying additional conditions that allow for the construction of Grobner bases for A -modules. When A is commutative, a finite reduced Grobner basis for an A -module can always be obtained, but when A is not commutative this is not the case; in fact in many cases a Grobner basis theory for A may not even exist.
E. Hinson has used position-dependent weights, encoded in so-called admissible arrays, to partially order words in the free associative algebra in a way which produces a length-dominant admissible order on a particular quotient of the free algebra, where the ideal by which the quotient is taken is an ideal generated by pure homogeneous binomial differences and is determined by the array A.
This dissertation investigates the properties of two large classes of admissible arrays A. We prove that weight ideals associated to arrays in the first class are finitely generated and we describe the generating sets. We exhibit instances of trivial and nontrivial finitely generated weight ideals associated to arrays in the second class and we partially characterize the corresponding arrays. We also exhibit instances of weight ideals associated to arrays in the second class which do not admit a finite generating set. We identify an algebro-combinatorial property on weight ideals, which we call saturation, that is connected to finite generation. In addition, we look at actions of the multiplicative monoid generated by the set of transvections and diagonal matrices with non-negative entries on the set of equivalence classes of admissible arrays under order-isomorphism and we analyze the stabilizers and orbits of these actions
Gr\"obner-Shirshov bases and their calculation
In this survey, we formulate the Gr\"{o}bner-Shirshov bases theory for
associative algebras and Lie algebras. Some new Composition-Diamond lemmas and
applications are mentioned.Comment: 91 page
- …