An Introduction to Constant Curvature Spaces in the Commutative (Segre) Quaternion Geometry

It is known that complex numbers can be associated with plane Euclidean geometry and their functions are successfully used for studying extensions of Euclidean geometry, i.e., non-Euclidean geometries and surfaces differential geometry. In this paper we begin to study the constant curvature spaces associated with the geometry generated by commutative elliptic-quaternions and we show how the “mathematics” they generate allows us to introduce these spaces and obtain the geodesic equations without developing a complete mathematical apparatus as the one developed for Riemannian geometry.     

A non‐commutative Landau‐Zener formula

We study semi‐classical measures of families of solutions to a 2 × 2 Dirac system with 0 mass, which presents bands crossing. We focus on constant electro‐magnetic fields. The fact that these fields are orthogonal or not leads to different geometric situations. In the first case, one reduces to some well‐understood model problem. For studying the second case, we introduce some two‐scale semi‐classical measures associated with symplectic submanifold. These measures are operator‐valued measures and the transfer of energy at the crossing is described by a non‐commutative Landau‐Zener formula for these measures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Categories of Projective Geometries with Morphisms and Homomorphisms

Projective geometries studied as Pasch geometries possess morphisms and homomorphisms. A homomorphic image of a projective geometry is shown to be projective. A projective geometry is shown to be Desarguesian iff it is a homomorphic image of a higher dimensional one, which in a sense is dual to the classical imbedding theorem. Semi-linear maps induce morphisms which are homomorphisms iff the associated homomorphisms of skewfields are isomorphisms. Projective geometries form categories with morphisms as well as homomorphisms and Desarguesian ones form a subcategory with Desarguesian homomorphisms.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Noncommutative structures

We propose a method for constructing noncommutative analogs of objects from classical calculus, differential geometry, topology, dynamical systems, etc. The standard (commutative) objects can be obtained from noncommutative ones by natural projections (a set of canonical homomorphisms). The approach is ideologically close to the noncommutative geometry of A. Connes but differs from it in technical details. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 203–242.     

The affine representation theorem for abstract convex geometries

A convex geometry is a combinatorial abstract model introduced by Edelman and Jamison which captures a combinatorial essence of “convexity” shared by some objects including finite point sets, partially ordered sets, trees, rooted graphs. In this paper, we introduce a generalized convex shelling, and show that every convex geometry can be represented as a generalized convex shelling. This is “the representation theorem for convex geometries” analogous to “the representation theorem for oriented matroids” by Folkman and Lawrence. An important feature is that our representation theorem is affine-geometric while that for oriented matroids is topological. Thus our representation theorem indicates the intrinsic simplicity of convex geometries, and opens a new research direction in the theory of convex geometries.     

On a new generalization of Fibonacci quaternions

In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k -Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Binet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative.     

Two-dimensional hypercomplex numbers and related trigonometries and geometries

All the commutative hypercomplex number systems can be associated with a geometry. In two dimensions, by analogy with complex numbers, a general system of hypercomplex numbers can be introduced and can be associated with plane Euclidean and pseudo-Euclidean (space-time) geometries. In this paper we show how these systems of hypercomplex numbers allow to generalise some well known theorems of the Euclidean geometry relative to the circle and to extend them to ellipses and to hyperbolas. We also demonstrate in an unusual algebraic way the Hero formula and Pytaghoras theorem, and show that these theorems hold for the generalised Euclidean and pseudo-Euclidean plane geometries.     

Projective mappings between projective lattice geometries

The concept of projective lattice geometry generalizes the classical synthetic concept of projective geometry, including projective geometry of modules.In this article we introduce and investigate certain structure preserving mappings between projective lattice geometries. Examples of these so-calledprojective mappings are given by isomorphisms and projections; furthermore all linear mappings between modules induce projective mappings between the corresponding projective geometries.     

Circle Geometries Modeled in Projective Lines over {{\mathbb{R}}^2} -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $ {{\mathbb{R}}^2} $ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.     

Abstract Hilbert Schemes

In analogy with classical projective algebraic geometry, Hilbert functors can be defined for objects in any Abelian category. We study the moduli problem for such objects. Using Grothendieck's general framework. We show that with suitable hypotheses the Hilbert functor is representable by an algebraic space locally of finite type over the base field. For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme. For the projective scheme corresponding to a suitable noncommutative graded algebra, the Hilbert functor is represented by a countable union of commutative projective schemes.     

On Linear Functionals and Hahn-Banach Theorems for Hyperbolic and Bicomplex Modules

We consider modules over the commutative rings of hyperbolic and bicomplex numbers. In both cases they are endowed with norms which take values in non–negative hyperbolic numbers. The exact analogues of the classical versions of the Hahn–Banach theorem are proved together with some of their consequences. Linear functionals on these modules are studied and their relations with the corresponding hyperplanes are established. Finally, we introduce the notion of hyperbolic convexity for hyperbolic modules (in analogy with real, not complex, convexity) and establish its relation with hyperplanes.     

A proof of the arithmetic-geometric mean inequality using non-Euclidean geometry

This note presents a proof of the arithmetic-geometric mean inequality that uses basic facts about the upper half-plane model of hyperbolic plane geometry. This material could find use as enrichment material in any model-oriented course on the classical geometries.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

Lattice theory and metric geometry

K. Menger and G. Birkhoff recognized 70 years ago that lattice theory provides a framework for the development of incidence geometry (affine and projective geometry). We show in this article that lattice theory also provides a framework for the development of metric geometry (including the euclidean and classical non-euclidean geometries which were first discovered by A. Cayley and F. Klein). To this end we introduce and study the concept of a Cayley–Klein lattice. A detailed investigation of the groups of automorphisms and an algebraic characterization of Cayley–Klein lattices are included. The authors would like to thank an unknown referee for his helpful suggestions.     

Chapter 8: invariant preserving linear matrix mappings over rings and related topics

This paper surveys the ideas involved in the theory of invariant preserving linear mappings of matrixrings where the scalar ring is not necessarily a field. Section 1 provides several historical examples of the origins of these problems. Section 2 discusses the basic context when the vector space over a field is replaced by a projective module over a commutative ring. Section 3 sketches the classification of the rank one preserving linear mappings using the approach of McDonald, Marcus, and Moyls. Section 4 continues the discussion of Section 3 by placing the problem within the context of group schemes and the invariant preserving theory of Waierhousc. Section 5 begins a sketch of the evolution of these ideas to a context where the scalar ring h not necessarily commutative with a discussion of some classical results of Hua concerning coherence, projective geometry, and matrices over division rings. Trie concluding section, Section 6, discusses the results developed by Wong of linear preserving maps over noncommutative scaiar rings.     

Algebraic properties of separated power series

We study the commutative algebra of rings of separated power series over a ring E and that of their extensions: rings of separated (and more specifically convergent) power series from a field K with a separated E-analytic structure. Both of these collections of rings already play an important role in the model theory of non-Archimedean valued fields and we establish their algebraic properties. This will make a study of the analytic geometry over such fields through the classical methods of algebraic geometry possible.